The Parabola in Universal. Hyperbolic Geometry

Transcription

1 The Parabola in Universal Hyperbolic Geometry Ali Alkhaldi School of Mathematics and Statistics University of New South Wales Supervisor: A. /Prof Norman Wild berger A thesis in fulfilment of the requirements for the degree of Doctor of Philosophy 9th, January 2014

2 Contents Acknowledgements vii 1 Introduction 1 2 Universal Hyperbolic Geometry and Projective Geometry Classical hyperbolic geometry Beltrami-Poincaré model Beltrami-Klein model The Beltrami-Poincaré upper half plane model The hyperboloid model Universal hyperbolic geometry Projective geometry UHG from a synthetic projective view Metrical projective linear algebra Hyperbolic trigonometry in UHG Sydpoints and hyperbolic triangle geometry Introduction Midpoints of a side Sydpoints of a side The construction of Sydpoints Twin circles Constructions of twin circles Circumlinear coordinates and the Orthocenter hierarchy Change of coordinates and the main example Altitudes, Orthocenter and Orthic triangle Desargues points and the Orthoaxis Parallels and the Double triangle The Circumcenter hierarchy Circumcenters, medians and centroids CircumCentroids Twin Circumcircles of a Triangle CircumDual points, Tangent lines and Sound points Jay and Wren lines CircumMeets and reflections i

3 CONTENTS ii Sound conics The parabola and its Euclidean properties Introduction The parabola and its construction Basic definitions Construction with a dynamic geometry program Dual conics and the connection with sydpoints Standard Coordinates and duality The four basis null points The Fundamental theorem and standard coordinates A rational parabola Formulas for directrices, vertex lines, base points and base lines Quadrance and spread in standard coordinates Duality with respect to a conic and tangents The j, h and d points and lines The sydpoints of a parabola Focal and base lines Parallels between the Euclidean and hyperbolic parabolas Some congruent triangles The optical property The s points and S circles Focal chords and conjugates Quadrance cross ratios Spreads related to chords of a parabola Normals to the parabola P Conjugate normals and conics Concurrent normals, symmetric functions, and four point conics The four normal conic A n and finding normals Normal conjugate points The evolute and centers of curvature Formula for the evolute Further properties of the hyperbolic parabola The Twin Parabola Alternate views of a parabola The parabola as a midpoint locus The parabola as a quadrance product locus Structures associated to a variable point p 0 on P The e-points of p The perpendicular points m 1, m 2 of p The conjugate points n 1, n 2 of p The focal and base lines of the twin point p The σ points of p The vertex perpendicular points q 1 and q 2 of p The h-points of p

4 CONTENTS iii The γ and µ-points, and the 14-points conic G Midpoints on Directrices Canonical structures on the hyperbolic parabola The y-points and the Y conic Metrical relations for the hyperbolic parabola Conics related to a point p 0 on the parabola P Thaloids associated to the parabola Conclusion 158

5 List of Figures 1.1 A hyperbolic parabola P 0 with foci f 1 and f Construction of sydpoints of ab Constructing the twin circle D of C Circumcircles and CircumMeet points The four basis points v 1, v 2, α 0 and α A focal chord p 0 q 0 with the polar point z on directrix Four points on a parabola with given normal point Twin foci and the twin parabola Midpoint locus property of a parabola The σ points A quadrangle and some associated points and lines Reflection r m in m sends a to b Constructing midpoints m and n of the side ab Circles centered at a, for an interior point Circles centered at a, for an exterior point Two Thaloids, with diameters ab and cd Defining quadrance and spread via cross ratios Midpoints, Midlines, Circumlines, Circumcenters and Circumcircles A non-classical triangle with both midpoints and sydpoints Four twin circumcircles of a non-classical triangle Construction of sydpoints of ab Constructing r and a or r and a) from s and b Constructing the twin circle D of C Another construction of a twin circle Basic example triangle with coordinates Altitudes, Orthocenter, Orthic triangle and Base center b Desargues points, Orthic axis S and Orthoaxis A The Double triangle, Orthoaxis A, and the points z, b, x, h and s Circumlines, Circumcenters, Medians and Centroids Circumcenters of a triangle outside the null circle Twin circumcircles for a classical triangle Twin Circumcircles and Tangent lines CircumDual points and Sound points iv

6 LIST OF FIGURES v 3.17 Jay lines J, Wren lines W, T, U, V axes and new points a, u, t Circumcircles and CircumMeet points Sound conics The Euclidean parabola A parabola P 0 with foci f 1 and f Various examples of parabolas Dual and tangent lines, twin point and focal lines Construction of a Euclidean parabola Construction of a hyperbolic parabola P The parabola P 0 and its twin P The four basis points v 1, v 2, α 0 and α A standard coordinate view of a parabola Some basic points associated to a parabola P The j and h points and lines Null tangents and d 0, d 0 points Sydpoints and the twin foci f 1 and f 2 of P The r, s, t and w points of p 0 on P Two pairs of congruent triangles The j 0 and h 0 points Two chord midpoints The Parallel line spread relation The S 1 and S 2 circles A focal chord p 0 q 0 with the polar point z on directrix Focal conjugates n 1 and n The polar point z of the chord p 0 q Chord directrix meets x 1 and x Tangent directrix meets s 1 and s Two points and polar circles Opposite triangle spreads Evolute of a Euclidean parabola Conjugate normal meets h 1 and h 2 and conics Normal conjugate collinearities Four points p with normals through l and associated conic A Construction of points p on P 0 with normals through n The normal conjugate conic K Evolute of a parabola Normals to a parabola The parabola P 0 and its twin P Midpoint locus property of a parabola e 0 and e 0 points Perpendicular points collinearities Focal conjugates n 1, n 2 and the ω 0 conic W Perpendicular conjugate relation The σ points

7 LIST OF FIGURES vi 5.8 Vertex perpendicular points The h-points The 14-points conic G The o-points and l-points Canonical structures The Y conic The conics F 1 and F Four Thaloids T 0, T 0, W 0 and W 0 and meet points x 0 and x

8 Acknowledgements It is my pleasure to thank my supervisor A/Prof. N. J. Wildberger whose help, advice and supervision was invaluable. I would like to acknowledge his generous help in editing and improving the structure and style of this thesis, along with the appearance of diagrams. He has always shared with me his ideas about this work. Many thanks go to my parents, my wife Norah, my son Thamer and my daughter Reetal for their continued support and encouragement throughout my PhD study. I could never have done this without the support of all my brothers and sisters and friends, so thanks to all of them. Finally, I would like also to thank the University of King Khalid in Saudi Arabia for a financial grant for my PhD studies, and also UNSW for giving me the opportunity to be one of its students. vii

9 Chapter 1 Introduction This thesis aims to develop a new and broader theory of the parabola in hyperbolic geometry which we often refer to as the hyperbolic parabola by working in the algebraic setting of Universal Hyperbolic Geometry UHG), introduced by N. J. Wildberger. We will show that the hyperbolic parabola not only has a rich theory that connects intimately with the classical work in Euclidean geometry, but also that there are many new and remarkable features that have no classical parallel, and that open a wide avenue for further research directions. Along the way we will develop the novel theory of sydpoints of a side: these are analogs of midpoints that allow us to extend much of hyperbolic triangle geometry beyond the interior of the disk in the Cayley Klein projective model. Hopefully this will augment the paper of [53] in setting out a new framework for triangle geometry in non-euclidean settings. Two of the chapters of this thesis are now joint papers with N. J. Wildberger; both have already appeared, see [2] and [55]. Despite the antiquity and wealth of results on conic sections, the subject has remained an interesting and active topic for researchers, connecting with algebraic geometry, projective geometry, linear algebra and dynamics. It gives rise to many applications in science, for instance to optics, special relativity, and astronomy. Conics have been studied extensively and systematically since the ancient Greeks, who used them in their investigations on the famous problems of the trisection of an angle, duplication of a cube, and squaring of a circle. The ancient Greeks viewed conics as curves resulting from intersecting a right circular cone with a plane, with the type of conic dependent on the inclination of the plane to the cone. Apollonius BC), who was the greatest geometer of the ancient world, gathered and extended many previous results and wrote eight books about conic sections, consisting of about 487 Propositions. Apollonius did not work with three-dimensional space too much, but he took a conic and made it planar and studied it using pla- 1

10 1. Introduction 2 nar techniques; the names ellipse, parabola, and hyperbola are accredited to him. In Descartes coordinate framework, a conic can be defined as the quadratic equation ax 2 + by 2 + cxy + dx + fy + g = 0. In fact, there are many interesting and remarkable properties of conic sections which have been found over the years: see for example [1], [6], [7], [8], [22], [33]. Most of this occurs in the context of Euclidean geometry; the story in hyperbolic geometry is different, with far fewer works compared to the Euclidean case, a point also made by O. Avcıoğlu and O. Bizim in [4]. The theory of conics and more general curves in hyperbolic geometry has also been the subject of considerable investigation. In 1882, E. Story published some works on the properties of non-euclidean conics, see [44]. He classified conics into several types an ellipse, hyperbola, semi-hyperbola, elliptic parabola, hyperbolic parabola, semi-circular parabola, circular parabola and a circle according to their meets with the absolute conic. In this work he considered the outside of the absolute conic as well as the interior, and proved interesting properties using the Cayley-Klein framework for projective measurement. In 1957, K. Fladt also classified conics in hyperbolic geometry in his work [16]. In 1971, E. Molnár studied the asymptotic properties of conics in hyperbolic geometry in [27]. Recently, Avcıoğlu and Bizim in [4], working in the upperhalf plane model, gave geometric definitions of ellipses, hyperbolas, and parabolas following the Euclidean definitions, and gave some properties of these curves. M. Henle discussed the interesting question of what the right definition of the hyperbolic parabola ought to be: either as a curve satisfying the locus property or as a curve satisfying the reflection property, see [21]. Wildberger in [52] stated some properties for the hyperbolic parabola in universal hyperbolic geometry but no proofs were given. In addition, G. Csima and J. Szirmai investigated isoptic curves of the conic sections in hyperbolic geometry and gave formulas for ellipses, hyperbolas, and parabolas [14]. In [36], N. Sönmez studied the images of hyperbolic ellipses under Möbius and harmonic Möbius transformations. In this thesis, we will give a novel definition of a hyperbolic parabola, and investigate many properties of it: some are similar to the Euclidean situation, but quite a few will be unfamiliar to the reader. For example, we will see that the hyperbolic parabola P 0 shown in Figure 1.1 is determined by two points f 1 and f 2, called the foci. In this investigation, we found that we needed a new notion analogous to that of midpoint, but applying to pairs of points sides) both inside and outside the null conic C. This new notion of sydpoints allows us, perhaps surprisingly, to extend hyperbolic triangle geometry in new ways; this work has appeared in [55]. Triangle geometry has seen surprisingly limited development in the hyperbolic

11 1. Introduction 3 Figure 1.1: A hyperbolic parabola P 0 with foci f 1 and f 2 setting when compared to the very rich Eucldiean case, the latter which has been prominently summarized and laid out by the heroic efforts of C. Kimberling [23]. The 2010 book of A. A. Ungar [47] was, in our understanding, the first large scale attempt at setting out hyperbolic triangle geometry. Ungar proceeds from the basis of gyrogroups, modelled on the algebraic relations satisfied by relativistic velocities under Einstein s composition, see also [45] and [46]. He explores some centers of a triangle in hyperbolic geometry which are analogous to Euclidean triangle centers. Also Sönmez studied some features of triangle hyperbolic geometry, for example he gave a proof of the Steiner-Lehmus theorem and the Euler theorem in [37] and [38]. The beginnings of triangle geometry in Universal hyperbolic geometry were laid out by Wildberger in [53]. In Wildberger s set-up of universal hyperbolic geometry, the setting is quite wider, working in a Cayley-Klein geometry both inside and outside the null circle absolute), over a general field. One of the challenges is to extend basic notions, such as midpoints, angle bisectors/bilines, perpendicular bisectors/midlines etc. to triangles which are rather general. This is one of our main contributions here: we show that the notions of midpoints and bilines do have extensions, but we must be prepared to substitute the study of sydpoints and silines instead. In this Introduction, we give some motivation and overview of the thesis as a whole. In the second chapter, we give a brief history and summary of the basic models of classical hyperbolic geometry, and then introduce and review Wildberger s universal hyperbolic geometry via projective geometry and projective linear algebra. Some fundamental facts of projective geometry are reviewed, in particular the Fundamental Theorem in the planar case, which we will use on several occasions to simplify coordinate systems. We also review some of the main constructions and formulas which

12 1. Introduction 4 we will need in the thesis. The basic measurements of quadrance q a 1, a 2 ) between points a 1 and a 2, and spread S L 1, L 2 ) between lines L 1 and L 2 are defined. To connect with the more familiar classical distance d a 1, a 2 ) between points, and angle θ L 1, L 2 ) between lines in the Klein projective model, we have the relations q a 1, a 2 ) = sinh 2 d a 1, a 2 )) and S L 1, L 2 ) = sin 2 θ L 1, L 2 )) valid when the points and lines are inside the null circle C. These quantities turn out to be definable purely algebraically in terms of the projective) bilinear form which determines the absolute, or null circle, hence they extend also to points and lines outside the null circle. This is a main feature of UHG. Figure 1.2: Construction of sydpoints of ab In the third chapter, we introduce the new notion of sydpoints into universal hyperbolic geometry and use them to extend projective triangle geometry with respect to a general bilinear form. Here is the definition: a sydpoint of a non-null side ab is a point s lying on the line ab which satisfies qa, s) = qb, s). These are analogs of midpoints, and share some of their properties. For example, the existence of sydpoints of a side ab requires q a, b) 1 to be a square, while for midpoints to exist Wildberger showed that 1 q a, b) must be a square; and sydpoints generally come in pairs, just as do midpoints in Cayley Klein geometries. Sydpoints allow us to extend much of triangle geometry to non-classical triangles with points inside and outside of the null conic. For example in Figure 1.2 we see a synthetic construction of the sydpoints s and r of the side ab, using the midpoints k and l of the side cs. We introduce Circumlinear coordinates to build up the Orthocenter hierarchy of a general hyperbolic triangle, including the Orthoaxis, Ortholines, the Double Triangle centers, and the five canonical points x, w, z, s and b on the Orthoaxis, introduced in [53]. These coordinates are created by starting with three points a 1, a 2 and a 3 of

13 1. Introduction 5 a triangle, and then sending them to e 1 = 1, 0, 0), e 2 = 0, 1, 0) and e 3 = 0, 0, 1) respectively via a projective transformation. With respect to this new basis, the original hyperbolic bilinear form is then given by a new matrix of the form 1 a b c 2 ε aε bc b ac C = a 1 c with adjugate D = aε bc b 2 ε c ab 1.1) b c ε b ac c ab a 2 1 where ε 2 = 1. This allows us to develop formulas for points and lines in terms of the variables a, b, c and ε. We use these to develop the Circumcenter hierarchy of a triangle, treating midpoints and sydpoints uniformly, discussing Circumlines, Medians, Circumcentroids, Circumcircles, Sound points, Tangent points, Jay and Wren lines, and more. We are able to extend many of Wildberger s results and also discover new theorems, for example the existence of Sound conics. Figure 1.3: Constructing the twin circle D of C Twin circles are defined to be circles with the same center, whose quadrances sum to 2. Figure 1.3 shows a construction of twin circles C and D centered at c. These turn out to be intimately related to the notion of sydpoints. Surprising analogs of circumcircles may also be defined, involving the appearance of pairs of twin circles, yielding in general eight Circumcircles of a triangle, with interesting intersection properties. Figure 1.4 shows a triangle a 1 a 2 a 3 with two interior points and one exterior point, and so two midpoints and four sydpoints, and the four pairs of twin circles that act as circumcircles for it. In chapter four, we introduce a novel definition of a parabola into the framework of universal hyperbolic geometry sometimes we refer to a hyperbolic parabola). We begin by defining a parabola in terms of two points f 1 and f 2, called the foci, as the locus P 0 of a point p 0 satisfying qp 0, f 1 ) + qp 0, f 2 ) = 1.

14 1. Introduction 6 Figure 1.4: Circumcircles and CircumMeet points This is seemingly different from the Euclidean definition, but after introducing the directrices f 1 = F 1 and f 2 = F 2 respectively, this defining equation can be also be rewritten either as q p 0, f 1 ) = q p 0, F 2 ) or as q p 0, f 2 ) = q p 0, F 1 ). Over the rational numbers, there are quite a few different kinds of parabola, depending on the positions of the foci with respect to the null circle. We verify that the parabola is indeed a conic, and then introduce some of the basic definitions such as the axis line f 1 f 2 A with dual axis point a, vertices v 1 and v 2, the base points F 1 A = b 1 and F 2 A = b 2, with dual base lines b 1 = B 1 and b 2 = B 2. Associated to a point p 0 on P 0 is the tangent line P 0 to P 0 at a point p 0 with its dual tangent point p 0, and the dual line P 0 of p 0 with respect to the null conic C), which of course passes through p 0, as well as the focal lines R 1 f 1 p 0 and R 2 f 2 p 0 and the altitude base points t 1 R 1 F 1 and t 1 R 1 F 1. We give a construction for the hyperbolic parabola similar to the usual construction of the Euclidean parabola. In Figure 1.1 we see the parabola P 0 in red with foci f 1 and f 2, with typical point p 0 on it, along with these above points and lines. In this example we have q p 0, f 1 ) = 3.92 and q p 0, f 2 ) = 2.92, approximately. During the course of this investigation, many other points and lines will be studied! One of our aims in this thesis is to establish the main properties of a hyperbolic parabola in a uniform way across all types, with systematic and clear formulas for all main points and lines. The main idea for doing this is to allow flexibility in our field and to carefully choose an optimum coordinate framework. For this we utilize four important points associated to the parabola: a pair of opposite null points α 0, α 0 lying on P 0, and the vertices v 1, v 2.

15 1. Introduction 7 Figure 1.5: The four basis points v 1, v 2, α 0 and α 0 These four points α 0, α 0, v 1, v 2 will be sent to [1 : 1 : 1], [1 : 1 : 1], [0 : 0 : 1] and [1 : 0 : 0] by a suitable projective transformation. The main Parabola standard coordinates theorem then shows that the original hyperbolic bilinear form now has transformed matrix C = α α for some parameter α. This parameter plays a similar role as did the constants a, b, c and ε in the Circumlinear coordinates for a triangle: almost all subsequent formulas for points, lines and curves will involve α. Crucially, this system allows the parabola P 0 to have equation y 2 = xz so that it can easily be parametrized and worked with. As will become apparent, this choice of coordinates reveals many beautiful algebraic relations many of the formulas turn out to have interesting factorizations occurring in the coeffi cients. In Figure 1.5 we see the four main points, along with some pleasant collinearities associate to them. Figure 1.6: A focal chord p 0 q 0 with the polar point z on directrix

16 1. Introduction 8 At this point, we develop many results for the hyperbolic parabola which are analogous to the Euclidean case. These are too numerous to list here, but let us give a few representative examples. Figure 1.6 shows a focal chord p 0 q 0 of the parabola through f 2, with the corresponding tangent lines meeting perpendicularly at z on the directrix F 1. This is of course exactly the same phenomenon that occurs with a traditional parabola. Figure 1.7 addresses the delicate issue of how many normal lines pass through a given point l; the answer in the hyperbolic case is in general a maximum of four, rather than three in the Euclidean case, with the four points constructed with the aid of an auxiliary conic which we call the Four normal conic A l. This connect relates in an interesting way to the symmetric functions of four points. Figure 1.7: Four points on a parabola with given normal point In the fifth chapter, we look at further properties of the hyperbolic parabola which go beyond the Euclidean case. Prominently this includes discussion of the twin parabola P 0 : which is the locus of the twin point p 0 as p 0 moves on P 0, in other words the dual conic of P 0 with respect to the null circle. Figure 1.8: Twin foci and the twin parabola

17 1. Introduction 9 The symmetry between P 0 and P 0 is a main aspect of the theory, and motivates many of our definitions and indeed notation for points and lines. Remarkably, the twin P 0 is also a parabola, with foci the sydpoints f 1 and f 2 of f 1 f 2! Figure 1.8 shows a number of pleasant collinearities that relate the two conics. In our standard coordinates the twin parabola has equation y 2 = 4α2 α 2 1) 2 xz. It is very interesting that there are alternative metrical definitions of the parabola P 0 : for instance it can also be viewed as part of the locus of midpoints between f 1 and F 2, or as the locus of midpoints between f 2 and F 1, that is q f 1 ), p 0 = q p0, t 2) and q f 2 ), p 0 = q p0, t 1). This latter situation is illustrated in Figure 1.9. Figure 1.9: Midpoint locus property of a parabola We then define associated points to a variant point p 0 on the parabola P 0 ; such as e points, m-points, ε points, q points, h points, γ points and µ points. Many collinearities are determined by previous points and lines already defined, and give rise to new points and lines. In Figure 1.10 we see the corresponding R-lines R 1, R 2 for p 0 and the corresponding R-lines for p 0, namely R 1 f 2 p 0 and R 2 f 1 p 0, with their dual points r 1 and r 2. Somewhat remarkably, the two sets of R lines for p 0 and p 0 meet at four null points: namely σ1 1 R 1R 1, σ1 2 R 1R 2, σ2 1 R 2R 1 and σ2 2 R 2R 2. We show also the dual lines of the σ points, which are necessarily null lines passing through the corresponding r points. Specializing these constructions for the special cases of p 0 = α 0, α 0 and using the various key points on the axis such as the foci, base points etc. yields an intricate

18 1. Introduction 10 Figure 1.10: The σ points canonical structure, featuring in particular the Y conic, passing through ten distinguished meets of lines. The Y conic carries additional interesting symmetry and gives rise to many more points, by considering diagonals obtained from these ten points. We introduce some metrical relations for the hyperbolic parabola in UHG. For example, let P 0 be a parabola and p 0 be a point lying on it. If m 1 and m 2 on P 0 are the perpendicular points of p 0 the meets of the dual line P 0 with P 0, then q f 1, m 2) q f 2, m 1) = q P 0 ). Our final results show some remarkable relations involving the Thaloids variants of circles in the UHG setting) associated to various pairs of canonical points on the axis A.

19 Chapter 2 Universal Hyperbolic Geometry and Projective Geometry 2.1 Classical hyperbolic geometry In this chapter we review classical hyperbolic geometry, then introduce universal hyperbolic geometry UHG), first pictorially via projective geometry, and then more analytically using projective linear algebra. In an effort to comprehend Euclid s axiomatic basis for geometry, Lobachevsky, Bolyai and Gauss discovered the concept of hyperbolic geometry by the first half of the 19th century. This is a non-euclidean geometry which goes beyond the parallel postulate of Euclid. As Milnor states in [26], Lobachevsky was the first mathematician to publish on hyperbolic geometry, where he showed the existence of a natural unit distance in this new geometry. In 1831, another work on non-euclidean geometry was published independently by Bolyai, while Gauss revealed that he had studied this subject some years earlier, see [28]. The theory was more fully developed by Beltrami, Klein and Poincaré who introduced models which make this geometry part of ordinary mathematics [40], [41]). In the 20th century, Einstein and Minkowski realized that an understanding of physical time and space necessitated understanding of more general geometries, both that of Riemann as well as non-euclidean geometry; both subjects then became important for many students of mathematics and physics in the following decades. With the work of Thurston and others, it has been recognized that the negatively-curved geometries, of which hyperbolic geometry is a prototype, are in some sense generic forms of geometry, and crucial for low-dimensional topology, see for example [9]. Non-Euclidean geometries also have applications in different fields such as complex variables, number theory, Lie theory, infinite discrete groups following the more recent work of Gromov) 11

20 2. Universal Hyperbolic Geometry and Projective Geometry 12 and even optics. Although Euclidean geometry has a single standard model, various models may describe hyperbolic geometry, prominently the Beltrami-Poincaré disk and upper half plane models, the Beltrami-Klein projective model, and the hyperboloid model. These models differ from each other in certain aspects and some properties are more evident in one rather than the other. The Beltrami-Poincaré models naturally connect to complex analysis, while the Beltrami-Klein model connects naturally to the projective geometry and more general Cayley-Klein geometries Beltrami-Poincaré model In this model the underlying space is an open disk ζ, the interior of a unit circle C : x 2 + y 2 = 1, in the complex plane C given by ζ = {x + iy : x 2 + y 2 < 1}. The points on the unit circle C itself are assumed to be points "at infinity" instead of being part of the hyperbolic plane: these points also are referred to in the literature sometimes as ideal points, omega points, vanishing points or null points whatever their name, they still play a key role in the theory. Lines of ζ are represented by either circular arcs, which are parts of circles orthogonal to C, or Euclidean lines which are diameters of C. The hyperbolic distance between any two given hyperbolic points a and b in this model is defined in terms of the complex structure by ) d a, b) = tanh 1 b a 1 ab. Measuring angles between hyperbolic lines in this model is the same as measuring Euclidean angles, making it a conformal model of the hyperbolic plane, see [11] and [39] Beltrami-Klein model The underlying space in this projective model is similar to the previous model; that is the open disk ζ in C, where now lines are actual Euclidean lines instead of being circular arcs. The hyperbolic distance between two points a 1 and a 2 is d a 1, a 2 ) = 1 2 log R a 1, a 2 : a 3, a 4 ). where R a 1, a 2 : a 3, a 4 ) is the cross ratio of the points a 1, a 2 and the intersection points a 3, a 4 of the line a 1 a 2 and the null circle C. This model is not conformal, so the hyperbolic angles do not have the same measurement as the Euclidean angles, see [32] and [13].

21 2. Universal Hyperbolic Geometry and Projective Geometry 13 This model may be viewed also as an example of a Cayley-Klein geometry, where however the underlying space becomes the entire projective plane, with a distinguished conic, usually called the absolute, playing the same role as the unit circle does here The Beltrami-Poincaré upper half plane model This model can be obtained from the Beltrami-Poincaré disk model by a Cayley transformation. The underlying space of this model is the upper half plane H = {z C : Im z) > 0} in the complex plane C, and notions such as point and angle will remain the same as in the complex plane. For instance, the angle between two curves in H is the angle in C between the tangent lines to the two curves. Lines are either the intersection of H with a Euclidean line in C which is orthogonal to the x axis; or the intersection of H with a Euclidean circle whose center is on the x axis, see [3]. Given any two hyperbolic points a and b, we can measure the hyperbolic distance between them as ) d a, b) = cosh b a 1 ab The hyperboloid model There is another model coming from a three-dimensional vector space with quadratic form Q x, y, z)) x 2 + y 2 z ) Here the hyperbolic space is represented by the upper sheet of the hyperboloid of two sheets with equation x 2 + y 2 z 2 = 1, which turns out to be a Riemannian manifold with the induced metric. Lines or geodesics are given by intersections of the hyperboloid with planes through the origin. This model is quite similar in spirit to the usual spherical model of elliptic geometry, where antipodal points on a sphere are identified, see [3] and [31]. There are well-known projections from the hyperboloid model to the Beltrami- Klein and Beltrami-Poincaré disk models, see for example [39]. 2.2 Universal hyperbolic geometry Universal hyperbolic geometry UHG) is a new model of hyperbolic geometry introduced and developed by Wildberger in [49], [50], [51] and [52]. In this new model, the Beltrami-Klein model has been extended to the entire projective plane, so instead of working in the interior of a disk, we are allowed to consider exterior points including points at infinity), and also points on the boundary of the disk which are called null

22 2. Universal Hyperbolic Geometry and Projective Geometry 14 points. These play a very important role, at least as important as the interior and exterior points. The lines in this geometry are complete projective lines rather than straight line segments. All measurements and theorems ultimately are projective; and work more generally with an underlying projective plane together with a distinguish conic in it. The symmetry between points and lines of the projective plane is inherited by this hyperbolic geometry; the main measurements of quadrance and spread which replace the hyperbolic distance and the hyperbolic angle respectively become completely dual in nature. The introduction of these metrical concepts grants new perspectives to this geometry because it gives a purely algebraic approach to Cayley-Klein geometries, emphasizing a projective metrical formulation without transcendental functions, valid both inside and outside the usual null circle or absolute), and working over a general field, generally not of characteristic two. Because of the projective nature of the measurements, this geometry extends to the case of a general conic in the projective plane. From the point of view of the hyperboloid model, we are looking at all onedimensional and two-dimensional subspaces of the three-dimensional vector space with quadratic form 2.1); these form the points and lines of the geometry, and the quadrance and spread measurements are intimately and naturally linked to the associated bilinear form. From the point of view of projective linear algebra, it is straightforward to extend this to more general quadratic forms. Since projective geometry is thus key to understanding this model of hyperbolic geometry, the next section reviews some of the main facts for it Projective geometry Projective geometry arose as a result of attempting to present 3D figures in the plane properly; Renaissance artists were concerned about giving their drawings a more realistic resemblance similar to actual scenes. In the 17-th century, the work of artists had been expressed mathematically by Desargues but unfortunately that work was largely ignored for about two hundred years; perhaps due to the wide interest generated by the analytic geometry of Descartes and Fermat, which relates algebra to classical geometry. Desargues is considered the founder of the subject, although of course Pappus theorem had been discovered much earlier; in 1636 Desargues published a paper on perspective which gave the first view of projective geometry as an independent discipline. Pascal also contributed to the subject; in particular his celebrated theorem extending Pappus theorem to conics [41]. Projective geometry, also called the geometry of the straightedge because there is

23 2. Universal Hyperbolic Geometry and Projective Geometry 15 no need for a compass, is the study of properties which are invariant under projective transformations, such as incidence of points, concurrency of lines, and the cross ratio. On the other hand, some familiar and fundamental concepts from classical geometry are not preserved in projective geometry; for instance, the usual notion of parallelism has no meaning, since any two lines always meet at a point; and length between points and angles between lines are variable quantities under projection, so, they lose their applicability in this kind of geometry. Although quite different from the usual classical geometry, projective geometry has many advantages when dealing with the study and classification of curves; for example the exceptions to Bezout s theorem can been removed; in this becomes the foundational subject for modern algebraic geometry [41]. One of Wildberger s main points is that projective geometry is also the natural framework in which to develop hyperbolic geometry, see [49], [50], [51] and [52], and this is why the subject is important for us in this thesis. So far we dealt almost exclusively with situations in which only points and lines were involved. Large parts of classical Euclidean geometry deal of course also with constructions involving circles. While circles are not intrinsically a concept of projective geometry, Steiner showed that conic sections have a natural place in this framework. It is the purpose of this section to review some basic projective geometry and discuss a purely projective treatment of conics. This will form the basis of our treatment of Universal hyperbolic geometry; in fact the main aim of this thesis is to develop a holistic theory of the parabola in the setting of hyperbolic geometry. The projective plane The projective plane can be obtained from the usual affi ne plane by augmenting it by a new additional line called the line at infinity. This can be done by introducing new points called points at infinity. For every set of parallel lines, we add a new point which represents the meet of these parallel lines. Therefore, we can say that every two parallel lines meet at a point. More precisely, we can introduce the projective plane using linear algebra as follows; consider a fixed affi ne plane in three-dimensional space; and a fixed point O corresponding to the origin) not on this plane. Then each point on the plane is represented by a line which passes through O and through that point. The remaining lines through O which are parallel to the affi ne plane represent the points at infinity and the plane containing them is then the line at infinity. In other words, the projective plane is identified with the set of lines in space which pass through a fixed point, or equivalently as the one-dimensional subspaces of

24 2. Universal Hyperbolic Geometry and Projective Geometry 16 a vector space. Lines are then the two-dimensional subspaces of that vector space, which typically meet the fixed plane in a line. If our fixed plane is the plane z = 1 in x, y, z space, then we can specify a projective point by its homogeneous coordinates [x : y : z] with the convention that this is the same as [λx : λy : λz] where λ 0. A point [x : y : 1] represents an actual point on the affi ne plane, and a point at infinity has the form [x : y : 0]. Dually, a line is determined by the equation of a plane, such as lx + my + nz = 0 and so is represented by the proportion l : m : n which again is identical to λl : λm : λn where λ 0. The line at infinity is 0 : 0 : 1, while any other line is represented on the fixed affi ne plane as a usual line, see [20] and [32]. Given any two points a and b in the projective plane, there is then exactly one line ab passing through them both, and given any two lines L and M there is exactly one point LM which lies on both of them. At this point, the projective plane has no intrinsic metrical structure. However there is still one important quantity that can be measured! The symmetries of the projective plane in this model are just the linear transformations of the vector space, which naturally preserve both one and two dimensional subspaces, hence points and lines. So the projective group here is the projective general linear group, since two linear transformations which are multiples of each other give the same map on subspaces. The Cross Ratio The Cross ratio is a fundamental measurement in projective geometry, which has many applications in different directions with beautiful algebraic properties. As mentioned earlier, the usual length and angle are not invariant under projections, but the cross ratio compensates, and plays a big role in UHG. Let a 1, a 2, a 3, a 4 be four collinear projective) points on a projective) line L, and let λ 1, λ 1 ), λ 2, λ 2 ), λ 3, λ 3 ), λ 4, λ 4 ) be the corresponding affi ne parameters of these four points respectively with respect to another two points on that line. This means that in our three-dimensional vector space, we have chosen two basis vectors e and e of the two-dimensional subspace L, and that representative vectors of the four projective points can be chosen as v i = λ i e + λ i e for i = 1, 2, 3 and 4. Then the cross ratio of the projective) points a 1, a 2, a 3, a 4 is given by λ 1 λ 3 λ 2 λ 3 λ 1 λ 3 λ 2 λ 3 R a 1, a 2 ; a 3, a 4 ) = λ 1 λ / 4 λ 2 λ. 4 λ 1 λ 4 λ 2 λ 4 This is independent of the choice of basis vectors e and e.

25 2. Universal Hyperbolic Geometry and Projective Geometry 17 Dually, the cross ratio of four concurrent projective) lines can be defined in a similar way. An important fact is that if a line A meet concurrent lines A 1, A 2, A 3, A 4 in four distinct points a 1, a 2, a 3, a 4, then R A 1, A 2 ; A 3, A 4 ) = R a 1, a 2 ; a 3, a 4 ). In addition, if four collinear points a 1, a 2, a 3, a 4 are projective with four collinear points a 1, a 2, a 3, a 4, then R a 1, a 2 ; a 3, a 4 ) = R a 1, a 2 ; a 3, a 4 ). We call four collinear points a 1, a 2, a 3, a 4 a harmonic range when R a 1, a 2 ; a 3, a 4 ) = 1. In such a case a 1 and a 2 are harmonic conjugates with respect to the pair a 3, a 4 and vice versa. Quadrangles and quadrilaterals In projective geometry, four points such that no three of them are collinear are called a quadrangle, while four lines such that no three are concurrent are called a quadrilateral. In a quadrangle, there exist six lines joining the four points, called the sides of the quadrangle. Any two of these sides which do not have a common point from the quadrangle are called opposite sides and their meet is a new point called a diagonal point. Since we have three pairs of opposite sides, there are three diagonal points, which also give a rise to three new lines called diagonal lines. It is a well known fact that the diagonal points are never collinear. Dually, in a quadrilateral, the four lines meet in six points, called the vertices of the quadrilateral. Any two of these points which do not lie on a common line from the quadrangle are called opposite vertices and their join is a new line called a diagonal line. Since we have three pairs of opposite vertices, there are three diagonal lines, which also give a rise to three new points called diagonal points. In Figure 2.1, the points a 1, a 2, a 3 and a 4 form a quadrangle with diagonal triangle x 1 x 2 x 3. The diagonal lines intersect the six sides in six new distinct points y i, i = 1,..., 6. We get harmonic ranges along each side, for example Ra 1, a 2 ; x 2, y 2 ) = 1. In addition the six points y i are themselves lying on four lines, forming the opposite quadrilateral to the original quadrangle. Figure 2.1: A quadrangle and some associated points and lines

26 2. Universal Hyperbolic Geometry and Projective Geometry 18 The Fundamental Theorem of Projective Geometry Suppose that we are given four projective points a = [a 1 : a 2 : a 3 ], b = [b 1 : b 2 : b 3 ], c = [c 1 : c 2 : c 3 ] and d = [d 1 : d 2 : d 3 ] no three collinear) and we are interested in sending them to the projective points a = [a 1 : a 2 : a 3 ], b = [b 1 : b 2 : b 3 ], c = [c 1 : c 2 : c 3 ], d = [d 1 : d 2 : d 3 ] no three collinear). The fundamental theorem of projective geometry asserts that there exists a unique linear transformation which sends the points a, b, c, d to the points a, b, c, d respectively. This can be done by firstly, mapping the points e 1 = [1 : 0 : 0], e 2 = [0 : 1 : 0], e 3 = [0 : 0 : 1] to points a, b, c using the transformation a 1 a 2 a 2 T v) = vm where M = b 1 b 2 b 3. In fact, there are infinitely many such transformations since any projective point a can be represented by λa where λ 0. c 1 c 2 c 3 This means that the image of any fourth point is changing depending on the representatives of the three projective points a, b, c. To make things under our control, we shall choose specific representative for a, b, c by mapping the point e 4 = [1 : 1 : 1] to the point d using T v). So, we have that αa 1 αa 2 αa 2 1, 1, 1) βb 1 βb 2 βb 3 = d 1, d 2, d 3 ) γc 1 γc 2 γc 3 which yields a system of linear equations in the three variables α, β, γ. Solving this for α, β, γ gives us the required transformation T v) = va mapping e 1, e 2, e 3, e 4 to a, b, c, d respectively. Now, if we follow the same technique, the points e 1, e 2, e 3, e 4 can be sent to a, b, c, d respectively by new transformation T v) = va where α a 1 α a 2 α a 2 A = β b 1 β b 2 β b 3. γ c 1 γ c 2 γ c 3 Therefore, the transformation T T 1 v) = v A 1 A ) sends the four points a, b, c, d to the points a, b, c, d respectively, see [11] and [56]. This theorem is very useful in projective geometry since calculation using the four standard points is often more elegant and effi cient. We will use this theorem to change coordinates appropriately in several places in this thesis UHG from a synthetic projective view Universal Hyperbolic Geometry can be approached from either a synthetic projective geometry view or an analytic linear algebra point of view; both are useful, and they

27 2. Universal Hyperbolic Geometry and Projective Geometry 19 shed light on each other. In this section we give a synthetic introduction useful for dynamic geometry packages such as GSP, C.a.R., Cabri, GeoGebra and Cinderella. We work in the projective plane over a field, which in our pictures will be the rational numbers, with a distinguished conic, called the null circle, but elsewhere also the absolute. In our pictures, this will be the familiar unit circle, always in blue, with points lying on it called null points. We may consider this to be the affi ne circle X 2 + Y 2 = 1 or the projective version x 2 + y 2 z 2 = 0. The key duality, or polarity, between points and lines induced by the null circle which goes back to Apollonius) allows a notion of perpendicularity: two points a and b are perpendicular, written a b, precisely when b lies on the dual a of a, or conversely a lies on the dual b of b these are equivalent. Similarly two lines L and M are perpendicular, written L M, precisely when L passes through the dual of M, or conversely M passes through the dual of L. In Figure 2.2 we see a construction for the dual of a point d; this is the line D formed by the other two diagonals n and m of any null quadrangle for which d is a diagonal point. Then d is perpendicular to any point on D nm, and any line through d is perpendicular to D. To construct the dual of a line L, take the meet of the duals of any two points on it. Figure 2.2: Reflection r m in m sends a to b The basic isometries in such a geometry are reflections in points or reflections in lines these two notions turn out to be the same). If m is not a null point, the reflection r m in m interchanges the two null points on any line through m, should there be such. In Figure 2.2 for example, r m interchanges x and w, and interchanges y and z. It is then a remarkable and fundamental fact that r m extends to a projective transformation: to find the image of a point a, construct any line through a which meets the null circle at two points, say x and y, then find the images of x and y under r m, namely w and z, and then define r m a) = b am) wz) as shown. Perpendicularity of both points and lines is preserved by r m. The notion of reflection allows us to define midpoints without metrical measurements: if r m a) = b then we may say that m is a midpoint of the side ab. To construct the midpoints of a side ab, when they exist this is essentially a quadratic

28 2. Universal Hyperbolic Geometry and Projective Geometry 20 condition), we essentially invert the above construction. Figure 2.3: Constructing midpoints m and n of the side ab Recall from [51] that a side a 1 a 2 = {a 1, a 2 } is a set of two points and a vertex L 1 L 2 = {L 1, L 2 } is a set of two lines. Figure 2.3 shows two situations where we can construct midpoints m and n of the side ab, at least approximately over the rational numbers, which is the orientation of Geometer s Sketchpad and other dynamic geometry packages. In the left diagram, we take the dual c of the line ab, and if the lines ac and bc meet the null circle we take the other two diagonal points of this null quadrangle. This is also the case in Figure 2.2. In the right diagram, the lines ac and bc do not meet the null circle, but the dual lines A and B of a and b, which necessarily pass through c, do meet the null circle in a quadrangle, whose other diagonal points are the required midpoints m and n. Figure 2.4: Circles centered at a, for an interior point To define a circle C in this projective setting, suppose that c and p are points; then the locus of the reflections r x p) as x runs along the dual line of c is the circle with center c through p. This projective definition immediately gives a correspondence between a circle and a line. Of course there is also a metrical definition, once we have set up quadrance and spread. In Figure 2.4 we see examples of hyperbolic circles centered at a, for two different choices of a inside the null circle. These appear in our diagrams as ellipses, parabolas or hyperbolas.

29 2. Universal Hyperbolic Geometry and Projective Geometry 21 Figure 2.5: Circles centered at a, for an exterior point In Figure 2.5 we see examples of hyperbolic circles centered at a, for a choice of a outside the null circle. The circles that result here are usually called curves of constant width in classical hyperbolic geometry, at least for those inside the null circle. Notice that all such circles are tangent to the null circle at the two points where the dual line a of the center meets the null circle. The numbers on these diagrams are for future reference; they are the quadrances of the circles. These two images are taken from [49] by permission. Figure 2.6: Two Thaloids, with diameters ab and cd There is a closely related conic to a circle in universal hyperbolic geometry that seems not to have been too much studied. It was introduced in Wildberger [53]. Given two points a and b, the Thaloid with diameter ab is the locus of the point x satisfying ax bx. In Figure 2.6 we see two Thaloids, with diameters ab and cd. Thaloids have some properties of circles. Now we show how the main metrical notions of quadrance and spread introduced by Wildberger can be framed purely projectively. In the Beltrami-Klein model, the hyperbolic distance between two points a and b may be defined by a log, together with an absolute value of a cross ratio of the points a, b and the intersection points of the line ab and the null circle. However in universal hyperbolic geometry, the quadrance

30 2. Universal Hyperbolic Geometry and Projective Geometry 22 between two points a and b may be defined to be the cross ratio qa, b) = Ra, d : b, c) where c ab) a and d ab) b. This is the content of the Quadrance as Quadrance cross ratio theorem of [51]. This hyperbolic quadrance is preferable to the usual Figure 2.7: Defining quadrance and spread via cross ratios hyperbolic distance in several ways. First of all for two general points a, b the existence of the conjugate points c, d is always guaranteed, and so the quadrance extend to general points. In addition, there is no ambiguity of ordering the four points when applying the cross-ratio, while this is not true in the Beltrami Klein model. It does not have any complicated transcendental function such as a logarithm or inverse circular functions. This is much more easier analytically, and allows the theory to extend to finite fields. In addition, the purely projective nature of the cross ratio guarantees that the same results will hold if we apply a projective transformation to change our distinguished conic into a more general one. The quadrance takes negative values when a and b are both interior or exterior to the null circle, while it is positive when one of a or b is interior and the other is exterior. Dually, we define the spread between the lines A and B to be the cross-ratio S A, B) R A, D : B, C) where C and D are the dual lines of the conjugate points c and d. Notice that Figure 2.7 shows important points and lines used in both the quadrance and spread definitions. From basic facts about a cross-ratio, q a, b) = S A, B). This means that the fundamental projective duality between points and lines extends to the metrical notions of quadrance and spread. Note also that it is not necessary for the line ab to meet the null circle, in fact these metrical notions are valid for all points and lines, except when null points or null lines are involved, when the cross-ratio becomes infinite.

31 2. Universal Hyperbolic Geometry and Projective Geometry 23 Recall from [51], that it is possible to translate the theorems in UHG to formulas of classical hyperbolic trigonometry in the special case of interior points and lines using the following relations; q a, b) = sinh 2 d a, b)) and S A, B) = sin 2 θ A, B)). Hence all formulas in this thesis can be interpreted and reformulated in case the points and lines are interior to the null circle, yielding relations between classical distances and angles via these relations, if such are desired. While we could proceed with this synthetic point of view, we prefer to work in an analytic environment using projective linear algebra Metrical projective linear algebra While the synthetic framework is attractive, for explicit computations and formulas it is useful to work with analytic geometry in the context of projective) linear algebra. We now proceed to explain this. We begin with establishing some notation and basic results in the affi ne setting, although the projective setting is the main interest. The three-dimensional vector space V = F 3 over a field F, of characteristic ) not two, consists of row vectors v = x, y, z) or equivalently 1 3 matrices x y z. A metrical structure is determined by a symmetric bilinear form v u = vu vcu T 2.2) where C is an invertible symmetric 3 3 matrix. Note in particular our use of the algebraic notation vu. The dual vector space V may be viewed as column vectors f = l, m, n) T or equivalently 3 1 matrices. Vectors v, u are perpendicular precisely when v u = vu = 0. The affi ne) quadrance of a vector v is the number Q v v v = v 2. A vector v is null precisely when Q v = v 2 = 0. A variant of the following also appears in [25]. Theorem 1 Parallel vectors) If vectors v and u are parallel then Q v Q u = vu) ) Conversely if 2.3) holds then either v and u are parallel, or the bilinear form restricted to the span of v and u is degenerate. Proof. Consider a two-dimensional space containing ) v and u and the bilinear form restricted to it, given by a matrix C a b = with respect to some basis. If in this b c

32 2. Universal Hyperbolic Geometry and Projective Geometry 24 basis v = x 1, y 1 ) and u = x 2, y 2 ), then we may calculate that Q v Q u vu) 2 x 1 y 2 y 1 x 2 ) 2 ac b 2) = ) ax bx 2 y 2 + cy2 2 2 ax bx 1 y 1 + cy1) 2 2. So if v and u are parallel, the left hand side is zero, and conversely if the left hand side is zero, then either ac b 2 = 0 in which case the bilinear form restricted to the span of v and u is degenerate, or x 1 y 2 y 1 x 2 = 0, meaning that the vectors v and u are parallel. The previous result motivates the following measure of the non-parallelism of two vectors. The affi ne) spread between non-null vectors v and u is the number s v, u) 1 vu)2 Q v Q u. The spread is unchanged if either v or u are multiplied by a non-zero number. One-dimensional and two-dimensional subspaces of V = F 3 may be viewed as the basic objects forming the projective plane, with metrical notions coming from the affi ne notions of quadrance and spread in the associated vector space, but we prefer to give independent definitions so that logically neither the affi ne nor projective settings have priority. In general our notation in the projective setting is opposite to that in the affi ne setting, in the sense that the roles of small and capital letters are reversed throughout. Recall that vectors and matrices are considered to be the main objects in ordinary linear algebra in three-dimensional space the only dimension we deal with here), and we obtain different objects when multiply a vector or matrix by a scalar. In projective linear algebra, when multiplying a vector or matrix by a scalar we obtain the same objects meaning that vectors and matrices are defined only up to non-zero scalar multiples. Here, in order to differentiate between ordinary linear algebra and projective linear algebra, round brackets will be used for the usual vectors and matrices, and square brackets will be used to indicate vectors and matrices in the projective setting. Notice that the operations of addition and subtraction of projective vectors or matrices are undefined, while the operations of multiplication, transposes and inverses are welldefined. Pleasantly, computing inverses in the projective setting is very easy since common denominators can be scaled away. In particular over the rational numbers, integer arithmetic is usually enough to deal with projective linear algebra. A projective) point is a proportion [ a ] = [x : y : z] in square brackets, or equivalently a projective row vector a = x y z, where the square brackets in the latter are interpreted projectively: when multiplying by a non-zero number this is unchanged. A projective) line is a proportion L = l : m : n in pointed brackets, or equivalently

33 2. Universal Hyperbolic Geometry and Projective Geometry 25 a projective column vector l L = m. n When the context is clear, projective points and projective lines are referred to simply as points and lines. The incidence between the point a = [x : y : z] and the line L = l : m : n is given by the relation al = [ ] l x y z m = lx + my + nz = 0. In such a case we say a lies on L, or L passes through a. n The join a 1 a 2 of distinct points a 1 [x 1 : y 1 : z 1 ] and a 2 [x 2 : y 2 : z 2 ] is the line a 1 a 2 [x 1 : y 1 : z 1 ] [x 2 : y 2 : z 2 ] y 1 z 2 y 2 z 1 : z 1 x 2 z 2 x 1 : x 1 y 2 x 2 y ) This is the unique line passing through a 1 and a 2. The meet L 1 L 2 of distinct lines L 1 l 1 : m 1 : n 1 and L 2 l 2 : m 2 : n 2 is the point L 1 L 2 l 1 : m 1 : n 1 l 2 : m 2 : n 2 [m 1 n 2 m 2 n 1 : n 1 l 2 n 2 l 1 : l 1 m 2 l 2 m 1 ]. This is the unique point lying on L 1 and L ) Three points a 1, a 2, a 3 are collinear precisely when they lie on a line L; in this case we will also write L = [[a 1 a 2 a 3 ]]. Similarly three lines L 1, L 2, L 3 are concurrent precisely when they pass through a point a; in this case we will also write a = [[L 1 L 2 L 3 ]]. It will be convenient to connect the affi ne) and projective frameworks by the following conventions. If v = x, y, z) = x y z is a vector, then a = [v] = [x : y : z] = [ ] x y z is the associated projective point, and v is a representative vector [ T for a. If f = l, m, n) T is a dual vector, then L = [f] = l : m : n = l m n] is the associated projective line, and f is a representative dual vector for L. Projective quadrance and spread If C is a symmetric invertible 3 3 matrix, with entries in F, and D is its adjugate matrix the inverse, up to a multiple), then we denote by C and D the corresponding projective matrices, each defined up to a non-zero multiple. This pair of projective matrices determine a metrical structure on projective points and lines, as follows. The projective) points a 1 and a 2 are perpendicular precisely when a 1 Ca T 2 = 0, written a 1 a 2. This is a symmetric relation, and is well-defined. Similarly

34 2. Universal Hyperbolic Geometry and Projective Geometry 26 projective) lines L 1 and L 2 are perpendicular precisely when L T 1 DL 2 = 0, written L 1 L 2. The point a and the line L are dual precisely when L = a Ca T or equivalently a = L L T D. 2.6) Then two points are perpendicular precisely when one is incident with the dual of the other, and similarly for two lines. So a 1 a 2 precisely when a 1 a 2, because of the projective relation ) Ca T T ) 1 D Ca T 2 = a1 C T ) D Ca T ) 2 = a1 CD) Ca T ) 2 = a1 Ca T 2. A point a is null precisely when it is perpendicular to itself, that is, when aca T = 0, and a line L is null precisely when it is perpendicular to itself, that is, when L T DL = 0. absolute. The null points determine the null conic, sometimes also called the Hyperbolic and elliptic geometries arise respectively from the special cases C = J = D and C = I = D. 2.7) In the hyperbolic case, which is the main object of interest for us in this thesis, and which forms the basis for almost all examples, the point a = [x : y : z] is null precisely when x 2 + y 2 z 2 = 0 and dually the line L = l : m : n) is null precisely when l 2 + m 2 n 2 = 0. This is the reason we can picture the null circle in affi ne coordinates X x/z and Y y/z as the blue) circle X 2 + Y 2 = 1. Note that in the elliptic case the null circle, over the rational numbers, has no points lying on it. This is why visualizing hyperbolic geometry is often easier than elliptic geometry! In the general setting, the bilinear forms determined by C and D can be used to define the metrical structure in the associated projective plane. The dual notions of projective) quadrance q a 1, a 2 ) between points a 1 and a 2, and projective) spread S L 1, L 2 ) between lines L 1 and L 2, are q a 1, a 2 ) 1 a1 Ca T ) 2 2 ) ) a1 Ca T 1 a2 Ca T 2 and ) L T 2 S L 1, L 2 ) 1 1 DL 2 ) ). L T 1 DL 1 L T 2 DL 2 2.8)

35 2. Universal Hyperbolic Geometry and Projective Geometry 27 While the numerators and denominators of these expressions depend on choices of representative vectors and matrices for a 1, a 2, C, L 1, L 2 and D, the quotients are independent of scaling, so the overall expressions are indeed well-defined projectively. If a 1 = [v 1 ], a 2 = [v 2 ], and L 1 = [f 1 ], L 2 = [f 2 ], then we may write q a 1, a 2 ) = 1 v 1 v 2 ) 2 v 1 v 1 ) v 2 v 2 ) and S L 1, L 2 ) = 1 f 1 f 2 ) 2 f 1 f 1 ) f 2 f 2 ) where we use 2.2) and introduce the dual bilinear form on column vectors by f 1 f 2 f T 1 Df 2. Clearly q a, a) = 0 and S L, L) = 0 for any point a and any line L, while q a 1, a 2 ) = 1 precisely when a 1 a 2, and dually S L 1, L 2 ) = 1 precisely when L 1 L 2. Then using 2.6) we see that ) S a 1, a 2 = q a 1, a 2 ). In [50], Wildberger showed that these metrical notions agree with the projective formulation using suitable cross ratios that we described in the previous section. Example 2 In the hyperbolic case, the quadrance between points and spread between lines are given by essentially similar formulas: q [x 1 : y 1 : z 1 ], [x 2 : y 2 : z 2 ]) = 1 S l 1 : m 1 : n 1, l 2 : m 2 : m 2 ) = 1 x 1 x 2 + y 1 y 2 z 1 z 2 ) 2 x y1 2 ) z2 1 x y ) 2) z2 l 1 l 2 + m 1 m 2 n 1 n 2 ) 2 l m 2 1 ) n2 1 l m 2 2 2). n2 Example 3 The Figures 2.5 and 2.4 show quadrances from a fixed point a in the hyperbolic plane. It is interesting to observe that if a is interior to the null circle over the rational numbers, then q a, x) can take on negative values, or values greater than or equal to one, but not values in the range 0, 1) Hyperbolic trigonometry in UHG The following formula, introduced in [48], is given in a more general setting in [49]. We will give a proof which utilizes our basic strategy of setting up appropriate coordinates. Theorem 4 Projective Triple quad formula) Suppose that a 1, a 2, a 3 are collinear points, with quadrances q 1 q a 2, a 3 ), q 2 q a 1, a 3 ) and q 3 q a 1, a 2 ). Then q 1 + q 2 + q 3 ) 2 = 2 q1 2 + q2 2 + q3) 2 + 4q1 q 2 q )

36 2. Universal Hyperbolic Geometry and Projective Geometry 28 Proof. We may assume at least two of the points distinct, as otherwise the relation is trivial. Suppose that representative vectors are then v 1,v 2 and v 3 kv 1 + lv 2, with v 1 and v 2 linearly independent. Consider just the two-dimensional subspace spanned by v 1 and v 2. The bilinear form restricted to the subspace ) spanned by the ordered basis v 1,v 2 is given by some symmetric matrix C a b =. Then in this basis v 1 = 1, 0), b c v 2 = 0, 1) and v 3 = k, l), and we may compute that q 3 = s v 1, v 2 ) = and q 1 = s v 2, v 3 ) = Then 2.10) is an identity. ac b2 ac k 2 ac b 2) c ak 2 + 2bkl + cl 2 ). q 2 = s v 1, v 3 ) = l 2 ac b 2) a ak 2 + 2bkl + cl 2 ) Here are a few useful consequences of the Triple quad formula. quadrances is q 3 = 1, then q 1 + q 2 = 1; this is a consequence of the identity q 1 + q 2 + 1) 2 2q 2 1 2q q 1 q 2 = q 1 + q 2 1) 2. If one of the Also if two of the quadrances are equal, say q 1 = q 2 = r, then q 3 = 0 or q 3 = 4r 1 r); this is a consequence of the identity Here is a useful theorem that relates harmonic ranges defined projective and metrically. 2r + q 3 ) 2 4r 2 2q 2 3 4r 2 q 3 = q 3 q3 4r + 4r 2). Theorem 5 Quadrance cross ratio) Suppose that a, b, c, d are a harmonic range of points on a line L. Then q a, c) q b, c) = q a, d) q b, d). Proof. We know from projective geometry that a harmonic range of points a, b, c, d in the projective space can be realized as [v], [u], [αv + βu], [αv βu] for two vectors v and u and two scalars α and β. Then using again the short hand notation v 2 v v and uv = u v, we calculate that q [v], [αv + βu]) = 1 v αv + βu)) 2 v v) αv + βu) αv + βu)) = v2 α 2 v 2 + 2αβ uv) + β 2 u 2) αv 2 + βuv ) 2 v 2 α 2 v 2 + 2αβ uv) + β 2 u 2 ) β 2 u 2 v 2 uv) 2) = v 2 α 2 v 2 + 2αβ uv) + β 2 u 2 )

37 2. Universal Hyperbolic Geometry and Projective Geometry 29 and similarly It follows that q [u], [αv + βu]) = 1 u αv + βu)) 2 u u) αv + βu) αv + βu)) = u2 α 2 v 2 + 2αβ uv) + β 2 u 2) α uv) + βu 2) 2 u 2 α 2 v 2 + 2αβ uv) + β 2 u 2 ) α 2 u 2 v 2 uv) 2) = u 2 α 2 v 2 + 2αβ uv) + β 2 u 2 ). q a, c) q b, c) = q [v], [αv + βu]) q [u], [αv + βu]) = β2 u 2 α 2 v 2. But this quantity is then unchanged if we replaced α with α, or β with β. Here now is a more complete list of the main trigonometric laws in the subject, taken from Wildberger s paper UHG I [51]. These completely algebraic rules incorporate and extend the more familiar, transcendental ones found in the Klein and Poincaré models. Recall also from [51] that a triangle a 1 a 2 a 3 = {a 1, a 2, a 3 } is a set of three points which are not collinear and trilateral L 1 L 2 L 3 = {L 1, L 2, L 3 } is a set of three lines which are not concurrent. Every triangle a 1 a 2 a 3 has three sides, namely a 1 a 2, a 2 a 3 and a 1 a 3, and similarly any trilateral L 1 L 2 L 3 has three vertices, namely L 1 L 2, L 2 L 3 and L 1 L 3. Theorem 6 Triple quad formula) If a 1, a 2 and a 3 are collinear points then q 1 + q 2 + q 3 ) 2 = 2 q q q 2 3) + 4q1 q 2 q 3. Theorem 7 Triple spread formula) If L 1, L 2 and L 3 are concurrent lines then S 1 + S 2 + S 3 ) 2 = 2 S S S 2 3) + 4S1 S 2 S 3. Theorem 8 Pythagoras) If L 1 and L 2 are perpendicular lines then q 3 = q 1 + q 2 q 1 q 2. Theorem 9 Pythagoras dual) If a 1 and a 2 are perpendicular points then S 3 = S 1 + S 2 S 1 S 2. Theorem 10 Spread law) For a triangle a 1 a 2 a 3 with quadrances q 1, q 2, q 2 and spreads S 1, S 2, S 3 S 1 q 1 = S 2 q 2 = S 3 q 3.

38 2. Universal Hyperbolic Geometry and Projective Geometry 30 Theorem 11 Cross law) For a triangle a 1 a 2 a 3 with quadrances q 1, q 2, q 2 and spreads S 1, S 2, S 3 q 1 q 2 S 3 q 1 + q 2 + q 3 ) + 2) 2 = 4 1 q 1 ) 1 q 2 ) 1 q 3 ). Theorem 12 Cross dual law) For a triangle a 1 a 2 a 3 with quadrances q 1, q 2, q 2 and spreads S 1, S 2, S 3 S 1 S 2 q 3 S 1 + S 2 + S 3 ) + 2) 2 = 4 1 S 1 ) 1 S 2 ) 1 S 3 ). Theorem 13 Menelaus ) Suppose that a 1 a 2 a 3 is an arbitrary triangle and let b 1, b 2 and b 3 three points on a 2 a 3, a 1 a 3 and a 1 a 2 respectively, then b 1, b 2 and b 3 are collinear if and only if qa 1, b 3 ) qa 2, b 1 ) qa 3, b 2 ) qb 3, a 2 ) qb 1, a 3 ) qb 2, a 1 ) = 1. Theorem 14 Napier s Rules) Suppose that a 1 a 2 a 3 is a right triangle which has quadrances q 1,q 2 and q 3, and spreads S 1, S 2 and S 3 = 1. Then knowing any two of the five quantities S 1, S 2, q 1, q 2 and q 3 will be suffi cient in order to determine the other three, only by the three basic equations from Thales theorem and Pythagoras theorem: S 1 = q 1 q 3, S 2 = q 2 q 3 and q 3 = q 1 + q 2 q 1 q 2.

39 Chapter 3 Sydpoints and hyperbolic triangle geometry 3.1 Introduction In this chapter we introduce the new notion of the sydpoints of a side in hyperbolic geometry, and use it to continue and extend the study of hyperbolic triangle geometry. This is parallel to, but with different features to the Euclidean case laid out in [23] and [24], and in a related but different direction from [45], [46] and [47]. The sydpoints s of a side ab are analogous and somewhat complementary to the more familiar notion of midpoints m; and allow much of the triangle geometry introduced by Wildberger in [53] to extend to triangles with some interior and exterior points. In particular we are able to define the idea of twin circumcircles of a triangle. We will introduce circumlinear coordinates to build up the Circumcenter hierarchy of a triangle, treating midpoints and sydpoints uniformly. Many formulas become algebraically elegant. While the present chapter only treats sydpoints and triangle geometry, it may appear unrelated to the main study of parabola, but when we do begin the study of parabola in the next chapter, we will see that sydpoints play a crucial role. Furthermore, the twin symmetry appearing here has a clear parallel with the notion of twin parabolas investigated in the final chapter. In [53] it was shown that if each of the three sides of a triangle in UHG) has midpoints m, then these six points lie three at a time on four circumlines C, whose duals are the four circumcenters c. These are the centers of the four circumcircles which pass through the three points of the triangle. This is shown for a classical triangle inside the null circle in Figure 3.1. The duals of the midpoints m are the midlines M traditionally called the perpendicular bisectors of the sides. While the red circumcircle is a classical circle in the Cayley- 31

40 3. Sydpoints and hyperbolic triangle geometry 32 Figure 3.1: Midpoints, Midlines, Circumlines, Circumcenters and Circumcircles Beltrami-Klein model of hyperbolic geometry, the other three are usually described as curves of constant width, but in UHG they are all just circles. This is the start of the Circumcenter hierarchy as studied in [53]. Remarkably, much of this extends also to triangles with points both interior and exterior to the null circle, in the process of studying this we also find new phenomenon relating to circumcircles that suggest a reconsideration of the classical case above. The fundamental metrical notion between points in UHG is, as we have seen, the quadrance q, and a midpoint of ab is a point m on ab satisfying q a, m) = q b, m). Our key new concept is the following: a sydpoint of ab is a point s on ab satisfying q a, s) = q b, s). As shown in [51] the existence of midpoints is equivalent to 1 q a, b) being a square in the field. We will see that the existence of sydpoints is equivalent to q a, b) 1 being a square. As with midpoints, if sydpoints exist there are generally two of them. Figure 3.2: A non-classical triangle with both midpoints and sydpoints In Figure 3.2, the non-classical triangle a 1 a 2 a 3 has one side a 1 a 2 with midpoints

41 3. Sydpoints and hyperbolic triangle geometry 33 m whose duals are midlines M, and two sides a 1 a 3 and a 2 a 3 with sydpoints s whose duals are sydlines S. Somewhat remarkably, the six midpoints and sydpoints lie three at a time on four circumlines C, whose duals are the four circumcenters c. The connection between these new circumcenters and the idea of circumcircles is particularly interesting, since in this case it is impossible to find any circles which pass through all three points of the triangle a 1 a 2 a 3. In UHG circles can often be paired: two circles are defined to be twins if they share the same center and their quadrances sum to 2. The circumcenters c are the centers of twin circumcircles passing through collectively the three points of the triangle. This notion extends our understanding even in the classical case. The four pairs of twin circumcircles give eight generalized circumcircles even for the classical case), and these meet in a surprising way in the CircumMeet points, some of which pleasantly depend only on the side of the triangle on which they lie. Figure 3.3: Four twin circumcircles of a non-classical triangle In Figure 3.3 we see the twin circumcircles of the triangle of the previous Figure; some of these appear in this model as hyperbolas tangent to the null circle these are invisible in classical hyperbolic geometry, but have a natural interpretation in terms of hyperboloids of one sheet in three-dimensional space DeSitter space). The other main contribution of this chapter to hyperbolic triangle geometry is in setting up circumlinear coordinates. UHG is more algebraic than the classical theory, emphasizing a projective metrical formulation without transcendental functions for Cayley-Klein geometries, valid both inside and outside the usual null circle or absolute), and working over a general field, generally not of characteristic two. In [53], Wildberger studied triangle geometry in the more general setting of a projective plane over a field, with a metrical structure induced by a symmetric bilinear form on the associated three-dimensional vector space, or equivalently a general conic playing the

42 3. Sydpoints and hyperbolic triangle geometry 34 role of the null circle or absolute. That paper focussed on ortholinear coordinates, and gave derivations for many initial constructions in the Incenter hierarchy, and only dual statements for the corresponding results for the Circumcenter hierarchy. In this chapter we introduce the complementary circumlinear coordinates, which are well suited for studying midpoints and sydpoints simultaneously. Finding formulas for key points and lines is, as always, a main aim. If the triangle a 1 a 2 a 3 has either midpoints or sydpoints for each of its sides, a change of coordinates allows us to write a 1 = [1 : 0 : 0], a 2 = [0 : 1 : 0] and a 3 = [0 : 0 : 1], with the bilinear form given by a matrix 1 a b C = a 1 c 3.1) b c ε where ε 2 = ±1. We reformulate formulas of the Orthocenter hierarchy of [53] using circumlinear coordinates, including Wildberger s Orthoaxis A with his five important points h, s, b, x and z, and then turn to the Circumcenter hierarchy, studying Medians, Centroids, CircumCentroids, CircumDual points, Tangent lines, Jay lines, Wren lines, CircumMeet points and some new associated points and lines, and finish with a nice correspondence between the Circumcenters and four Sound conics passing two at a time through the twelve Sound points. Note that when we study a particular triangle, we adopt the convention of Capitalizing major points and lines of that Triangle Midpoints of a side Recall from [51] that a midpoint of a non-null side ab is a point m lying on ab which satisfies q a, m) = q b, m). We exclude null sides because the quadrance between any two points on such a side is 0. We now restate the midpoints theorem in [51]. Theorem 15 Side midpoints) Suppose that a and b are distinct non-null points and ab is a non-null side. Then ab has a midpoint precisely when the quantity 1 qa, b) is a square number. In this case we may find representative vectors v and u for a and b respectively satisfying v 2 = u 2, and then there are exactly two midpoints of ab, namely m = [u + v] and n = [u v]. These two midpoints are perpendicular. Furthermore a, m, b, n form a harmonic range. Proof. Suppose that a = [v] and b = [u] so that 1 qa, b) = v u)2 Q v Q u = vu)2 v 2 u )

43 3. Sydpoints and hyperbolic triangle geometry 35 A general point m on ab has representative non-zero vector w = kv + lu. We first compute v w) 2 q a, m) = 1 v v) w w) = 1 v kv + lu)) 2 v 2 kv + lu) kv + lu)) k 2 v 2) 2 + 2lkv 2 vu) + l 2 vu) 2) = 1 v 2 k 2 v 2 + 2lk uv) + l 2 u 2 ) u w) 2 q b, m) = 1 u u) w w) = 1 u kv + lu)) 2 u u) kv + lu) kv + lu)) k 2 uv) 2 + 2lku 2 uv) + l 2 u 2) ) 2 = 1 u 2 k 2 v 2 + 2lk uv) + l 2 u 2 ). So the equation q a, m) = q b, m) means u 2 k 2 v 2) 2 + 2lkv 2 vu) + l 2 vu) 2) = v 2 k 2 uv) 2 + 2lku 2 vu) + l 2 u 2) 2 ) or k 2 v 2) 2 u 2 + l 2 uv) 2 u 2 = k 2 uv) 2 v 2 + l 2 u 2) 2 v 2. This can be rearranged as v 2 u 2 uv) 2) k 2 v 2 l 2 u 2) = 0. If v 2 u 2 = vu) 2 then by the Parallel vectors theorem either v and u are parallel, which is impossible since a and b are distinct, or the bilinear form restricted to [v, u] is degenerate, which implies that the side ab is null. So a midpoint m exists precisely when k 2 v 2 = l 2 u 2. In this case since a and b are non-null, v 2 and u 2 are non-zero, so k and l are also, since by assumption w = kv + lu is non-zero, and we may renormalize v and u so that v 2 = u 2 by for example setting ṽ = kv and ũ = lu, and then replacing ṽ and ũ by v and u again). After this renormalization 1 qa, b) = vu) 2 / v 2) 2 is then a square, and there are two midpoints [v + u] and [v u]. Since v + u) v u) = v 2 u 2 = 0, these midpoints are perpendicular. It is well-known that for any two vectors v and u, the four lines [v], [v + u], [u], [v u] form a harmonic range. Conversely suppose that 1 qa, b) = vu) 2 / v 2 u 2) is a square, say r 2. Then the ratio of v 2 to u 2 is a square, so v and u can be renormalized so that v 2 = u 2, at which point the above calculations show that [v + u] and [v u] are both midpoints. Note that from 3.2) we can restate the condition for midpoints in terms of the quantity Q v Q u = v 2 u 2 being a square. We can also relate this to hyperbolic trigonometry as in [50]. If qa, b) = r 0, and m is a midpoint of the side ab with qa, m) =

44 3. Sydpoints and hyperbolic triangle geometry 36 qb, m) = q, then {r, q, q} satisfies the Triple quad formula. So as we observed earlier, r = 4q 1 q), and in particular 1 r = 1 4q 1 q) = 2q 1) 2 is a square number. The dual lines M and N of the midpoints m and n of a side are called the midlines of the side. Since m and n are perpendicular, these each pass through the other midpoint, and so might also be called the perpendicular bisectors of the side. The dual concept of a midpoint of a side is a biline, so a biline of a non-null vertex AB is a line L passing through AB which satisfies SA, L) = SB, L). By duality the vertex AB has a biline precisely when the quantity 1 SA, B) is a square number, and in this case we have exactly two bilines which are perpendicular. The symmetry between midpoints and bilines is reflected in the duality between the Incenter and Circumcenter hierarchies in UHG. This notion of symmetry is absent in classical hyperbolic geometry, since there we always have only one midpoint of a side and two bilines usually called angle bisectors); the number-theoretic considerations with the existence of these are generally invisible Sydpoints of a side We now come to the crucial definition. Definition 16 A sydpoint of a non-null side ab is a point s lying on ab which satisfies qa, s) = qb, s). Note both the similarities and differences between the following theorem and the Side midpoints theorem. Theorem 17 Side sydpoints) Suppose that a and b are distinct non-null points and ab is a non-null side. Then ab has a sydpoint precisely when qa, b) 1 is a square number. In this case we can find representative vectors v and u for a and b respectively satisfying v 2 = u 2, and then there are exactly two sydpoints of ab, namely s = [v + u] and r = [v u]. In such a case, a and b are also sydpoints of the side sr, and while s and r are not in general perpendicular, we do have qa, s) = qb, r) and qa, r) = qb, s). Furthermore a, s, b, r form a harmonic range. Proof. Suppose that a = [v] and b = [u] so that qa, b) 1 = vu)2 v 2 u )

45 3. Sydpoints and hyperbolic triangle geometry 37 A general point s = [w] on ab has representative vector w = kv + lu. Then following the notation and computations of the previous theorems, v w) 2 q a, s) = 1 v v) w w) = 1 v kv + lu)) 2 v 2 kv + lu) kv + lu)) k 2 v 2) 2 + 2lkv 2 vu) + l 2 vu) 2) = 1 v 2 k 2 v 2 + 2lk uv) + l 2 u 2 ) u w) 2 q b, s) = 1 u u) w w) = 1 u kv + lu)) 2 u u) kv + lu) kv + lu)) k 2 uv) 2 + 2lku 2 uv) + l 2 u 2) ) 2 = 1 u 2 k 2 v 2 + 2lk uv) + l 2 u 2 ). Suppose s is a sydpoint of ab. Then from the relation q a, s) = q b, s) we get k 2 v 2) 2 + 2lkv 2 vu) + l 2 vu) 2) k 2 uv) 2 + 2lku 2 uv) + l 2 u 2) ) 2 1 v 2 k 2 v 2 + 2lk uv) + l 2 u 2 ) = 1 + u 2 k 2 v 2 + 2lk uv) + l 2 u 2 ) or 2u 2 v 2 k 2 v 2 + 2lk uv) + l 2 u 2) u 2 k 2 v 2) 2 + 2lkv 2 vu) + l 2 vu) 2) = v 2 k 2 uv) 2 + 2lku 2 uv) + l 2 u 2) ) 2. This can be rearranged as k 2 u 2 v 2) 2 + l 2 u 2) 2 v 2 k 2 v 2 + l 2 u 2) uv) 2 = 0 or v 2 u 2 uv) 2) k 2 v 2 + l 2 u 2) = 0. If v 2 u 2 = vu) 2 then by the Parallel vectors theorem either v and u are parallel, which is impossible since a and b are distinct, or the bilinear form restricted to [v, u] is degenerate, which implies that the side ab is null. So a sydpoint s exists precisely when k 2 v 2 = l 2 u 2. In this case we may renormalize v and u so that v 2 = u 2, so that s [v + u] and r [v u] are sydpoints. If q a, s) = q b, s) = d, q a, r) = q b, r) = e and also q r, s) = f, then the Triple quad formula applied to {a, r, s} and {b, r, s} implies that both f + d + e) 2 = 2 f 2 + d 2 + e 2) +4fde and f d e) 2 = 2 f 2 + d 2 + e 2) +4fde which implies that f + d + e = ± f d e). Since f 0, we conclude that d = e, which shows that qa, s) = qb, r) and qa, r) = qb, s).

46 3. Sydpoints and hyperbolic triangle geometry 38 Now v + u) v u) = v 2 u 2 = 2v 2, so the two sydpoints s and r are not in general perpendicular. However v + u) 2 = v 2 + 2uv + u 2 = 2uv and v u) 2 = v 2 2uv + u 2 = 2uv so that v + u) 2 = v u) 2. By symmetry this implies that [v + u) + v u)] = [2v] = a and [v + u) v u)] = [ 2u] = b are sydpoints of rs. Note that from 3.3) we can restate the condition for sydpoints in terms of the quantity Q v Q u = v 2 u 2 being a square. For a fixed q 0 there is at most one sydpoint s of ab for which q a, s) = q; the other sydpoint r then satisfies q a, r) = q q since q is non-zero. Example 18 In the usual hyperbolic case, suppose that a = [x : 0 : 1] and b = [y : 0 : 1]. Then from [50], Ex. 6 x y) 2 q a, b) = 1 x 2 ) 1 y 2 ) and so midpoints m = [w : 0 : 1] and sydpoints s = [z : 0 : 1] of ab exist precisely when x 2 1 ) y 2 1 ) = r 2 and x 2 1 ) y 2 1 ) = t 2 respectively, in which cases w = xy + 1 ± r x + y and z = 1 xy) x + y) ± t x y) x 2 + y 2. 2 So we see that algebraically sydpoints are somewhat more complicated than midpoints in general. It is very important to make the following simple observation: over the rational numbers, any non-null side either approximately has midpoints or approximately has sydpoints but not both, since being a square is approximately the same as being positive, and exactly one of 1 qa, b) and qa, b) 1 is always positive, if a and b are not perpendicular. There are a few related notions which are useful to define. The duals S and R of the sydpoints s and r of a side ab are the sydlines of the side ab. They do not in general pass through the sydpoints themselves. There is also a dual notion to that of sydpoints of a side which applies to vertices; this is an alternative to the idea of a biline. Definition 19 A siline of a vertex AB is a line L which passes through AB and satisfies SA, L) = SB, L). Again by duality we deduce that a vertex AB has a siline precisely when the quantity SA, B) 1 is a square number, and in this case there are exactly two silines L and K of the vertex AB. Then also A, B, L and K are a harmonic pencil of lines. The duals of the silines are the sipoints of a vertex AB.

47 3. Sydpoints and hyperbolic triangle geometry 39 Example 20 Over F 13 and with the usual hyperbolic metric given by C = J, suppose that a [1 : 1 : 1] and b [3 : 1 : 0]; then Q a Q b = 10 is a square number. Therefore, the midpoints of the side ab exist, and they are m = [7 : 3 : 1] and n = [8 : 12 : 1]. Then qa, m) = qb, m) = 3 and qa, n) = qb, n) = 11. Also, notice that Q a Q b = 3 is also square. Thus, sydpoints of the side ab also exist which are s = [10 : 4 : 1] and r = [5 : 11 : 1] where qa, s) = qb, r) = 1 and qa, r) = qb, s) = 1. Example 21 In F 13, if a = [1 : 1 : 1] and c = [0 : 1 : 2] then Q a Q c = 10 is a square number. Therefore, the midpoints of the side ac exist which are m = [4 : 9 : 1] and n = [8 : 11 : 1] where qa, m) = qc, m) = 6 and qa, m ) = qb, m ) = 8. Also, notice that Q a Q c = 3 is also square. Thus, sydpoints of the side ac exist which are s = [2 : 8 : 1] and r = [5 : 3 : 1] where qa, s) = qc, r) = 6 and qa, r) = qc, s) = 7. Example 22 In F 13, if b = [3 : 1 : 0] and c = [0 : 1 : 2] then Q b Q c = 9 is a square number. Therefore, the midpoints of the side bc exist which are m = [5 : 0 : 1] and n = [8 : 1 : 1] where qb, m) = qc, m) = 9 and qb, n) = qc, n) = 5. Also, notice that Q b Q c = 4 is also square. Thus, sydpoints of the side bc exist which are s = [1 : 3 : 1] and r = [12 : 11 : 1] where qb, s) = qc, r) = 11, and qb, r) = qc, s) = 2. Example 23 The triangle abc with vertices a = [1 : 1 : 1], b = [3 : 1 : 0] and c = [0 : 1 : 2] is an interesting triangle since every side has midpoints and sydpoints as well. There are two circumcenter hierarchies; One with respect to the midpoints only and the other with respect to two sydsides and one midside The construction of Sydpoints The following theorem is helpful in constructing sydpoints using a dynamic geometry package. Theorem 24 Sydpoints null points) Suppose that the non-null side ab has sydpoints s and r, and that ac has midpoints m and n, where c = ab). Then x mr) bc) = ns) bc) and y ms) bc) = nr) bc) are null points. Proof. Suppose that a = [v], b = [u] and c = [w]. Then vw = uw = 0, since c = ab), and also since ab is not null v, u and w are independent. If ac has midpoints, in which case we may assume that v 2 = w 2, these are then m [v + w] and n [v w]. If also ab has sydpoints, then we may assume that v 2 = u 2, these are then s = [v + u] and r = [v u]. Note that this renormalization can be made independent of the previous one. Now consider x mr) bc). This is a point with a representative vector of the form k v + w)+l v u) for some numbers k and l. Since x has a representative vector

48 3. Sydpoints and hyperbolic triangle geometry 40 which is also in the span of u and w, it must be a multiple of v + w) v u) = u+w. But then u + w) 2 = u 2 + 2uw + w 2 = 0 since uw = 0 and u 2 = w 2. So x is a null point, and similarly for y. Figure 3.4: Construction of sydpoints of ab We use this theorem to give practical constructions of sydpoints using Geometer s Sketchpad, C.a.R., Cabri, GeoGebra or Cinderella etc. To construct the sydpoints r and s of ab as in Figure 3.4, first construct the dual c = ab), then the midpoints m and n of ac, and then use the null points x and y lying on bc as shown. The required sydpoints are s = nx) ab) = my) ab) and r = ny) ab) = mx) ab). Similarly, given the sydpoints r and s of ab, a and b can be constructed as the sydpoints of rs using the null points w and z lying on rc and the midpoints k and l of cs, the required points are a = lz) rs) = kw) rs) and b = lw) rs) = kz) rs). So the construction of sydpoints can be reduced, at least in this kind of situation, to computations of midpoints. Once we establish the Circumlines theorem, it is interesting that Figure 3.4 can be viewed as a limiting case applied to the triangle abc the null points x and y act as midpoints of bc, so mrx acts as a circumline. Another useful construction is to find, given the point b and one of the sydpoints s, the other point a and the other sydpoint r as in Figure 3.5. First construct the dual c = bs), then find the midpoints k and l of cs. Use the null points u, t lying on bk and the null points v, w lying on bl to construct r = cuv)bs) and a = lu) bs) = kv) bs). However by symmetry there is a second solution: r = cwt)bs) and a = lt) bs) = kw) bs). Thus, we can think of s and r as being the sydpoints of the side ab, and s and r as the sydpoints of the side ab. Notice also that b is a midpoint of the side rr and similarly s is a midpoint of the side aa, and in fact q b, r) = q b, r) = q s, a) = q s, a).

49 3. Sydpoints and hyperbolic triangle geometry 41 Figure 3.5: Constructing r and a or r and a) from s and b Twin circles In the geometry we are studying, a circle C may be defined as an equation of the form q c, x) = k, for a fixed point c called the center, and a fixed number k called the quadrance of the circle. We also write C k c for this circle, and say that a point a lies on the circle precisely when q c, a) = k. Since in this case the circle is also determined by c and a, we write Cc k = C c a). The bracket reminds us that a is not unique. Definition 25 Two circles C 1 and C 2 with the same center c and quadrances q 1 and q 2 are twins precisely when q 1 + q 2 = 2. We now show that the notion of twin circles are naturally connected with sydpoints. Theorem 26 Sydpoint twin circle) If s is a sydpoint of ab, and c lies on S s, then the circles C c a) and C c b) are twins. Conversely if C c a) and C c b) are twins, then s c ab) is a sydpoint of ab. Proof. If s is a sydpoint of ab then q a, s) = q = q b, s) for some q. Then since c and s are perpendicular, q c, s) = 1. Let d = s ab). Then since d and s are perpendicular, q d, s) = 1, and then q a, d) = 1 q a, s) = 1 q and q b, d) = 1 q b, s) = 1 + q. So q a, d) + q b, d) = 2. Now suppose that q c, d) = r. Then by Pythagoras theorem in the right triangle cda we have while in the right triangle cdb we have q c, a) = r + 1 q) r 1 q) q c, b) = r q) r 1 + q).

50 3. Sydpoints and hyperbolic triangle geometry 42 Then q c, a) + q c, b) = r + 1 q) r 1 q) + r q) r 1 + q) = 2. The argument can be reversed to show the converse. We note that the theorem has another possible, quite interesting, interpretation: the locus of a point c such that q a, c) + q b, c) = 2 is a line Constructions of twin circles The Sydpoint twin circle theorem assists us to construct twin circles; we generally expect this to reduce to finding midpoints, but there are also some simpler scenarios. Suppose we are given a circle C in brown) with center c as in Figure 3.6. Choose an arbitrary point a on the circle C and construct C c, then let s be the meet of ac and C, and t the meet of A a and C. Figure 3.6: Constructing the twin circle D of C Now, we can apply the construction of Figure 3.5; suppose that the side st has midpoints m and n, and that x and y are null points on am, and z and w are null points on an. Then b mz) ac) = ny) ac) and e mw) ac) = nx) ac) lie on the twin circle D to C. Symmetry implies that we could also use d mw) ct) = ny) ct) and f mz) ct) = nx) ct). Figure 3.7 shows another example of constructing the twin D of a given circle C in brown) with center c. In this case c is outside the null circle, so its dual line C passes through null points x and y approximately remember that a dynamic geometry package usually only deals with decimal approximations, so the number-theoretical subtlety is diminished). Choose a point a on C with dual line A = a. Then the twin circle D in red) is the locus of the point b = ax) A or the point d = ay) A as a moves along C. The fact that q a, c)+q b, c) = 2 follows by applying either the Nil Cross law [50, Thm 80]) or the Null subtended quadrance theorem [50, Thm 90]) to the triangle abc.

51 3. Sydpoints and hyperbolic triangle geometry 43 Figure 3.7: Another construction of a twin circle Similarly, given the red circle D, its twin circle C in brown) can be constructed as the locus of the point a = bx) b when moving the point b on D. It should also be noted that we have not at all established that the twin of any circle necessarily exists. In fact over the rational numbers, the twin circle of a given circle does not always exist. For example over the rational numbers, if c is inside the null circle, then q c, a) never takes on values in the range 0, 1), but it can take on values in the range 1, 2). 3.2 Circumlinear coordinates and the Orthocenter hierarchy In the paper [52]) Wildberger focussed on ortholinear coordinates, emphasizing that the Orthocenter is arguably the most important point in hyperbolic triangle geometry, and then secondly on the Incenter hierarchy. In this chapter we want to complement that work by focussing primarily on the Circumcenter hierarchy, and we introduce circumlinear coordinates to work effi ciently with both midpoints and sydpoints simultaneously. This allows us to not only introduce new ideas related to sydpoints, but also to derive new formulas for the classical situation of a hyperbolic triangle and its circumcenters. While triangle geometry involving sydpoints will be new and somewhat unfamiliar, the natural beauty and elegance of this theory is very compelling indeed. Suppose the bilinear form v u = vau T in the associated three-dimensional vector space V = F 3 is given by a symmetric matrix A, and that T : V V is a linear transformation given by an invertible 3 3 matrix M, so that T v) = vm = w, with inverse matrix N, so that wn = v. The new bilinear form defined by w 1 w 2 w 1 N) w 2 N) = w 1 N) A w 2 N) T = w 1 NAN T )w2 T 3.4) has matrix C = NAN T.

52 3. Sydpoints and hyperbolic triangle geometry 44 So let us start with three projective) points a 1, a 2 and a 3 such that each of the three sides of the triangle a 1 a 2 a 3 has either midpoints or sydpoints. That means we can find representative vectors v 1, v 2 and v 3 in V so that for any i and j, vi 2 = ±v2 j. There are two possibilities up to relabelling and re-scaling: 1) v1 2 = v2 2 = v2 3 = 1 this corresponds to three midsides) and 2) v1 2 = v2 2 = v2 3 = 1 this corresponds to one midside and two sydsides). We can incorporate it by supposing that v1 2 = v2 2 = εv3 2 = 1 where ε = ±1. Now we can find a linear transformation to map v 1, v 2 and v 3 to the basis vectors e 1 = 1, 0, 0), e 2 = 0, 1, 0) and e 3 = 0, 0, 1) respectively. With respect to this new basis, the bilinear form is then given by a new matrix of the form 1 a b c 2 ε aε bc b ac C = a 1 c with adjugate D = aε bc b 2 ε c ab 3.5) b c ε b ac c ab a 2 1 where the diagonal entries of C ensure that e 2 1 = e2 2 = 1 and e2 3 = ε, and otherwise e 1 e 2 = a, e 1 e 3 = b and e 2 e 3 = c are arbitrary. So the metrical structure depends on the numbers a, b and c and the sign of) ε. Note that 1 a b det a 1 c = ε a 2 ε b 2 c 2 + 2abc. b c ε This quantity appears as a common factor in several of the derivations of proportions in this chapter, and since it is by assumption non-zero, we simply cancel it without mention. We now reformulate some of the formulas of the Orthocenter hierarchy of [53]) using circumlinear coordinates, maintaining the convention of using capital letters for various constructions associated to a base triangle. The projective matrices corresponding to C and D are denoted C and D respectively. Our starting point is that the basic Triangle a 1 a 2 a 3 has been projectively transformed so that its Points are a 1 = [1 : 0 : 0] a 2 = [0 : 1 : 0] a 3 = [0 : 0 : 1]. 3.6) The Lines of the Triangle are then L 1 = 1 : 0 : 0 L 2 = 0 : 1 : 0 L 3 = 0 : 0 : 1. The main assumption is that each of the three sides is either a midside or a sydside, or possibly both, which we have seen allows us to write the bilinear form using the

53 3. Sydpoints and hyperbolic triangle geometry 45 matrices 3.5). The Triangle will have three midsides if ε = 1, and two sydsides and one midside if ε = 1. The computations are based on two basic operations: finding joins and meets, which essentially amounts to taking cross products as in 2.4) and 2.5); and finding duals, either by multiplying transposes of points by C on the left, or transposes of lines by D on the right as in 2.6). Our goal is to establish formulas for important points and lines to facilitate determining relationships between them. Occasionally we simplify a proportion by cancelling a common factor: naturally this factor should not be zero, so we state this as a condition Change of coordinates and the main example Most of the diagrams in this chapter deal with the particular triangle in Figure 3.8 created with GSP, with affi ne points a 1 [ 0.039, 0.152], a 2 [ 0.203, 0.780] and a 3 [ 1.753, 0.197], and representative vectors v , 0.914, 5.985), v , 7.806, 10) and v , 0.805, 4.065). These have been normalized so that Q v1 = Q v2 = Q v3 with respect to the hyperbolic bilinear form v u vju T, where J is defined in 2.7). Figure 3.8: Basic example triangle with coordinates We now show how to explicitly change coordinates, following Section 1.5 of [53]. The linear transformation T v) = vn, where N is N = ,

54 3. Sydpoints and hyperbolic triangle geometry 46 sends e 1 = 1, 0, 0), e 2 = 0, 1, 0) and e 3 = 0, 0, 1) to v 1, v 2 and v 3 respectively. The inverse matrix M = N 1 sends the vectors v 1, v 2 and v 3 to e 1, e 2 and e 3. Following 3.4), after we apply the linear transformation T, J is replaced by the matrix We get the constants C = NJN T D = with adjugate a = b = c = ε = 1. As an example of how to explicitly apply the theorems of this chapter to our specific triangle, consider the midpoints of the side a 1 a 2 in standard coordinates which are m = n 1+ = [1 : 1 : 0] and m = n 1 = [1 : 1 : 0]. Multiply by N and then renormalize so that z = 1, to find these midpoints in the original triangle to be [ ] n 1+ = [1 : 1 : 0] N = [ 0.142, 0.546] [ ] n 1 = [1 : 1 : 0] N = [ 0.448, 1. 72]. As another example, using the formulas from the Circumlines/Circumcenter theorem, we may similarly compute that the circumcenters c, in agreement with Figure 3.8, are c 0 = [0.268, 0.653] c 1 = [ 0.997, ] c 2 = [0.249, ] c 3 = [ , 0.241] Altitudes, Orthocenter and Orthic triangle We now follow Wildberger [53], but recompute the formulas in our Circumlinear coordinates. We state the results only, as the computations are straightforward, with occasional algebraic manipulations and simplifications required. The Dual lines are A 1 a 1 = Ca T 1 = 1 : a : b A 2 a 2 = Ca T 2 = a : 1 : c A 3 a 3 = Ca T 3 = b : c : ε.

55 3. Sydpoints and hyperbolic triangle geometry 47 The Dual points are l 1 L T 1 D = [ c 2 ε : εa bc : b ac ] l 2 = [ εa bc : b 2 ε : c ab ] l 3 = [ b ac : c ab : a 2 1 ]. The Altitudes are N 1 a 1 l 1 = 0 : ac b : εa bc N 2 a 2 l 2 = c ab : 0 : bc εa N 3 a 3 l 3 = ab c : b ac : 0 and the Altitude dual points are n 1 A 1 L 1 = [0 : b : a] n 2 A 2 L 2 = [c : 0 : a] n 3 A 3 L 3 = [ c : b : 0]. The Base points are b 1 N 1 L 1 = [0 : εa bc : b ac] b 2 N 2 L 2 = [εa bc : 0 : c ab] b 3 N 3 L 3 = [b ac : c ab : 0] and the Base lines are B 1 n 1 l 1 = b 2 2abc + a 2 ε : a ε c 2) : b ε c 2) B 2 n 2 l 2 = a ε b 2) : c 2 2abc + a 2 ε : c ε b 2) B 3 n 3 l 3 = b 1 a 2) : c 1 a 2) : b 2 2abc + c 2. Figure 3.9: Altitudes, Orthocenter, Orthic triangle and Base center b

56 3. Sydpoints and hyperbolic triangle geometry 48 Assuming aε bc 0, b ac 0 and c ab 0, the Orthic lines are C 1 b 2 b 3 = ab c : b ac : εa bc C 2 b 1 b 3 = c ab : ac b : εa bc C 3 b 1 b 2 = c ab : b ac : bc aε. The Orthic points are c 1 B 2 B 3 = [ 2ca 2 ba c ) ε + c 2b 2 + c 2 3abc ) : ac b) b 2 ε ) : bc aε) a 2 1 )] c 2 B 1 B 3 = [ ab c) c 2 ε ) : 2ba 2 ca b ) ε + b b 2 + 2c 2 3abc ) : bc aε) a 2 1 )] c 3 B 1 B 2 = [ ab c) c 2 ε ) : ac b) b 2 ε ) : a a 2 1 ) ε + 2ab 2 3a 2 bc + 2ac 2 bc )]. The Orthocenter is, according to Wildberger, possibly the most important point in triangle geometry, it is h N 1 N 2 = N 2 N 3 = N 1 N 3 = [b ac) aε bc) : c ab) aε bc) : ac b) ab c)]. The dual line is the Ortholine H n 1 n 2 = n 1 n 3 = n 2 n 3 = ab : ac : bc. The Orthic triangle b 1 b 2 b 3 is perspective with the Triangle a 1 a 2 a 3 with center of perspectivity the Orthocenter h. The Triangle Base center theorem in [53] states that the Orthic dual triangle c 1 c 2 c 3 is perspective with the Triangle a 1 a 2 a 3. The center of perspectivity is the Base center b = [ ab c) c 2 ε ) : ac b) b 2 ε ) : bc εa) a 2 1 )] with dual line the Base axis B = c + ab : b + ac : εa + bc. In Figure 3.9 we see the Altitudes, Orthocenter h and the dual Ortholine H, the Orthic triangle b 1 b 2 b 3, Orthic dual triangle c 1 c 2 c 3, Base center b and Base axis B Desargues points and the Orthoaxis The Desargues points are the meets of corresponding Orthic lines and Lines: g 1 C 1 L 1 = [0 : bc εa : b ac] g 2 C 2 L 2 = [bc εa : 0 : c ab] g 3 C 3 L 3 = [b ac : ab c : 0]

57 3. Sydpoints and hyperbolic triangle geometry 49 and the dual Desargues lines are G 1 = b 2 a 2 ε : 2bc ac 2 aε : bc 2 + bε 2acε G 2 = 2bc ab 2 aε : c 2 a 2 ε : b 2 c + cε 2abε G 3 = b + a 2 b 2ac : 2ab c a 2 c : b 2 c 2. Figure 3.10: Desargues points, Orthic axis S and Orthoaxis A Desargues theorem implies that the Desargues points g 1, g 2, g 3 are collinear. They lie on the Orthic axis S = ab c : ac b : bc aε. 3.7) Dually the Desargues lines G 1, G 2, G 3 are concurrent, passing through the Orthostar 2ca 2 3ba + c ) ε + c 2b 2 c 2 abc ) : s = 2ba 2 3ca + b ) ε b b 2 2c 2 + abc ) : a 1 a 2) ε + 2ab 2 a 2 bc + 2ac 2 3bc ). The Orthoaxis A sh, introduced of [53], is arguably the most important line in hyperbolic triangle geometry; it and its dual the Orthoaxis point a are A sh = ab c) a 2 ε b 2) : b ac) a 2 ε c 2) : bc aε) b 2 c 2) a SH = [ c a 2 ε b 2) : b c 2 εa 2) : a b 2 c 2)]. The Base center on Orthoaxis theorem of [53] asserts that the Orthoaxis A passes through the Base center b Parallels and the Double triangle Recall from [50] that the parallel line P through a point a to a line L is the line through a perpendicular to the altitude from a to L. This motivates the definition of the Double triangle of a Triangle. The Parallel lines P 1 a 1 n 1 = 0 : a : b P 2 a 2 n 2 = a : 0 : c P 3 a 3 n 3 = b : c : 0 and

58 3. Sydpoints and hyperbolic triangle geometry 50 are the joins of corresponding Points a and Altitude points n, and their duals are the Parallel points p 1 = [ b 2 2abc + a 2 ε : bc aε : ac b ] p 2 = [ bc aε : c 2 2abc + a 2 ε : ab c ] p 3 = [ ε ac b) : ε ab c) : b 2 2abc + c 2]. Assuming a 0, b 0 and c 0, the meets of Parallel lines are the Double points d 1 P 2 P 3 = [ c : b : a] d 2 P 1 P 3 = [c : b : a] d 3 P 1 P 2 = [c : b : a] and their duals are the Double lines D 1 p 2 p 3 = 2ab c : b : εa D 2 p 1 p 3 = c : 2ac b : εa D 3 p 1 p 2 = c : b : 2bc εa. Figure 3.11: The Double triangle, Orthoaxis A, and the points z, b, x, h and s We give here another proof of the following result, involving a simpler computation than in [52]. Theorem 27 Double triangle midpoint) The Points a 1, a 2, a 3 are midpoints of the Double triangle d 1 d 2 d 3. Proof. We compute qd 1, a 3 ) = b2 c 2 + 2abc a 2 b 2 c 2 + 2abc = qd 2, a 3 ). Similarly, a 1 is a midpoint of d 2 d 3, and a 2 is a midpoint of d 1 d 3.

59 3. Sydpoints and hyperbolic triangle geometry 51 The Double triangle perspectivity theorem of [53] states that the Double triangle d 1 d 2 d 3 and the Triangle a 1 a 2 a 3 are perspective from a point, the Double point, or x point x = [c : b : a] which lies on the Orthoaxis A. The proof is very simple in these coordinates: we compute that a 1 d 1 = 0 : a : b a 2 d 2 = a : 0 : c a 3 d 3 = b : c : 0 and then observe that these lines meet at x. The dual of the x point is the X line X = 2ab + c : 2ac + b : 2bc + aε. The Double dual triangle perspectivity theorem of [53] asserts that the Double triangle d 1 d 2 d 3 and the Dual triangle l 1 l 2 l 3 are perspective from a point, the Double dual point, or z point [ ca 2 2ba + c ) ε + c b 2 c 2) : ba 2 2ca + b ) ε b b 2 c 2) ] z = : a 1 a 2) ε + ab 2 2bc + ac 2. Its dual is the Z line Z = c : b : εa. The z point lies on the Orthoaxis A, or equivalently the Orthoaxis point a lies on the Z line. 3.3 The Circumcenter hierarchy We now begin the study of the Circumcenter hierarchy. The basic assumption that we used to set up circumlinear coordinates was that each side of the triangle was either a midside or a sydside. We wish to treat both cases symmetrically, hence we introduce the notion that a smydpoint n of the side ab is either a midpoint or a sydpoint or possibly both). Smydpoints exists precisely when 1 q a, b) is either a square or the negative of a square or possibly both). We introduce consistent labelling to bring out the four-fold symmetry in this situation. Our diagrams will illustrate the case when one side has midpoints and the other two sides have sydpoints.

60 3. Sydpoints and hyperbolic triangle geometry Circumcenters, medians and centroids By the Side midpoints and Side sydpoints theorems, in Circumlinear coordinates the smydpoints are n 1+ = [0 : 1 : 1] and n 1 = [0 : 1 : 1] on a 2 a 3 n 2+ = [1 : 0 : 1] and n 2 = [1 : 0 : 1] on a 1 a 3 n 3+ = [1 : 1 : 0] and n 3 = [1 : 1 : 0] on a 1 a 2. Note that the indices of our labelling reflect the positions and relative signs of the non-zero entries. Theorem 28 Circumlines/Circumcenters) The six Smydpoints lie three at a time on four Circumlines C 0 n 1 n 2 n 3 = 1 : 1 : 1 C 1 n 1 n 2+ n 3+ = 1 : 1 : 1 C 2 n 2 n 1+ n 3+ = 1 : 1 : 1 C 3 n 3 n 1+ n 2+ = 1 : 1 : 1. The duals are the Circumcenters c 0 = C0 = [a 1) ε c a + b c) + b : a 1) ε b a b + c) + c : a 1) a b c + 1)] c 1 = C1 = [a + 1) ε c a + b + c) + b : c a + 1) ε b a b c) : a + 1) a b + c 1)] c 2 = C2 = [b a + 1) ε c a b c) : a + 1) ε b a + b + c) + c : a + 1) a + b c 1)] c 3 = C3 = [b a 1) ε c a b + c) : c a 1) ε b a + b c) : a 1) a + b + c + 1)]. Proof. The formulas for the Circumlines can be checked immediately, the Circumcenter formulas are computations using duality. Figure 3.12: Circumlines, Circumcenters, Medians and Centroids

61 3. Sydpoints and hyperbolic triangle geometry 53 Median lines or just medians) are joins of Points a and Smydpoints n which lie on the opposite lines: D 1 a 1 n 1 = 0 : 1 : 1 D 1+ a 1 n 1+ = 0 : 1 : 1 D 2+ a 2 n 2+ = 1 : 0 : 1 D 2 a 2 n 2 = 1 : 0 : 1 D 3 a 3 n 3 = 1 : 1 : 0 D 3+ a 3 n 3+ = 1 : 1 : 0. Figure 3.12 shows the six Medians and their meets. Theorem 29 Centroids) The Median lines D are concurrent in threes, meeting at four Centroid points g 0 D 1+ D 2+ D 3+ = [1 : 1 : 1] g 1 D 1+ D 2 D 3 = [ 1 : 1 : 1] g 2 D 1 D 2+ D 3 = [1 : 1 : 1] g 3 D 1 D 2 D 3+ = [1 : 1 : 1]. The dual Centroid lines are G 0 = a + b + 1 : a + c + 1 : b + c + ε G 1 = a + b 1 : c a + 1 : c b + ε G 2 = b a + 1 : a + c 1 : b c + ε G 3 = a b + 1 : a c + 1 : b + c ε. Proof. Straightforward CircumCentroids While many aspects of the Circumcenter hierarchy are independent of ε, there are some that are not. The following is an extension of the similarly named result in [52]. Theorem 30 CircumCentroid axis) The meets of corresponding Circumlines and Centroid lines are collinear precisely when either b = ±c or ε = 1. If ε = 1, the common line is the Z axis c : b : εa, and the joins of corresponding Circumcenters and Centroid points meet at the z point. If b = c, then the common line is b : b : a + ε 1, while if b = c, then the common line is b : b : a ε + 1. Proof. The meets of Circumlines C 0, C 1, C 2, C 3 and corresponding Centroid lines G 0, G 1, G 2, G 3 are the four CircumCentroid points z 0 C 0 G 0 = [a b ε + 1 : c a + ε 1 : b c] z 1 C 1 G 1 = [b a ε + 1 : 1 a c ε : b + c] z 2 C 2 G 2 = [1 a b ε : c a ε + 1 : b + c] z 3 C 3 G 3 = [a + b ε + 1 : ε a c 1 : b c].

62 3. Sydpoints and hyperbolic triangle geometry 54 The determinants a b ε + 1 c a + ε 1 b c det b a ε a c ε b + c = 4 b 2 c 2) ε 1) 1 a b ε c a ε + 1 b + c a b ε + 1 c a + ε 1 b c det 1 a b ε c a ε + 1 b + c = 4 b 2 c 2) ε 1) a + b ε + 1 ε a c 1 b c show that the CircumCentroid points are collinear precisely when ε = 1 or b = ±c. If ε = 1 the common line is c : b : a which in this case agrees with Z = c : b : εa. If b = c we can check that the common line is b : b : a + ε 1, and if b = c the common line is b : b : a ε Twin Circumcircles of a Triangle If a triangle has three midsides, then corresponding Circumcenters will be centers of circles which pass through all three points, as in the classical triangle in Figure 3.1. This situation also holds for a triangle such as a 1 a 2 a 3 in Figure 3.13, lying outside the null circle still in blue) shown with three of its Midpoints m, the other three are off the page), six Midlines M, three of the four Circumlines C, the four Circumcenters c, and the corresponding Circumcircles. Figure 3.13: Circumcenters of a triangle outside the null circle But what happens if a triangle has some points inside and some outside the null circle? In that case it turns out that we need to consider special pairs of circles, which collectively play the role of circumcircles. We do not know of any classical precedents for this phenomenon. Definition 31 Twin circles C and C are twin circumcircles for a triangle a 1 a 2 a 3 precisely when each of a 1, a 2, a 3 lie on either C or C. Theorem 32 Twin circumcircles) If a triangle a 1 a 2 a 3 has smydpoints on all three sides, then the four circumcenters c 0, c 1, c 2, c 3 are each the center of twin circumcircles for a 1 a 2 a 3.

63 3. Sydpoints and hyperbolic triangle geometry 55 Proof. If n is a smydpoint of the side a k a l then its dual n passes through two circumcenters, say c i and c j. Let s consider just c i. If n is a sydpoint of a k a l then the Sydpoint twin circle theorem shows that the circles C a k) c i and C a l) c i are twin circles. If n is a midpoint of a k a l then the reflection r n interchanges a k and a l and fixes both c i and c j, so that C a k) c i and C a l) c i coincide. Since c i is perpendicular to two smydpoints on different lines of the triangle a 1 a 2 a 3, the argument can be repeated, so that either there is one circle with center at c i that passes through all three points, or one of the twin circles C a k) c i and C a l) c i also passes through the third point of the triangle, in which case these are twin circumcircles. Now let s introduce some labelling and explicit formulas. Consider the circles C i = C a 3) centered at c i i and passing through a 3, for i = 0, 1, 2, 3. Their equations qp, c i ) = qc i, a 3 ) in a variable point p = [x : y : z], can be written, after factoring a common term ε + a 2 ε + b 2 + c 2 2abc, as C 0 : 1 ε) x 2 + y 2) + 2 a ε) xy + 2 b ε) xz + 2 c ε) yz = 0 C 1 : 1 ε) x 2 + y 2) + 2 a + ε) xy + 2 b + ε) xz + 2 c ε) yz = 0 C 2 : 1 ε) x 2 + y 2) + 2 a + ε) xy + 2 b ε) xz + 2 c + ε) yz = 0 C 3 : 1 ε) x 2 + y 2) + 2 a ε) xy + 2 b + ε) xz + 2 c + ε) yz = 0. The respective twin circles C i with equations qp, c i ) = 2 qc i, a 3 ) can be written as C 0 : 1 + ε) x 2 + y 2) + 2εz a + ε) xy + 2 b + ε) xz + 2 c + ε) yz = 0 C 1 : 1 + ε) x 2 + y 2) + 2εz a ε) xy + 2 b ε) xz + 2 c + ε) yz = 0 C 2 : 1 + ε) x 2 + y 2) + 2εz a ε) xy + 2 b + ε) xz + 2 c ε) yz = 0 C 3 : 1 + ε) x 2 + y 2) + 2εz a + ε) xy + 2 b ε) xz + 2 c ε) yz = 0. If ε = 1, then each of the four circumcircles C i passes through all three points of the triangle, while their twins C i pass through none of the points of the triangle; even so, their presence is felt. Figure 3.14: Twin circumcircles for a classical triangle

64 3. Sydpoints and hyperbolic triangle geometry 56 In Figure 3.14 we see a triangle a 1 a 2 a 3 with all three points inside the null circle, together with its four pairs of twin circumcircles, each pair with the same colour CircumDual points, Tangent lines and Sound points If ε = 1, then the circumcircles C i pass only through c 3, while the twins C i pass through c 1 and c 2. In each case we have four twin circumcircle pairs of the Triangle. These eight circles are shown for our standard example Triangle in Figure 3.15, along with the Tangent lines, which we now introduce. Figure 3.15: Twin Circumcircles and Tangent lines The CircumDual point p ij is the meet of the Dual line A i and the Circumline C j, for i = 1, 2, 3 and j = 0, 1, 2, 3. Then p 10 = [a b : b 1 : a + 1] p 20 = [c 1 : a c : a + 1] p 30 = [ε c : b ε : b + c] p 11 = [a b : b 1 : a + 1] p 21 = [1 c : a c : a + 1] p 31 = [c ε : b ε : b + c] p 12 = [a + b : b 1 : a 1] p 22 = [c + 1 : c a : a 1] p 32 = [ c ε : b ε : b + c] p 13 = [ a b : b + 1 : 1 a] p 23 = [ c 1 : a + c : a 1] p 33 = [ c ε : b + ε : b c]. The Tangent line T ij is the join of the CircumDual point p ij and the point a i. This line is indeed tangent to the circumcircle C i at the point a i if this circle passes through a i. The twelve Tangent lines are: T 10 = 0 : a 1 : b 1 T 20 = a 1 : 0 : c 1 T 30 = b ε : c ε : 0 T 11 = 0 : a + 1 : b + 1 T 21 = a + 1 : 0 : c 1 T 31 = b + ε : c ε : 0 T 12 = 0 : a + 1 : b 1 T 22 = a + 1 : 0 : c + 1 T 32 = b ε : c + ε : 0 T 13 = 0 : a 1 : b + 1 T 23 = a 1 : 0 : c + 1 T 33 = b + ε : c + ε : 0.

65 3. Sydpoints and hyperbolic triangle geometry 57 The Sound point s ij is the meet of the Tangent line T ij with the opposite Line L i. The twelve Sound points are: s 10 = [0 : 1 b : a 1] s 20 = [1 c : 0 : a 1] s 30 = [ε c : b ε : 0] s 11 = [0 : b 1 : a + 1] s 21 = [1 c : 0 : a + 1] s 31 = [ε c : b + ε : 0] s 12 = [0 : 1 b : a + 1] s 22 = [ 1 c : 0 : a + 1] s 32 = [c + ε : ε b : 0] s 13 = [0 : b + 1 : 1 a] s 23 = [1 + c : 0 : 1 a] s 33 = [ c ε : b + ε : 0]. Figure 3.16: CircumDual points and Sound points Jay and Wren lines In this section we begin to see more divergence between the ε = 1 and ε = 1 cases. Our terminology continues to follow Wildberger [53], but now our results are more general. In the latter case a symmetry emerges between the Circumcenters c 0 and c 3, and between c 1 and c 2. Theorem 33 Jay lines) If ε = 1 then the sets of Sound points {s 10, s 20, s 30 }, {s 11, s 21, s 31 }, {s 12, s 22, s 32 } and {s 13, s 23, s 33 } are each collinear, while if ε = 1 then the sets of Sound points {s 10, s 20, s 33 }, {s 11, s 21, s 32 }, {s 12, s 22, s 31 } and {s 13, s 23, s 30 } are each collinear. In both cases the common lines are respectively the four Jay lines J 0 = a 1) b 1) : a 1) c 1) : c 1) b 1) J 1 = a + 1) b + 1) : a + 1) c 1) : c 1) b + 1) J 2 = a + 1) b 1) : a + 1) c + 1) : c + 1) b 1) J 3 = a 1) b + 1) : a 1) c + 1) : c + 1) b + 1).

66 3. Sydpoints and hyperbolic triangle geometry 58 Proof. The forms of the Sound points and Jay lines make verifying these incidences almost trivial. Note that changing the sign of ε interchanges s 30 with s 33, and s 31 with s 32. This explains why the two lists appear different in these two cases. In the case of ε = 1 we associate each triple of Sound points to the Circumline which is involved in each term. In the case of ε = 1 we associate each triple to the Circumline which is involved in two of the three elements of the triple. There are four meets of Circumlines and associated Jay lines called CircumJay points, namely t 0 C 0 J 0 = [c 1) a b) : b + 1) a c) : a 1) b c)] t 1 C 1 J 1 = [c 1) a b) : b + 1) a + c) : a + 1) b + c)] t 2 C 2 J 2 = [c + 1) a + b) : 1 b) a c) : a + 1) b + c)] t 3 C 3 J 3 = [ c + 1) a + b) : b + 1) a + c) : a 1) b c)]. Note that these formulas are independent of ε. Theorem 34 CircumJay) The four CircumJay points t 0, t 1, t 2, t 3 are collinear and lie on the line T = c + ab : b + ac : a + bc. When ε = 1 this coincides with the Base axis B. When ε = 1, this is a new line which we call the T axis. In the case of ε = 1, T, B and L 3 are concurrent at a new point t = [ b + ac) : c + ab : 0]. Proof. The CircumJay point t 0 lies on T since c 1) a b) c + ab) + b + 1) a c) b + ac) + a 1) b c) a + bc) = 0 and similarly for the other points. The T axis agrees with the Base axis B = c + ab : b + ac : εa + bc if ε = 1. For ε = 1, the verification of t = T B is also straightforward, and clearly it lies on L 3. Theorem 35 Wren lines) If ε = 1 then the sets of Sound points {s 11, s 22, s 33 }, {s 10, s 32, s 23 }, {s 31, s 20, s 13 } and {s 21, s 12, s 30 } are each collinear, while if ε = 1 then the sets of Sound points {s 11, s 22, s 30 }, {s 10, s 23, s 31 }, {s 13, s 20, s 32 } and {s 12, s 21, s 33 } are each collinear. In both cases the common lines are respectively the four Wren

67 3. Sydpoints and hyperbolic triangle geometry 59 lines W 0 = a + 1) b + 1) : a + 1) c + 1) : b + 1) c + 1) W 1 = a 1) b 1) : c + 1) a 1) : c + 1) b 1) W 2 = b + 1) a 1) : a 1) c 1) : b + 1) c 1) W 3 = a + 1) b 1) : a + 1) c 1) : b 1) c 1). Proof. Again, with the formulas for Sound points and Wren lines, it is straightforward to check incidences. As with the Jay lines, changing the sign of ε interchanges s 03 with s 33, and s 13 with s 23. Notice that each set of collinear Sound points is associated to the Circumcenter which is not involved in the indices of that group. CircumWren points are the meets of Circumlines and associated Wren lines. These points are u 0 C 0 W 0 = [c + 1) a b) : b + 1) a c) : a + 1) b c)] u 1 C 1 W 1 = [c + 1) a b) : b + 1) a + c) : a 1) b + c)] u 2 C 2 W 2 = [c 1) a + b) : b + 1) a c) : a + 1) b + c)] u 3 C 3 W 3 = [ c + 1) a + b) : b 1) a + c) : a + 1) b c)]. Theorem 36 CircumWren) The four CircumWren points u 0, u 1, u 2, u 3 are collinear and lie on the line U ab c : ac b : bc a. When ε = 1 this coincides with the Orthic axis S. When ε = 1, this is a new line which we call the U axis. In case ε = 1, S, U and L 3 are concurrent in a new point u = [ac b : c ab : 0]. Proof. We may compute that v 0 lies on U since c + 1) a b) ab c) b + 1) a c) ac b) + a + 1) b c) bc a) = 0. The other incidences are similar. From 3.7) we recall that the Orthic axis has equation S = ab c : ac b : bc aε which agrees with U precisely when ε = 1. Again the formula for u is easy. In Figure 3.17 we see the CircumJay points t j dark blue) on T, the CircumWren points u j purple) on U, and the JayWren points v j yellow) on V. Theorem 37 CircumJayWren) The lines U, T and H are concurrent, and pass through a [ c a 2 b 2) : b c 2 a 2) : a b 2 c 2)]. 3.8) If ε = 1 then a agrees with the Orthoaxis point a = [ c a 2 ε b 2) : b c 2 εa 2) : a b 2 c 2)].

68 3. Sydpoints and hyperbolic triangle geometry 60 Figure 3.17: Jay lines J, Wren lines W, T, U, V axes and new points a, u, t Proof. The concurrence of these lines follows from ab c ac b bc a det c + ab b + ac a + bc = 0. ab ac bc The common incidence with 3.8) is also readily checked. The last statement is selfevident. There are four JayWren points which are the meets of associated Jay lines and Wren lines: v 0 = J 0 W 0 = [ c 2 1 ) a b) : b 2 1 ) c a) : a 2 1 ) b c) ] v 1 = J 1 W 1 = [ c 2 1 ) a b) : b 2 1 ) a + c) : 1 a 2) b + c) ] v 2 = J 2 W 2 = [ c 2 1 ) a + b) : b 2 1 ) a c) : 1 a 2) b + c) ] v 3 = J 3 W 3 = [ c 2 1 ) a + b) : 1 b 2) a + c) : a 2 1 ) b c) ]. Theorem 38 JayWren) The four JayWren points v 0, v 1, v 2, v 3 are collinear and lie on the JayWren axis, or the V line V = c b 2 1 ) a 2 1 ) : b c 2 1 ) a 2 1 ) : a c 2 1 ) b 2 1 ). Proof. The JayWren point v 0 lies on V since 0 = c 2 1 ) a b) c b 2 1 ) a 2 1 ) b 2 1 ) a c) b c 2 1 ) a 2 1 ) + a 2 1 ) b c) a c 2 1 ) b 2 1 ). Checking the other incidences is similar.

69 3. Sydpoints and hyperbolic triangle geometry CircumMeets and reflections One of the interesting features of this situation concerns the meets of the eight generalized circumcircles forming the four twin circumcircles of a triangle with six smydpoints. We establish easily a basic fact. Figure 3.18: Circumcircles and CircumMeet points Theorem 39 Smydpoint reflection) Suppose that a generalized circumcircle C has center c j perpendicular to a smydpoint n. If C passes through a point a k of the Triangle, then it also passes through the reflection r n a k ). Proof. If n is perpendicular to c j, then the reflection r n in n fixes the center c j of C, and so fixes C. Thus if C passes through a k, it also passes through r n a k ). This theorem helps explain why in Figure 3.18 the meets of the generalized circumcircles lie either on the lines of the Triangle, or on the Medians. We see that reflections of Points in Sydpoints are also interesting points of the Triangle in fact somewhat surprisingly these CircumMeet points are independent of the third Point of the Triangle, and depend only on the particular side on which they lie! This may be verified with a dynamic geometry package: as we vary one point of the Triangle, the generalized circumcircles move, but their meets on the opposite Line do not. In general, meets of circles are complicated by number-theoretical issues circles do not have to meet, after all). We conjecture that whenever generalized Circumcircles meet, they do so either on Lines or Medians.

70 3. Sydpoints and hyperbolic triangle geometry Sound conics The twelve Sound points are quite interesting, supporting the linear structures of Jay and Wren lines. They also are connected with four special conics in an interesting way, each conic naturally also associated with a Circumcenter. Figure 3.19: Sound conics Theorem 40 Each of the four sextuples of sound points {s 12, s 13, s 21, s 23, s 31, s 32 }, {s 12, s 13, s 20, s 22, s 30, s 33 }, {s 10, s 11, s 21, s 23, s 30, s 33 } and {s 10, s 11, s 20, s 22, s 31, s 32 } lies on a conic. Each of these four Sound conics K j is associated to a Circumcenter c j. Proof. We compute the coeffi cients of the equation of the blue) conic K 0 : a 1 x 2 + a 2 y 2 + a 3 z 2 + a 4 xy + a 5 xz + a 6 yz = 0 passing through points s 12, s 13, s 21, s 23, s 31 by solving the linear system 1 b) 2 a 2 + a + 1) 2 a b) a + 1) a 6 = b) 2 a a) 2 a b) 1 a) a 6 = 0 1 c) 2 a 1 + a + 1) 2 a c) a + 1) a 5 = c) 2 a a) 2 a c) 1 a) a 5 = 0 ε c) 2 a 1 + b + ε) 2 a 2 + ε c) b + ε) a 4 = 0. This results in the values a 1 = c + ε) b ε) b 2 1 ) a 2 1 ) a 2 = c + ε) b ε) c 2 1 ) a 2 1 ) a 3 = c + ε) b ε) c 2 1 ) b 2 1 ) a 4 = 2 a 2 1 ) bc + 1) bε cε + bc 1) a 5 = 2 c + ε) b ε) b 2 1 ) ac + 1) a 6 = 2 c + ε) b ε) ab + 1) c 2 1 ).

71 3. Sydpoints and hyperbolic triangle geometry 63 When substituting the coordinates of s 32 in the above equation with these coeffi cients, we obtain equality precisely when ε 2 1 ) a 2 1 ) b c) 4bc + b 2 + c ) ε + bc 1) b 2 + c 2 2 )) = 0 which is true since ε 2 = 1. By following the same argument, we can obtain the equations of the red) conic K 1 : b 1 x 2 + b 2 y 2 + b 3 z 2 + b 4 xy + b 5 xz + b 6 yz = 0 through s 12, s 13, s 20, s 22, s 30, s 33 with coeffi cients b 1 = b + ε) c + ε) b 2 1 ) a 2 1 ) b 2 = b + ε) c + ε) c 2 1 ) a 2 1 ) b 3 = b + ε) c + ε) c 2 1 ) b 2 1 ) b 4 = 2 bc 1) a 2 1 ) bε + cε + bc + 1) b 5 = 2 b + ε) c + ε) ac 1) b 2 1 ) b 6 = 2 b + ε) c + ε) ab + 1) c 2 1 ), the green) conic K 2 : c 1 x 2 + c 2 y 2 + c 3 z 2 + c 4 xy + c 5 xz + c 6 yz = 0 through s 10, s 11, s 21, s 23, s 30, s 33 with coeffi cients c 1 = c + ε) b + ε) b 2 1 ) a 2 1 ) c 2 = c + ε) b + ε) c 2 1 ) a 2 1 ) c 3 = c + ε) b + ε) c 2 1 ) b 2 1 ) c 4 = 2 bc 1) bε + cε + bc + 1) a 2 1 ) c 5 = 2 c + ε) b + ε) ac + 1) b 2 1 ) c 6 = 2 c + ε) b + ε) ab 1) c 2 1 ), and the brown) conic K 3 : d 1 x 2 + d 2 y 2 + d 3 z 2 + d 4 xy + d 5 xz + d 6 yz = 0 through s 10, s 11, s 20, s 22, s 31, s 32 with coeffi cients d 1 = c + ε) b ε) b 2 1 ) a 2 1 ) d 2 = c + ε) b ε) c 2 1 ) a 2 1 ) d 3 = c + ε) b ε) c 2 1 ) b 2 1 ) d 4 = 2 bc + 1) bε cε + bc 1) a 2 1 ) d 5 = 2 c + ε) b ε) ac 1) b 2 1 ) d 6 = 2 c + ε) b ε) ab 1) c 2 1 ). We associate each Sound conic K j to the Circumcenter c j not involved in any of the six Sound points lying on it. Clearly there are many more developments awaiting discovery here. We have clearly established that the new notion of sydpoints is both natural and important for hyperbolic triangle geometry. Now we turn to parabola, where this notion will also play a major role.

72 Chapter 4 The parabola and its Euclidean properties 4.1 Introduction This chapter concentrates on basic definitions and properties of the hyperbolic parabola which are analogous to those of the Euclidean parabola. The next chapter deals with more unusual properties not shared by the Euclidean parabola. We will see that this investigation opens up a door to many new phenomenon, and hints again at the inexhaustible beauty of conic sections! In Euclidean geometry, the parabola plays several distinguished roles. It is the graph resulting from a quadratic function f x) = a + bx + cx 2, and so familiar as the second degree Taylor expansion of a general function. The parabola is also a conic section in the spirit of Apollonius, obtained by slicing a cone with a plane which is parallel to one of the generators of a cone. In affi ne geometry the parabola is the distinguished conic which is tangent to the line at infinity. In everyday life, the parabola occurs in reflecting mirrors and automobile head lamps, in satellite dishes and radio telescopes, and in the trajectories of comets. Of course the ancient Greeks also studied the familiar metrical formulation of a parabola: it is the locus of a point which remains equidistant from a fixed point F, called the focus, and a fixed line f, called the directrix. We have a good reason for using the same letters for both concepts, with only case separating them). Such a conic P has a line of symmetry: the axis a through F perpendicular to f. It also has a distinguished point V called the vertex, which is the only point of the parabola lying on the axis a, aside from the point at infinity. The vertex V is the midpoint between the focus F and the base point B af. For such a classical parabola P hundreds of facts are known, see for example 64

73 4. The parabola and its Euclidean properties 65 Figure 4.1: The Euclidean parabola [1], [7], [12], [15], [22], [33], [34]; quite a few of them going back to Archimedes and Apollonius. Of particular importance are theorems that relate to an arbitrary point P on the conic and its tangent line p. In particular the construction of the tangent line p itself is important: there are two common ways of doing this. One is to take the foot T of the altitude from P to the directrix f, and connect P to the midpoint M of T F ; so that p = P M. Another is to take the perpendicular line t to P F through F, and find its meet S with the directrix; this gives p = P S. The point S is equidistant from T and F, and the circle S with center S through F is tangent to both the lines P F and P T. A related and useful fact is that a chord P N is a focal chord meaning that it passes through F precisely when the meet of the two tangents at P and N lies on the directrix f, and in this case the two tangents are perpendicular. These facts are illustrated in Figure 4.1. Another result, which figures often in calculus, is that if P and Q are arbitrary points on the parabola with Z the meet of their tangents p and q, and T, U and W are respectively the feet of the altitudes from P, Q and Z to the directrix, then W is the midpoint of T U. So when we investigate hyperbolic geometry, some natural questions are: what is the analog of a parabola in this context, what properties of the Euclidean case carry over in this setting, and what additional properties might the hyperbolic parabola have that do not hold in the Euclidean case? These issues have of course been studied by quite a few authors, such as [4], [43], [21]. In this chapter we answer these questions in a new and more general way, using the wider framework of UHG, and allowing the beginnings of a much deeper investigation. There is a very natural analog of a parabola in this hyperbolic setting, and many, but certainly not all, properties of the Euclidean parabola hold or have reasonable analogs for it. But there are also many interesting aspects which have no Euclidean counter-

74 4. The parabola and its Euclidean properties 66 part, such as the existence of a dual or twin parabola, and an intimate connection with the theory of sydpoints from the previous chapter. We shall see also a rich canonical structure on a hyperbolic parabola, with lovely collinearities and concurrences. The outline of the chapter is as follows. We first define the parabola in the hyperbolic setting we often refer simply to the hyperbolic parabola), give a dynamic geometry package construction for it, introduce some basic points associated to it, and use some of these and the Fundamental theorem of Projective Geometry to define standard coordinates, involving a single parameter α, in which the parabola has the convenient equation xz = y 2. This allows a simple parametrization for the curve, as well as pleasant explicit formulas for many interesting points, lines, conics and higher degree curves associated to it. In our study of the basic points and lines associated with the parabola P 0, concrete and explicit formulae are key objectives, because they allow us a firm foundation for deeper investigations. The main thrust of the chapter is then to show how the hyperbolic parabola shares many similarities with the Euclidean parabola. The highlights include the duality leading to the twin parabola, a straightedge construction of the evolute of the parabola, and a conic construction of four points on the parabola whose normals pass through a fixed point in the Euclidean case there are at most three points with this property). 4.2 The parabola and its construction In this section we introduce definitions and some basic results for a parabola in universal hyperbolic geometry. We will illustrate the theory in the familiar setting of universal hyperbolic geometry, with our null circle as usual as the unit circle C in the plane. The situation is richer than in the Euclidean setting because of duality: whenever we define an important point x, its dual line X = x with respect to C is also likely to be important, and vice versa. Recall that we will consistently employ small letters for points and capital letters for lines, with the convention that if x i is a point, then X i = x i is the corresponding dual line and conversely. So what is a parabola in the hyperbolic setting? As already discussed in [21], the definition is not entirely obvious: there are several different possible ways of trying to generalize the Euclidean theory. Recall that if a is a point and L is a line, then the quadrance q a, L) is defined to be the quadrance between a and the foot t of the altitude line from a to L. Definition 41 Suppose that f 1 and f 2 are two non-perpendicular points such that f 1 f 2 is a non-null line. The parabola P 0 with foci f 1 and f 2 is the locus of a point p 0

75 4. The parabola and its Euclidean properties 67 satisfying q f 1, p 0 ) + q p 0, f 2 ) = ) The lines F 1 f 1 and F 2 f 2 are the directrices of the parabola P 0. Figure 4.2: A parabola P 0 with foci f 1 and f 2 This definition is likely surprising to the classical geometer. In Euclidean geometry, such a relation defines a circle; if F 1 = [a 1, b 1 ] and F 2 = [a 2, b 2 ] then the corresponding equation Q F 1, P ) + Q F 2, P ) = 1 becomes x a1 + a 2 2 )) 2 + y b1 + b 2 2 )) 2 = 1 2 a 1 a 2 ) 2 b 1 b 2 ) 2). 4 So at this point it is not clear what justification we have for the definition of a parabola above. The following connects our theory with the more traditional approach in [28] and [14]. Theorem 42 Parabola focus directrix) A point p 0 satisfies 4.1) precisely when either of the following hold: q f 1, p 0 ) = q p 0, F 2 ) or q f 2, p 0 ) = q p 0, F 1 ). Proof. If f 1 p 0 ) F 1 t 1 and f 2 p 0 ) F 2 t 2 are the feet of the altitudes from a point p 0 on the parabola P 0 with foci f 1 and f 2 to the directrices F 1 and F 2, then f 1 and t 1 are perpendicular points, as are f 2 and t 2. It follows that q f 1, p 0 ) + q p 0, t 1 ) = 1 and q f 2, p 0 ) + q p 0, t 2 ) = 1. But then 4.1) is equivalent to q f 1, p 0 ) = q p 0, F 2 ) or to q f 2, p 0 ) = q p 0, F 1 ). In this way we recover the ancient Greek metrical definition of the parabola, but we note now that there are two foci-directrix pairs: f 1, F 2 ) and f 2, F 1 ). This is a main

76 4. The parabola and its Euclidean properties 68 feature of the hyperbolic theory of the parabola: a fundamental symmetry between the two foci-directrix pairs. The reason for the index 0 on the point p 0 and the parabola P 0 will become clearer when we introduce the twin point p 0 on the twin parabola P 0. We observe that the foci f 1 and f 2 do not lie on the parabola P 0, since for example if f 1 lies on P 0, then q f 1, f 1 ) + q f 2, f 1 ) = 1, which would imply that q f 1, f 2 ) = 1, contradicting the assumption of non-perpendicularity of f 1 and f 2. In Figure 4.2 we see an example of a parabola P 0, in red, with foci f 1 and f 2, and directrices F 1 and F 2, also in red. Figure 4.3: Various examples of parabolas In Figure 4.3 we see some different examples of parabolas over the rational numbers, at least approximately. When the foci f 1 and f 2 are both interior points of the null circle C, there is no interior point p satisfying the condition qp, f 1 )+qp, f 2 ) = 1, since the quadrance between any two interior points is always negative and the quadrance between an interior and exterior points is always greater than or equal 1. So the parabola with both foci inside the null circle is an empty conic. While this does not mean that it is without interest, this chapter deals with non-empty parabolas, by extending the field if necessary. In the next theorem we find the equation of a parabola with given foci. Theorem 43 Parabola conic) The parabola P 0 with foci f 1 and f 2 is a conic. Proof. Suppose that f 1 = [x 1 : y 1 : z 1 ] and f 2 = [x 2 : y 2 : z 2 ]. Then the point p = [x : y : z] lies on P 0 precisely when ) ) xx 1 + yy 1 zz 1 ) 2 1 x 2 + y 2 z 2 ) xx x y2 1 ) 2 + yy 2 zz 2 ) z2 1 x 2 + y 2 z 2 ) x y2 2 ) = 1 z2 2

77 4. The parabola and its Euclidean properties 69 which yields the quadratic equation x 2 + y 2 z 2) x y1 2 z1 2 ) x y2 2 z2) 2 = xx 1 + yy 1 zz 1 ) 2 x y 2 2 z 2 2) + xx2 + yy 2 zz 2 ) 2 x y 2 1 z 2 1) Basic definitions We now define some basic points and lines associated to a parabola P 0 with foci f 1 and f 2, and directrices F 1 f 1 and F 2 f 2 as in Figure 4.4. The axis of the parabola P 0 is the line A f 1 f 2. The axis point of P 0 is the dual point a A. By assumption the axis A is a non-null line, so that a does not lie on A. If the axis A has null points, we shall call these the axis null points of P 0, and denote them by η 1 and η 2, in no particular order. The axis point and line will generally be in black in our diagrams, while the axis null points will be in yellow. Figure 4.4: Dual and tangent lines, twin point and focal lines Theorem 44 Axis symmetry) The axis A = f 1 f 2 of a hyperbolic parabola P 0 is a line of symmetry, and its dual point a is a center. Proof. We denote the reflection of an arbitrary point p 0 lying on P 0 in the axis line A by p 0 r A p 0 ). Then we need to prove that p 0 also lies on P 0. Recall that the reflection in a line or equivalently the reflection in the dual point of that line) is an isometry, so for any two points a and b, q a, b) = q r A a), r A b)). Thus, since f 1, f 2 are fixed by r A they lie on A), 1 = q f 1, p 0 ) + q f 2, p 0 ) = q r A p 0 ), r A f 1 )) + q r A p 0 ), r A f 1 )) = q p 0, f 1 ) + q p 0, f 2 ).

78 4. The parabola and its Euclidean properties 70 This shows that p 0 lies on the parabola P 0. Since reflecting p 0 in A is the same as reflecting p 0 in a, the point a is also the center of the parabola. The base points of P 0 are the points b 1 AF 1 and b 2 AF 2. The dual lines B 1 af 1 and B 2 af 2 are the base lines of P 0. Both base points and base lines will be shown in blue in our diagrams. The vertices v 1 and v 2 are the points, if they exist, where the parabola meets the axis; they are in no particular order. The duals of the vertices are the vertex lines V 1 v1 and V 2 v2. The vertices and vertex lines will be shown in black. A generic point on P 0 will be denoted p 0, and its dual line denoted P 0. Both are shown in black in our diagrams, with often a small circle drawn around p 0 to highlight it. The tangent line to P 0 at p 0 will be denoted P 0, and its dual point p 0 will be called the twin point of p 0. Both p 0 and P 0 will be shown in grey. The focal lines of p 0 are R 1 p 0 f 1 and R 2 p 0 f 2, and the altitude base points of p 0 are t 1 R 1 F 1 and t 2 R 2 F 2. The duals of the focal lines are the focal points r 1 R1 and r 2 R2 of p 0. The duals of the focal base points are the altitude base lines T 1 t 1 and T 2 t 2 of p 0. The focal lines and points will be shown in green in our diagrams. Figure 4.4 shows these various basic points and lines associated to the parabola P 0 and a point p 0 on it Construction with a dynamic geometry program It is helpful to have a construction of a hyperbolic parabola that can be used with a dynamic geometry package, such as Geometer s Sketchpad, GeoGebra, C.a.R., Cinderella, Cabri etc., which can create loci. For this it is helpful to refresh our minds about the construction of the Euclidean parabola, because a similar technique applies to construct a hyperbolic parabola. We also mention some related facts that will have analogs in the hyperbolic setting. Figure 4.5: Construction of a Euclidean parabola

79 4. The parabola and its Euclidean properties 71 Firstly, we choose a point F focus), and a line f directrix), not passing through F. Draw the perpendicular line a axis) to f through F. Using an arbitrary point T on the directrix f, construct the midpoint M of the side T F, and draw the perpendicular line p to T F through M. Finally, the intersection of the altitude r to f through T and the line p is a point P on the parabola P, which is then the locus of the point P as T moves on f. This is shown in Figure 4.5. It is a well-known classical fact that the line p = P M is the tangent to P at P. Another construction of the tangent arises from considering the altitude to the focal line P F through F. If this line meets the directrix f at S, then p = P S is also the tangent line to P. The circle S with center S through F also passes through T, and is tangent at these points to P F and P T respectively. Furthermore the tangent n to the focal conjugate N, which is the other point of the parabola on the focal line P F, also meets the directrix at S, and is perpendicular to the tangent p at P there. Figure 4.6: Construction of a hyperbolic parabola P 0 To construct a hyperbolic parabola P 0 from a pair of foci f 1 and f 2 with axis A, we proceed as in the Euclidean case, but we must be aware that the existence of midpoints is more subtle they may not exist, and when they do, there are generally two of them! The situation is illustrated in Figure 4.6; choose a point t 1 on the directrix F 1 f 1 with the property that the side t 1 f 2 has midpoints, call them m 1 and p 0, with corresponding midlines M 1 = m 1) and P 0 = p 0). One way of choosing such a point t 1 is to first choose an arbitrary point a 1 on F 1 and then reflect b 1 F 1 A in a 1 to obtain t 1. In the triangle b 1 t 1 f 2, two sides now have midpoints, so by Menelaus theorem [51]) the third side t 1 f 2 will also have midpoints. Now construct the meets p 0 P 0 R 1 and n 1 M 1 R 1, where R 1 = t 1 f 1. Then p 0 and n 1 will both be points on the parabola P 0. The Figure also shows the symmetry available here: it is equally possible to choose a point t 2 on the other directrix F 2

80 4. The parabola and its Euclidean properties 72 f 2 with the property that the side t 2 f 1 has midpoints, call them m 2 and p 0, with corresponding midlines M 2 = m 2) and P 0 = p 0). In that case the points p0 P 0 R 2 and n 2 M 2 R 2, where R 2 = t 2 f 2 lie on the parabola P 0. In Figure 4.6, the two points t 1 and t 2 are related by the fact that t 1 t 2 meets the axis A at the same point j 0 as does P 0 ; this accounts for the fact that t 1 f 2 and t 2 f 1 have a common midpoint p 0. The justification for this construction will be available later, after we establish a suitable framework for coordinates and derive formulas for all the relevant points Dual conics and the connection with sydpoints The theory of the hyperbolic parabola connects strongly with the notion of sydpoints as developed in the last chapter. Figure 4.7: The parabola P 0 and its twin P 0 The reason is that the sydpoints f 1 and f 2 of the side f 1 f 2, should they exist and our assumptions on our field will guarantee that they do) are naturally determined by the geometry of the parabola P 0, and then they become the foci for the twin parabola P 0 in orange in our diagrams), which turns out to be the dual of the conic P 0 with respect to the null circle C. The sydpoint symmetry between the sides f 1 f 2 and f 1 f 2 is key to understanding many aspects of these conics. Although we will be studying the twin parabola more in the next chapter, it will be useful to be aware of it initially, as it explains some of our notational conventions. In Figure 4.7, we see the parabola P 0 with foci f 1, f 2 and a point p 0 on it, as well as the twin parabola P 0 with foci f 1, f 2 and the twin point p 0 on it, which is the dual of the tangent P 0 to P 0 at p 0. Reciprocally the dual of p 0 is the tangent to P 0 at p 0.

81 4. The parabola and its Euclidean properties 73 Note that the tangents to both the parabola P 0 and the null circle C at their common meets, namely the null points α 0 and α 0, pass through the foci of the twin parabola P 0! Dually, note that the tangents to both the parabola P 0 and the null circle C at their common meets, namely the null points δ 0 and δ 0, pass through the foci of P 0! This Figure also shows the twin directrices F 1 and F 2, and the twin base points b 1 and b Standard Coordinates and duality The four basis null points In order to bring a systematic treatment to the study of the hyperbolic parabola P 0, we would like an appropriate coordinate system to bring P 0 into as simple a form as possible. Although there is a great deal of choice for such an attempt, the one that we present here is the simplest and most elegant we could find; in it the beauty of the parabolic theory is reflected in an elegance and coherence in the corresponding formulae. The key point is that aside from the two foci f 1 and f 2 which we used to define the parabola, there are four other points which naturally lie on the parabola and which can be used effectively as a basis for projective coordinates: the two vertices v 1 and v 2, together with two null points α 0 and α 0 which are symmetrically placed with respect to the axis. We need to say some words about the existence of four such points. A priori there is no guarantee that the axis A meets the parabola; it will do so when the corresponding quadratic equation formed by meeting the line with the conic has a solution. The existence of the vertices is then an assumption that we may justify by adjoining an algebraic square root, if required, to our field. We will use the four points v 1, v 2, α 0 and α 0, no three which are collinear, as a basis of a new projective coordinate system. Theorem 45 Parabola vertex) If there is a non-null point v 1 lying both on the axis A and the parabola P 0, then the perpendicular point v 2 v1 A also lies on both the axis and the parabola, and these then are the only two points with this property. Proof. Suppose that v 1 lies on the axis A f 1 f 2 and on the parabola. Then if v 1 is not a null point, q f 1, v 1 ) + q v 1, f 2 ) = 1. Define v 2 v1 A, so that q v 1, v 2 ) = 1. Now recall that if a, b and c are collinear points with q a, b) = 1, then q a, c) + q c, b) = 1. So q v 1, f 1 ) + q f 1, v 2 ) = 1 and q v 1, f 2 ) +

82 4. The parabola and its Euclidean properties 74 q f 2, v 2 ) = 1. Combining all three equations we see that q f 1, v 2 ) + q v 2, f 2 ) = 1, showing that v 2 also lies on the parabola. Since a line meets a conic at most at two points, there can be no other points on the axis and on P 0. We can see from the green parabola in Figure 4.3 that a parabola need not necessarily meet its axis. However any given line will meet a given conic if we are allowed to augment our field to an appropriate quadratic extension. So by possibly extending our field, we will henceforth assume that the parabola P 0 meets the axis A = f 1 f 2. By the theorem above, it then meets this axis in exactly two points, which we call the vertices of the parabola, and denote by v 1 and v 2. What about the existence of null points on P 0? The meet of any two conics might have from zero to four points. We are going to assume that P 0 has at least one null point α 0, not lying on the axis. In that case the reflection α 0 of α 0 is also a null point on P 0. In order for this assumption to hold, we may well have to extend our field. Figure 4.8: The four basis points v 1, v 2, α 0 and α 0 The parabola P 0 with foci f 1 and f 2 need not meet the null conic C. However for most examples, especially those of interest to a classical geometer working in the Klein model in the interior of the unit disk, we do have such an intersection at least approximately over the rational numbers. So by possibly extending our field to a quartic extension, we will henceforth assume that our parabola P 0 passes through at least one null point α 0. By the assumption in the previous theorem such a null point α 0 cannot lie on the axis, so if we reflect it in the axis we get a second null point α 0 r a α 0 ) which also lies on P 0, since P 0 is invariant under r a. Clearly no three of the four basis points v 1, v 2, α 0 and α 0 are collinear, since they all lie on the parabola.

83 4. The parabola and its Euclidean properties The Fundamental theorem and standard coordinates We now invoke the Fundamental Theorem of Projective Geometry as described in Chapter 2, which allows us to make a unique projective change of coordinates so that the four basis points become v 1 = [0 : 0 : 1] v 2 = [1 : 0 : 0] α 0 = [1 : 1 : 1] α 0 = [1 : 1 : 1]. It follows that A = v 1 v 2 = [0 : 0 : 1] [1 : 0 : 0] = 0 : 1 : 0. These new coordinates will be called standard coordinates for the parabola P 0, or parabolic standard coordinates. Note carefully that the introduction of such new coordinates via a projective transformation will necessarily change the form of the quadrance and spread! We now define, as in Figure 4.8, the points obtained by reflecting α 0 and α 0 in v 2 : namely β 0 r v2 α 0 ) and β 0 r v2 α 0 ). Because reflection is an isometry, these are also null points. Our notation with the overbar is something we will employ consistently: α 0 and α 0 are reflections in the point a, or equivalently in the dual line A, and so similarly for β 0 and β 0. Theorem 46 β null points) We have β 0 = α 0 v 2 ) α 0 v 1 ) and β 0 = α 0 v 2 ) α 0 v 1 ). Furthermore in the new coordinate system β 0 = [ 1 : 1 : 1] and β 0 = [ 1 : 1 : 1], Proof. The quadrangle of null points α 0 α 0 β 0 β 0 has one diagonal point v 2, obviously from the definition of β 0 and β 0. It has another diagonal point a, because both α 0 α 0 and β 0 β 0 pass through it; the first by construction and the second because it is obtained from the first by reflection in v 2, which lies on A = a. So the third diagonal point is the dual of av 2, which is v 1 by the previous theorem. It follows that β 0 = α 0 v 2 ) α 0 v 1 ) and β 0 = α 0 v 2 ) α 0 v 1 ). Now we can calculate that β 0 = [1 : 1 : 1] [1 : 0 : 0]) [1 : 1 : 1] [0 : 0 : 1]) = 0 : 1 : 1 1 : 1 : 0 = [ 1 : 1 : 1] β 0 = [1 : 1 : 1] [1 : 0 : 0]) [1 : 1 : 1] [0 : 0 : 1]) = 0 : 1 : 1 1 : 1 : 0 = [ 1 : 1 : 1]. When we apply a general projective transformation of the projective plane to get the four points v 1, v 2, α 0 and α 0 into standard position, the metrical structure will change. While we started with the symmetric matrix J for the form, the new

84 4. The parabola and its Euclidean properties 76 symmetric matrix is of the form C = MJM T for some invertible matrix M. However this matrix C is not arbitrary; since we require that the four points lie on the parabola P 0. We now arrive at the crucial result which sets up our coordinate system, and is the basis for all subsequent calculations. This is the fact that the new matrix C, and its adjugate D, have a particularly simple form, depending on a single parameter α which subsequently appears in almost all our formulas. Theorem 47 Parabola standard coordinates) The symmetric bilinear form in standard coordinates is given by v 1 v 2 = v 1 Cv2 T where α C = 0 1 α 2 0 and D = adj C) = for some number α. In terms of α, the parabola P 0 has equation and its foci are xz y 2 = 0 α α α 2 1 α 2) f 1 = [α + 1 : 0 : α α 1)] and f 2 = [1 α : 0 : α α + 1)]. 4.2) Proof. Suppose that our new bilinear form in standard coordinates is given by v 1 v 2 = v 1 Cv2 T where a d f bc g 2 fg cd dg bf C = d b g and D = adj C) = fg cd ac f 2 df ag. f g c dg bf df ag ab d 2 The fact that the four points α 0 = [1 : 1 : 1], α 0 = [1 : 1 : 1], β 0 = [ 1 : 1 : 1] and β 0 = [ 1 : 1 : 1] must all be null points means α 0 Cα T 0 = α 0 C α 0 ) T = β 0 Cβ T 0 = β 0 C β 0 ) T = 0. These conditions lead to the following linear system of equations involving the entries of C: a + b + c + 2d + 2f + 2g = 0 a + b + c 2d + 2f 2g = 0 a + b + c 2d 2f + 2g = 0 a + b + c + 2d 2f 2g = 0.

85 4. The parabola and its Euclidean properties 77 From this we deduce that d = f = g = 0, and a = b + c). So the matrices have the form, up to scaling, of: a 0 0 C = 0 1 a and D = a a a a 1) But there is also the condition that P 0 is a parabola with foci f 1 and f 2, passing through all four basis points v 1 = [0 : 0 : 1], v 2 = [1 : 0 : 0], α 0 = [1 : 1 : 1] and α 0 = [1 : 1 : 1]. Since the foci lie on the axis A = v 1 v 2, we can write f 1 = [m 1 : 0 : 1] and f 2 = [m 2 : 0 : 1] for some m 1, m 2. Then recall that the quadrance and spread are determined by the matrices C and D by the rules 2.8). We then compute q f 1, f 2 ) = 1. am 1 m 2 1) 2 am ) am 2 2 1) = a m 1 m 2 ) 2 am ) am 2 1 1). Since f 1 and f 2 are by assumption not perpendicular, Also v 1 and v 2 lie on P 0, so that am 1 m ) 0 = q [m 1 : 0 : 1], [0 : 0 : 1]) + q [m 2 : 0 : 1], [0 : 0 : 1]) 1 = am 1m 2 1) am 1 m 2 + 1) am ) am 2 and 1 1) 0 = q [m 1 : 0 : 1], [1 : 0 : 0]) + q [m 2 : 0 : 1], [1 : 0 : 0]) 1 = am 1m 2 1) am 1 m 2 + 1) am ) am 2 2 1). Both these conditions, given 4.3), are equivalent to the relation which we henceforth assume, implying that we may write am 1 m = 0 4.4) m 1 = m and m 2 = 1 am for some non-zero number m. In addition we must ensure that α 0 and α 0 lie on P 0, but since these are both null points, the quadrances q f 1, α 0 ) and q f 2, α 0 ) etc. are undefined, and we must rather work with the general equation of the parabola. This is 0 = q [m : 0 : 1], [x : y : z]) + q [ 1 am : 0 : 1 ], [x : y : z] ) 1 = 4amxz y2 a 1) am 2 1 ) am 2 1) ax 2 ay 2 + y 2 z 2 )

86 4. The parabola and its Euclidean properties 78 which shows the equation of the parabola to be 4amxz y 2 a 1) am 2 1 ) = ) Now the condition that α 0 = [1, 1, 1] and α 0 = [1, 1, 1] lie on P 0 is that 4am a 1) am 2 1 ) = a 1 a) m 2 + 4am + a 1) = ) Given that we started out with the existence of f 1 and f 2 assumed, we see that the discriminant of this quadratic equation 4.6), namely 4a) 2 4a 1 a) a 1) = 4a a + 1) 2 must be a square. But this occurs precisely when a is a square, say a = α 2. In this case the quadratic equation 4.6) becomes α 2 1 α 2) m 2 +4α 2 m+ α 2 1 ) = 0 with solutions m 1 = 1 + α α α 1) and m 2 = 1 α α α + 1). Combining these with 4.5), the identity 2 α + 1) 0 = 4α α α 1) xz y2 α 2 1 ) ) 1 + α 2 α 2 1) α α 1) = 4 xz y 2) α α + 1) α 1 shows that the equation of the parabola pleasantly simplifies to be The foci may now be expressed as xz y 2 = ) f 1 = [m 1 : 0 : 1] = [α + 1 : 0 : α α 1)] f 2 = [m 2 : 0 : 1] = [1 α : 0 : α α + 1)]. and Notice that α det 0 1 α 2 0 = α 2 α 1) α + 1) so α 0, ±1, since C is an invertible projective matrix. Figure 4.9 shows a view in the standard coordinate plane, where [x : y : 1] is represented by the affi ne point [x, y]. This corresponds roughly to a value of α = 0.3.

87 4. The parabola and its Euclidean properties 79 Figure 4.9: A standard coordinate view of a parabola While it is both interesting and instructive to see different views of such a standard coordinate plane, this is somewhat unfamiliar to the classical geometer, so we will stick mostly to the Universal Hyperbolic Geometry model for our diagrams, where the unit circle always appears in blue as the unit circle x 2 + y 2 = 1. Theorem 48 Parabola parametrization) The parabola P 0 is parametrized, using an affi ne parameter t, by p 0 = [ t 2 : t : 1 ] p t) or by using a projective parameter [t : r] as p 0 = [ t 2 : tr : r 2] p t : r). Proof. The simple form of the equation xz = y 2 makes the parametrization immediate. The projective parametrization of P 0 has the advantage that it includes the important point at infinity p 1 : 0) = [1 : 0 : 0] = v 2. We can recover the affi ne parametrization by setting r = 1, and we can go from the affi ne to the projective parametrization by replacing t with t/r and clearing denominators. In practice we will generally use the affi ne parametrization, since it is requires only one variable, not two. The existence of this simple parametrization will be extremely useful for us: giving us the same amount of control over the hyperbolic parabola as we have over the much simpler Euclidean parabola which of course can be positioned to have exactly the same equation!) Theorem 49 Parabola quadrance) The quadrance of the parabola is q P0 = q f 1, f 2 ) = α ) 2 4α 2.

88 4. The parabola and its Euclidean properties 80 Proof. We compute that q P0 = q [α + 1 : 0 : α α 1)], [1 α : 0 : α α + 1)]) = 1 α 4α 2 α 1)2 α + 1) ) = 4α 2. We note that q P0 is a square. This is a reflection of the fact that the assumption of the existence of vertices implies that the sides f 1 b 2 and f 2 b 1 have midpoints. The conditions for points and lines to be null, in other words the equation for the null circle, are the following in standard coordinates. Theorem 50 Null points/ lines) The point p = [x : y : z] in standard coordinates is a null point precisely when α 2 x α 2) y 2 z 2 = 0. The line L = l : m : n is a null line precisely when 1 α 2 ) l 2 + α 2 m 2 + α 2 α 2 1 ) n 2 = 0. Proof. These follow by using 4.2) to expand the respective conditions [x : y : z] C [x : y : z] T = 0 and l : m : n T D l : m : n = 0. In order for the parabola y 2 = xz to have a null point p t), the parameter t must satisfy [ t 2 : t : 1 ] C [ t 2 : t : 1 ] T = 0, which yields t 2 1 ) t 2 α ) = 0. Over the rational field, the values t = ±1 agree with the null points α 0 = [1 : 1 : 1] and α 0 = [1 : 1 : 1] with which we begun our work. However, there are also another two solutions which are invisible over the rational field, but in an extension field obtained by adjoining a square root i of 1, they do exist. These points are ζ 1 [ 1 : iα : α 2] and ζ 2 [ 1 : iα : α 2]. In this thesis we will not mention these points too much, as we are mostly interested in the rational case A rational parabola If we work in the original setting of universal hyperbolic geometry, with metrical structure given by 2.7) and 2.9), then we can illustrate the above transformations explicitly with a two parameter family of rational hyperbolic parabolas. The conic P 0 with equation t 2 1 t ) x t 2 1t ) x + t 2 1 t 2 ) 2 y 2 + t 2 1t ) = 0

89 4. The parabola and its Euclidean properties 81 meets the null circle at the null points α 0 = [ 1 t 2 1 : 2t 1 : t ] and α 0 = [ 1 t 2 1 : 2t 1 : t ]. This is a parabola with foci f 1 = [ t 1 + t 2 t 1 t t 2 1t 2 : 0 : t 1 + t 2 + t 1 t 2 2 t 2 1t 2 ] f 2 = [ t 1 t 2 t 1 t 2 2 t 2 1t 2 : 0 : t 1 t 2 + t 1 t t 2 1t 2 ], and axis A = 0 : 1 : 0, and t-foci f 1 = [ t : 0 : t )] and f 2 = [ t : 0 : t )]. The null points β 0, β 0 are β 0 = [ 1 t 2 2 : 2t 2 : t ] and β 0 = [ 1 t 2 2 : 2t 2 : t ], and the vertices are v 1 = [t 1 t 2 1 : 0 : t 1 t 2 + 1)] and v 2 = [t 1 t : 0 : t 1 t 2 1)]. Note that is a square. q f 1, f 2) 1 = q [ t : 0 : t )], [ t : 0 : t )]) 1 = 1 4 t 1 t 2 ) 2 t 1 + t 2 ) 2 t 2 1 t2 2 We now send the points α 0, α 0, β 0, β 0 respectively to the points [1 : 1 : 1], [1 : 1 : 1], [ 1 : 1 : 1], [ 1 : 1 : 1], using a projective transformation. Firstly, we send [1 : 1 : 1], [1 : 0 : 0], [0 : 1 : 0] and [0 : 0 : 1] to α 0, α 0, β 0, β 0 respectively by the linear transformation T 1 v) = vn where N is t 2 t ) 2t 1 t 2 t 2 t ) N = t 1 t ) 2t 1 t 2 t 1 t ) t 1 t ) 2t 1 t 2 t 1 t ). Its inverse sends α 0, α 0, β 0, β 0 back to [1 : 1 : 1], [1 : 0 : 0], [0 : 1 : 0], [0 : 0 : 1] by T v) = vr where R is the adjugate of N : 2t 1 t ) t 1 t 2 ) t 1 t 2 1) t 1 t 2 + 1) t 1 + t 2 ) R = 0 t 2 1 t2 2 t 2 1 t2 2 2t 1 t ). t 1 t 2 ) t 1 t 2 + 1) t 1 t 2 1) t 1 + t 2 ) Secondly, the linear transformation T 2 v) = vm, where M is M =

90 4. The parabola and its Euclidean properties 82 sends [1 : 1 : 1], [1 : 0 : 0], [0 : 1 : 0], [0 : 0 : 1] to [1 : 1 : 1],[ 1 : 1 : 1],[ 1 : 1 : 1],[ 1 : 1 : 1] respectively. Thus, the required transformation is T v) = v RM) where RM is t 1 t 2 + 1) t 1 + t 2 ) 0 t 1 t 2 ) t 1 t 2 1) RM = 0 t 1 t 2 ) t 1 + t 2 ) 0. t 1 t 2 1) t 1 + t 2 ) 0 t 1 t 2 ) t 1 t 2 + 1) After applying this linear transformation, the matrix J is replaced by C = RM) 1 J RM) 1) t 1 t 2 t 1 t 2 ) T = 0 4t 2 1 t t 1 t 2 t 1 + t 2 ) 2 D = RM) T J RM) = 4t 1 t 2 t 1 + t 2 ) t 2 1 t 2 ) t 1 t 2 t 1 t 2 ) 2 and we get α = t 1 t 2 t 1 +t 2. In this new coordinate system, the parabola is indeed y 2 = xz with foci f 1 = [t 1 t 1 + t 2 ) : 0 : t 2 t 1 t 2 )] and f 2 = [t 2 t 1 + t 2 ) : 0 : t 1 t 1 t 2 )]. Example 51 For a completely explicit example, we take t 1 = 1/2 and t 2 = 3. Then the parabola P 0 has equation 26x+5x 2 35y 2 +5 = 0 which meets the null circle at the null points α 0 = [3 : 4 : 5] and α 0 = [3 : 4 : 5]. It has axis A = 0 : 1 : 0, foci f 1 = [ 1 : 0 : 29] and f 2 = [ 31 : 0 : 11], and t-foci f 1 = [5 : 0 : 3] and f 2 = [4 : 0 : 5]. So the null points β 0, β 0 are β 0 = [ 4 : 3 : 5] and β 0 = [4 : 3 : 5]. The parabola P 0 has also vertices v 1 = [1 : 0 : 5] and v 2 = [5 : 0 : 1] Formulas for directrices, vertex lines, base points and base lines Returning to the general case, we can now augment our formulas using standard coordinates. The directrices are F 1 = f 1 = C [α + 1 : 0 : α α 1)] T = α α + 1) : 0 : 1 α F 2 = f 2 = C [1 α : 0 : α α + 1)] T = α α 1) : 0 : 1 + α. The base points are the meets of the directrices and the axis line. They are b 1 = F 1 A = α 2 α + 1) : 0 : α 1 α) 0 : 1 : 0 = [α 1 : 0 : α α + 1)] b 2 = F 2 A = α 2 α 1) : 0 : α 1 + α) 0 : 1 : 0 = [α + 1 : 0 : α 1 α)]. The duals are the base lines B 1, B 2, which are the altitudes to the axis A through the foci f 1, f 2 of the parabola: B 1 = b 1 = C [α 1) : 0 : α α + 1)] = α α 1) : 0 : α + 1 B 2 = b 2 = C [α + 1) : 0 : α 1 α)] = α α + 1) : 0 : α 1. and

91 4. The parabola and its Euclidean properties 83 The vertex lines V 1, V 2 are the altitudes to the axis A through the vertices v 1, v 2 of the parabola: V 1 = v 1 = C [0 : 0 : 1] = [0 : 0 : 1] and V 2 = v 2 = C [1 : 0 : 0] = [1 : 0 : 0]. Figure 4.10: Some basic points associated to a parabola P Quadrance and spread in standard coordinates We can now give explicit formulas for quadrance and spread in standard coordinates, using the matrices C and D appearing in the parabola standard coordinates theorem. Theorem 52 Quadrance formula) The quadrance between the points p 1 = [x 1 : y 1 : z 1 ] and p 2 = [x 2 : y 2 : z 2 ] in parabolic standard coordinates is α 4 ϕ 2 1 q p 1, p 2 ) = + α2 ϕ 2 2 ϕ2 3 1) ϕ2 + ϕ 2 3 α 2 x 2 1 α2 1) y1 2 ) z2 1 α 2 x 2 2 α2 1) y2 2 2) z2 where ϕ 1 = x 1 y 2 x 2 y 1 ), ϕ 2 = x 1 z 2 x 2 z 1 ) and ϕ 3 = y 1 z 2 y 2 z 1 ). Proof. From 2.8) and the parabola standard coordinates theorem [x 1, y 1, z 1 ] C [x 2, y 2, z 2 ] T = α 2 x 1 x 2 α 2 1 ) y 1 y 2 z 1 z 2. the formula follows using an identity calculation. Theorem 53 Spread formula) The spread between the lines L 1 = l 1 : m 1 : n 1 and L 2 = l 2 : m 2 : n 2 is S L 1, L 2 ) = α l 2 1 n 2 l 2 n 1 ) 2 m 1 n 2 m 2 n 1 ) 2) + l 1 m 2 l 2 m 1 ) 2 l 2 n 1 l 1 n 2 ) 2) α 2 1) l1 2 α2 m 2 1 ) α2 α 2 1) n 2 1 α 2 1) l2 2 α2 m 2 2 α2 α 2 1) n2) 2.

92 4. The parabola and its Euclidean properties 84 Proof. From 2.8) and [l 1, m 1, n 1 ] D [l 2, m 2, n 2 ] T = l 1 l 2 α 2 1 ) α 2 m 1 m 2 α 2 α 2 1 ) n 1 n 2 the formula follows using an identity calculation. The fundamental symmetry of the parabola is expressed very nicely in our coordinates. Theorem 54 Axis reflection) The reflection r a standard coordinates r a [x : y : z]) = [x : y : z]. in the point a has the form in Proof. We use the standard formula for reflection of vectors in space u v) v ucv T ) v r v u) = 2 u = 2 v v vcv T u. With the matrix C above, we get [0, 1, 0] C [x, y, z]t r [0,1,0] [x, y, z]) = 2 [0, 1, 0] [x, y, z] = [ x, y, z] = [x, y, z]. T [0, 1, 0] C [0, 1, 0] Duality with respect to a conic and tangents Let s recall some basic facts from the general theory of points and tangents to a projective conic. Suppose that a general conic C is given by the projective symmetric 3 3 matrix A, with adjugate B, so that a general point p = [x : y : z] lies on C precisely when pap T = 0. The tangent line P to a point p lying on C is then P = p Ap T. Dually, the point at which a tangent line L meets the conic is l = L L T B. While a point p on the conic satisfies the equation pap T = 0, a line L on the conic that is, a tangent line to the conic at some point) satisfies the dual equation L T BL = 0 where we regard lines as column vectors). More generally, we can regard the projective matrix A as determining a projective bilinear form, which is equivalent to a duality between points and lines. For a general point p, not necessarily lying on C, its dual with respect to C is the line p = Ap T, while for a general point L, its dual with respect to C is the point L = L T B. These are inverse procedures. These notions go back to Apollonius, and it could be argued that this duality between points and lines is the essential feature or characteristic of a conic. But this modern formulation in the language of projective linear algebra and matrices makes many of its aspects much easier to understand, see [5], [32], [7], [8]. In this thesis, the main example of duality is with respect to the null circle C, for which we will stick with the notation that if x is a point, then X = Cx T refers to the

93 4. The parabola and its Euclidean properties 85 dual line and conversely. However the secondary duality with respect to the parabola P 0 will also be involved, as we now see. The equation 4.7) for the parabola P 0 in standard coordinates, namely p x, y, z) = xz y 2 = 0, can be expressed in homogeneous matrix form as pap T = 0 or [ ] [ T x y z A x y z] = 0 where A = and adj A) B = Theorem 55 Parabola tangent parametrization) The tangent line P 0 to the parabola at p 0 = [ t 2 : t : 1 ] is P 0 = 1 : 2t : t 2 P t) or projectively the tangent to p 0 = [ t 2 : tr : r 2] is P 0 = r 2 : 2rt : t 2 P t : r). A line L = l : m : n is tangent to the parabola precisely when m 2 = 4nl. Proof. The formula for the tangent line is a direct application of the general fact mentioned in the previous section, so that P 0 Ap T 0 = or using projective parameters P 0 Ap T 0 = [ t 2 : t : 1 ] T = 1 : 2t : t [ t 2 : rt : r 2] T = r 2 : 2rt : t The relation m 2 = 4nl is exactly satisfied by those lines of this form. Theorem 56 Tangent meets) If p 0 p t) and q 0 p u) are two distinct points on P 0, then their tangents P 0 and Q 0 meet at the polar point z P 0 Q 0 = [2tu : t + u : 2] while Z p 0 q 0 = 1 : t + u) : tu. Proof. We compute that z P 0 Q 0 = 1 : 2t : t 2 1 : 2u : u 2 = [2tu : t + u : 2] and Z p 0 q 0 = [ t 2 : t : 1 ] [ u 2 : u : 1 ] = 1 : t + u) : tu.

94 4. The parabola and its Euclidean properties 86 The dual of the point p 0 = [ t 2 : t : 1 ] on P 0 is P 0 = Cp T 0 = α α 2 0 [ t 2 : t : 1 ] T = t 2 α 2 : t 1 α 2) : and the dual of the tangent line P 0 = 1 : 2t : t 2 is p 0 = P 0) T D = 1 : 2t : t 2 α = [ α 2 1 : 2tα 2 : t 2 α 2 α 2 1 )]. 0 α α 2 1 α 2) We will say that p 0 is the twin point to p 0. Later we will see that the locus of p 0 is also a parabola P 0, whose foci f 1 and f 2 are the sydpoints of f 1 f 2. We now check that the expected polarity between foci and directrices with respect to P 0 holds. Theorem 57 Focus directrix polarity) The focus f 1 is the pole of the directrix F 2 with respect to the parabola P 0, and similarly the focus f 2 is the pole of the directrix F 1. Proof. We check that F T 2 B = α α 1) : 0 : α + 1 B = [α + 1 : 0 : α α 1)] = f 1 Af T 1 = A [α + 1 : 0 : α α 1)] T = α α 1) : 0 : α + 1 = F 2. or Similarly, F T 1 B = α α + 1) : 0 : 1 α B = [ α 1) : 0 : α α + 1)] = f 2 Af T 2 =A [1 α : 0 : α α + 1)] T = α α + 1) : 0 : 1 α = F 1. or The j, h and d points and lines We now introduce some other secondary points and lines associated to a generic point p 0 on the parabola P 0. The reflection of p 0 = [ t 2 : t : 1 ] in the axis is the opposite point p 0 = r a p 0 ) = [ t 2 : t : 1 ]. Clearly p 0 also lies on the parabola. We define the axis null points to be the meets of the axis A and the null conic C. These points exist under our assumptions, and are η 1 AC = [1 : 0 : α] and η 2 = AC = [ 1 : 0 : α].

95 4. The parabola and its Euclidean properties 87 The meet of the dual line P 0 with the axis A is the j-point with dual the J-line j 0 P 0 A = t 2 α 2 : t 1 α 2) : 1 0 : 1 : 0 = [ 1 : 0 : t 2 α 2] J 0 = ap 0 = [0 : 1 : 0] [ t 2 : t : 1 ] = 1 : 0 : t 2. By duality J 0 is the altitude from p 0 to the axis, and so also J 0 = p 0 p 0. The meet of the J-line with the axis is the foot of this altitude; it is the h-point h 0 AJ 0 = 0 : 1 : 0 1 : 0 : t 2 = [ t 2 : 0 : 1 ] and its dual is the H-line H 0 h 0 = aj 0 = [0 : 1 : 0] [ 1 : 0 : t 2 α 2] = t 2 α 2 : 0 : 1. Figure 4.11: The j and h points and lines The meet of the tangent line P 0 with the axis is the twin j-point j 0 P 0 A = 1 : 2t : t 2 0 : 1 : 0 = [ t 2 : 0 : 1 ] with dual the twin J-line J 0 = ap 0 = [0 : 1 : 0] [ α 2 1 : 2tα 2 : t 2 α 2 α 2 1 )] = t 2 α 2 : 0 : 1. The meet of the twin J-line with the axis is the twin h-point h 0 AJ 0 = 0 : 1 : 0 t 2 α 2 : 0 : 1 = [ 1 : 0 : t 2 α 2] and its dual is the twin H-line H 0 h 0) = aj 0 = [0 : 1 : 0] [ t 2 : 0 : 1 ] = 1 : 0 : t 2.

96 4. The parabola and its Euclidean properties 88 Theorem 58 Null tangent) The tangent P 0 to the parabola P 0 at p 0 is a null line precisely when p 0 lies on a directrix, and in this case the twin point p 0 is a null point lying on the other directrix, j 0 coincides with a focus, and j 0 with the other focus. Proof. If the tangent P 0 = 1 : 2t : t 2 at p 0 = [ t 2 : t : 1 ] is a null line, then by the Null points/lines theorem 1 α 2 ) + 4α 2 t 2 + α 2 α 2 1 ) t 4 = 0. This factors as α α + 1) t 2 α 1) ) α α 1) t 2 + α + 1) ) = 0 so that t 2 = α 1 α α + 1) or t 2 = α + 1 α α 1). 4.8) Now p 0 = [ t 2 : t : 1 ] is on the directrix F 1 or F 2, precisely when [ t 2 : t : 1 ] [ α 2 α + 1) : 0 : α 1 α) ] T = 0 or [ t 2 : t : 1 ] [ α 2 α 1) : 0 : α 1 + α) ] T = 0 and similarly, the point p 0 = [ α 2 1 : 2tα 2 : t 2 α 2 α 2 1 )] is on the directrix F 1 or F 2, precisely when [ α 2 1 : 2tα 2 : t 2 α 2 α 2 1 )] [ α 2 α + 1) : 0 : α 1 α) ] T = 0 or [ α 2 1 : 2tα 2 : t 2 α 2 α 2 1 )] [ α 2 α 1) : 0 : α 1 + α) ] T = 0. These conditions are exactly the same as 4.8). Using 4.8) we get either j 0 = [ 1 : 0 : t 2 α 2] = [α + 1 : 0 : α α 1)] = f 1 and j 0 = [ t 2 : 0 : 1 ] = [1 α : 0 : α α + 1)] = f 2 or j 0 = [1 α : 0 : α α + 1)] = f 2 j 0 = [α + 1 : 0 : α α 1)] = f 1. and We introduce the points d 0 and d 0 to be the meets of the directrix F 2 with the parabola P 0, should they exist, and the corresponding twin null points d 0 δ 0 and δ 0 lying on the directrix F 1. These are important canonical points associated with the parabola. Since their existence requires solutions to 4.8), and so a number τ satisfying τ 2 = α α 2 1 ), we may write d 0 = F 2 P 0 = [α 1 : τ : α α + 1)] and d 0 = F 2 P 0 = [α 1 : τ : α α + 1)]

97 4. The parabola and its Euclidean properties 89 Figure 4.12: Null tangents and d 0, d 0 points and [ ] d 0 δ 0 = α 1) 2 α + 1) : 2αiτ : α α + 1) 2 α 1) [ ] d 0 = δ 0 = α 1) 2 α + 1) : 2αiτ : α α + 1) 2 α 1) and where iτ) 2 = α α 2 1 ). In Figure 4.12, notice that the directrix F 1 meets the null circle at the two points δ 0 and δ 0, while the lines f 1 δ 0 and f 1 δ 0 are joint tangents to both C and the parabola ) ) P 0, touching P 0 at the points d 0 = F 2 f1 δ 0 and d0 = F 2 f1 δ The sydpoints of a parabola It is a remarkable fact that the theory of sydpoints that we developed in the previous chapter plays a crucial role in the theory of the hyperbolic parabola. Define the lines F 2 α 0 α 0 = [1 : 1 : 1] [1 : 1 : 1] = 1 : 0 : 1 B 1 β 0 β 0 = [ 1 : 1 : 1] [ 1 : 1 : 1] = 1 : 0 : 1 with corresponding axis meets b 2 F 2 A = 1 : 0 : 1 0 : 1 : 0 = [1 : 0 : 1] f 1 = B 1 A = 1 : 0 : 1 0 : 1 : 0 = [ 1 : 0 : 1]. The duals of these points and lines are f 2 F 2) [ = 1 : 0 : 1 D = 1 : 0 : α 2 ] b 1 = B 1) [ = 1 : 0 : 1 D = 1 : 0 : α 2 ] B 2 = b 2) = C [1 : 0 : 1] T = α 2 : 0 : 1 F 1 = f 1) = C [ 1 : 0 : 1] T = α 2 : 0 : 1.

98 4. The parabola and its Euclidean properties 90 The points f 1 and f 2 are the twin foci, or t-foci for short, of the parabola P 0. They will play a major role in the theory. The dual lines of f 1 and f 2, namely F 1 and F 2 respectively, are the t-directrices of P 0. The meets of the t-directrices and the axis A are F 1 A b 1 and F 2 A b 2 respectively; these are the t-base points of P 0. The dual lines of b 1 and b 2, namely B 1 and B 2 respectively, are the t-base lines of P. These are all shown in Figure Figure 4.13: Sydpoints and the twin foci f 1 and f 2 of P 0 Theorem 59 Parabola sydpoints) The points f 1 and f 2 are the sydpoints of the side f 1 f 2. Proof. We calculate that Clearly q f 1, f 1) = q [α + 1 : 0 : α α 1)], [1 : 0 : 1]) α α 1) + α 2 α + 1) ) 2 α ) 2 = 1 + 4α 3 4α 5 = 4α α 2 1) q f 2, f 1) = q [1 α : 0 : α α + 1)], [1 : 0 : 1]) α α + 1) α 2 α 1) ) 2 α ) 2 = 1 4α 3 4α 5 = 4α α 2 1) q f 1, f 2) = q [α + 1 : 0 : α α 1)], [ 1 : 0 : α 2]) α 2 α + 1) α 3 α 1) ) 2 = 1 4α 5 4α 7 = 1 α ) 2 4 α α 2 1) q f 2, f 2) = q [1 α : 0 : α α + 1)], [ 1 : 0 : α 2]) α 2 α 1) + α 3 α + 1) ) 2 = 1 + 4α 5 4α 7 = 1 α ) 2 4 α α 2 1). q f 1, f 1) = q f 2, f 1) and q f 1, f 2) = q f 2, f 2)

99 4. The parabola and its Euclidean properties 91 so f 1 and f 2 are the sydpoints of the side f 1 f 2. Theorem 60 Parabola null tangents) The tangents to the null circle at α 0 and α 0 meet at f 2. The tangents to P 0 at α 0 and α 0 meet at f 1. Proof. The tangents to the null circle at α 0 and α 0 are the dual lines α0 = C [1 : 1 : 1] T = α 2 : 1 α 2 : 1 and α 0 ) = C [1 : 1 : 1] T = α 2 : α 2 1 : 1 and these meet at α0 α 0 ) = α 2 : 1 α 2 : 1 α 2 : α 2 1 : 1 = [ 1 : 0 : α 2] = f 2. The tangents to the parabola P 0 at α 0 and α 0 are the lines A [1 : 1 : 1] T = 1 : 2 : 1 and A [1 : 1 : 1] T = 1 : 2 : 1 and these meet at 1 : 2 : 1 1 : 2 : 1 = [ 1 : 0 : 1] = f Focal and base lines We now define some other fundamental points and lines associated with a point p 0 [ t 2 : t : 1 ] on the parabola P 0. It will be convenient to introduce the quantities 1 t) α t 2 α t 2 α 2 and 2 t) α 1 + t 2 α + t 2 α 2 3 t) α + 1 t 2 α + t 2 α 2 and 4 t) α 1 t 2 α 2 t 2 α which depend on p 0, and which will appear in many formulas to follow. Notice that = 4α t 2 α 2 1 ) t ) = 4αt 2 α 2 1 ) = 4α t 4 α 2 1 ) = 4α t 4 α 2 1 ) = 4t 2 α α 2 1 ) = 4α t 2 1 ) t 2 α ). The focal lines R 1, R 2 and the dual focal line points r 1, r 2 are defined by, and calculated as: R 1 f 1 p 0 = [α + 1 : 0 : α α 1)] [ t 2 : t : 1 ] = tα α 1) : 1 : t α + 1) R 2 f 2 p 0 = [1 α : 0 : α α + 1)] [ t 2 : t : 1 ] = tα α + 1) : 2 : t α 1) r 1 R1 = F 1 P 0 = [t α 1) 2 α + 1) : α 1 : tα α 1) α + 1) 2] [ ] r 2 R2 = F 2 P 0 = t α 1) α + 1) 2 : α 2 : tα α 1) 2 α + 1).

100 4. The parabola and its Euclidean properties 92 Since R 1, R 2 and P 0 are concurrent at p 0, dually we see that r 1, r 2 and p 0 are collinear on P 0. The altitude base points t 1 and t 2 and the dual altitude base lines T 1, T 2 are defined by, and calculated as: t 1 F 1 R 1 = [ α 1) 1 : 4tα 2 : α α + 1) 1 ] t 2 F 2 R 2 = [ α + 1) 2 : 4tα 2 : α α 1) 2 ] T 1 t 1 = f 1 r 1 = α α 1) 1 : 4tα α 2 1 ) : α + 1) 1 T 2 t 2 = f 2 r 2 = α α + 1) 2 : 4tα α 2 1 ) : α 1) 2. The focal lines R 1 and R 2 also meet the directrices at the second altitude base points u 1, u 2, with dual lines U 1, U 2 : u 1 R 2 F 1 = [ α 1) 2 : 2tα α 2 1 ) : α α + 1) 2 ] u 2 R 1 F 2 = [ α + 1) 1 : 2tα α 2 1 ) ] : α α 1) 1 U 1 u 1 = α α + 1) 1 : 2t α 2 1 ) 2 : α 1) 1 U 2 u 2 = α α 1) 2 : 2t α 2 1 ) 2 : α + 1) 2. Figure 4.14: The r, s, t and w points of p 0 on P 0 The t-base lines S 1, S 2 and their duals the t-base points s 1, s 2 are defined by, and calculated as: S 1 f 1 t 2 = 2tα 2 α 1) : α 2 1 ) 2 : 2tα α + 1) S 2 f 2 t 1 = 2tα 2 α + 1) : α 2 1 ) 1 : 2tα α 1) s 1 S1 = F 1 T 2 = [2t α 1) : 2 : 2tα α + 1)] s 2 S2 = F 2 T 1 = [2t α + 1) : 1 : 2tα α 1)].

101 4. The parabola and its Euclidean properties 93 Theorem 61 T-base) Both s 1 and s 2 lie on the tangent P 0. Dually the lines S 1 and S 2 meet at p 0. Proof. We verify that s 1 and s 2 lie on the tangent P 0 = 1 : 2t : t 2 by computing [2t α 1) : 2 : 2tα α + 1)] [ 1 : 2t : t 2] T = 0 [2t α + 1) : 1 : 2tα α 1)] [ 1 : 2t : t 2] T = 0. The statement that S 1 and S 2 meet at p 0 follows from duality. as: The w-points w 1 and w 2, and their duals W 1 and W 2, are defined and computed w 1 F 1 S 1 = [ α 2 1 ) α 1) 2 : 8tα 3 : α α 2 1 ) α + 1) 2 ] w 2 F 2 S 2 = [ α 2 1 ) α + 1) 1 : 8tα 3 : α α 2 1 ) α 1) 1 ] W 1 f 1 s 1 = α α 1) 2 : 8tα 2 : α + 1) 2 ) W 2 f 2 s 2 = α α + 1) 1 : 8tα 2 : α 1) 1 ). Theorem 62 J points collinearity) We have collinearities [[ t 1 t 2 j 0]], [[j 0 u 1 u 2 ]] and [[w 1 w 2 j 0 ]]. Proof. Using the various formulas above, we compute α 1) 1 4tα 2 α α + 1) 1 det α + 1) 2 4tα 2 α α 1) 2 = 0, t and 1 0 t 2 α 2 det α + 1) 1 2tα α 2 1 ) α α 1) 1 = 0 α 1) 2 2tα α 2 1 ) α α + 1) 2 α 2 1 ) α 1) 2 8tα 3 α α 2 1 ) α + 1) 2 det α 2 1 ) α + 1) 1 8tα 3 α α 2 1 ) α 1) 1 = t 2 α 2 Define the line T 0 as T 0 = t 1 t 2 = 4tα 2 : 1 2 : 4t 3 α 2. From the collinearity [[ t 1 t 2 j 0]], we can obtain an easy construction for the tangent at the point p 0 : join p 0 to the foci and determine t 1 and t 2 on the directrices F 1 and F 2. Then determine the meet j 0 of this line T 0 and the axis A. Finally, the tangent is the line p 0 j 0.

102 4. The parabola and its Euclidean properties 94 Theorem 63 Null focal line) The focal line R 1 of a point p 0 on the parabola P 0 is a null line precisely when 3 = 0. Similarly, the focal line R 2 is a null line precisely when 4 = 0. Proof. By the Null points/lines theorem, the focal line R 1 = tα α 1) : 1 : t α + 1) of p 0 = [ t 2 : t : 1 ] is a null line precisely when tα α 1) : 1 : t α + 1) D tα α 1) : 1 : t α + 1) T = 0 or α 2 α + t 2 α 2 t 2 α + 1 ) 2 = 0. Since α 0, this is equivalent to 3 = 0. Similarly R 2 = tα α + 1) : 2 : t α 1) is a null line precisely when α 2 α + t 2 α 2 + t 2 α + 1 ) 2 = 0 or 4 = Parallels between the Euclidean and hyperbolic parabolas Some congruent triangles Having established quite a few important points, lines and their formulas in parabolic standard coordinates, we are now in a position to effi ciently derive numerous properties of the hyperbolic parabola which are analogous to these of the Euclidean parabola. Recall that the focal line R 1 p 0 f 1 meets the directrix F 1 in the point t 1. We will assume that the p 0 is such that the focal lines R 1 and R 2 are non-null lines, so that from the previous theorem we have 3 0 and 4 0. Theorem 64 Congruent triangles) Suppose that the tangent P 0 to P 0 at p 0 meets S 2 = t 1 f 2 at the point m 1. Then the triangles p 0 t 1 m 1 and p 0 f 2 m 1 are congruent. In particular i) q p 0, t 1 ) = q p 0, f 2 ); ii) q t 1, m 1) = q m 1 ), f 2 ; iii) S2 P 0 ; iv) the tangent P 0 is a bisector of the vertex R 1 R 2 ; v) S S 2, R 1 ) = S S 2, R 2 ); and vi) the tangent P 0 is a midline of the side t 1 f 2. The same statements are true by f 1 f 2 symmetry if we interchange the indices 1 and 2. Proof. i) The first statement q p 0, t 1 ) = q p 0, f 2 ) comes from the definition of the parabola P 0, and we can also calculate quadrances to obtain q p 0, t 1 ) = q [ t 2 : t : 1 ], [ α 1) 1 : 4tα 2 ]) : α α + 1) 1 = = q [ t 2 : t : 1 ], [1 α : 0 : α α + 1)] ) = q p 0, f 2 ). 2 3

103 4. The parabola and its Euclidean properties 95 Figure 4.15: Two pairs of congruent triangles ii) Calculate m 1 = P 0 S 2 = 1 : 2t : t 2 2tα 2 α + 1) : α 2 1 ) 1 : 2tα α 1) ] [ = [t 2 α 1) 2 4 : 2tα 4 : α + 1) 2 4 = t 2 α 1) 2 : 2tα : α + 1) 2]. Here we have used that the focal line R 2 is non-null so that 4 is nonzero. Thus q t 1, m 1) [α = q 1) 1 : 4tα 2 ] : α α + 1) 1, [ t 2 α 1) 2 : 2tα : α + 1) 2]) α 2 1 ) 4 = 1 4 α [ 3 = q t 2 α 1) 2 : 2tα : α + 1) 2] ), [1 α : 0 : α α + 1)] = q m 1 ), f 2. iii) Since the tangent line P 0 passes through s 2, which is the dual of the line S 2 = t 1 f 2, the tangent P 0 is perpendicular to the line S 2 ; and we can also check that 1 : 2t : t 2 D 2tα 2 α + 1) : α 2 1 ) 1 : 2tα α 1) T = 0. iv) The tangent P 0 is a bisector of the vertex R 1 R 2 since S R 1, P 0) = S tα α 1) : 1 : t α + 1), 1 : 2t : t 2 ) α 2 1 ) 2 3 = 4) 2 4α 4 3 = S tα α + 1) : 2 : t α 1), 1 : 2t : t 2 ) = S R 2, P 0).

104 4. The parabola and its Euclidean properties 96 v) Now calculate the spreads S S 2, R 1 ) = S 2tα 2 α + 1) : α 2 1 ) 1 : 2tα α 1), tα α 1) : 1 : t α + 1) ) = 4t 2 α α ) 2 16t 2 α α2 1) = S 2tα 2 α + 1) : α 2 1 ) 1 : 2tα α 1), tα α + 1) : 2 : t α 1) ) = S S 2, R 2 ). vi) It is obvious that the tangent P 0 is a midline of the side t 1 f 2, since P 0 is perpendicular to the line S 2 = t 1 f 2 through the point m 1 which is, from ii), a midpoint of t 1 f 2. The symmetry between f 1 and f 2 in the definition of the parabola P 0 ensures that all these statements hold also if we interchange the indices 1, 2. In Figure 4.15 we see also the point m 2 = P 0 S 1 and the congruent triangles p 0 t 2 m 2 and p 0 f 1 m 2. We call m 1 and m 2 the t-perpendicular points of p 0. Note that the theorem allows us a simple construction of the tangent P 0 at p 0 : drop the perpendicular to the line t 1 f 2, or symmetrically to t 2 f 1. Corollary 65 We have i) the triangles m 1 t 1 j 0 and m 1 f 2 j 0 are congruent, and ii)the triangles p 0 t 1 j 0 and p 0 f 2 j 0 are congruent. The same statements are true by f 1 f 2 symmetry if we interchange the indices 1 and 2. Proof. The triangles m 1 f 2 j 0 and m 1 t 1 j 0 are right triangles since P 0 is perpendicular to S 2 ; we also have q t 1, m 1) = q m 1, f 2 ) and m 1 j 0 is a common side. i) We calculate the quadrances q t 1, j 0) = q [ α 1) 1 : 4tα 2 : α α + 1) 1 ], [ t 2 : 0 : 1 ]) 2 4 = = q [1 α : 0 : α α + 1)], [ t 2 : 0 : 1 ]) = q j 0 ), f 2 and spreads S t 1 m 1, t 1 j 0) = q m 1, j 0) q t 1, j 0 ) = 16t 2 α 3 16t 2 α α2 1) = q m 1, j 0) q j 0, f 2 ) = S f 2 j 0, f 2 m 1) S j 0 m 1, j 0 ) q m 1 ), t 1 α 2 t 1 = q t 1, j 0 ) = 1 ) t 4 α 2 1 ) = q m 1 ), f q j 0, f 2 ) = S j 0 m 1, j 0 ) f 2. Therefore, the triangles m 1 t 1 j 0 and m 1 f 2 j 0 are congruent.

105 4. The parabola and its Euclidean properties 97 ii) The triangles p 0 f 2 j 0 and p 0 t 1 j 0 have one common side p 0 j 0. Using the Spread law and the congruences above, S t 1 p 0, t 1 j 0) = S p 0 t 1, p 0 j 0) q p 0, j 0) q t 1, j 0 ) = S p 0 f 2, p 0 j 0) q p 0, j 0) q f 2, j 0 = ) 2 4 Therefore, the triangles p 0 f 2 j 0 and p 0 t 1 j 0 are congruent. = S f 2 p 0, f 2 j 0). Theorem 66 Tangent base symmetry) Let j 0 AP 0 be the meet of the axis A and the tangent P 0, and h 0 the base of the altitude from p 0 to A. Then i) q b 1, j 0) = q f 2, h 0 ) and ii) q v 1, j 0) = q v 1, h 0 ). The same statements are true if we interchange the indices 1 and 2 by f 1 f 2 symmetry. Figure 4.16: The j 0 and h 0 points Proof. i) We calculate the quadrances q b 1, j 0) = q [ α α 1) : 0 : α 2 α + 1) ], [ t 2 : 0 : 1 ]) 2 2 = 2 4 = q [1 α : 0 : α α + 1)], [ t 2 : 0 : 1 ]) = q f 2, h 0 ). 2 1 ii) Similarly, we calculate the quadrances q v 1, j 0) = q [0 : 0 : 1], [ t 2 : 0 : 1 ]) = t4 α 2 t 4 α 2 1 = q [0 : 0 : 1], [ t 2 : 0 : 1 ]) = q v 1, h 0 ). Theorem 67 Two chord midpoints) Let p 0 p t), q 0 p u) be two points on a hyperbolic parabola P 0, with p 0 the opposite point of p 0 with respect to the axis A. Suppose that the chords p 0 q 0 and q 0 p 0 meet A at x and y respectively. Then the vertices v 1, v 2 of P 0 are the midpoints of the side xy.

106 4. The parabola and its Euclidean properties 98 ft Figure 4.17: Two chord midpoints Proof. Suppose p 0 = [ t 2 : t : 1 ] and q 0 = [ u 2 : u : 1 ]. The line p 0 q 0 = 1 : t + u) : tu meets the axis A = [0 : 1 : 0] at x = [ tu : 0 : 1]. The chord p 0 q 0 = 1 : t u : tu intersects A = 0 : 1 : 0 at y = [tu : 0 : 1]. Thus q v 1, x) = q [0 : 0 : 1], [ tu : 0 : 1]) = The optical property = q [0 : 0 : 1], [tu : 0 : 1]) = qv 1, y) α 2 t 2 u 2 t 2 u 2 α 2 1) which shows v 1 is a midpoint of the side xy. The other midpoint will be perpendicular to v 1, which must be v 2 without calculation. Recall the famous optical property of a parabola P in Euclidean geometry: if P is a point lying on P, and light emanates from the focus F heading towards the point P, then the light will be reflected to be parallel to the axis. An analogous result in the hyperbolic case is the statement iv) of the Congruent triangles theorem: that the tangent line P 0 to a point p 0 is a biline bisector) of the vertex R 1 R 2. So reflecting the focal line R 1 f 1 p 0 in the tangent P 0 results in the other focal line R 2, which is perpendicular to the directrix F 2. Recall from [50] that in Universal Hyperbolic Geometry there is an important notion of parallelism, which is quite different from the usage in classical hyperbolic geometry. We say rather generally that the parallel line P through a point a to a line L is the line through a perpendicular to the altitude from a to L. Now recall that given a point p 0 on the hyperbolic parabola P 0, the perpendicular to the axis A through p 0 is J 0 j 0 ) = ap 0 = 1 : 0 : t 2 with dual point j 0 =

107 4. The parabola and its Euclidean properties 99 P 0 A = [ 1 : 0 : t 2 α 2]. Therefore, the parallel line to the axis A through the point p 0 is L 0 = j 0 p 0 = t 3 α 2 : t 4 α 2 1 : t. Here then is another analog of the optical property, dealing with the relationship between two spreads formed by the tangent line P 0. Recall that the quadrance of the parabola was defined as q P0 q f 1, f 2 ). Theorem 68 Parallel line spread relation) Let p 0 be a point on the hyperbolic parabola P 0. If T is the spread between the tangent line P 0 at p 0 and the parallel line L 0 to the axis through p 0, and Ŝ is the common spread between the tangent P 0 and the lines R 1 and R 2, then Ŝ T )Ŝ 1 T = 1 q P0. Figure 4.18: The Parallel line spread relation Proof. Using the Spread formula, we compute that α 2 1 ) 2 3 4) 2 and So now Ŝ = S R 1, P 0) = 4α 4 3 T = S L 0, P 0) = α 2 1 ) t 4 α ) 2 t 4 α 2 1) 3 4. Ŝ T )Ŝ 1 T = 1 4 α 2 1 ) 2 α 2 = 1 q P0. Note that 1 q P0 = q b 1, f 2 ) since b 1 and f 1 are perpendicular points. So in the limiting Euclidean case when b 1 is very close to f 2, it follows that Ŝ is very close to T.

108 4. The parabola and its Euclidean properties The s points and S circles Recall that s 1 F 1 P 0 and s 2 F 2 P 0. Figure 4.19: The S 1 and S 2 circles Theorem 69 The S 1 and S 2 circles) The circle S 1 with center s 1 passing through f 2 also passes through t 1, and is tangent at these points to R 2 and R 1 respectively. In particular i) q s 1, t 1 ) = q s 1, f 2 ); ii) R 1 F 1 ; iii) R 2 T 2 and iv) S s 1 t 1, t 1 f 2 ) = S s 1 f 2, f 2 t 1 ). The same statements are true if we interchange the indices 1 and 2, giving also a circle S 2 with center s 2. Proof. i) Calculate q s 1, t 1 ) = q [2t α 1) : 2 : 2tα α + 1)], [ α 1) 1 : 4tα 2 ]) : α α + 1) 1 α 2 1 ) 2 4 = 16t 2 α α2 1) = q s 1, f 2 ). ii) The line R 1 = f 1 p 0 is clearly perpendicular to the directrix F 1 since it passes through the focus f 1 = F 1. iii) Since t 2 = R 2 F 2, S 1 = f 1 t 2, the lines R 2, F 2 and S 1 are concurrent at t 2, so the line T 2 = t 2 R 2. iv) Calculate passes through r 2, f 2 and s 1. Therefore T 2 is perpendicular to the line S t 1 s 1, t 1 f 2 ) = S α α + 1) : 0 : 1 α, 2tα 2 α + 1) : α 2 1 ) 1 : 2tα α 1) ) α 2 1 ) 2 3 = 16t 2 α α2 1) = S f 2 s 1, f 2 t 1 ).

109 4. The parabola and its Euclidean properties 101 In particular, property iii) provides us with an important alternate construction of the tangent P 0 to the parabola P 0 at p 0 : namely we construct the altitude T 2 to p 0 f 2 through f 2, and obtain s 1 = F 1 T 2, giving P 0 = p 0 s 1 or similarly construct p 0 s 2 ). In Figure 4.19 we see the circles S 1 and S 2. Note that S 2 looks like a hyperbola tangent to the null circle, in fact it is tangent at exactly the points where S 2 meets the null circle C see the discussion in [52] Focal chords and conjugates A chord p 0 q 0 is a focal chord precisely when p 0 q 0 passes through a focus. Such chords play an important role both in the Euclidean and the hyperbolic theory. Figure 4.20: A focal chord p 0 q 0 with the polar point z on directrix Theorem 70 Focal tangents perpendicularity) If p 0 p t) and q 0 p u) are two points on P 0 then p 0 q 0 is a focal chord precisely when the respective tangents P 0 and Q 0 are perpendicular; and precisely when the polar point z P 0 Q 0 lies on a directrix. Proof. i) Suppose p 0 = [ t 2 : t : 1 ] and q 0 = [ u 2 : u : 1 ] lie on P 0. Then p 0 q 0 = 1 : t + u) : tu is a focal line precisely when it passes through either f 1 of f 2, in other words precisely when 1 : t + u) : tu) [α + 1 : 0 : α α 1)] T = α tuα α 1) = 0 or 1 : t + u) : tu) [1 α : 0 : α α + 1)] T = α tuα α + 1) = 0.

110 4. The parabola and its Euclidean properties 102 On the other hand the tangents P 0 = 1 : 2t : t 2 and Q 0 = 1 : 2u : u 2 are perpendicular precisely when 0 = 1 : 2t : t 2 D 1 : 2u : u 2 T = α 2 4tuα 2 t 2 u 2 α 2 α 2 1 ) 1 = α tuα α 1)) α 1 tuα α + 1)). Thus the two conditions are equivalent. As in the Tangent meets theorem, the tangents P 0 and Q 0 meet at z = [2tu : t + u : 2]. This point lies on F 1 = α α + 1) : 0 : 1 α or F 2 = α α 1) : 0 : α + 1 precisely when [2tu : t + u : 2] α α + 1) : 0 : 1 α) T = 2 α + tuα + tuα ) = 0 [2tu : t + u : 2] α α 1) : 0 : α + 1) T = 2 α tuα + tuα ) = 0. or Since we work over a field not of characteristic two, the conditions are equivalent to the previous ones. Figure 4.21: Focal conjugates n 1 and n 2 Given a point p 0 on the parabola P 0, we define the conjugate points n 1, n 2 as the second meets of the focal lines R 1 and R 2 with the parabola P 0 respectively. Since one meet is known, solving the quadratic equations is straightforward and yields [ n 1 = α + 1) 2 : tα 1 α 2) : t 2 α 2 α 1) 2] and [ n 2 = α 1) 2 : tα α 2 1 ) : t 2 α 2 α + 1) 2]. 4.9) The dual lines are the conjugate lines; N 1 n 1 = α α + 1) 2 : t α 2 1 ) 2 : t 2 α α 1) 2 N 2 n 2 = α α 1) 2 : t α 2 1 ) 2 : t 2 α α + 1) 2 and

111 4. The parabola and its Euclidean properties 103 Theorem 71 Conjugate points parameter) Let p 0 p t) be a point on the parabola P 0, then the point p u) is the conjugate point n 1 of p 0 with respect to the focus f 1 precisely when u = α+1 αtα 1), while p u) is the conjugate point n 2 of p 0 with respect to the focus f 2 precisely when u = α 1 αtα+1). Proof. Let p 0 = [ t 2 : t : 1 ] and p u) = [ u 2 : u : 1 ] lie on P 0. Then, the line p 0 q 0 = 1 : t + u) : tu is a focal line with respect to the focus f 1 when it passes through the focus f 1, and then we have [1 : t + u) : tu] [α + 1 : 0 : α α 1)] T = 0 so that α tuα + tuα = 0. This gives the condition u = α+1 αtα 1). Similarly, the other direction is straightforward. When the line p 0 q 0 = 1 : t + u) : tu is a focal line with respect to the focus f 2, then, the focal line passes through the focus f 2 and we have [1 : t + u) : tu] [1 α : 0 : α α + 1)] T = 0 so that α + tuα + tuα = 0. This gives the condition u = α 1 αtα+1). Similarly, the other direction is straightforward Quadrance cross ratios Theorem 72 Let p 0 be a point on the parabola P 0, with n 1 and n 2 the focal conjugates and u 1 and u 2 the meets of R 2 and R 1 with the directrices F 1 and F 2 respectively. Then q p 0, f 1 ) q f 1, n 1 ) = q p 0, u 2 ) q u 2, n 1 ) and qp 0, f 2 ) qf 2, n 2 ) = qp 0, u 1 ) qu 1, n 2 ). Proof. From the Focus directrix polarity theorem, we know that f 2 and F 1 are a polepolar pair with respect to the parabola P 0. Hence f 1, u 2 ; p 0, n 1 is a harmonic range. From the Quadrance cross relation theorem in Chapter 2, this implies that q p 0, f 1 ) q f 1, n 1 ) = q p 0, u 2 ) q u 2, n 1 ). The other relation follows similarly since f 2, u 1 ; p 0, n 2 is also a harmonic range Spreads related to chords of a parabola Theorem 73 Polar point spreads) If the tangents P 0 and Q 0 at the points p 0 p t) and q 0 p u) lying on the parabola P 0 meet at the polar point z, then S f 1 p 0, f 1 z) = S f 1 q 0, f 1 z) and S f 2 p 0, f 2 z) = S f 2 q 0, f 2 z).

112 4. The parabola and its Euclidean properties 104 Figure 4.22: The polar point z of the chord p 0 q 0 Proof. Suppose that p 0 [ t 2 : t : 1 ] and q 0 [ u 2 : u : 1 ] are on the parabola P 0. Then z = [2tu : t + u : 2] and we calculate and α t u) 2 α 2 1 ) S f 1 p 0, f 1 z) = α + α 2 t 2 αt 2 + 1) α + α 2 u 2 αu 2 + 1) = α t u)2 α 2 1 ) = S f 1 q 0, f 1 z) 3 t) 3 u) Theorem 74 Chord directrix meets) Let p 0 p t) and q 0 p u) be two points on a parabola P 0. Let z be the polar point of the chord p 0 q 0, and x 1 F 1 p 0 q 0 ) and x 2 F 2 p 0 q 0 ). Then i) f 1 z f 1 x 2, ii) f 2 z f 2 x 1 and iii) S x 1 z, zf 2 ) = S x 2 z, zf 1 ). Proof. We suppose as usual that p 0 = [ t 2 : t : 1 ] and q 0 = [ u 2 : u : 1 ]. Also α α 2 1 ) t u) 2 S f 2 p 0, f 2 z) = α u 2 α 2 u 2 α 1) α + t 2 α 2 + t 2 α + 1) = α α 2 1 ) t u) 2 = S f 2 q 0, f 2 z). 4 t) 4 u) i) We compute that x 2 F 2 p 0 q 0 ) = α 2 α 1) : 0 : α 1 + α) 1 : t + u) : tu = [ α + 1) t + u) : α + tuα tuα : α α 1) t + u) ]. f 1 z = α α 1) t + v) : 2 α tvα + tvα 2 1 ) : α + 1) t + v) f 1 x 2 = α α 1) α tvα + tvα 2 1 ) : 2α α 2 1 ) t + v) : α + 1) α + tvα tvα )

113 4. The parabola and its Euclidean properties 105 Figure 4.23: Chord directrix meets x 1 and x 2 and so we may verify that 0 = α α 1) t + v) : 2 α tvα + tvα 2 1 ) : α + 1) t + v) D α α 1) α tvα + tvα 2 1 ) : 2α α 2 1 ) t + v) : α + 1) α + tvα tvα ) T. Thus Sf 1 z, f 1 x 2 ) = 1. ii) Similarly and the lines x 1 F 1 p 0 q 0 ) = α α + 1) : 0 : 1 α 1 : t + u) : tu = [ α 1) t + u) : α + tuα + tuα 2 1 : α α + 1) t + u) ] f 2 z = α α + 1) t + u) : 2 α + tuα + tuα 2 1 ) : α 1) t + u) f 2 x 1 = α α + 1) α + tuα + tuα 2 1 ) : 2α α 2 1 ) t + u) : α 1) α + tuα + tuα 2 1 ) are perpendicular, so that Sf 2 z, f 2 x 1 ) = 1. iii) Another calculation shows that S x 1 z, zf 2 ) = 1 α 2 1 ) 2tu) 2 t + u) 2) ) α 2 + t + u) 2 4 ) 4 t 2 u 2 ) α 4 + t + u) 2 t 2 u 2 + 1) α = S x 2 z, zf 1 ). In Figure 4.23 we see the two triangles f 1 zx 2 and f 2 zx 1, which are both right triangles sharing a common spread.

114 4. The parabola and its Euclidean properties 106 Theorem 75 Tangent directrix meets) If the two tangents P 0 and Q 0 to a parabola P 0 at p 0 p t) and q 0 p u) respectively meet the directrix F 1 at s 1 and s 1 respectively, and meet F 2 at s 2 and s 2 respectively, then S f 1p 0, f 1 q 0 ) = S f 1 s 2, f 1 s 2 ) and S f 2 p 0, f 2 q 0 ) = S f 2 s 1, f 2 s 1 ). Figure 4.24: Tangent directrix meets s 1 and s 2 Proof. Suppose that p 0 [ t 2 : t : 1 ] and q 0 [ u 2 : u : 1 ] are on the parabola P 0. Then S f 1 p 0, f 1 q 0 ) = 4α α 2 1 ) t u) 2 α + α 2 tu αtu + 1 ) t) 2 3 u) = S f 1 s 2, f 1 s ) 2. Also, we have that S f 2 p 0, f 2 q 0 ) = 4α α 2 1 ) t u) 2 α + tuα + tuα ) t) 2 4 u) = S f 2 s 1, f 2 s ) 1. Recall that in universal hyperbolic geometry, a triangle may have four circumcircles, see [53]. Theorem 76 Two tangents circumcircle) Suppose that the two points p 0 p t) and q 0 p u) on a parabola P 0 have respective altitude base points t 1, t 2 and t 1, t 2 on F 1, F 2 respectively, and that their tangents meet at the polar point z. Then z is a circumcenter of both the triangles t 1 f 2 t 1 and t 2f 1 t 2. In particular q t 1, z) = q t 1, z) = q z, f 2 ) and q t 2, z) = q t 2, z) = q z, f 1).

115 4. The parabola and its Euclidean properties 107 Figure 4.25: Two points and polar circles Proof. Suppose that p 0 [ t 2 : t : 1 ] and q 0 [ u 2 : u : 1 ] are on the parabola P 0, then, qz, f 2 ) = q [2tu : t + u : 2], [1 α : 0 : α α + 1)]) 4 t) 4 u) = α 4t 2 u 2 t + u) 2) ) α 2 + t + u) 2 4 = qz, t 1 ) = qz, t 1). Hence z is a circumcenter of the triangle t 1 f 2 t 1. Similarly, z is the circumcenter of the triangle t 2 f 1 t 2 since qz, f 1 ) = q [2tu : t + u : 2], [α + 1 : 0 : α α 1)]) 3 t) 3 u) = α 4t 2 u 2 t + u) 2) ) α 2 + t + u) 2 4 = qz, t 2 ) = qz, t 2). In Figure 4.25 we see the polar point of p 0 q 0 together with the two polar circles Z 1, Z 2 centered at z through the foci f 1 and f 2 respectively. Corollary 77 If the tangents at p 0 p t) and q 0 p u) on P 0 meet at z then the line f 1 z is a midline of the side t 1 t 1 and similarly f 2z is a midline of the side t 2 t 2. Proof. This follows immediately from the previous theorem, since f 1 z is the altitude from z to the directrix F 1, so it bisects the chord t 1 t 1.

116 4. The parabola and its Euclidean properties 108 Figure 4.26: Opposite triangle spreads Theorem 78 Opposite triangle spreads) If the tangents at p 0 p t) and q 0 p u) on P 0 meet at z, then Szp 0, zf 1 ) = Szq 0, zf 2 ) and Szp 0, zf 2 ) = Szq 0, zf 1 ). Proof. Using the Spread formula, we obtain α 2 1 ) 4t 2 u 2 t + u) 2) α 2 + t + u) 2 4) Szp 0, zf 2 ) = 4 4 u) 3 t) = Szq 0, zf 1 ) and α 2 1 ) 4t 2 u 2 t + u) 2) α 2 + t + u) 2 4) Szp 0, zf 1 ) = 4 3 u) 4 t) = Szq 0, zf 2 ). 4.5 Normals to the parabola P 0 In the Euclidean case, it is well known that the evolute of the parabola, which is defined as the locus of the centers of curvature of the curve namely the meet of adjacent normals, as Huygens or Newton would have said is a semi-cubical parabola. For the curve y = x 2, shown in Figure 4.27, the evolute has equation y 1 ) 3 = x2. This formula suggests that there is no Euclidean ruler and compass construction for the center of curvature C 0 of the parabola for a general point P 0 on it. We will see that in the hyperbolic case, the situation is in some ways simpler, and indeed we will show how to give a straightedge construction for the center of curvature! In Figure 4.27 we see a point P 0 on the Euclidean parabola, with its tangent p 0, obtained by finding the meet S of the directrix f with the altitude to the focal line r = F P 0 through the focus F. The center of curvature is the point C 0 on the evolute

117 4. The parabola and its Euclidean properties 109 Figure 4.27: Evolute of a Euclidean parabola E. The figure shows also that for points L above the evolute, there are three normals that meet there; we exhibit also the other two points marked P whose normals also pass through L. Below the evolute only one normal passes through any fixed point. For a point p 0 on the hyperbolic parabola P 0, the altitude line P to the tangent P 0 through p 0 is called the normal line at p 0. Since the dual of P 0 is the twin point p 0, we see that P p 0 p 0 = [ t 2 : t : 1 ] [ α 2 1 : 2tα 2 : t 2 α 2 α 2 1 )] = tα 2 t 2 α 2 t ) : α 2 1 ) t 4 α ) : t 2t 2 α 2 α ). 4.10) By symmetry, this means that P is both the normal line to the parabola P 0 at p 0 as well as the normal line to the twin parabola P 0 at p 0. Hence we have no index on P. The meet of P and the axis A is the point n P A = tα 2 t 2 α 2 t ) : α 2 1 ) t 4 α ) : t 2t 2 α 2 α ) 0 : 1 : 0 = [ t 2t 2 α 2 α ) : 0 : tα 2 t 2 α 2 t )] = [ 2t 2 α 2 α : 0 : α 2 t 2 α 2 t )] provided that t 0. Since the normal P of p 0 is perpendicular to the tangent P 0, and since P 0 is a biline of the vertex R 1 R 2, the normal P is the other biline for the vertex R 1 R 2. In fact we may calculate that SR 1, P ) = SP, R 2 ) = t2 α )

118 4. The parabola and its Euclidean properties Conjugate normals and conics Recall that the conjugate points n 1, n 2 of p 0 are the second meets of the focal lines R 1 f 1 p 0 and R 2 f 2 p 0 with the parabola P 0 respectively. They are given in 4.9). The normal lines to P 0 at the conjugate points n 1 and n 2 can then be computed using the formula 4.10): tα α 1) 2α 2 α 1) t 2 + α + 1) 3) : α 2 α 1) 4 t 4 + α + 1) 4 : P 1 ) tα α + 1) 2 α + 1) α 1) 3 and t 2 tα α + 1) 2α 2 α + 1) t 2 + α 1) 3) : α 2 α + 1) 4 t 4 + α 1) 4 : P 2 ) tα α 1) 2 α 1) α + 1) 3. t 2 We will call these the conjugate normal lines of p 0. Theorem 79 Conjugate normal conics) Given the parabola P 0, there are two conics H 1 and H 2 with the following properties. Let h 1 be the meet of the normal P and the conjugate normal P 1 of a point p 0 on P 0. Then h 1 lies on H 1, which passes through f 2 and is tangent to B 1 there. Similarly if h 2 is the meet of P and P 2 at p 0, then h 2 lies on H 2, which passes through f 1 and is tangent to B 2 there. Furthermore we have collinearities [[f 1 s 2 h 2 ]] as well as [[f 2 s 1 h 1 ]]. In addition H 1 passes through the points d 0 and d 0. Figure 4.28: Conjugate normal meets h 1 and h 2 and conics Proof. The conjugate normal P 1 will meet the normal P at [ α 2 α 1) 3 t 4 + 4α 2 α + 1) t 2 α 1) α + 1) 2 : tα α ) 1 : h 1 P P 1 = α α 2 α + 1) α 1) 2 t 4 + 4α 2 α 1) t 2 + α + 1) 3) ].

119 4. The parabola and its Euclidean properties 111 A computation shows this point always lies on the conic H 1 with equation 0 = α 2 α 2 1 ) 1 + 4α + α 2) x 2 + 2α 1 2α α 2) 1 + 2α α 2) xz + 32α 3 y 2 + α 2 1 ) 1 4α + α 2) z 2. The conjugate normal P 2 will meet the normal P at [ α 2 α + 1) 3 t 4 4α 2 α 1) t 2 + α + 1) α 1) 2 : tα α ) 2 : h 2 P P 2 = α α 2 α 1) α + 1) 2 t 4 + 4α 2 α + 1) t 2 + α 1) 3) ]. This point always lies on the conic H 2 with equation 0 = α 2 α 2 1 ) 1 4α + α 2) x 2 2α 2α + α 2 1 ) 2α + α 2 1 ) xz 32α 3 y 2 + α 2 1 ) 1 + 4α + α 2) z 2. The collinearity [[f 1 s 2 h 2 ]] is established by checking that the determinant formed by the respective vectors is indeed 0, and similarly for the collinearity [[f 2 s 1 h 1 ]]. We can also check with a computer package) that both of the points d 0 and d 0 identically satisfy the equation of H 1. The normal P at p 0 meets the parabola P 0 again at a second point [ 2t p 0 = 2 α 2 α ) 2 : tα 2 t 2 α 2 + t 2 2 ) 2t 2 α 2 α ) : t 2 α 4 t 2 α 2 t ) ] 2 and similarly the conjugate normals P 1, P 2 at n 1, n 2 meet P 0 respectively also at 2 t 2 α α 2 1) 3 t 2 2 α + 1)) : ) n 1 = tα 2 α + 1) α 1) 3 t 2 2α 2 α 1) t 2 + α + 1) 3) : and : 2α 2 α 1) t 2 + α + 1) 3) 2 2 t 2 α α 2 + 1) 3 t α)) : n 2 = 2α 2 α + 1) t 2 + α 1) 3) ) α α + 1) 3 t 3 2α α 1) t : 2α 2 α + 1) t 2 + α 1) 3). 2 Theorem 80 Normal conjugate colliearities) Let p 0, n 1 and n 2 be the second meets of the normals and conjugate normals P, P 1 and P 2 of p 0 with the parabola P 0 respectively, and t 1, t 2 the altitude base points of p 0. Then we have collinearities [[p 0 n 1 t 1]] and [[p 0 n 2 t 2]]. Proof. Since the forms of all the points involved are known, it is straightforward with a computer package) to verify that the corresponding determinants for both collinearities do evaluate identically to 0. These collinearities are illustrated in Figure 4.29.

120 4. The parabola and its Euclidean properties 112 Figure 4.29: Normal conjugate collinearities Concurrent normals, symmetric functions, and four point conics In the Euclidean case, finding the three points P on the parabola whose normals pass through a given point L above the evolute is not straightforward, see [15]. We will show that in the hyperbolic case there is an interesting conic, related to the elementary symmetric functions of four variables t 1, t 2, t 3, t 4, that allows us to find four such points. Theorem 81 Four parabola normals) If l is a point in the hyperbolic plane, then there are at most four points p on the parabola P 0 whose normals pass through l. Proof. We know that the normal to p 0 = [ t 2 : t : 1 ] is the line P = tα 2 t 2 α 2 + t 2 2 ) : α 2 1 ) t 4 α ) : t 2t 2 α 2 α ). If P passes through a point l = [x 0 : y 0 : z 0 ], then lp = 0, which after rearranging is the equation α 2 α 2 1 ) y 0 t 4 +α 2 1 α 2) ) x 0 + 2z 0 t α 2) z 0 2α 2 ) x 0 t+ α 2 1 ) y 0 = 0. This is a polynomial of degree four in t, so it has at most four solutions. 4.11) Theorem 82 Quadratic normal meets) Suppose p 0 = p t) and q 0 = p u) are two points on the parabola, whose respective normals P and Q meet at a point l and suppose that α ) 0. Then there are 0, 1 or 2 other points on the parabola whose normals pass through l precisely when = t 2 u 2 α ) 2 4tuα 2 t + u) 2 is not a square, is zero, or is a non-zero square respectively.

121 4. The parabola and its Euclidean properties 113 Proof. The meet of the two normals is α 2 1 ) tu 2t 2 u 2 tu t 2 u 2)) α 4 + tu 2) tu + t 2 + u 2) + 1 ) α 2 1 ) : l P Q = tuα 2 α ) 2 t + u) : α 2 α 2 1 ) t 3 u 3 α 4 + 2tu 1) tu + t 2 + u 2) t 3 u 3) α 2 + t 2 + tu + u 2 2 )) and we need to check when a third point r 0 p v) on P 0 has a normal R also passing through l. This is equivalent to lr = 0 which yields, after remarkable simplification, α 2 α 2 1 ) α ) 2 u v) t v) t + u + v + tu 2 v 2 α 2 + t 2 uv 2 α 2 + t 2 u 2 vα 2) = 0. Since α 0, ±1 and u, t, v are disjoint, this condition reduces to the quadratic equation tuα 2 t + u) v 2 + t 2 u 2 α ) v + t + u) = 0 in v with discriminant = t 2 u 2 α ) 2 4tuα 2 t + u) 2. The question of the existence of four points on the parabola P 0 with a common normal point is closely related to an interesting conic associated to four points on the parabola; namely the conic A through those four points and the axis point a, which has independent interest due to its form. We call this conic A p 1, p 2, p 3, p 4 ) the four-point conic through p 1, p 2, p 3 and p 4. Theorem 83 Four point conic) For any four points p 1 p t), p 2 p u), p 3 p v) and p 4 p w) lying on P 0, the four-point conic A p 1, p 2, p 3, p 4 ) has equation 0 = x 2 t + u + v + w) xy + tu + tv + tw + uv + uw + vw) xz 4.12) tuv + tuw + tvw + uvw) yz + tuvwz 2. Proof. We use a standard technique, see for example [17]), for computing a conic through five given points: by taking a combination of the degenerate line products formed by pairs of four points p 1, p 2, p 3 and p 4. Now p 1 p 2 = 1 : t + u) : tu) p 1 p 3 = 1 : t + v) : tv) p 3 p 4 = 1 : v + w) : vw) p 2 p 4 = 1 : t + w) : tw) so the general conic in the pencil through p 1, p 2, p 3 and p 4, has the form 0 = p x, y, z) = x t + u) y + tuz) x v + w) y + vwz) + λ x t + v) y + tvz) x u + w) y + uwz). Now since also p 0, 1, 0) = 0, we can solve for λ to get t + u) v + w) λ = t + v) u + w).

122 4. The parabola and its Euclidean properties 114 Figure 4.30: Four points p with normals through l and associated conic A Substituting back and simplifying, we find that the equation of the required conic is 4.12). There is a clear similarity between the form of this conic A and the familiar identity x t 1 ) x t 2 ) x t 3 ) x t 4 ) = x 4 t 1 + t 2 + t 3 + t 4 ) x 3 + t 1 t 2 + t 1 t 3 + t 1 t 4 + t 2 t 3 + t 2 t 4 + t 3 t 4 ) x 2 t 1 t 2 t 3 + t 1 t 2 t 4 + t 1 t 3 t 4 + t 2 t 3 t 4 ) x + t 1 t 2 t 3 t 4 relating the coeffi cients of a degree four polynomial and the elementary symmetric functions of its zeros. This may be explained by noting that if p = [x : y : z] = [ t 2 : t : 1 ] is a point on the parabola, then the quantities x 2, xy, xz, yz and z 2 are respectively exactly t 4, t 3, t 2, t and 1, while the condition that the conic passes through a ensures that the coeffi cient of y 2 is necessarily The four normal conic A n and finding normals Theorem 84 four normal conic) Suppose that the normal lines at four points p 1, p 2, p 3, p 4 lying on P 0 are concurrent at a point l = [x 0 : y 0 : z 0 ] not lying on the axis A. Then the conic A l with equation α 2 α 2 1 ) y 0 x 2 + α 2 x 0 + 2z 0 x 0 α 2) xy + z 0 z 0 α 2 2x 0 α 2) yz + α 2 1 ) y 0 z 2 = ) passes through the six points p 1, p 2, p 3, p 4, a and l, so in particular A l = A p 1, p 2, p 3, p 4 ). Proof. The condition 4.11) on t for p = [ t 2 : t : 1 ] on P 0 to have a normal line passing through l [x 0, y 0, z 0 ] may be rewritten, since y 0 0, as t 4 + α2 x 0 1 α 2 ) ) + 2z 0 α 2 α 2 t 3 z0 1 α 2 ) 2x 0 α 2) + 1) y 0 α 2 α 2 t + 1 1) y 0 α 2 = 0.

123 4. The parabola and its Euclidean properties 115 If we have four distinct solutions t, u, v, w of this equation, then t + u + v + w = α2 x 0 1 α 2 ) + 2z 0 ) α 2 y 0 α 2 1) tu + tv + tw + uv + uw + vw = 0 tuv + tuw + tvw + uvw = z 0 1 α 2 ) 2x 0 α 2 α 2 y 0 α 2 1) tuvw = 1 α 2. From the previous theorem, the conic passing through the five points p 1 = p t), p 2 = p u), p 3 = p v), p 4 = p w) and a then has the form x 2 + α2 x 0 + 2z 0 x 0 α 2) α 2 α 2 1) y 0 xy + z0 2x 0 α 2 z 0 α 2) α 2 α 2 1) y 0 yz + 1 α 2 z2 = 0 which we can rewrite as the conic A l 4.13). But now we can check that also l lies on this conic, since identically α 2 α 2 1 ) y 0 x 2 0+α 2 x 0 1 α 2 ) + 2z 0 ) x0 y 0 + z 0 1 α 2 ) 2x 0 α 2) y 0 z 0 + α 2 1 ) y 0 z 2 0 = 0. We might call A l the four normal conic of l, with respect to P 0. Theorem 85 Conic construction of common normals) Let l be a point of the hyperbolic plane with the property that the dual line L of l meets P 0 at two points x and y. Then the meet z of the tangent lines to P 0 at x and y, the meet x of the tangent line at x and the dual line of x, and the meet y of the tangent line at y and the dual line of y, all lie on the conic A l. Proof. Suppose that the dual line L of l meets P 0 at two points x = [ t 2 : t : 1 ] and y = [ u 2 : u : 1 ]. Then the meets of the tangent lines is z = [2tu : t + u : 2] from the Tangent meets theorem. Also L = 1 : t + u) : tu and l = [ α 2 1 : α 2 t + u) : α 2 tu α 2 1 )]. to In this case the equation 4.13) for the conic A l simplifies, after some cancellation, α 2 t + u) x tuα 2 α 2) xy + tuα 2 tu 2 ) yz + t + u) z 2 = ) The dual line of x meets the tangent line at x at x = [ t α 2 t 2 t ) : α 2 t : t 2α 2 t 2 α )]

124 4. The parabola and its Euclidean properties 116 Figure 4.31: Construction of points p on P 0 with normals through n and the dual line of y meets the tangent line at y at y = [ u α 2 u 2 u ) : α 2 u : u 2α 2 u 2 α )]. We check that both of these points identically satisfy the equation 4.14). This also provides us with an elegant method to find all normals through a given point l. Firstly, find the dual line L of the point l and then find the meets x, y of this line L with the parabola P 0. Construct the tangents P x, P y to P 0 at x and y and find their meet z. Construct the dual lines X and Y of x and y, then find the meet of the tangent at x and the dual line of x, that is x = P x X and the meet of the tangent at y and the dual line of y, that is y = P y Y. According to the above theorem, the five points l, x, y, z, a lie on a conic A l which may meet the parabola P 0 in at most four points which have the property that their normals meet at l. We see that the number of normals passing through l is determined by the meet of the conic A l with the parabola P 0. So if we can find the meets of these two conics, we have the normals which pass through l. This construction shows that some aspects of hyperbolic geometry are, surprisingly, more simple than in Euclidean geometry. In the latter, finding normals to points on a parabola from a particular point is quite cumbersome, as shown in [15]. Furthermore, the four normals drawn from a particular point are also the normals to four points on the twin parabola P 0. These points are the dual points of the tangents to four points on the original parabola P 0. This observation is the result of duality between lines and points Normal conjugate points If p 0 is a point on P 0 with tangent line P 0 and normal line P, then the other meet of P with the parabola gives a point p 0, which we call the normal conjugate point of

125 4. The parabola and its Euclidean properties 117 p 0. Then the tangent line P 0 to p 0 meets with P 0 at the point k 0 = P 0 P 0 = t 2 α 4 t 2 α 2 t ) 2 : 2tα 2 t 2 α 2 t ) 2t 2 α 2 α ) : 2t 2 α 2 α ) 2 1 : 2t : t 2 = [ 2t 2t 2 α 2 α ) : α 2 1 ) t 4 α ) : 2tα 2 t 2 α 2 t )]. Figure 4.32 shows in green the normal conjugate curve K 0 : the locus of k 0 as p 0 moves. This a higher degree curve which passes through a as well as d 0 and d 0, and is tangent to P 0 at those latter two points. It seems an interesting future direction to investigate more fully such associated algebraic curves connected with P 0. Figure 4.32: The normal conjugate conic K The evolute and centers of curvature Recall that the evolute of a curve is the envelope of the normals to that curve, or equivalently the locus of the centers of curvature. Following the technique described in [7], here is a pleasant construction of the center of curvature c 0 to the hyperbolic parabola P 0 at the point p 0. Theorem 86 Center of curvature construction) Let P be the normal at p 0 to the parabola P 0, and construct the altitude line Q to P through n = AP. Suppose that the meets of Q with the focal lines R 1 and R 2 are respectively x 1 and x 2. Then the meet of the perpendicular line to R 1 through x 1 and the perpendicular line to R 2 through x 2 is the required center of curvature c 0 to P 0 at the point p 0.

126 4. The parabola and its Euclidean properties 118 Figure 4.33: Evolute of a parabola Proof. Let p 0 = [ t 2 : t : 1 ] and n = [ 2t 2 α 2 α : 0 : α 2 t 2 α 2 t )], then the perpendicular to P through l = n is Q pn = α 2 t 4 α ) t 2 α 2 t ) : t 2α t 2 α 2t 2 α 2 + t 2 α 3 + α 2 1 ) 2α + t 2 α 2t 2 α 2 t 2 α 3 + α 2 1 ) : t 4 α ) 2t 2 α 2 + α 2 1 ). This line Q will meet R 1 at x 1 = 2α 4 t 6 + α 5 + 3α 4 3α 2 α ) t 4 + 2α 3 α 4 + 4α 2 + 2α 1 ) t α 2) : tα α ) t 4 α ) : α α 3 α 2 1 ) t 6 + α 2α 4α 2 + 2α 3 + α ) t 4 α 2 1 ) 3α + α ) t 2 + 2α ) and the line R 2 at 2α 4 ) t 6 + α 5 3α 4 + 3α 2 α ) t 4 + α 4 + 2α 3 4α 2 + 2α + 1 ) t 2 + α 2 1 ) : x 2 = tα α ) t 4 α ) : α α 3 α 2 1 ) t 6 + 2α 4 α 5 + 4α 3 + 2α 2 α ) t 4 + 3α 3α 3 α ) t 2 2α ). The perpendicular line to R 1 through x 1 is X 1 = x 1 r 1 and the perpendicular line to R 2 through x 2 is X 2 = x 2 r 2 which meet at [ α 2 1 ) 2α 4 t 6 + 3α 2 1 α 2) t 4 6α 2 t 2 + α 2 1 )) : 2t 3 α 2 α ) 2 : c 0 = X 1 X 2 = α 2 α 2 1 ) α 2 α 2 1 ) t 6 + 6α 2 t α 2) t 2 2 ) ].

127 4. The parabola and its Euclidean properties 119 To evaluate the center of curvature, we note that two normals, say at p t) and p r), meet at f t, r) = α 2 1 ) 2t 2 α 2 + α 2 1 ) r 3 tα 2 1 ) + rα 2 t 2 α 2 t ) r + t) ) : rtα 2 r + t) α ) 2 : α 2 α 2 1 ) t 2 α 2 t ) r 3 tα 2 1 ) + r 2t 2 α 2 α ) r + t) ) where we have removed a common factor of r t. Now if we let r = t we find that f t, t) = c 0. Figure 4.34: Normals to a parabola Formula for the evolute Can we get a formula for the evolute? Working with affi ne coordinates setting z = 1), we need eliminate t from the equations 2t 6 α 4 3t 4 α 4 + 3t 4 α 2 6t 2 α 2 + α 2 1 ) x = α 2 t 6 α 4 t 6 α 2 + 6t 4 α 2 3t 2 α 2 + 3t 2 2) 2t 3 α ) 2 y = α 2 1) t 6 α 4 t 6 α 2 + 6t 4 α 2 3t 2 α 2 + 3t 2 2). We could use a Gröbner basis to calculate this, but the polynomials are small enough to do it by hand with classical elimination. We get, after some calculation,

128 4. The parabola and its Euclidean properties 120 that x and y satisfy the affi ne equation 0 = h x, y) = 32α 8 α 2 1 ) 3 x 6 256α 2 α 2 1 ) 6 y 6 + 3α 4 8α + 6α 2 8α 3 + 3α ) 8α + 6α 2 + 8α 3 + 3α ) α 1) 2 α + 1) 2 x 4 y α 4 α 2 1 ) 5 x 2 y α 6 2α + α 2 1 ) 2α + α 2 1 ) α 2 1 ) 2 x 5 192α 4 2α + α 2 1 ) 2α + α 2 1 ) α 2 1 ) 3 x 3 y α 2 2α + α 2 1 ) 2α + α 2 1 ) α 2 1 ) 4 xy α 4 α 2 1 ) 2α 6α 2 + 2α 3 + α ) 2α 6α 2 2α 3 + α ) x 4 384α 2 α 2 1 ) 5 y 4 + 6α 2 196α 2 378α α 6 + α ) α 2 1 ) 2 x 2 y 2 + 4α 2 2α + α 2 1 ) 2α + α 2 1 ) 36α α 4 36α 6 + α ) x α 2 2α + α 2 1 ) 2α + α 2 1 ) α 2 1 ) 3 xy 2 24α 2 α 2 1 ) 2α 6α 2 2α 3 + α ) 2α 6α 2 + 2α 3 + α ) x α + 6α 2 + 8α 3 + 3α ) 8α + 6α 2 8α 3 + 3α ) α 2 1 ) 2 y α 2 2α + α 2 1 ) 2α + α 2 1 ) α 2 1 ) 2 x 32α 2 α 2 1 ) 3. So the evolute is a six degree curve, with coeffi cients that depend in a pleasant way on α. Note that all the coeffi cients are divisible by α 2 1, with the exception of the coeffi cient of x 3.

129 Chapter 5 Further properties of the hyperbolic parabola In this final chapter we will show that there are many new and completely unexpected aspects of the hyperbolic parabola that have no prior Euclidean analogs. The outline of the chapter is as follows. We first define the twin parabola P 0 of the parabola P 0 and look at alternate definitions of a parabola: remarkably the hyperbolic parabola can be viewed as the locus of midpoints between a t-focus and a point on the dual line of the other t-focus, and is also characterized by a novel metrical condition. Then we introduce some interesting points associated to a variable point p 0 on P 0 ; such as the e-points, the perpendicular points m 1 and m 2 ; the conjugate points n 1 and n 2 ; the σ-points, and others. These lead to many fascinating collinearities and concurrences. Then we investigate canonical structures on the hyperbolic parabola, including prominently the rich Y conic. Then we state some metrical relations between points associated to the parabola. Finally, we show some additional special conics, and in particular describe some beautiful properties of Thaloids associated to P The Twin Parabola We will introduce here a fundamental symmetry into the theory of the hyperbolic parabola which has no known parallel in the Euclidean case. Recall that every nondegenerate conic in a Cayley-Klein geometry has a dual conic with respect to the absolute or null conic, obtained by taking the duals of tangent lines. Definition 87 The twin P 0 of the parabola P 0 is the locus of p 0 = P 0) where P 0 is the tangent to the parabola P 0 at the point p

130 5. Further properties of the hyperbolic parabola 122 It is remarkable that the twin P 0 of the parabola P 0 is another hyperbolic parabola, yielding a fundamental symmetry in the theory of the hyperbolic parabola. The following important result establishes why the theory of sydpoints developed in Chapter 3 is crucial for understanding the hyperbolic parabola. Theorem 88 Twin parabola) The twin of the parabola P 0 with foci f 1, f 2 is a hyperbolic parabola P 0 whose foci f 1, f 2 are the sydpoints of f 1 f 2. Proof. The dual point of the tangent P 0 = 1 : 2t : t 2 to P 0 at p 0 = [ t 2 : t : 1 ] is p 0 = [ α 2 1 : 2tα 2 : t 2 α 2 α 2 1 )]. The parabola with respect to the foci f 1 = [1 : 0 : 1] and f 2 = [ 1 : 0 : α 2] recall from the last chapter that f 1 and f 2 are the t-foci of P 0 ) has equation which simplifies to 0 = q [x : y : z], [1 : 0 : 1]) + q [x : y : z], [ 1 : 0 : α 2]) 1 α 2 1 ) 2 y 2 + 4α 2 xz = α 2 1) α 2 x 2 α 2 1) y 2 z 2 ) y 2 = 4α2 α 2 2 xz. 5.1) 1) It is easy to see that p 0 satisfies the equation 5.1 and therefore, the locus of the dual of the tangent P 0 is indeed a hyperbolic parabola. Notice that when α 2 = 1 the twin parabola will be the same as P 0 with equation y 2 = xz. The twin parabola P 0 meets its axis A = f 1 f 2 = f 1 f 2 at the same vertices v 1 = [0 : 0 : 1] and v 2 = [1 : 0 : 0] as P 0. In Figure 5.1), we see several aspects of the duality between P 0 and its twin P 0. The t-directrices of P 0 are the directrices of P 0 and conversely. Also the twin P 0 meets the directrix F 1 of P 0 at the null points δ 0 and δ 0. Similarly, the parabola P 0 meets the directrix F 2 of P 0 at the null points α 0 and α 0. Notice that the null lines at α 0 and α 0 pass through f 2 ; the tangent lines to P 0 at α 0 and α 0 pass through f 1 ; the null lines at δ 0 and δ 0 pass through f 1 ; and that the tangent lines to P 0 at δ 0 and δ 0 pass through f 2. Recall from Chapter 3 that the points f 1, f 2, f 1, f 2 form a harmonic range, and that the relation between the sides f 1 f 2 and f 1 f 2 is reciprocal: f 1 and f 2 are also sydpoints of the side f 1 f 2. It follows that the twin of P 0 is the parabola P 0. Recall also from the definition of sydpoints that q f 1, f 1) = q f 2, f 1) and q f 1, f 2) = q f 2, f 2). Here are some more metrical relations involving the four points f 1, f 2, f 1, f 2 and some related ones;

131 5. Further properties of the hyperbolic parabola 123 Figure 5.1: The parabola P 0 and its twin P 0 Theorem 89 Sydpoint base relations) We have the following relations; q f 1, f 1) = q b 1, b 1), q f 1, b 1) = q f 1, b 1 ) and q v 1, f 1) = q v 1, b 1). Proof. We calculate that α ) 2 q f 1, f 1) = q [α + 1 : 0 : α α 1)], [1 : 0 : 1]) = 1 4 α α 2 1) = q [α + 1 : 0 : α 1 α)], [1 : 0 : 1]) = q b 1, b 1) 2α + α 2 1 ) 2 q f 1, b 1) = q [α + 1 : 0 : α α 1)], [1 : 0 : 1]) = 1 4 α α 2 1) = q [1 : 0 : 1], [α + 1 : 0 : α 1 α)]) = q f 1 ), b 1 and q v 1, f 1) = q [0 : 0 : 1], [1 : 0 : 1]) = 5.2 Alternate views of a parabola α2 α 2 1 It is quite surprising that the parabola as we have been studying it also allows quite diff erent definitions. = q [0 : 0 : 1], [1 : 0 : 1]) = q v 1, b 1).

132 5. Further properties of the hyperbolic parabola The parabola as a midpoint locus In Euclidean geometry, the locus of the midpoints between a fixed point F and a variable point T lying on a line l is another line, parallel to l and half-way between it and F. However in hyperbolic geometry, this situation is quite different; in general such a midpoint locus algebraically gives a quartic curve, but the following theorem reveals a lovely factorization. Theorem 90 Midpoints locus) The locus of midpoints between f 1 = [ 1 : 0 : 1] and F 2 = 1 : 0 : 1 is xz y 2 ) α 2 x 2 + 2α 2 xz + α 2 1 ) 2 y 2 + α 2 z 2) = 0. In particular, the parabola P 0 is part of this locus. Proof. Let m = [x : y : z] be a midpoint between the point f 1 = [ 1 : 0 : 1] and a point on the line F 2 = 1 : 0 : 1. The line f 1 m = [x : y : z] [ 1 : 0 : 1] = y : x + z) : y meets F 2 = 1 : 0 : 1 at a point f 1 m ) F 2) = y : x + z) : y 1 : 0 : 1 = [ x + z) : 2y : x + z)]. Now applying the midpoint condition q [x : y : z], [ 1 : 0 : 1]) q [x : y : z], [ x + z) : 2y : x + z)]) = 0 leads to the equation xz y 2 ) x 2 α 2 2y 2 α 2 + y 2 α 4 + z 2 α 2 + y 2 + 2xzα 2) 4 α 2 1) x 2y + z) x + 2y + z) x 2 α 2 y 2 α 2 + y 2 z 2 ) = 0 which gives the quartic curve xz y 2 ) x 2 α 2 2y 2 α 2 + y 2 α 4 + z 2 α 2 + y 2 + 2xzα 2) = 0. Now observe that the factor xz y 2 is the equation of the parabola P 0. By symmetry we can deduce that also the parabola P 0 is part of the locus of midpoints between f 2 and F 1. In terms of the points t 1 F 1 f 2 ) p 0 and t 2 F 2 f 1 ) p 0, we may restate these conditions as the equations q f 1 ), p 0 = q p0, t 2) and q f 2 ), p 0 = q p0, t 1). Figure 5.2 shows a variable point p 0 on P 0 as a midpoint between f 1 and t 2 on F 2, or as a midpoint between f 2 and t 1 on F 1. Now we give another quite different description of the same parabola P 0.

133 5. Further properties of the hyperbolic parabola 125 Figure 5.2: Midpoint locus property of a parabola The parabola as a quadrance product locus Here is a quite different characterization of a parabola using the metrical structure. Theorem 91 Quadrance product) For any point p 0 on the parabola P 0 with foci f 1 and f 2, and t-foci f 1 and f 2, we have that 1 q f 1 )), p 0 1 q f 2 )) 1, p 0 = 4 1 q f 1, f 2 )) = 1 1 q f 1, f 2)) Proof. Computing 1 q f 1 ) [, p 0 = 1 q [ 1 : 0 : 1], t 2 : t : 1 ]) t α = α 2 1) t 2 1) 1 q f 2 ) [, p 0 = 1 q 1 : 0 : α 2 ], [ t 2 : t : 1 ]) α 2 t 2 1 ) = α 2 1) t 2 α 2 + 1) Thus 1 q f 1 )), p 0 1 q f 2 )) α 2, p 0 = α 2 1) 2 = 1 ) 4 1 α2 +1) 2 4α 2 1 = 4 1 q f 1, f 2 -- )) = 1 α ) ) α 2 1) 2 = 1 1 q f 1, f 2)). 4 Note that this theorem reveals a new -fact about sydpoints. Namely if f 1 and f 2 are the sydpoints of the side f 1 f 2, then we have the identity 1 q f 1, f 2 )) 1 q f 1, f 2)) =

134 5. Further properties of the hyperbolic parabola 126 Recall that 1 q x, y) equals the quadrance between x and the line y. So this yields yet a quite different interpretation of what a parabola is. Now we turn to the pleasant existence of a rich and canonical structure for a parabola. 5.3 Structures associated to a variable point p 0 on P 0. In this section, we will generate several interesting and important points related to a general point p 0 on the parabola P 0, and note some pleasant concurrences The e-points of p 0 If p 0 = [ t 2 : t : 1 ] lies on P 0, then the reflection p 0 r a p 0 ) = [ t 2 : t : 1 ] is the opposite point of p 0. It also lies on P 0 and has dual line P 0 = t 2 α 2 : t 1 α 2) : 1. Join p 0 and p 0 to the null axis points to get the lines E 1 η 1 p 0 = [ tα : t 2 α 1 : t ] and E 2 η 2 p 0 = [ tα : t 2 α + 1 ) : t ] E 1 η 1 p 0 = [ tα : t 2 α 1 : t ] and E 2 η 2 p 0 = [ tα : t 2 α + 1 : t ]. Then define the e-points of p 0 as e 0 E 1 E 2 = [ 1 : tα : t 2 α 2] and e 0 E 2 E 1 = [ 1 : tα : t 2 α 2]. Figure 5.3: e 0 and e 0 points Theorem 92 e-points) For an arbitrary point p 0 on the parabola P 0, i) the associated e-points e 0, e 0 always lie on P 0 ; ii) the line e 0 e 0 always passes through the point j 0 ; and iii) the tangents to P 0 at e 0 and e 0 are respectively the lines e 0 h 0 and e 0 h 0.

135 5. Further properties of the hyperbolic parabola 127 Proof. i) This is straightforward since the parabola P 0 has equation xz = y 2. ii) The collinearity [[e 0 e 0 j 0 ]] follows from iii) We have that Similarly, 1 tα t 2 α 2 det 1 tα t 2 α 2 = t 2 α 2 e 0 h 0 = [ 1 : tα : t 2 α 2] [ 1 : 0 : t 2 α 2] = [ t 2 α 2 : 2tα : 1 ] = [ : tα : t 2 α 2] = e 0 A e 0 h 0 = [ 1 : tα : t 2 α 2] [ 1 : 0 : t 2 α 2] = [ t 2 α 2 : 2tα : 1 ] = [ : tα : t 2 α 2] = e 0 A The perpendicular points m 1, m 2 of p 0 Recall that a point is perpendicular to every point lying on its dual line. So a point p 0 on the parabola P 0 has perpendicular points lying on P 0 precisely when the dual line P 0 meets the parabola. A priori it is not clear that this should always occur. Theorem 93 Perpendicular points) The dual line P 0 of a point p 0 = [ t 2 : t : 1 ] on the parabola P 0 meets the parabola at exactly two points m 1 P 0 P 0 = [ 1 : t : t 2] and m 2 P 0 P 0 = [ 1 : tα 2 : t 2 α 4] except when t = 0 and α 2 = 1 in which case the dual line meets the parabola at one point [ 1 : t : t 2], or when t =, in which case the dual line meets the parabola at one point [0 : 0 : 1]. Proof. We can easily find the meets of the dual line P 0 = t 2 α 2 : t α 2 1 ) : 1 of a point p 0 and the parabola P 0 by solving the system t 2 α 2 x t α 2 1 ) y z = 0 y 2 xz = 0

136 5. Further properties of the hyperbolic parabola 128 which yields the required points m 1 = [ 1 : t : t 2] and m 2 = [ 1 : tα 2 : t 2 α 4]. Notice that m 1 = m 2 = [ 1 : t : t 2] when t = 0 or α 2 = 1; while when t takes the projective value then m 1 and m 2 evaluate to [0 : 0 : 1]. Recall from the last chapter that we defined p 0 = p t) [ t 2 : t : 1 ]. We call the points m 1 and m 2 the perpendicular points of p 0. Their dual lines are M 1 m 1 = α 2 : t α 2 1 ) : t 2 and M 2 m 1 = 1 : t α 2 1 ) : t 2 α 2. Corollary 94 If p 0 = p t) is a point on the parabola, then its perpendicular points are given as m 1 = p ) 1 t and m 2 = p 1 ) tα 2. Proof. This is just an algebraic restatement of the forms of m 1 and m 2 from the Theorem. Motivated by the t-base theorem in the last chapter, define the lines S 1 f 1 p 0 = t : t ) : t and S 2 f 2 p 0 = tα 2 : t 2 α 2 1 : t with dual points s 1 F 1 P 0 = [ t α 2 1 ) : α 2 t ) : tα 2 α 2 1 )] s 2 F 2 P 0 = [ t α 2 1 ) : t 2 α 2 1 : t α 2 1 )]. and Theorem 95 Perpendicular collinearities) The perpendicular points of p 0 satisfy m 1 = S 1 P 0 and m 2 = S 2 P 0, provided that t 2 α and t Dually under the same conditions M 1 = s 1 p 0 and M 2 = s 2 p 0. Proof. We have that S 1 P 0 = t : t ) : t t 2 α 2 : t α 2 1 ) : 1 = [ t 2 α ) : t t 2 α ) : t 2 t 2 α )] = [ 1 : t : t 2] = m 1 if t 2 α and S 2 P 0 = tα 2 : t 2 α 2 1 : t t 2 α 2 : t α 2 1 ) : 1 = [ t 2 1 ) : tα 2 t 2 1 ) : t 2 α 4 t 2 1 )] = [ 1 : tα 2 : t 2 α 4] = m 2 if t The reflection m 1 = r a m 1 ) and m 2 = r a m 2 ) are the perpendicular points of the opposite point p 0, we call them the opposite perpendicular points of p 0. They are m 1 = [ 1 : t : t 2] and m 2 = [ 1 : tα 2 : t 2 α 4]

137 5. Further properties of the hyperbolic parabola 129 and their duals are M 1 = α 2 : t α 2 1 ) : t 2 and M 2 = 1 : t α 2 1 ) : t 2 α 2. Figure 5.4: Perpendicular points collinearities The basic reflection r a = r A in the axis is an isometry that fixes the parabola and all its properties, as a result, many collinearities and concurrences have immediate reflections, which we will generally not mention. Theorem 96 Perpendicular points collinearities) For an arbitrary point p 0 on the parabola P 0, we have collinearities [[ m 1 b 2 ]] [[ p 0, m2 b 1 ]] [[ p 0 and m1 h 0 ]] m 2. Proof. We compute the determinants: 1 t t 2 det = 0 t 2 t 1 1 tα 2 t 2 α 4 det 1 0 α 2 = 0 t 2 t 1 1 t t 2 det 1 0 t 2 α 2 = 0. 1 tα 2 t 2 α 4 and Motivated by the previous theorem, we can define the L and L lines as L 1 p 0 b 1 = tα 2 : t 2 α ) : t and L 1 p 0 b 1 = tα 2 : t 2 α : t L 2 p 0 b 2 = t : t 2 1 : t and L 2 p 0 b 2 = t : t 2 1 : t.

138 5. Further properties of the hyperbolic parabola 130 This provides us with another simple way for constructing the perpendicular points m 1 and m 2 of p 0 given the t-foci of a parabola; just join p 0 to the t-foci f 1 and f 2 and determine the meets of P 0 with the lines f 1 p 0 and f 2 p 0. Following the perpendicular collinearities theorem, define the associated perpendicular points of the twin point p 0 as [ m 1 P 0 S 2 = t 2 α 1) 2 : 2tα : α + 1) 2] [ m 2 P 0 S 1 = t 2 α + 1) 2 : 2tα : α 1) 2]. and The dual lines are M 1 m 1) = p 0 s 2 = t 2 α 2 α + 1) 2 : 2tα α 2 1 ) : α 1) 2 M 2 m 2) = p 0 s 1 = t 2 α 2 α 1) 2 : 2tα α 2 1 ) : α + 1) 2. and The conjugate points n 1, n 2 of p 0 Recall from the last chapter that the points [ n 1 R 1 P 0 = α + 1) 2 : tα 1 α 2) : t 2 α 2 α 1) 2] and n 2 R 2 P 0 = [α 1) 2 : tα α 2 1 ) : t 2 α 2 α + 1) 2] were defined as the conjugate points of p 0 on the parabola P 0. Here we are going to construct them easily as the meets of lines. Joining p 0 to the base points b 1 and b 2 gives the K-lines K 1 b 1 p 0 = tα α + 1) : 4 : t α 1) K 2 b 2 p 0 = tα α 1) : 3 : t α + 1) and with dual k-points k 1 = [ tα α + 1) u 2 1 ) : u 2 4 : tu 2 α 1) u 2 1 )] k 2 = [ tα α 1) u 2 1 ) : u 2 3 : tu 2 α + 1) u 2 1 )]. and Also, define the opposite K-lines as K 1 b 1 p 0 = tα α + 1) : 4 : t α 1) K 2 b 2 p 0 = tα α 1) : 3 : t α + 1). and Theorem 97 Conjugate points construction) The conjugate points n 1 and n 2 of p 0 satisfy n 1 = R 1 K 2 and n 2 = R 2 K 1.

139 5. Further properties of the hyperbolic parabola 131 Proof. We check that and R 1 K 2 = tα α 1) : 1 : t α + 1) tα α 1) : 3 : t α + 1) [ = α + 1) 2 : tα 1 α 2) : t 2 α 2 α 1) 2] = n 1 R 2 K 1 = tα α + 1) : 2 : t α 1) tα α + 1) : 4 : t α 1) [ = α 1) 2 : tα α 2 1 ) : t 2 α 2 α + 1) 2] = n 2. Define the opposite conjugates of p 0, using the axis reflection theorem in the last chapter, to be respectively [ n 1 = α + 1) 2 : tα α 2 1 ) : t 2 α 2 α 1) 2] and n 2 = [α 1) 2 : tα α 2 1 ) : t 2 α 2 α + 1) 2] with duals the opposite conjugate lines N 1 n 1 = α α + 1) 2 : t α 2 1 ) 2 : t 2 α α 1) 2 N 2 n 2 = α α 1) 2 : t α 2 1 ) 2 : t 2 α α + 1) 2. and Theorem 98 Conjugate collinearities) For an arbitrary point p 0 on the parabola P 0, we have collinearities [[n 1 j 0 n 2 ]] and [[ n 1 h 0 n 2 ]]. Proof. We compute that α + 1) 2 tα 1 α 2) t 2 α 2 α 1) 2 det 1 0 t 2 α 2 α 1) 2 tα α 2 1 ) = 0 t 2 α 2 α + 1) 2 α + 1) 2 tα 1 α 2) t 2 α 2 α 1) 2 det 1 0 t 2 α 2 α 1) 2 tα = 0. 1 α 2) t 2 α 2 α + 1) 2 and Theorem 99 Conjugate vertex meets) Let n 1, n 2 be the conjugate points of a point p 0 on the parabola P 0. Then the points z 1 = v 2 n 1 ) v 1 n 2 ) and z 2 = v 1 n 1 ) v 2 n 2 ) lie on the line J 0. Proof. The meet of the lines v 2 n 1 = 0 : tα α 1) : α + 1 and v 1 n 2 = tα α + 1) : α 1) : 0 is the point z 1 = [ α 1) : tα α + 1) : t 2 α 2 α 1) ]

140 5. Further properties of the hyperbolic parabola 132 Figure 5.5: Focal conjugates n 1, n 2 and the ω 0 conic W 0 and the meet of the lines v 1 n 1 = tα α 1) : α + 1 : 0 and v 2 n 2 = 0 : tα α + 1) : α 1) is the point z 2 = [ α + 1) : tα α 1) : t 2 α 2 α + 1) ]. which both lie on the line J 0 = t 2 α 2 : 0 : 1. Define the focal conjugates meets of p 0 to be [ w 0 N 1 N 2 = t 2 α 2 1 ) 2 : 4tα 2 : α 2 1 ) ] 2 [ w 0 N 1 N 2 = t 2 α 2 1 ) 2 : 4tα 2 : α 2 1 ) ] 2. Theorem 100 Focal conjugates conic) The locus of the point w 0 as p 0 moves along P 0 is a conic W 0 which shares with P 0 the vertices v 1 and v 2. Its equation is and y 2 = 16α4 α 2 4 xz. 5.2) 1) Proof. We verify that the points w 0 and w 0 lie on 5.2). Clearly the vertices v 1 = [0 : 0 : 1] and v 2 = [1 : 0 : 0] also lie on W 0. Theorem 101 Focal conjugate collinearity) The points w 0, w 0 satisfy w 0 w 0 = J 0. In particular, w 0, w 0 and h 0 are collinear. Proof. We compute that w 0 w 0 = [ 1 : 0 : t 2] = J 0. Now recall that J 0 passes through h 0. These results are illustrated in Figure 5.5.

141 5. Further properties of the hyperbolic parabola 133 Similarly, motivated by the conjugate points construction theorem, we can define the conjugate points of the twin point p 0 as with dual lines n 1 R 1 K 2 = [ t 2 α 2 α 2 1 ) : 2tα 2 : α 2 1 ] n 2 R 2 K 1 = [ t 2 α 2 1 ) : 2tα 2 : α 2 α 2 1 )] N 1 n 1) = t 2 α 4 : 2tα 2 : 1 and N 2 n 2) = t 2 : 2t : 1. Here is a main theorem relating the the perpendicular points m 2 and m 1 of p 0 and the conjugate points n 1 and n 2 of p 0 ; illustrated in Figure 5.6. Theorem 102 Perpendicular conjugate relation) The duals of the perpendicular points m 2 and m 1 of p 0 are respectively the tangents to P 0 at the conjugate points n 1 and n 2 of p 0. Dually, the duals of the conjugate points n 1 and n 2 of p 0 are respectively the tangents to P 0 at the perpendicular points m 2 and m 1 of p 0. Proof. The tangent lines to P 0 at n 1 and n 2 can be calculated as n 1 B = [ t 2 α 2 α 2 1 ) : 2tα 2 : α 2 1 ] 0 0 2α 2 0 α 2 1 ) 2 0 and 2α = [ 1 : t α 2 1 ) : t 2 α 2] = M 2 and n 2 B = [ t 2 α 2 1 ) : 2tα 2 : α 2 α 2 1 )] 0 0 2α 2 0 α 2 1 ) 2 0 = [ α 2 : t α 2 1 ) : t 2] = M 1. 2α In addition, the tangent lines to P 0 at m 1 and m 2 can be calculated as m 1 A = [ : t : t 2] = t 2 : 2t : 1 = N 2 and m 2 A = [ : tα 2 : t 2 α 4] = t 2 α 4 : 2tα 2 : 1 = N Dually, the dual lines of the perpendicular points m 1 and m 2 of p 0 are the tangents of the parabola P 0 at the conjugate points n 2 and n 1 of p 0 respectively. The duals of the conjugate points n 1 and n 2 of the point p 0 on parabola P 0 are the tangents to the twin parabola P 0 at m 2 and m 1 respectively. This configuration is shown in Figure 5.6.

142 5. Further properties of the hyperbolic parabola 134 Figure 5.6: Perpendicular conjugate relation The focal and base lines of the twin point p 0 Define additional points t 1 and t 2 as t 1 F 1 S 2 = [ t 2 α 2 1 : 2tα 2 : α 2 t 2 α 2 1 )] and t 2 F 2 S 1 = [ t : 2t : t ] with dual lines T 1 f 1 s 2 = t 2 α 2 1 : 2t α 2 1 ) : t 2 α 2 1 T 2 f 2 s 1 = α 2 t ) : 2t α 2 1 ) : t and Now introduce the lines R 1 and R 2 as R 1 f 1 t 1 = [ 2tα 2 : α 2 1 ) t 2 α 2 1 ) : 2tα 2] R 2 f 2 t 2 = [ 2tα 2 : α 2 1 ) t ) : 2t ] and with dual points r 1 F 1 T 1 = [ 2t : t 2 α 2 1 : 2tα 2] and r 1 F 2 T 2 = [ 2t : t : 2t ]. Theorem 103 Altitude base foci collinearities) We have collinearities [[p 0 f 1 t 1 ]] and [[p 0 f 2 t 2 ]].