Singular Frégier Conics in Non-Euclidean Geometry

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1 Singular Frégier onics in on-euclidean Geometry Hans-Peter Schröcker University o Innsbruck, Austria arxiv: v1 [math.mg] 24 May 2016 May 25, 2016 The hyotenuses o all right triangles inscribed into a ixed conic with ixed right-angle vertex are incident with the Frégier oint to and. As varies on the conic, the locus o the Frégier oint is, in general, a conic as well. We study conics whose Frégier locus is singular in Euclidean, ellitic and hyerbolic geometry. The richest variety o conics with this roerty is obtained in hyerbolic lane while in ellitic geometry only three amilies o conics have a singular Frégier locus. Keywords: Frégier oint, Frégier conic, Thales theorem, hyerbolic geometry, ellitic geometry, singular conic 2010 Mathematics Subject lassiication: 51M09, Introduction Many theorems o Euclidean elementary geometry have their counterarts in ellitic or hyerbolic geometry, ossibly ater a suitable reormulation. An examle o this is a version o Pythagoras Theorem [2, 3]. By contrast, we know o no convincing non-euclidean version or Thales Theorem or its converse (comare [4]). This aer will not rovide one either but we will resent some interesting geometric coniguration that are at least reminiscent o Thales classical theorem. Thales Theorem talks about right triangles with the same hyotenuse and imlies that the hyotenuses o all right triangles inscribed into a circle contain the circle center. In articular, the hyotenuses o all right triangles with right angle vertex ixed on that circle all ass through one ixed oint. This statement remains true i the circle is relaced by an arbitrary regular conic (Frégier s Theorem). In this situation, the oint is called the Frégier oint to and. It is not diicult to rove Frégier s theorem by means o basic rojective geometry. We resent two well-known roos: One has the beneit to clearly demonstrate the relation between Thales and Frégier s theorems. The other emloys the theory o rojective transormations on conics and immediately imlies the validity o Frégier s Theorem in ellitic and hyerbolic geometry. The Frégier oint deends on and but the locus F o all Frégier oints or varying only deends on. This locus turns out to be a conic which, by construction, shares the symmetry grou with. In Euclidean geometry, and F are even similar. We are interested in regular 1

2 conics whose Frégier conic F is singular. In Euclidean geometry, these are circles and right hyerbolas. In the ormer case, F degenerates to the circle center, in the latter, it is the line at ininity. The same question in ellitic and hyerbolic geometry calls or a more involved answer and gives rise to a number o interesting geometric conigurations, even when viewed with the eyes o a Euclidean observer. y 1 m x 0 x 1 D A y 0 2 Frégier onics in Euclidean geometry Figure 1: Proo o Theorem 1 by homology to circle We begin by recalling a ew well-known results and roos on Frégier oints and conics in the Euclidean lane. They introduce some basic concets and set standards that later will be comared with the non-euclidean situation. Theorem 1 (Frégier). Given a regular conic in the Euclidean lane and a oint, the hyotenuses o all right triangles inscribed into and with right angle at intersect in a common oint. Deinition 1. The oint o Theorem 1 is called the Frégier oint o and. We resent two roos o Theorem 1, both having their own merits. The irst roo shows how to derive Frégier s Theorem rom Thales Theorem by means o a homology to a circle (Figure 1). First roo o Theorem 1. Take an arbitrary circle D, tangent to at. There exist a homology η with center that mas D to. (Its axis A is the Desargues axis o two triangles that corresond in η and are inscribed into D and, resectively.) By Thales Theorem, the Frégier oint is then = η(m) where m is the circle center. Second roo o Theorem 1. For a right triangle inscribed into and with right angle at, denote the other vertices by q and r. The ma ϕ :, q r (with aroriate conventions i coincides with q or r) rojects to the orthogonal involution in the line bundle around. Hence, it is an involution in and there exists a oint, the Frégier oint, which is collinear with all airs o corresonding oints [1, Theorem 8.2.8]. This second roo aeals to a more roound knowledge o rojective geometry but has the beneit o retaining its validity in non-euclidean geometries: orollary 1. Frégier s Theorem is true in the ellitic and hyerbolic lane. The deining roerty o the Frégier oint also allows its comutation. However, the arbitrariness o the inscribed right triangle is somewhat awkward. We thereore look or alternatives. A useul observation, yet insuicient to nail down, is the act that it is located on the conic normal at. As suggested in [1, Exercise 10.12], we are led to consider the ix oints o the involution ϕ in our second roo o Theorem 1. They are the intersection oints i, i dierent rom o and the isotroic lines through. Their tangents intersect in. These arguments equally aly to Euclidean and non- Euclidean geometries. Figure 2 dislays this construction in the seudo-euclidean lane with absolute oints I, I and in the hyerbolic lane with absolute conic. 2

3 I i the Frégier locus o. In general, it is a regular conic section but excetions may occur. I i F Figure 2: onstruction o the Frégier oint via the isotroic lines through (seudo- Euclidean and hyerbolic geometry) Proosition 1. The Frégier oint to a conic and a oint in Euclidean, ellitic or hyerbolic geometry is the ole o the line i i where i and i are the rojections o onto via the isotroic lines through. Remark 1. At this oint, a remark on our view on hyerbolic geometry seems aroriate. In the sense o Wildberger s universal hyerbolic geometry [5 7], we treat oints in- and outside the absolute conic equally. This leads to simliied statements and comutations. Arguably, the resulting theory is richer and more comrehensive. It also allows a real Figure 2 to illustrate Proosition 1. ote however that universal hyerbolic geometry ails to be a model or the axiomatic geometry obtained by relacing Euclid s arallel ostulate with its hyerbolic counterart. Denoting the Frégier oint to conic and oint by (, ), we call the set i i F F := { (, ) } Theorem 2. Generically, the Frégier locus o a conic is a conic F. It is our aim in this aer to characterize regular conics whose Frégier locus F is not a regular conic. In doing so, we will derive the algebraic equation o the Frégier locus or dierent relative ositions o and and thus rove Theorem 2. This discussion also makes the word generically recise. I the Frégier locus is a conic, we call it the Frégier conic to. Otherwise, we seak o the singular Frégier locus. 3 Singular Frégier Loci ow we have a closer look at conics in the Euclidean, ellitic and hyerbolic lane whose Frégier locus is singular. The Euclidean discussion is straightorward and yields known results. The non-euclidean discussion requires the distinction between relative osition o and, that is, dierent tyes o encils o conics. We shall see that singular Frégier loci are o greater variety in a non-euclidean setting. 3.1 Frégier onics in the Euclidean Plane Using homogeneous coordinates [x 0,x 1,x 2 ], an ellise or hyerbola in the Euclidean lane can always be described by an equation o the shae : bx ax 2 2 = x 2 0 with non-zero real numbers a, b that are not both negative. The Frégier conic, comuted via Proosition 1, has equation F : b(a + b) 2 x 1 + a(a + b) 2 x 2 = ab(a b) 2 x 0. We see that, in general, F is obtained rom by a scaling with actor (a b)/(a + b) about the common center o and F. This statement is not true i the conic is a circle (a 2 b 2 = 0) 3

4 or a right hyerbola (a 2 +b 2 = 0). In the ormer case, the Frégier conic consists o a single oint, in the latter, the Frégier conics degenerates to the line at ininity. It is noteworthy that in this case the ma F is a double cover o the line at ininity but only the ideal oints on normals o arise as real Frégier oints. Thus, the Frégier locus is a rojective line segment. A arabola in the Euclidean lane may be described by the equation : x 0 x 2 = ax 2 1 with a R \ {0}. Its Frégier conic F : x 0 x 2 = ax a x2 0 is just a translate o and we can summarize: Proosition 2. I the Frégier locus in the Euclidean lane is not a regular conic then either is a circle and F is its center or is a right hyerbola and F consists o those ideal oints that belong to normal directions o. 3. Bitangent encils where recisely two airs o base oints coincide. We call the conics in this case circles. 4. Double contact or osculating encils where three o the our base oints coincide. Here, we seak o osculating arabolas. 5. Trile contact or hyerosculating encils where all our base oints coincide. The corresonding conics are called horocycles General onics The equation o a general conic may be written as : bx ax 2 2 = x 2 0, a,b R \ {0} 3.2 Frégier onics in the Hyerbolic Plane Our investigation o Frégier conics in the hyerbolic lane is based on a discussion o the relative osition o the absolute conic and the conic in the comlex rojective lane, that is, encils o conics in that lane. This is justiied because the Frégier locus to a real conic is always real, even i some elements in the construction o Figure 2 aear as conjugate imaginary airs. The line F and its ole are always real. The conics o a encil share our dierent oints ( base oints ), some o which may coincide and thus result in tangency or contact o higher order [1, Section 9.6]. Deending on the number o coinciding oints, one can distinguish ive dierent cases (Figure 3): 1. General encils with our dierent base oints. 2. Simle contact encils with a single air o coinciding base oints. In analogy to the Euclidean situation, we call the corresonding conics arabolas. with non-zero real numbers a, b that are not both negative and not both equal to 1. This case also comrises circles or a = 1, b 1 or b = 1, a 1. Its Frégier conic has the equation F : (a 2 b 2 a 2 b 2 ) 2 (a 2 b 2 + a 2 b 2 ) 2 a 2 x2 1 It is singular i and only i b 2 = + (a2 b 2 a 2 b 2 ) 2 (a 2 b 2 a 2 + b 2 ) 2 b 2 x2 2 = x 2 0. a2 a 2 + 1, b2 = a2 a 2 1, or b2 = a2 a 2 1. The Frégier locus is, in that order, a rojective line segment on the irst, the second or the third axis o the underlying rojective coordinate rame (Figure 4) Parabolas A arabola that is tangent to at [1, 1,0] admits an equation o the shae : λ(x x 2 2 x 2 0) + (x 0 + x 1 )(µ(x 0 + x 1 ) + x 1 ) = 0. 4

5 general arabola circle osculating arabola horocycle Figure 3: Pencils o conics, conics in the hyerbolic lane F F Figure 4: Three conics with singular Frégier locus (F 3 is art o the line at ininity) The arameters λ and µ range in R but λ = 0 and λ = 2 1 are rohibited in order to ensure regular conics. The limiting case µ yields a horocycle and will be treated later. The Frégier conic has equation F : (x1 2 + x2 2 x2 0 )λ 4 + (x 0 + x 1 )Λ where F Λ = (x 0 +x 1 )(5µλ 3 +12µλ 2 +9µλ +2µ +1) + λ(4(λ + 1)x 0 + (5λ 2 + 8λ + 5)x 1 ). Figure 5: Parabola with singular Frégier locus. It is singular recisely or λ = 0, λ = 1 2, or λ = 1. Because only the last value is admissible, we obtain a one-arametric amily o hyerbolic arabolas with singular Frégier conic: : x 2 0 x 2 1 x (x 0 + x 1 )(µ(x 0 + x 1 ) + x 1 ) = 0 and µ ranges in R (Figure 5). 5

6 3.2.3 Osculating Parabolas Writing the equation o an osculating arabola as : λ(x x 2 2 x 2 0) + (x 0 + x 1 )x 2 = 0, with λ R \ {0}, the Frégier conic becomes F : 2λ(x x 2 2 x 2 0) + (x 0 + x 1 )(10x 2 λ + 8(x 0 + x 1 )) = 0. It is never singular ircles ircles in hyerbolic geometry are characterized by having double contact with. The oints o contact may be both real or both conjugate comlex. It is ossibly to discuss these two cases at once but it is robably easier to consider them searately. I the oints o tangency are real, we may write the circle equation as : λ(x x 2 2 x 2 0) + (x 1 x 0 )(x 1 + x 0 ) = 0, λ R \ {0,1}. This is the arabola case or µ = 2 1. Frégier conic is F : λ 3 (x x 2 2 x 2 0) The + (5λ 2 + 8λ + 4)(x 1 x 0 )(x 1 + x 0 ) = 0. (1) Unless λ = 2 (see below), it is again a circle. Obviously, it shares the symmetries o. Because the signs o λ + 1 and λ 3 /(5λ 2 + 8λ + 4) agree, it lies in the interior o i and only i does. The conic (1) is singular i λ 3 (λ + 2) 4 (λ + 1) 2 = 0. Because λ / {0,1}, the Frégier locus is singular recisely or λ = 2. It degenerates to the line with equation x 2 = 0, that is, the san o the two oints o tangency. More recisely, only interior oint o occur as real Frégier oints (Figure 6). The lines connecting any conic oint with the oints o tangency are erendicular. F Figure 6: Singular Frégier locus to a hyerbolic circle Remark 2. The situation o Figure 6 may also be described by saying that Thales Theorem in the hyerbolic lane holds true or ininite line segments. Remark 3. Figure 6 is also remarkable rom a Euclidean viewoint. The ellise with semiaxis ratio 1 : 1/ 2 is inscribed into the Thales circle over major axis in such a way that or any oint, the ole o the ellise tangent in with resect to, the rojection o onto the major axis and itsel are collinear. I both oints o tangency are conjugate comlex, the circle equation becomes : λ(x x 2 2 x 2 0) + x x 2 2 = 0, λ R \ {0, 1} and the Frégier conic is F : λ 3 (x x 2 2 x 2 0) (5λ 2 + 8λ + 4)(x x 2 2) = 0. 6

7 a comlex rojective transormation, all results o the latter also hold in the rojective extension o the ormer. In order to make statements on Frégier conics in the ellitic lane over the real numbers, we only need to discuss the reality o the involved geometric objects. The only relevant case is that o a general conic and its secialization to a circle: : bx ax 2 2 = x 2 0, a,b R \ {0}. Keeing things short, we only mention the inal result. In the ellitic lane, the Frégier conic to is singular i and only i a and b satisy one o Figure 7: Singular Frégier locus to a hyerbolic circle Unless λ = 2, symmetry is shared between and F and one lies in the interior o i and only i the other does. The Frégier conic is singular recisely or λ = 2 (Figure 7) and it is the san o the two oints o tangency. Viewing Figure 7 with the eyes o a Euclidean observer, the Frégier locus is a rojective line segment on the line at ininity Horocycles I the intersection oints o and all coincide, is called a horocycle. Its equation may be written as : λ(x 2 1 +x 2 2 x 2 0)+(x 2 x 0 ) 2 = 0, whence the the Frégier conic becomes b 2 = a2 a 2 + 1, b2 = a2 a 2 1, or b2 = a2 a The last case is never real and none o these amilies contains real circles. 4 onclusion We recalled some well-known acts about Frégier conics in the Euclidean lane and transerred them to ellitic and hyerbolic geometry. It turned out that the situation between Euclidean and non-euclidean geometry is dierent when it comes to singular Frégier loci. In articular, we saw that the Frégier locus to a regular conic can be singular i is an ellise, a hyerbola, a arabola, or a circle but not i it is an osculating arabola or a horocycle. Another notable dierence is that a singular Frégier locus in non-euclidean geometry is always a rojec- λ R\{0} tive line segment while it may be a single oint in Euclidean geometry. F : λ(x x 2 2 x 2 0) + 5(x 2 x 0 ) 2 = 0. We see that and F are related by the ma λ 1 5λ. o singularities arise. 3.3 Frégier onics in the Ellitic Plane The situation in ellitic geometry is algebraically equivalent to universal hyerbolic geometry. Via Reerences [1] Eduardo asas-alvero. Analytic Projective Geometry. Euroean Mathematical Society, ISB [2] Maria Teresa Familiari-alaso. Sur une classe de triangles et sur le théorème de ythagore en géométrie hyerbolique.. R. 7

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