NOTES ON NON-EUCLIDEAN GEOMETRIES. Gábor Moussong Budapest Semesters in Mathematics. Contents. 0 Introduction 2

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1 NOTES ON NON-EUCLIDEAN GEOMETRIES Gábor Moussong Budapest Semesters in Mathematics 2018 Contents 0 Introduction 2 Part One: Preparatory Chapters 5 1 Affine Geometry 5 2 Spherical Geometry 15 3 Inversive Geometry 22 Part Two: Projective Geometry 38 4 Projective Space and Incidence 38 5 Coordinates in Projective Geometry 43 6 Cross-ratio and Projective Geometry of the Line 48 7 Conics 56 Part Three: Hyperbolic Geometry 64 8 The Projective Model 64 9 The Poincaré Models The Hyperboloid Model The Hyperbolic Plane 82 1

2 2 GÁBOR MOUSSONG 0 Introduction 0.1 What is geometry? In very general and vague terms, geometry is the study of space and shapes with mathematical rigor. It is a very broad subject within mathematics, and in the framework of a one-semester course we can only touch upon some selected topics, and present these from a specific point of view. One of the goals in this course is to show how geometry finds its place among the many abstract structures of modern mathematics. We shall treat geometry through the powerful methods of other chapters of abstract mathematics: linear algebra, calculus, and groups. This point of view opens up several channels through which classical geometry is linked with today s research in advanced mathematics, notably in differential geometry, in topology, and in group theory. The history of mathematics has produced many different geometric systems, the oldest of which is the familiar Euclidean geometry. Typically, such a geometric system Euclidean or non-euclidean has the following types of characteristic features: basic objects, like points, lines, planes, circles, etc., transformations which move around these objects, and measurements, like distance, area, angle, which remain invariant under these transformations. Our treatment of geometry will investigate these features in certain classical geometries. Special emphasis will be given to the role of transformations, and to the way how the structure of these transformations can distinguish between varios types of geometry. 0.2 What makes Euclidean geometry Euclidean? More than two thousand years ago Euclid presented the whole body of geometric knowledge of his era in a systematic way. His axiomatic treatment of geometry set a standard for treating mathematics ever since. The general structure of such an axiomatic theory starts with some undefined basic notions, called primitive terms (in geometry, such are ponts, lines, and the like), and with some statements, called axioms, about these objects, which are accepted as truth without proof. The theory is then developed through definitions, theorems, and rigorous proofs, all based either on the axioms or on earlier theorems. The theory built on Euclid s axioms eventually results in an essentially unique mathematical structure, the so-called Euclidean space. This is the mathematical counterpart of our physical space given by everyday experience. A geometric system is Euclidean if it is isomorphic to the one defined by Euclid s axioms. The most famous example of non-euclidean geometry was discovered in connection with one particular axiom of Euclidean geometry, the so-called parallel postulate (stated here in an equivalent form which is different from Euclid s original): Given any line L and any point P not on L, there exists only one line through P, within the plane containing L and P, which does not intersect L.

3 NOTES ON NON-EUCLIDEAN GEOMETRIES 3 It was naturally expected that Euclid s axioms express unquestionable truth about the physical space around us. Since this particular axiom seemed a lot less obvious than the others, many geometers through the ages tried to deduce the parallel postulate from the rest of the axioms, thereby making it unnecessary to use. All these attempts failed, and finally (in the last third of nineteenth century) it turned outthatsuchaproofisimpossible. Itsnegation, takenasasubstituteoftheparallel postulate, plus the rest of Euclid s axioms, form a new system of axioms, and define a meaningful geometric system, now called hyperbolic geometry. Clearly, the parallel postulate makes it possible to define the concept of parallel lines in Euclidean geometry. If it fails, then the whole issue of parallelism is completely different. Therefore, it is useful to be aware what parts, which theorems of traditional elementary Euclidean geometry really depend on the parallel postulate. Here are some notable examples: The angle sum theorem. Recall that the standard proof that the angles of a triangle add up to 180 degrees starts with drawing a line parallel to one side. In fact, the angle sum theorem turns out equivalent to the parallel postulate. Therefore, if angles make sense in a non-euclidean geometric system, then the sum of angles within a triangle is expected different from 180 degrees. The concept of vectors, more precisely, of free vectors. In Euclidean geometry, vectors are freely translated anywhere; this cannot be done without parallel lines. Translations in Euclidean geometry usually are defined using parallel lines, or simply by vectors. It is possible to define translations under non-euclidean circumstances but this must be done carefully avoiding any reference to vectors or parallelism. The resulting non-euclidean translations behave quite different from what we are used to in elementary geometry. The concept of similarity. In Euclidean geometry there exist similar figures of different size. The standard procedure to construct two similar but noncongruent triangles uses the intercept theorem (on two lines intercepted by a pair of parallel lines), therefore, existence of non-congruent similar figures also depends on parallelism. Use of Cartesian coordinates. Finding coordinates of a point involves drawing parallels to the axes, therefore coordinates cannot be used the same way if the geometry is non-euclidean. Some further theorems and methods of elementary Euclidean geometry depend on the above, therefore are not expected to work (or are expected to work differently) in non-euclidean geometry. Such are, for example, the theorem of Thales on angles inscribed in a semicircle, the theorem of Pythagoras, the trigonometric laws, and some area formulas for triangles. With all these concepts missing, or different from usual, non-euclidean geometries are quite different from the familiar Euclidean geometry. Even so, much of our work will be devoted to showing that large part of traditional geometry can be salvaged in the non-euclidean setting. 0.3 Topics covered in the course and in these notes Our main goal is to learn the methods, formalism, and mathematical machinery that is necessary to understand various geometric systems as mathematical structures.

4 4 GÁBOR MOUSSONG We shall not use the axiomatic method at all; actually, no axiom other than the parallel postulate will ever be mentioned. As a starting point, we rely on the intuitive concept of Euclidean space and plane, and we take the usual theorems of elementary Euclidean geometry for granted. Other types of geometries will later be based on concrete definitions and constructions carried out on the basis of, or directly within, Euclidean geometry. The first part of these notes basically stays in the realm of Euclidean geometry while introduces some background material for later use. Parts of this are affine geometry and inversive geometry, which themselves may be regarded as examples of non-euclidean geometry. Spherical geometry is briefly introduced as another easy non-euclidean example. The second part covers projective geometry which, besides being an interesting, entertaining, classical piece of mathematics on its own right, serves as the main technical background for hyperbolic geometry. The third part is devoted to hyperbolic geometry which is introduced through three types of models, each using a different type of mathematical apparatus. All these types of geometries could be worked out in arbitrary dimensions using essentially the same methods. For simplicity, we restrict ourselves to the twodimensional case, that is, we only work with the affine plane, the inversive plane, the two-dimensional sphere, the projective plane, and the hyperbolic plane.

5 NOTES ON NON-EUCLIDEAN GEOMETRIES 5 Part One: Preparatory Chapters 1 Affine Geometry Affine geometry collects those features of Euclidean space which are immediate applications of the linear algebra of vectors in space. Only the vector space operations (that is, addition of vectors, and multiplication of vectors by scalars) are considered. Most notably, distance and angle play no role whatsoever in affine geometry. 1.1 Vectors in Euclidean geometry We start with a brief review of the definition and the structure of vectors in Euclidean space. Let E denote Euclidean space as the set of points. Ordered pairs (A,B) of points (that is, elements of E E) are naturally identified with directed straight segments. Two such are equivalent if a suitable translation takes one to the other. A vector in E is an equivalence class of directed segments. Notation: write v = AB if the vector v is represented by (A,B). The set of vectors is denoted V. Operations on vectors: u+v and λu (where u,v V, λ R) are defined in a straightforward way (choosing representatives). Fact: these operations turn V into a real vector space of dimension 3. Remarks: (1) If E were a plane or a line only, we would get dimv = 2 or dimv = 1, respectively. (2) At the moment we are not interested in other operations on vectors (products, notably). We want to study the part of geometry which relies on the vector space structure of V only. Summary: We have a set E, a real vector space V, and a structure map E E V, denoted as (A,B) AB, subject to the following two conditions: (1) For any fixed O E the map E V, A OA, is bijective; (2) AB + BC = AC for all A,B,C E. Remark: These properties serve as the basis for the concept of an abstract affine space. The set E is called affine line, affine plane, or affine space according as dimv = 1, 2, or 3. Note that distance and angle play no role here. In what follows, for simplicity, everything is formulated, defined, stated, etc., for the plane. As an exercise, the reader can modify the relevant data and constructions to get analogous concepts and theorems for three-dimensional or one-dimensional geometry.

6 6 GÁBOR MOUSSONG 1.2 Coordinate systems Distance and angle should play no role. Therefore, when a coordinate system is introduced in the plane, the two axes need not be perpendicular, and the units may be chosen independently for each axis. Formally speaking, a coordinate system is a map x which assigns coordinates to each point: x : E R 2, P x(p) = (x 1,x 2 ). (We call R 2 the standard coordinate plane.) A coordinate system in E is equivalent to a choice of an origin in E plus a basis in V. In practice, when a coordinate system x is chosen, a point P usually is identified with its image x(p), i.e., we write P = (x 1,x 2 ). Given two coordinate systems x, y, either in the same plane or in two distinct planes, there is a natural map associated with them: two points correspond if they have the same coordinates. That is, the point P (with coordinates x(p)) of the first plane is mapped to the point Q of the second plane for which y(q) = x(p) holds. (In formal terms, this is the map y 1 x.) Exercise: if the two planes coincide (and x is the the coordinate system taken first), verify that in terms of x this map is given by a formula x Mx+v where M is a 2 2 invertible matrix, and v is a fixed vector (independent of x). Such maps serve as prototypical examples of the type of transformations we discuss next. 1.3 Affine transformations Definition (affinity). Let f be a map between two (not necessarily distinct) planes. The map f is called an affine transformation (or affinity) if f is expressed in terms of coordinates (that is, as a map f : R 2 R 2 ) as f(x) = Mx + v, where M is any invertible matrix, and v is any vector. (Strictly speaking, the formula Mx + v only defines R 2 R 2 affinities. If P and Q are planes equipped with coordinate systems x and y, respectively, then f : P Q is an affinity if the triple composition y f x 1 : R 2 R 2 is.) Note that an affine transformation f(x) = Mx + v has a linear part, given by the matrix M, and a translation part, given by v. The calculation f(b) f(a) = (Mb+v) (Ma+v) = M(b a) shows that the way f acts on vectors between pairs of points is determined only by the linear part, i.e., by the matrix M. Definition (affine group). It is routine to verify that the inverse of an affine transformation is an affine transformation, and that the composition of two affine transformations is an affine transformation. It follows that affine transformations of a plane E to itself form a group under compositon. This group is called the affine group of E, and is usually denoted as Aff(E). If E is the coordinate plane R 2, we simply write Aff(2) for Aff(R 2 ). Aff(1) and Aff(3) are defined analogously. Remark: The fact whether a map is an affine transformation or not does not depend on the choice of coordinates. This follows from the above observations about compositions.

7 NOTES ON NON-EUCLIDEAN GEOMETRIES 7 Examples of affine transformations: All Euclidean congruences are affine transformations. Indeed, if the map f between two planes is a Euclidean congruence (that is, a bijective map which preserves all relevant geometric properties such as collinearity, distance, angle, area, etc.), then a coordinate system is taken to another coordinate system by f, and the map f simply coincides with the natural map associated with this pair of coordinate systems. All homotheties are affine transformations. A homothety(or central similarity ) h : E E with center at point O E takes any point A E to the point A = h(a) for which OA = λ OA with a fixed nonzero scalar λ R. If the origin of a coordinate system is at O, then the formula for h is h(x) = λx. The linear part of h is given by the scalar matrix λi (where I denotes the identity matrix). Corollary: all similarity transformations are affine. Indeed, any similarity transformation can be represented as a composition of a Euclidean congruence with a homothety. Parallel projections: Given two planes E and E in three-dimensional space, and a line L not parallel to either, one can define a map E E as follows. For any point A E there exists a unique point A E with AA L. The map E E, A A (A E), is called parallel projection from E to E in the direction of L. It is not hard to verify that any parallel projection is an affinity. If E E, then this affinity in general is not a Euclidean congruence between the planes (not even a similarity). 1.4 Affine invariants Roughly speaking, affine geometry studies properties which are invariant (i.e., remain unchanged) under affine transformations. For example, distance, angle, area are not affine invariants, but collinearity, parallelism, simple ratio (see below) are. Theorem. Any affine transformation takes lines to lines, parallel lines to parallel lines. Proof: Any line has a parametric representation tu+a (where u 0). Under an affinity Mx+v its image is t(mu)+(ma+v) which is a line again. If two lines are parallel, then the same direction vector u can be used for both; then the image lines also have the same direction vector Mu. Definition (simple ratio). Let A, B, C be three collinear points with A B C. Write AC = λ CB, then this uniquely determined real number λ is called the simple ratio of points A, B, C. Notation: (ABC) = λ.

8 8 GÁBOR MOUSSONG Remark: If C is between A and B, then (ABC) is indeed the ratio in which C divides the segment AB. For instance, (ABC) = 1 if C is the midpoint. If C is outside the interval from A to B, then (ABC) is negative. Immediate observations: (1) If A and B are fixed, then (ABC) can take all real number values except 1. (2) (ABC 1 ) = (ABC 2 ) = C 1 = C 2. Both follow from the formula (ABC) = x/(1 x) where A = 0, B = 1, and C = x. Lemma. Let a = OA and b = OB be linearly independent, and consider c = αa+βb = OC. If C is collinear with A and B, then α+β = 1. If, further, C A, then (ABC) = β/α. Proof: AC = (α 1)a + βb, CB = αa + (1 β)b. If these two vectors are linearly dependent, then (α 1)(1 β) β( α) = 0, which implies α+β = 1. Writing 1 α for β we have AC = βa + βb and CB = αa + αb, which imply AC = (β/α) CB. Theorem. Any affine transformation preserves simple ratio. Proof: Given A, B, C, choose O off the line, and apply the lemma. Corollary. Let f be an affine transformation of a plane E to itself. (1) If f fixes two distinct points A and B, then it fixes all points of line AB. (2) If f fixes three non-collinear points, then f is the identity map of E. Proof: (1) follows immediately from the preceding theorem. (2) follows from (1) since by (1) all three pairwise connecting lines are pointwise fixed, and through any given point of E there can be drawn a line which intersects at least two of these lines. 1.5 The fundamental theorem Definition (affine frame). An affine frame in a plane E is a choice of three non-collinear points A 0, A 1, A 2 E. There is a straightforward bijective correspondence between affine frames and coordinate systems in E: let A 0 be the origin, A 0 A 1 and A 0 A 2 be the basis vectors: The following theorem is sometimes called the fundamental theorem of affine geometry (in dimension two). Theorem. If E and E are (not necessarily distinct) planes, A 0, A 1, A 2 E and A 0, A 1, A 2 E are affine frames in each, then there exists a unique affine transformation f : E E such that f(a i ) = A i (i = 0,1,2).

9 NOTES ON NON-EUCLIDEAN GEOMETRIES 9 Proof: Existence: the natural map associated with the two coordinate systems is just a required affinity. Uniqueness: suppose both f and g are as required, then g 1 f : E E is an affinity fixing three non-collinear points. By Corollary (2) above, g 1 f = id E, that is, f = g. Remark: In the case E = E the statement of the theorem is often phrased as follows: the affine group Aff(E) acts simply transitively on the set of affine frames in E. Two figures are called affinely equivalent if a suitable affine transformation maps one onto the other. In this case the two figures are not distinguishable by affine means. For example, any two triangles are affinely equivalent. Affinely equivalent figures are considered equal in affine geometry, just like congruent figures are considered equal in Euclidean geometry. 1.6 Example: axial affinities Throughout this section, E denotes a plane. We consider a special type of affine transformations in E. When an affinity E E is given, we follow the general practice that the symbol P denotes the image of any point P E. Definition (axis, axial affinity). Let f : E E be an affine transformation. A line L E is called an axis of f, if it is pointwise fixed, i.e., A = A for all A L. An affinity is called axial if it has an axis. By preservation of simple ratio, if an affinity of E fixes two distinct points, then it is axial. By the fundamental theorem, a non-identity axial affinity can have no fixed points outside its axis. (In particular, the axis is unique.) Lemma. Let f : E E be an axial affinity with axis L, f id E. Then all lines PP, where P / L, are parallel to each other. Proof: Suppose P,Q / L, P Q. If line PQ meets L at a point, say A, then by preservation of simple ratio, (APQ) = (AP Q ). Now PP QQ follows by elementary geometry (say, similar triangles). If P Q L, then use a third point R not on PQ or L to deduce PP RR QQ. Definition (shear, strain). Given a non-identity axial affinity, by the lemma above the parallelism class of lines PP does not depend on the actual choice of the point P, which means that it is characteristic of the transformation itself. This parallelism class is called the direction of the axial affinity. If this direction is parallel to the axis, then the axial affinity is called a shear, otherwise it is called a strain.

10 10 GÁBOR MOUSSONG If an axial affinity with axis L is a strain, then for P / L and B = L PP there is a uniquely defined λ R such that BP = λ BP. This λ does not depend on the choice of P (again, by elementary geometry), and is called the ratio of the strain. (Note that shears do not have a ratio.) If a coordinate system is chosen so that the first coordinate ( axis ) coincides with the 1 a axis of the affinity, then a shear is given by a matrix (where a R), and ( ) a strain with ratio λ is given by (where λ R, λ 0). As a familiar 0 λ example, any Euclidean reflection is a strain with ratio 1. In this example the direction of the strain is perpendicular to the axis. 1.7 Structure of the affine group Recall that Aff(2) denotes the affine group in dimension 2, that is, the group of all affine transformations R 2 R 2 under the operation of composition of maps. We can recognize two important subgroups of Aff(2). One is the subgroup of translations. An affinity f(x) = Mx+v is a translation if and only if M = I, the identity matrix. Translations form a subgroup isomorphic to the additive group R 2 of vectors. Under this isomorphism the translation f(x) = x+v corresponds to the vector v R 2. It is common practice to identify translations with vectors, therefore we may consider R 2 Aff(2) as the subgroup of translations. The other subgroup is formed by all linear maps within Aff(2), that is, affinities given by a matrix only (with zero translation part). These are precisely those elements of Aff(2) which keep the origin fixed. This subgroup simply is the multiplicative group of invertible 2 2 matrices, called the general linear group in dimension 2, denoted as GL(2, R). It is clear that these two subgroups are disjoint, that is, R 2 GL(2,R) consists only of the identity map. Further, since by definition any affinity is a product of a linear map and a translation, these two subgroups together generate the whole affine group. Claim: R 2 is a normal subgroup in Aff(2). Indeed, consider the map from Aff(2) to GL(2,R) which assigns its linear part to any affine transformation. Direct calculation shows that this map is a homomorphism of groups. The translation subgroup is the kernel of this homomorphism, therefore it is normal. (Exercise: check that selecting the translation part of an affinity is not a homomorphism of groups, and that GL(2,R) is not a normal subgroup in Aff(2).) Remark: These observations show that Aff(2) is a so-called semidirect product of the two subgroups R 2 and GL(2,R). In general, we say that a group G is a semidirect product of its subgroups N and H (notation: G = N H) if (i) N is a normal subgroup, (ii) N H = 1, and (iii) NH = G. (Note that a semidirect product specializes to the direct product if H is also normal.) Affine geometry yields a nontrivial example of a semidirect product: Aff(2) = R 2 GL(2,R). An affine transformation f is called orientation preserving if, when written as f(x) = Mx+v, the matrix M has positive determinant. (Orientation reversing otherwise.)

11 NOTES ON NON-EUCLIDEAN GEOMETRIES 11 Remark: Whether an affinity of a plane to itself is orientation preserving or not does not depend on the choice of the coordinate system. Indeed, if matrix T is the change of basis, then the linear part of the affinity with respect to the new basis is TMT 1, and det(tmt 1 ) = detm. Orientation preserving affinities form an index two subgroup in Aff(2). All translations, all homotheties, and all shears, are examples of orientation preserving affinities of the plane. Among Euclidean congruences translations and rotations are orientation preserving, reflections are orientation reversing. 1.8 Euclidean congruences as affinities Some affine transformations f(x) = Mx + v are actually Euclidean congruences. It is natural to ask how this fact is seen from the algebraic data of f, i.e., from M and v. Throughout this section we assume that the coordinate system x is Cartesian, that is, the associated basis in V is orthonormal. (Recall that a basis is called orthonormal if it consists of pairwise ortogonal, unit length, vectors.) Clearly the map f is a Euclidean congruence if and only if the image of the coordinate system under f is also a Cartesian system. Since the image basis vectors are precisely the column vectors of the matrix M, this condition simply means that the column vectors of M form an orthonormal basis. The following lemma gives several equivalent characterizations of this property of matrices. For our purposes only the n = 2 and n = 3 cases are necessary, but it is worthwhile to state te lemma for general n. Recall that for n-dimensional vectors x = (x 1,...,x n ), y = (y 1,...,y n ) R n dot product and norm are defined as x y = x 1 y x n y n, and x = x x = x x2 n. Lemma. Let M be an n n real matrix. The following conditions are equivalent: (i) Column vectors of M form an orthonormal basis; (ii) M M = I; (iii) M preserves dot products: (Mx) (My) = x y for all x, y R n ; (iv) M preserves norms: Mx = x for all x R n. The proof is mostly a routine exercise in linear algebra. Definition (orthogonal matrix). A real square matrix is called orthogonal if either one (therefore all) conditions of the lemma hold for M. The set of n n orthogonal matrices is denoted O(n). Clearly O(n) is a group under matrix multiplication. If no coordinate system is specified in a Euclidean vector space V, conditions (iii) or (iv) of the lemma can still be used to define the coordinate-free concept of orthogonal linear maps from V to V, and the orthogonal group O(V). A linear map is orthogonal if and only if, with respect to an orthonormal basis, it is represented by an orthogonal matrix. Using this terminology we can say that an affine transformation f(x) = Mx+v is a Euclidean congruence if and only if the matrix M is orthogonal.

12 12 GÁBOR MOUSSONG Remark: It follows that the group E(n) of Euclidean congruences of R n is a semidirect product of the translation subgroup and the orthogonal group: E(n) = R n O(n). If M is an orthogonal matrix, (detm) 2 = (detm )(detm) = det(m M) = deti = 1, that is, detm is either +1 or 1. Those with positive determinant are called special orthogonal, and form a subgroup SO(n) of index 2 in O(n). Orthogonal matrices in dimension 2: any 2 2 orthogonal matrix is either ( ) cosα sinα sinα cosα or ( ) cosβ sinβ sinβ cosβ where α,β R. By inspection, the first matrix (with determinant 1) is rotation about the origin through angle α, the second (with determinant 1) is reflection in the line through the origin which makes angle β/2 with the first axis. 1.9 Conics A general quadratic equation in two real variables x and y is Ax 2 +Bxy +Cy 2 +Dx+Ey +F = 0, where the first three of the six coefficients are not all zero. Curves defined by such equations are called quadratic curves in the plane. Fact: In a Cartesian system of coordinates a quadratic curve typically is a conic (i.e., an ellipse, a parabola, or a hyperbola). It may also degenerate to a line, to a pair of lines, to a point, or to the empty set. If the system of coordinates is moved so that it fits the conic the best, then the equation simplifies as either x 2 a 2 + y2 b 2 = 1 (ellipse), y2 = 2px (parabola), or x2 a 2 y2 b 2 = 1 (hyperbola). Suppose an affine transformation f is applied to a quadratic curve. Coordinates of a point x satisfy the equation of the transformed curve if and only if the coordinates of f 1 (x) satisfy the original equation. Therefore, the equation of the transformed curve is obtained by substituting the coordinates of f 1 (x) into the equation. Since f 1 is an affine transformation, (inhomogeneous) linear expressions of the variables are substituted into the original quadratic equation. So, the resulting equation is again quadratic. Corollary: affine transformations always transform quadratic curves into quadratic curves.

13 NOTES ON NON-EUCLIDEAN GEOMETRIES 13 A quadratic curve is a conic if it has at least two points, and contains no line. Therefore, the image of a conic under an affine transformation is always a conic. One may refine this even further by observing that among conics precisely the ellipses are bounded, and precisely the hyperbolas are disconnected. Both these properties are preserved by affinities, therefore we have the following corollary: Affinities transform ellipses to ellipses, parabolas to parabolas, and hyperbolas to hyperbolas. For instance, an ellipse is always mapped to an ellipse by any shear mapping of the plane, or by any parallel projection between two planes Complex affine line Let E be a Euclidean plane. We assume that E is equipped with orientation, that is, it is determined which rotations are considered positive. Note that this structure is precisely what is needed to call E an affine line over the complex field C. Indeed, any choice of two distinct points in E, called 0 and 1, identifies E with the field of complex numbers the usual way. (For instance, i C is obtained by rotating 1 about 0 through +90 degrees.) Vectors AB, with A,B E, become complex numbers. Thus, by fixing 0 and 1 (an affine frame), we introduce a one-dimensional coordinate system in E where any coordinate now is a complex number. Complex affinities of E = C are the maps f : C C of the form f(z) = mz +v where m,v C, m 0. Some special cases: When m = 1, the map f(z) = z +v is a translation. When v = 0, the map f(z) = mz is a complex homothety. If the coefficient m is real, then this transformation is a homothety of E in the Euclidean sense. If m C and m = 1, then f is a rotation. In the general case, f is a combination of the two, called a spiral similarity. It follows that all complex affinities of the complex affine line E are actually orientation preserving similarity transformations of the Euclidean plane E. It is not hard to see that the converse is also true. Remark: For a plane E we now have two different ways to interpret E in the framework of affine geometry: it may be a (two-dimensional) affine plane over the reals, or a complex affine line (which is one-dimensional over C). These two interpretations differ in several aspects. For example, there are considerably more affinities in the first case than in the second Exercises 1. An affine transformation f : R 2 R 2, (x,y ) = f(x,y), is given by the formulas x = 3x+y +2 y = x+y +1. Draw the image of the unit square of the coordinate system. What is the image of the rectangular grid formed by the lines x = integer, y = integer? 2. Given two affine ( transformations ) f(x) = Mx+v and g(x) ( = Nx+w, ) where M =, v = (3, 1) and N =, w = (2,0), find the linear part and translation part of the composition g f.

14 14 GÁBOR MOUSSONG 3. Given the affine transformation f(x) = Mx+v, where ( ) 1 1 M =, v = (3, 1), 3 2 find the linear part and translation part of f Determine which of the following types of objects is taken to the same type of objects by any affine transformation: (a) a half-line, (b) a half-plane, (c) an interval, (d) a right angle, (e) an acute angle, (f) a polygon, (g) a convex set, (h) a circle, (i) a rectangle, (j) a parallelogram, (k) a trapezoid, (l) a rhombus. 5. If an affinity is applied to a triangle, show that the centroid (a. k. a. median point) is mapped to the centroid of the image triangle. (Recall that a median line connects a vertex of the triangle to the midpoint of the opposite side, and the centroid is the common point of the three median lines, dividing all of them in ratio 2 : 1.) 6. Is the statement of Exercise 5 true for other notable points of the triangle(orthocenter, incenter, circumcenter)? 7. Suppose that the composition of three homotheties is the identity. Show that the three centers are collinear. 8. Let A, B, C be three distinct collinear points. Show (ABC)(BCA)(CAB) = Let P, Q, R, S be four distinct collinear points. Prove (PQS)(QRS)(RPS) = Let a = OA, b = OB, and c = OC be linearly independent vectors in three-dimensional Euclidean space, and consider a vector x = αa + βb + γc = OX. Prove that the point X is incident to the plane ABC if and only if α+β +γ = Suppose that, under an affinity of the plane, vertex A of triangle ABC is kept fixed, vertex B is mapped to C, and C is mapped to B. Show that this affinity is axial, and find its axis, direction, and ratio. 12. Given a parallelogram, find a shear mapping which takes it to a rhombus. (Define the shear through an affine frame and its image.) 13. Given a triangle ABC and a line L, show that there exists an axial affinity with axis L which takes ABC to an equilateral triangle. 14. An affine transformation of the coordinate plane is given by x = x+4y 2 y = 2x+5y 2. Prove that this is an axial affinity, and find its axis, direction, and ratio. 15. Find an affinity f of a plane to itself such that A, f(a), and f ( f(a) ) are non-collinear for any point A. 16. An affine transformation of the coordinate plane R 2 is given by the affine frame A 0, A 1, A 2 and its image A 0, A 1, A 2. Find the image of point P = ( 3, 5) if: A 0 = (1, 1), A 1 = (2,0), A 2 = ( 1,3), A 0 = ( 4,4), A 1 = ( 5,5), A 2 = (4,2). 17. Let f : R 2 R 2 be rotation of the plane through +90 degrees about the center (2,1). Write f in the form f(x) = Mx+v, and find the matrix M and the vector v. 18. Prove that a square matrix is orthogonal if and only if its row vectors form an orthonormal basis.

15 NOTES ON NON-EUCLIDEAN GEOMETRIES Fill in the missing entries of the following 3 3 matrices to get orthogonal matrices. (Several solutions may exist.) In the second case try to use rational numbers only Let u R n be any nonzero vector. Define the map σ : R n R n by the formula σ(x) = x 2 u x u u u (x Rn ). Show that σ is an orthogonal linear map. If n = 2 or 3, what well-known Euclidean congruence is σ? (Note that σ(x) = x whenever x u). 21. How many n n orthogonal matrices are there such that all entries are integers? Check that these matrices form a subgroup of O(n). 22. Show that the group O(n) contains S n, the symmetric group on n letters, as a subgroup. (Recall that S n is the group of all permutations of the set {1,2,...,n}.) 23. (a) Given two points A and B of Euclidean plane, and their images A and B under some affine transformation, construct (with ruler and compass) the image of an arbitrarily given further point of the line AB. (b) An affinity of Euclidean plane is given through an affine frame and its image. Find a way to construct the image of an arbitrarily given further point of the plane. 24. Prove that any orientation preserving similarity transformation of the plane to itself is either a translation or a spiral similarity. (Hint: Work in the complex affine line, and look for a fixed point.) 25. Let A, B, A, B be four points in Euclidean plane given in terms of Cartesian coordinates as A = (1,0), B = (2,0), A = (0,1), B = (0,3). Find the center of the spiral similarity of the plane which takes A to A and B to B. (Hint: Use complex numbers, write the map as mz + v, use the given points to determine m and v, and solve for the fixed point.) Spherical Geometry We study the intrinsic geometry of the surface of the unit sphere in Euclidean space. We use the center of the sphere as origin for the vectors in space. Thus, Euclidean space is identified with the three-dimensional vector space V. When coordinates are needed, we assume V = R 3 via a choice of a Cartesian system of coordinates. An essential element of structure now is the dot product (and norm) in V. 2.1 Basic Concepts Definition (unit sphere). The unit sphere in V is the set S = {x V : x = 1}. Circles in S are the subsets of the form S P where P is any plane such that the origin is less than distance 1 from P. A circle S P is called a great circle if 0 P ( radius is maximal (= 1)). Definition (spherical segment). Ifx,y S aredistinctandarenotantipodal (i.e., x ±y, or equivalently, the vectors x and y are linearly independent), then there exists a unique great circle C through x and y. The shorter of the two

16 16 GÁBOR MOUSSONG arcs into which x and y divide the great circle C is called the spherical segment connecting the points x and y. Remark. Great circles play the role of straight lines in spherical geometry. Note that if two points are antipodal, then a great circle passing through them is not unique. But if two(distinct) points are not antipodal, then they uniquely determine the great circle and the spherical segment connecting them. Definition (tangent vector, tangent plane). A vector v V is called tangent to S at the point x S if v x ( x v = 0). The tangent plane at x S is T x S = x V, a 2-dimensional linear subspace. Remark. When visualizing tangent vectors, T x S is customarily identified with its translate x+t x S which really is tangent to the sphere S. The advantage of using our definition for the tangent plane is that it is now a linear space with respect to the natural operations on vectors. Example. Given a spherical segment with endpoints x and y, there is a unique unit vector v T x S which is determined by the requirement that y = αx+βv with β > 0. This vector v is called the unit tangent vector at x to the segment from x to y. Definition (spherical distance). We define the spherical distance ( angle metric ) of two points by the formula d s (x,y) = cos 1 (x y) (x,y S). (Note that d s (x,y) equals the arc length of the spherical segment from x to y.) Remark. The distance function d s : S S R turns the set S into a metric space. Proof of the triangle inequality for d s is postponed to Section 2.2. Definition (angle). We define two kinds of angles in spherical geometry. First, the angle between two great circles is defined as the angle between tangent lines drawn at a point of intersection (which happens to equal the angle between the planes containing the circles).

17 NOTES ON NON-EUCLIDEAN GEOMETRIES 17 Second, the angle formed by two spherical segments with a common endpoint: for any x,y,z S, x y z we define xyz as the angle between tangent vectors drawn at y to the spherical segments from y to x and from y to z. Great circles in S can be conveniently parameterized as follows. A great circle C is uniquely determined by a point x C and a unit vector v T x S tangent to C at x. (Namely, C = S P where P is the linear 2-plane spanned by x and v.) Working in the plane P, and using coordinates with respect to the orthonormal basis x,v in P, one may parameterize the circle C as r(t) = (cost)x+(sint)v. This is arc length parameterization, i.e., for any t 1,t 2 (with t 1 t 2 π) we have d s (r(t 1 ),r(t 2 )) = t 1 t 2. This follows from the calculation cosd s (r(t 1 ),r(t 2 )) = r(t 1 ) r(t 2 ) = cost 1 cost 2 +sint 1 sint 2 = cos(t 1 t 2 ). 2.2 The spherical Law of Cosines We define the concept of spherical triangles, and prove the most basic formula of spherical trigonometry. Definition (spherical triangle). If x,y,z S are linearly independent ( do not all lie on a great circle), then they determine a spherical triangle, namely: = S (convex hull of the three rays from the origin through x, y and z), or equivalently, = S (intersection of the three half-spaces bounded by the linear 2-planes spanned by two of x, y, z and containing the third as interior point). Vertices of are the points x, y, z; sides of are the pairwise connecting spherical segments between the vertices. The angles of are formed by the pairs of its sides. Let α, β, γ denote the angles at vertices x, y, z, respectively, of a spherical triangle, and let a, b, c denote the side lengths opposite x, y, z, respectively. Theorem (spherical Law of Cosines) Proof: cosa = cosb cosc+sinb sinc cosα

18 18 GÁBOR MOUSSONG Choose unit tangent vectors u,v T x S at the endpoint x to the spherical segments from x to y and z, respectively. Parameterize the side xy and substitute the parameter value c, and get the formula y = (cosc)x+(sinc)u. Similarly, parameterize xz, substitute b, and get z = (cosb)x+(sinb)v. Dot product of the two yields y z = (cosb)(cosc)(x x)+(sinb)(sinc)(u v). Here y z = cosa, x x = 1, u v = cosα, and the Law of Cosines follows. Remark. If the size of the triangle converges to zero, then this formula converges to the ordinary (Euclidean) Law of Cosines. To see this, replace cosa, cosb, cosc, sinb and sinc with their Taylor series about 0, and ignore all terms of degree greater than two. Corollary. For any spherical triangle with side lengths a, b, c the strict triangle inequality a < b+c holds. Proof: In the Law of Cosines formula one has sinb > 0, sinc > 0 and cosα > 1; therefore, one gets the inequality cosa > cosb cosc sinb sinc. The right hand side is just cos(b + c). Using that the cosx function is strictly monotone decreasing on the interval [0, π], the triangle inequality a < b + c follows. Remark. This implies that the angle metric introduced in Section 2.1 is indeed a metric. 2.3 Area of spherical triangles Recallthatthetotalsurfaceareaoftheunitsphereequals4π. Thereisaremarkable formula (the so-called Girard formula, see below) which expresses the surface area of spherical triangles in terms of their angles. Our goal is to prove the Girard formula. Definition (spherical bigon). Connect a pair of antipodal points in S with two great semicircles. The union of the two semicircles divides S into two regions each of which is called a spherical bigon. Any spherical bigon is uniquely determined up to congruence by its angle ϕ (equal at both vertices). A hemisphere is a special case of a spherical bigon (with ϕ = π), and the whole S may also be regarded a bigon with ϕ = 2π.

19 NOTES ON NON-EUCLIDEAN GEOMETRIES 19 Lemma. The surface area of a spherical bigon with angle ϕ equals 2ϕ. Proof: The surface area in question only depends on ϕ, therefore we may denote it as A(ϕ). This is a monotone and additive, therefore linear, function of the variable ϕ; that is, A(ϕ) = cϕ with some constant c. Total surface area of S is 4π, therefore A(2π) = 4π and c = 2. Thus, A(ϕ) = 2ϕ for all 0 ϕ 2π. Theorem (Girard formula). The surface area of a spherical triangle with angles α, β, γ equals α+β +γ π. Proof: Let A denote the surface area of the spherical triangle in question. Draw the three great circles containing the three sides of the triangle. Each pair of them determine two congruent (oppositely placed) spherical bigons one of which containing the triangle. The angle of such a bigon equals an angle of the triangle. The six bigons thus obtained cover the whole sphere. The triangle itself and its antipodal (centrally symmetric) image is triply covered, and the rest of S is singly covered. This gives the equation 2A(α)+2A(β)+2A(γ) = 4π+4A. Taking into account that A(ϕ) = 2ϕ, the equation A = α+β +γ π follows. Corollary. Any spherical triangle has angle sum strictly greater than π. Remark. The concepts we introduced and the theorems we proved in Sections about the geometry of the sphere all have interesting counterparts in hyperbolic geometry, see 10.5, Our brief study of spherical geometry actually serves as a warmup for later parts of the course. 2.4 Congruences in spherical geometry Congruence transformations of S are restrictions to S of Euclidean congruences of V which keep the origin fixed. These are precisely the orthogonal linear transformations of V. Thus, the group of spherical congruences is O(V). If a Cartesian coordinate system is given, then this group is O(3). Orientation preserving congruences form the subgroup SO(V) (or SO(3)). For an example of a sperical congruence, consider a rotation of the ambient Euclidean space about an axis through origin, restricted to S. Such a map is called a spherical rotation. The special case when the rotation angle equals π is called a half turn. Theorem. All orientation preserving congruences of S are spherical rotations. Proof: Given M SO(3), we first show that 1 is an eigenvalue of M. All real eigenvalues of M have absolute value 1, since M preserves norms of vectors. The characteristic polynomial has odd degree, therefore M has at least one real

20 20 GÁBOR MOUSSONG eigenvalue. If all three are real, at least one of them must equal 1, since their product is detm = 1. If there are non-real eigenvalues, then they are a pair of complex conjugate numbers whose product is positive. Then the real eigenvalue must also be positive since the product of the three is 1. Let u be an eigenvector with eigenvalue 1, then the line L spanned by u is pointwise fixed by M. By preservation of dot product, M keeps the plane u invariant. Restriction of M to this plane must be an orientation preserving twodimensional Euclidean congruence, that is, a rotation about the origin within this plane. Therefore, M itself is a three-dimensional rotation about axis L. Remarks: (1) Since all elements of SO(3) are rotations, this group is often called the rotation group in dimension 3. (2) It is well-known that orientation preserving congruences of Euclidean plane are either rotations or translations. The theorem says that the second type is missing from spherical geometry. Actually, translations do exist, but they happen to coincide with rotations. This is explained the following way. Clearly one may translate great circles within themselves through any given spherical distance. It is intuitivelyclearthat, ifthesphereisattachedtosuchamovinggreatcircle, thenthe whole sphere moves along together with the great circle. This is how a translation can be defined in spherical geometry. It is easy to see that this congruence is actually a rotation about the axis orthogonal to the plane of the great circle. Our goal now is to show that the group of orientation preserving congruences in spherical geometry is a simple group. Recall that a group is called simple if it has no normal subgroups other than the trivial subgroup and the whole group. This feature of spherical geometry is in sharp contrast with affine or Euclidean plane geometry where translations form a proper normal subgroup (cf. 1.7, 1.8). Recall that a subgroup H of a group G is normal if and only if it is closed under conjugation, that is, ghg 1 H whenever h H and g G. (Two elements, h 1,h 2 G are called conjugate if there exists an element g G such that h 2 = gh 1 g 1.) Theorem. SO(3) is a simple group, that is, it has no nontrivial normal subgroups. Proof: We start with a series of claims about SO(3). Claim 1: All half turns are conjugate in SO(3). Indeed, it suffices to find a rotation that takes a given line through 0 to another given line through 0. Such rotations clearly exist, and any such rotation conjugates the half turn about the first line into the half turn about the second. Claim 2: SO(3) is generated by half turns. Indeed, it is easy to see that the product of two half turns about axes intersecting at angle ϕ is rotation through 2ϕ about the common perpendicular line of the axes. Clearly any given rotation is obtained this way. Claim 3: If an element of SO(3) reverses some line through the origin, that is, maps all points of this line to their negatives, then it is a half turn. Indeed, if the rotation angle is different from π, then clearly no line is reversed. Suppose now that 1 G SO(3). We prove that G contains at least one half-turn. Then, by Claim 1, all half turns are in G, and so G = SO(3) by Claim 2.

21 NOTES ON NON-EUCLIDEAN GEOMETRIES 21 Chooseanontrivialelementf G; iff itselfisnotahalfturn, then, byreplacing f with a suitable power, we may assume that f is rotation through angle α where π/2 α < π. Find a line L through 0 such that f(l) L. (Such a line exists by the intermediate value theorem, since the angle between L and f(l) (for various L) varies continuously between 0 and α.) Let g denote the half turn about the line f(l). Consider the isometry h = g 1 f 1 g f. Then g 1 f 1 g G because f G and G is a normal subgroup, and so h G. The map h reverses the line L, therefore h is a half turn by Claim Exercises 1. Given two distinct points x,y S, find the locus of all points of S which are at equal spherical distance from x and y. 2. Given a spherical line (= great circle) C in S and a point x S which is not a pole to C (i.e., the vector x is not orthogonal to the plane containing C), show that there exists a unique spherical line through x which intersects C at a right angle. Prove that the shortest spherical segment connecting x with C is an arc of this perpendicular great circle. 3. Given two distinct great circles in S, show that the locus of all points at equal spherical distance from them is the union of the two angle bisector great circles. 4. Given two distinct great circles in S, show that there exists a unique third great circle which intersects both at a right angle. 5. Define inscribed and circumscribed circles for spherical triangles. Show that their center is a common point of the three bisectors of interior angles, and the three perpendicular bisectors of sides, respectively. 6. Define the medians of a spherical triangle as the spherical segments connecting a vertex with the midpoint of the opposite side. Apply central projection with center at origin to the flat triangle spanned by the vertices of a spherical triangle, and show that medians of the flat triangle project to the medians of the spherical triangle. Deduce that the three medians of any spherical triangle are concurrent. 7. Suppose that a spherical triangle has two right angles. Show that two sides have length π/2. 8. Prove: if a spherical triangle has unique altitudes, then the three altitudes are concurrent. (Note that altitudes are not always uniquely defined: if two of the sides have spherical length π/2, then one can drop infinitely many perpendiculars to the third side from the opposite vertex (cf. 7). But if the triangle has at most one side of length π/2, then altitudes are unique.) 9. Prove the spherical Law of Sines: sinα sina = sinβ sinb = sinγ sinc, where a, b, c are the sides of a spherical triangle, and α, β, γ are the corresponding angles. (Hint: Express cosα from the Law of Cosines and get a formula for sin 2 α/sin 2 a which is invariant under permutations of a, b and c.) 10. Consider a right spherical triangle with legs a and b and hypotenuse c < π/2. Suppose that the altitude h divides c into parts p and q (with p next to a). Prove: sin 2 h = (tanp)(tanq) and tan 2 a = (tanp)(tanc). (Keeping in mind that sinx tanx x for small x, these are spherical analogues of well-known Euclidean formulas.) 11. Prove that the perimeter of any spherical triangle is less than 2π. (Hint: Extend the triangle to a spherical bigon.) 12. The side length of an equilateral spherical triangle is π/3. Find its area. (Use calculator.)