NOTES ON NON-EUCLIDEAN GEOMETRIES
                                            ´
                                          Gabor Moussong
                                   Budapest Semesters in Mathematics
                                                 2022
                                              Contents
                     0 Introduction                                              2
                    Part One: Warmup                                             5
                     1 Affine Geometry                                             5
                     2 Spherical Geometry                                       16
                    Part Two: Inversive Geometry                                23
                     3 Inversion in Euclidean Plane                             23
                     4 M¨obius Transformations                                  31
                     5 Inversive Geometry and Complex Numbers                   37
                    Part Three: Projective Geometry                             45
                     6 Projective Space and Incidence                           45
                     7 Coordinates in Projective Geometry                       50
                     8 Cross-ratio and Projective Geometry of the Line          55
                     9 Conics                                                   64
                    Part Four: Models of Hyperbolic Geometry                    71
                    10 The Projective Model                                     72
                    11 The Poincar´e Models                                     79
                    12 The Hyperboloid Model                                    90
                    13 Equivalence of the Models                               102
                    Part Five: The Hyperbolic Plane                            109
                    14 Parallelism and Transformations                         109
                    15 Trigonometry and Applications                           117
                    16 Some Special Questions                                  121
                                                   1
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             2                     GABOR 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
             manyabstract 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.
             Ourtreatment of geometry will investigate these features in certain classical geome-
             tries. 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.
             Themostfamousexample 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.
                                      NOTES ON NON-EUCLIDEAN GEOMETRIES                               3
                  Historically, 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 out that such a proof is impossible. Its negation, taken as a substitute
                  of the parallel 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 com-
                  pletely 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 non-
                     congruent 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 differ-
                     ently) 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
                  Ourmaingoalistolearnthemethods,formalism,andmathematicalmachinerythat
                  is necessary to understand various geometric systems as mathematical structures.
                                                 ´
                 4                             GABOR 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 two parts of these notes basically stay in the realm of Euclidean geome-
                 try. As a warmup, concepts of affine geometry and spherical geometry are briefly
                 introduced. These two chapters cover some background material for later use. At
                 the same time these two types of geometry may themselves be regarded as easy
                 examples of non-Euclidean geometry. The second part introduces and discusses
                 inversive geometry. By and large, this is still done within the realm of Euclidean
                 geometry, but the main theme here is a detailed analysis of a type of transforma-
                 tions (namely, M¨obius transformations) which play a determining role in one of the
                 classical models of hyperbolic geometry. The third part covers projective geometry
                 which, besides being an interesting, entertaining, classical piece of mathematics on
                 its own right, is a quintessentially non-Euclidean geometric system, and also serves
                 as technical background for hyperbolic geometry.
                 The fourth and fifth parts of these notes are devoted to hyperbolic geometry. In
                 part four hyperbolic plane is presented through three types of models, each using
                 a different type of mathematical apparatus. In the last part various topics in hy-
                 perbolic geometry are discussed which, by viewing it from several different aspects,
                 emphasize the non-Euclidean character of this geometry.
                 All these types of geometries could be worked out in arbitrary dimensions using
                 essentially the same methods. For simplicity, we restrict ourselves to the two-
                 dimensional case, that is, we only work with the affine plane, the two-dimensional
                 sphere, the inversive plane, the projective plane, and the hyperbolic plane. What
                 makes these geometric systems non-Euclidean, and mathematically interesting, is
                 well detectable already in the two-dimensional setting.