Geometry of Möbius Transformations

The following sections are included:

• Preface

• Contents

• List of Figures

The simplest objects with non-commutative (but still associative) multiplication may be 2 × 2 matrices with real entries. The subset of matrices of determinant one has the following properties:

• It is a closed set under multiplication since det(AB) = det A·det B.

• The identity matrix is the set.

• Any such matrix has an inverse (since det A ≠ 0).

Group theory and representation theory are themselves two enormous and interesting subjects. However, they are auxiliary in our consideration and we are forced to restrict our consideration to a brief overview.

Now we adopt the previous theoretical constructions for the particular case of the group SL₂(ℝ). We are going to describe all homogeneous spaces SL₂(ℝ)/H, where H is a subgroup of SL₂(ℝ), see Section 2.2.2. To begin with, we start from the two-dimensional subgroup.

Cycles — circles, parabolas and hyperbolas — are invariant families under their respective Möbius transformations. We will now proceed with a study of the invariant properties of cycles according to the Erlangen programme. A very powerful tool used in these notes is the representation of cycles by appropriate 2 × 2 matrices.

In the previous chapter, we represented cycles by points of the projective space ℙ³ or by lines of 2 × 2 matrices. The latter is justified so far only by the similarity formula (4.7). Now, we will investigate connections between cycles and vector space structure. Thereafter we will use the special form of FSCc matrices to introduce very important additional structure in the cycle space.

The invariant cycle product, defined in the previous chapter, allows us to define joint invariants of two or more cycles. Being initially defined in an algebraic fashion, they also reveal their rich geometrical content. We will also see that FSCc matrices define reflections and inversions in cycles, which extend Möbius maps.

So far, we have discussed only invariants like orthogonality, which are related to angles. However, geometry, in the plain meaning of the word, deals with measurements of distances and lengths. We will derive metrical quantities from cycles in a way which shall be Möbius-invariant.

So far, we have been interested in the individual properties of cycles and (relatively) localised properties of the point space. We now describe some global properties which are related to the set of cycles as a whole.

The Euclidean metric is not preserved by automorphisms of the Lobachevsky half-plane. Instead, one has only a weaker property of conformality. However, it is possible to find such a metric on the Lobachevsky half-plane that Möbius transformations will be isometries. Similarly, in Chapter 7, we described a variety of distances and lengths and many of them had conformal properties with respect to SL₂(ℝ) action. However, it is worth finding a metric which is preserved by Möbius transformations. We will now proceed do this, closely following [56].

The upper half-plane is a universal object for all three EPH cases, which was obtained in a uniform fashion considering two-dimensional homogeneous spaces of SL₂(ℝ). However, universal models are rarely well-suited to all circumstances. For example, it is more convenient, for many reasons, to consider the compact unit disk in ℂ rather than the unbounded upper half-plane:

… the reader must become adept at frequently changing from one model to the other as each has its own particular advantage. [5, Sec. 7.1]

Of course, both models are conformally-isomorphic through the Cayley transform.

One of the important advantages of the elliptic and hyperbolic unit disks introduced in Sections 10.1—10.2 is a simplification of isotropy subgroup actions. Indeed, images of the subgroups K and A’, which fix the origin in the elliptic and hyperbolic disks, respectively, consist of diagonal matrices — see (10.4) and (10.10). These diagonal matrices produce Möbius transformations, which are multiplications by hypercomplex unimodular numbers and, thus, are linear. In this chapter, we discuss the possibility of similar results in the parabolic unit disks from Section 10.3.

The following sections are included:

• Appendix A Supplementary Material
  • Dual and Double Numbers
  • Classical Properties of Conic Sections
  • Comparison with Yaglom’s Book
  • Other Approaches and Results
  • FSCc with Clifford Algebras

• Appendix B How to Use the Software
  • Viewing Colour Graphics
  • Installation of CAS
  • Using the CAS and Computer Exercises
  • Library for Cycles

• Bibliography

• Index

The cover of this book depicts the painting ‘A Vityaz at the Crossroads’ (1882) by V.M. Vasnetsov. It is based on the archetypal situation from Russian epics: a vityaz (knight) on a mission arrives at a crossroads with a telling stone. The scripture on the stone gives directions:

Continue straight and death will greet you, with no way to pass, drive or fly. Turn to the left and you will become rich. Turn to the right and you will be married.