Section 4.2 Möbius Geometry
Definition 4.2.1.
Möbius Geometry is the geometry (C+,M), where M denotes the group of all Möbius transformations.
Any two clines are congruent in Möbius Geometry (Theorem 3.4.15).
The set of all clines is a minimally invariant set of Möbius Geometry (Theorems 3.4.5 and Theorem 3.4.15).
The cross ratio is an invariant of Möbius Geometry (Theorem 3.4.12).
Angle measure is an invariant of Möbius Geometry (Theorem 3.4.5).
Any transformation in M is uniquely determined by the image of three points.
If T in M is not the identity map, then T fixes exactly 1 or 2 points.
Möbius transformations preserve symmetry points.
Exercises Exercises
1.
Which figures in Figure 4.2.2 are congruent in (C+,M).
\(A \cong F\text{;}\) \(B \cong G\text{;}\) \(C \cong E\text{.}\)
2.
Describe a minimally invariant set in (C+,M) containing the “triangle” comprised of the three vertices 0, 1, and i and the three Euclidean line segments connecting them. Be as specific as possible about the members of this set.
3.
Suppose p and q are distinct, finite points in C+. Let G consist of all elliptic Möbius transformations that fix p and q. We consider the geometry (C+,G).
- Show that G is a group of transformations.
- Determine a minimally invariant set in (C+,G) that contains the Euclidean line through p and q.
- Determine a minimally invariant set in (C+,G) that contains the perpendicular bisector of segment pq.
- For any point z≠p,q in C+, characterize all points in C+ congruent to z.
- Is (C+,G) homogeneous?
4.
Repeat the previous exercise for the set G consisting of all hyperbolic Möbius transformations that fix p and q.