Rigid Motions
Let be a hyperbolic plane and H its field of ends, as introduced above. In the plane, we have rigid motions and their effects on ends as follows:
- The reflection in sends to −x.
- The reflection in (1, −1) gives,
- Translation along that sends 1 to any, a > 0 is represented by
- For any, there is a rigid motion σ(1/2)a σ0, the composition of reflection in the line and reflection in the line, which is called rotation around is given by
- The rotation around the point O, which sends 0 to any given end, effects as
- on ends. The rotation around O sending 0 to gives
For a more extensive treatment than this article can give, confer.
Read more about this topic: Hilbert's Arithmetic Of Ends
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