Proof of Theorem 1

Theorem 1 If t is a hyperbolic transformation, then
  • t maps h-lines to h-lines,
  • t preserves angles between h-lines.

Since H(2) is generated by h-inversions, it is enough to consider the case
where t is an h-inversion.

An h-inversion t is the restriction of an inversion t*.

Suppose that H is an h-line.
Then H = LnD, where L is an i-line orthogonal to C.
Since t* is an inversion, it maps the i-line L to an i-line L*
As inversion preserves angles, L* is orthogonal to C,
so that t(H) = L*nD is an h-line.

Also, as t* is an inversion, it preserves angles. It follows that the
restriction t preserves angles at points within D.

return to hyperbolic inversion page