Hyperbolic Motion - Introduction of Metric in Upper Half-plane

Introduction of Metric in Upper Half-plane

The points of the upper half-plane model HP are given in Cartesian coordinates as {(x,y): y > 0} or in polar coordinates as {(r cos a, r sin a): 0 < a < π, r > 0 }.The hyperbolic motions will be taken to be a composition of three fundamental hyperbolic motions. Let p = (x,y) or p = (r cos a, r sin a), p ∈ HP. The fundamental motions are:

p → q = (x + c, y ), c ∈ R (left or right shift)

p → q = (sx, sy ), s > 0 (dilation)

p → q = ( r −1 cos a, r −1 sin a ) (inversion in unit semicircle).

Note: the shift and dilation are mappings from inversive geometry composed of a pair of reflections in vertical lines or concentric circles respectively.