case); points move on circles around the center; three degrees of freedom. "rotation" around an ideal point (horolation) — two reflections through lines leading to the ideal point; points move along horocycles centered on the ideal point; two degrees of freedom. translation along a straight line — two reflections through lines perpendicular to the given line; points off the given line move along hypercycles; three degrees of freedom. Orientation reversing reflection through a line — one reflection; two degrees of freedom. combined reflection through a line and translation along the same line — the reflection and translation commute; three reflections required; three degrees of freedom. Introduction of metric in the Poincaré half-plane model The points of the Poincaré 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. Use of semi-circle Z Consider the triangle {(0,0),(1,0),(1,tan a)}. Since 1 + tan2a = sec2a, the length of the triangle hypotenuse is sec a, where sec denotes the secant function. Set r = sec a and apply the third fundamental hyperbolic motion to obtain q = (r cos a, r sin a) where r = sec−1a = cos a. Now |q – (½, 0)|2 = (cos2a – ½)2 +cos2a sin2a = ¼ so that q lies on the semicircle Z of radius ½ and center (½, 0). Thus the tangent ray at (1, 0) gets mapped to Z by the third fundamental hyperbolic motion. Any semicircle can be re-sized by a dilation to radius ½ and shifted to Z, then the inversion carries it to the tangent ray. So the collection of hyperbolic motions permutes the semicircles with diameters on y = 0 sometimes with vertical