Hyperbolic motion

Such an approach to geometry was cultivated by Felix Klein in his Erlangen program.

One of the most prevalent contexts for inversive geometry and hyperbolic motions is in the study of mappings of the complex plane by Möbius transformations.

Every motion (transformation or isometry) of the hyperbolic plane to itself can be realized as the composition of at most three reflections.

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

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 rays, and vice versa.

For m and n in HP, let b be the perpendicular bisector of the line segment connecting m and n. If b is parallel to the abscissa, then m and n are connected by a vertical ray, otherwise b intersects the abscissa so there is a semicircle centered at this intersection that passes through m and n. The set HP becomes a metric space when equipped with the distance d(m,n) for m,n ∈ HP as found on the vertical ray or semicircle.

A complex matrix with aa* − bb* = 1, which is an element of the special unitary group SU(1,1).