![]() At most two edges can meet at any one point 1. All glide reflections with the same translation length are conjugate to one another. A hyperbolic polygon is a region of the hyperbolic plane whose boundary is decomposed into finitely many generalized line segments (recall that this includes circle segments), called edges, meeting only at their endpoints, called vertices. Just like with hyperbolic isometries, a glide reflection has exactly two fixed points, namely the endpoints at infinity of the geodesic corresponding to the reflection and translation.Īlso like hyperbolic isometries, a glide reflection is determined by (1) its two fixed points, or equivalently the geodesic it fixes, (2) a direction, or orientation of the geodesic it fixes, and (3) a positive translation length. This is the sense in which one could classify orientation-reversing isometries with just one type.) (If we were to allow translations of length zero, then a reflection would be a type of glide reflection. Glide reflectionĪ glide reflection is an isometry that results from composing a reflection with a non-trivial translation (aka hyperbolic isometry) along the geodesic corresponding to the reflection. A reflection is uniquely determined by its geodesic, and every reflection is conjugate to every other one by an orientation-preserving isometry (so there is just one conjugacy class). Throughout this article we use \(H\) to denote the hyperbolic plane and \(\overline\) for a reflection is exactly the set of points making up the geodesic, including the geodesic's endpoints at infinity. The first row uses the Klein or projective model, and the second row the Poincaré disk model. ![]() This figure shows an animation of the three types of orientation-preserving isometries of the hyperbolic plane (from left to right): hyperbolic, elliptic, and parabolic. We discuss orientation-preserving isometries first after introducing some preliminaries. The isometries of the hyperbolic plane form a group under composition.Īn isometry of the hyperbolic plane can be either orientation-preserving or orientation-reversing. Algebraically, it is isomorphic to the free product of three order-two groups (Schwartz 2001).An isometry of the hyperbolic plane is a mapping of the hyperbolic plane to itself that preserves the underlying hyperbolic geometry (e.g. The real ideal triangle group is the reflection group generated by reflections of the hyperbolic plane through the sides of an ideal triangle. Real ideal triangle group The Poincaré disk model tiled with ideal triangles Note that in the Beltrami-Klein model, the angles at the vertices of an ideal triangle are not zero, because the Beltrami-Klein model, unlike the Poincaré disk and half-plane models, is not conformal i.e. In the Beltrami–Klein model of the hyperbolic plane, an ideal triangle is modeled by a Euclidean triangle that is circumscribed by the boundary circle. In the Poincaré half-plane model, an ideal triangle is modeled by an arbelos, the figure between three mutually tangent semicircles. In the Poincaré disk model of the hyperbolic plane, an ideal triangle is bounded by three circles which intersect the boundary circle at right angles. Thin triangle condition The δ-thin triangle condition used in δ-hyperbolic spaceīecause the ideal triangle is the largest possible triangle in hyperbolic geometry, the measures above are maxima possible for any hyperbolic triangle, this fact is important in the study of δ-hyperbolic space. If the curvature is − K everywhere rather than −1, the areas above should be multiplied by 1/ K and the lengths and distances should be multiplied by 1/ √ K. R = ln 3 = 1 2 ln 3 = artanh 1 2 = 2 artanh ( 2 − 3 ) =, with equality only for the points of tangency described above.Ī is also the altitude of the Schweikart triangle.
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