Ideal point

This article is about ideal points in hyperbolic geometry. For similar points in other geometries, see Point at infinity. For the term as used in political science, see Spatial model of voting.
Three ideal triangles in the Poincaré disk model; the vertices are ideal points

In hyperbolic geometry, an ideal point, omega point[1] or point at infinity is a well-defined point outside the hyperbolic plane or space. Given a line l and a point P not on l, right- and left-limiting parallels to l through P converge to l at ideal points.

Unlike the projective case, ideal points form a boundary, not a submanifold. So, these lines do not intersect at an ideal point and such points, although well-defined, do not belong to the hyperbolic space itself.

The ideal points together form the Cayley absolute or boundary of a hyperbolic geometry. For instance, the unit circle forms the Cayley absolute of the Poincaré disk model and the Klein disk model. The real line forms the Cayley absolute of the Poincaré half-plane model.[2]

Pasch's axiom and the exterior angle theorem still hold for an omega triangle, defined by two points in hyperbolic space and an omega point.[3]

  1. ^ Sibley, Thomas Q. (1998). The geometric viewpoint : a survey of geometries. Reading, Mass.: Addison-Wesley. p. 109. ISBN 0-201-87450-4.
  2. ^ Struve, Horst; Struve, Rolf (2010), "Non-euclidean geometries: the Cayley-Klein approach", Journal of Geometry, 89 (1): 151–170, doi:10.1007/s00022-010-0053-z, ISSN 0047-2468, MR 2739193
  3. ^ Hvidsten, Michael (2005). Geometry with Geometry Explorer. New York, NY: McGraw-Hill. pp. 276–283. ISBN 0-07-312990-9.