Point at Infinity - Hyperbolic Geometry

Hyperbolic Geometry

In hyperbolic geometry, an ideal point is also called an omega point. Given a line l and a point P not on l, right- and left-limiting parallels to l through P are said to meet l at omega points. Unlike the projective case, omega points form a boundary, not a submanifold. So, these lines do not intersect at an omega point and such points, although well defined, do not belong to a hyperbolic space itself. In the Poincaré disk model and the Klein model of hyperbolic geometry, the omega points can be visualized since they lie on the boundary circle (which is not part of the model). 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.