Omega point (geometry)

In hyperbolic geometry, an ideal point, omega point 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. While the real line forms the Cayley absolute of the Poincaré half-plane 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.

if all vertices of a triangle are ideal points the triangle is an ideal triangle.

Ideal triangles have a number of interesting properties:

if all vertices of a quadrilateral are ideal points the quadrilateral is an ideal quadrilateral.

While all ideal triangles are congruent, not all quadrilaterals are, the diagonals can make different angles with each other resulting in noncongruent quadrilaterals having said this:

The ideal quadrilateral where the two diagonals are perpendicular to each other form an ideal square.

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