Ideal point


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.

Properties

Ideal triangles

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.
It was use by Ferdinand Karl Schweikart in his memorandum on what he called "astral geometry", one of the first publications acknowledging the possibility of hyperbolic geometry.

Ideal ''n''-gons

An ideal n-gon can be subdivided into ideal triangles, with area times the area of an ideal triangle.

Representations in models of hyperbolic geometry

In the Klein disk model and the Poincaré disk model of the hyperbolic plane. In both disk models the ideal points are on the unit circle or unit sphere which is the unreachable boundary of the hyperbolic plane.

When projecting the same hyperbolic line to the Klein disk model and the Poincaré disk model both lines go through the same two ideal points..

Klein disk model

Given two distinct points p and q in the open unit disk the unique straight line connecting them intersects the unit circle in two ideal points, a and b, labeled so that the points are, in order, a, p, q, b so that |aq| > |ap| and |pb| > |qb|. Then the hyperbolic distance between p and q is expressed as

Poincaré disk model

Given two distinct points p and q in the open unit disk then the unique circle arc orthogonal to the boundary connecting them intersects the unit circle in two ideal points, a and b, labeled so that the points are, in order, a, p, q, b so that |aq| > |ap| and |pb| > |qb|. Then the hyperbolic distance between p and q is expressed as
Where the distances are measured along the segments aq, ap, pb and qb.

Poincaré half-plane model

In the Poincaré half-plane model the ideal points are the points on the boundary axis. There is also another ideal point that is not represented in the half-plane model.

Hyperboloid model

In the hyperboloid model there are no ideal points.