Ideal point

This article is about ideal points inhyperbolic geometry.For similar points in other geometries, seePoint at infinity.For the term as used in political science, seeSpatial model of voting.

Inhyperbolic geometry,anideal point,omega pointorpoint at infinityis awell-definedpoint outside the hyperbolic plane or space. Given a lineland a pointPnot onl,right- and left-limiting parallelstolthroughPconvergetolatideal points.

Threeideal trianglesin thePoincaré disk model;theverticesare ideal points

Unlike the projective case, ideal points form aboundary,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 theCayley absoluteor boundary of ahyperbolic geometry. For instance, theunit circleforms the Cayley absolute of thePoincaré disk modeland theKlein disk model. The real line forms the Cayley absolute of thePoincaré half-plane model.

Pasch's axiomand theexterior angle theoremstill hold for an omega triangle, defined by two points in hyperbolic space and an omega point.

Properties

• The hyperbolic distance between an ideal point and any other point or ideal point is infinite.
• The centres ofhorocyclesandhoroballsare ideal points; twohorocyclesareconcentricwhen they have the same centre.

Polygons with ideal vertices

Ideal triangles

Main article:Ideal triangle

if all vertices of atriangleare ideal points the triangle is anideal triangle.

Some properties of ideal triangles include:

• All ideal triangles are congruent.
• The interior angles of an ideal triangle are all zero.
• Any ideal triangle has an infinite perimeter.
• Any ideal triangle has area$-\pi /K$where K is the (always negative) curvature of the plane.

Ideal quadrilaterals

if all vertices of aquadrilateralare ideal points, the quadrilateral is an ideal quadrilateral.

While all ideal triangles are congruent, not all convex ideal quadrilaterals are. They can vary from each other, for instance, in the angle at which their two diagonals cross each other. Nevertheless all convex ideal quadrilaterals have certain properties in common:

• The interior angles of a convex ideal quadrilateral are all zero.
• Any convex ideal quadrilateral has an infinite perimeter.
• Any convex ideal quadrilateral has area$-2\pi /K$where K is the (always negative) curvature of the plane.

Ideal square

The ideal quadrilateral where the two diagonals areperpendicularto each other form an ideal square.

It was used byFerdinand Karl Schweikartin his memorandum on what he called "astral geometry", one of the first publications acknowledging the possibility ofhyperbolic geometry.

Idealn-gons

An idealn-gon can be subdivided into(n− 2)ideal triangles, with area(n− 2)times the area of an ideal triangle.

Representations in models of hyperbolic geometry

In theKlein disk modeland thePoincaré disk modelof the hyperbolic plane the ideal points are on theunit circle(hyperbolic plane) orunit sphere(higher dimensions) which is the unreachable boundary of the hyperbolic plane.

When projecting the same hyperbolic line to theKlein disk modeland thePoincaré disk modelboth lines go through the same two ideal points (the ideal points in both models are on the same spot).

Klein disk model

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

$d(p,q)={\frac {1}{2}}\log {\frac {\left|qa\right|\left|bp\right|}{\left|pa\right|\left|bq\right|}},$

Poincaré disk model

Given two distinct pointspandqin the open unit disk then the unique circlearcorthogonal to the boundary connecting them intersects the unit circle in two ideal points,aandb,labeled so that the points are, in order,a,p,q,bso that |aq| > |ap| and |pb| > |qb|. Then the hyperbolic distance betweenpandqis expressed as

$d(p,q)=\log {\frac {\left|qa\right|\left|bp\right|}{\left|pa\right|\left|bq\right|}},$

Where the distances are measured along the (straight line) segments aq, ap, pb and qb.

Poincaré half-plane model

In thePoincaré half-plane modelthe ideal points are the points on the boundary axis. There is also another ideal point that is not represented in the half-plane model (but rays parallel to the positive y-axis approach it).

Hyperboloid model

In thehyperboloid modelthere are no ideal points.

See also

• Ideal triangle
• Ideal polyhedron
• Points at infinityfor uses in other geometries.