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 .
- The hyperbolic distance between an ideal point and any other point or ideal point is infinite.
- The centres of horocycles and horoballs are ideal points; two horocycles are concentric when they have the same centre.
Polygons with ideal vertices
Ideal triangles have a number of interesting properties:
- 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 where K is the (negative) curvature of the plane.
if all vertices of a quadrilateral are ideal points the triangle 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 interior angles of an ideal quadrilateral are all zero.
- Any ideal quadrilateral has an infinite perimeter.
- Any ideal quadrilateral has area where K is the (negative) curvature of the plane.
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 posssibility of hyperbolic geometry.
Ideal n- polygons
As n- polygons can be subdivided in (n-2) ideal triangles their area is (n-2) times the area of an ideal triangle.
Representations in models of hyperbolic geometry
Klein disk model
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 (hyperbolic plane) or unit sphere (higher dimensions) which is the unreachable boundary of the hyperbolic plane.
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 (straight line) segments aq, ap, pb and qb.
Poincaré half-plane model
In the hyperboloid model there are no ideal points.
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