In
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For a ...
, an ideal point, omega point or point at infinity is a
well-defined
In mathematics, a well-defined expression or unambiguous expression is an expression (mathematics), expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', ill defined ...
point outside the hyperbolic plane or space.
Given a line ''l'' and a point ''P'' not on ''l'', right- and left-
limiting parallel
In neutral or absolute geometry, and in hyperbolic geometry, there may be many lines parallel to a given line l through a point P not on line R; however, in the plane, two parallels may be closer to l than all others (one in each direction of R). ...
s to ''l'' through ''P''
converge
Converge may refer to:
* Converge (band), American hardcore punk band
* Converge (Baptist denomination), American national evangelical Baptist body
* Limit (mathematics)
In mathematics, a limit is the value that a function (or sequence) app ...
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 Cayley may refer to:
__NOTOC__ People
* Cayley (surname)
* Cayley Illingworth (1759–1823), Anglican Archdeacon of Stow
* Cayley Mercer (born 1994), Canadian women's ice hockey player
Places
* Cayley, Alberta, Canada, a hamlet
** Cayley/A. J. ...
or boundary of a
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For a ...
.
For instance, the
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
forms the Cayley absolute of the
Poincaré disk model
In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk t ...
and the
Klein disk model.
The real line forms the Cayley absolute of the
Poincaré half-plane model
In non-Euclidean geometry, the Poincaré half-plane model is a way of representing the hyperbolic plane using points in the familiar Euclidean plane. Specifically, each point in the hyperbolic plane is represented using a Euclidean point with co ...
.
Pasch's axiom and the
exterior angle theorem
The exterior angle theorem is Proposition 1.16 in Euclid's Elements, which states that the measure of an exterior angle of a triangle is greater than either of the measures of the remote interior angles. This is a fundamental result in absolute ge ...
still 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 of
horocycles and
horoballs are ideal points; two
horocycles are
concentric
In geometry, two or more objects are said to be ''concentric'' when they share the same center. Any pair of (possibly unalike) objects with well-defined centers can be concentric, including circles, spheres, regular polygons, regular polyh ...
when they have the same centre.
Polygons with ideal vertices
Ideal triangles
if all vertices of a
triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
are ideal points the triangle is an
ideal triangle
In hyperbolic geometry an ideal triangle is a hyperbolic triangle whose three vertices all are ideal points. Ideal triangles are also sometimes called ''triply asymptotic triangles'' or ''trebly asymptotic triangles''. The vertices are sometime ...
.
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
where K is the (always negative) curvature of the plane.
Ideal quadrilaterals
if all vertices of a
quadrilateral
In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...
are 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
where K is the (always negative) curvature of the plane.
Ideal square
The ideal quadrilateral where the two diagonals are
perpendicular
In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', � ...
to each other form an ideal square.
It was used by
Ferdinand Karl Schweikart
Ferdinand Karl Schweikart (1780–1857) was a German jurist and amateur mathematician who developed an ''astral geometry'' before the discovery of non-Euclidean geometry.
Life and work
Schweikart, son of an attorney in Hesse, was educated in th ...
in his memorandum on what he called "astral geometry", one of the first publications acknowledging the possibility of
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For a ...
.
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
In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk t ...
of the hyperbolic plane the ideal points are on the
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
(hyperbolic plane) or
unit sphere
In mathematics, a unit sphere is a sphere of unit radius: the locus (mathematics), set of points at Euclidean distance 1 from some center (geometry), center point in three-dimensional space. More generally, the ''unit -sphere'' is an n-sphere, -s ...
(higher dimensions) 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
In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk t ...
both 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 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
Poincaré half-plane model
In non-Euclidean geometry, the Poincaré half-plane model is a way of representing the hyperbolic plane using points in the familiar Euclidean plane. Specifically, each point in the hyperbolic plane is represented using a Euclidean point with co ...
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 (but rays parallel to the positive y-axis approach it).
Hyperboloid model
In the
hyperboloid model
In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of ''n''-dimensional hyperbolic geometry in which points are represented by points on the forward sheet ''S''+ of a two-sheeted hyperboloi ...
there are no ideal points.
See also
*
Ideal triangle
In hyperbolic geometry an ideal triangle is a hyperbolic triangle whose three vertices all are ideal points. Ideal triangles are also sometimes called ''triply asymptotic triangles'' or ''trebly asymptotic triangles''. The vertices are sometime ...
*
Ideal polyhedron
In three-dimensional hyperbolic geometry, an ideal polyhedron is a convex polyhedron all of whose vertices are ideal points, points "at infinity" rather than interior to three-dimensional hyperbolic space. It can be defined as the convex hull o ...
*
Points at infinity
In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.
In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. Ad ...
for uses in other geometries.
References
{{reflist
Hyperbolic geometry
Infinity