In
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P ...
an ideal triangle is a
hyperbolic triangle whose three vertices all are
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 '' ...
s. Ideal triangles are also sometimes called ''triply asymptotic triangles'' or ''trebly asymptotic triangles''. The vertices are sometimes called ideal vertices. All ideal triangles are
congruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
.
Properties
Ideal triangles have the following properties:
* All ideal triangles are congruent to each other.
* The interior angles of an ideal triangle are all zero.
* An ideal triangle has infinite perimeter.
* An ideal triangle is the largest possible triangle in hyperbolic geometry.
In the standard hyperbolic plane (a surface where the constant
Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point:
K = \kappa_1 \kappa_2.
The Gaussian radius of curvature is the reciprocal of .
...
is −1) we also have the following properties:
* Any ideal triangle has area π.
Distances in an ideal triangle
* The
inscribed circle
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incen ...
to an ideal triangle has radius
.
: The distance from any point in the triangle to the closest side of the triangle is less than or equal to the radius ''r'' above, with equality only for the center of the inscribed circle.
* The inscribed circle meets the triangle in three points of tangency, forming an equilateral
contact triangle with side length
where
is the
golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
.
: A circle with radius ''d'' around a point inside the triangle will meet or intersect at least two sides of the triangle.
* The distance from any point on a side of the triangle to another side of the triangle is equal or less than
, with equality only for the points of tangency described above.
:''a'' is also the
altitude
Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
of the
Schweikart triangle.
If the curvature is −''K'' everywhere rather than −1, the areas above should be multiplied by 1/''K'' and the lengths and distances should be multiplied by 1/.
Thin triangle condition
Because the ideal triangle is the largest possible triangle in hyperbolic geometry, the measures above are maxima possible for any
hyperbolic triangle, this fact is important in the study of
δ-hyperbolic space In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations (depending quantitatively on a nonnegative real number δ) between points. The definition, introduced by Mikhael Gromov, generalizes the metric properti ...
.
Models
In 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 ...
of the hyperbolic plane, an ideal triangle is bounded by three circles which intersect the boundary circle at right angles.
In the
Poincaré half-plane model
In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H = \, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry.
Equivalently the Poincar� ...
, an ideal triangle is modeled by an
arbelos, the figure between three mutually tangent
semicircle
In mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle. The full arc of a semicircle always measures 180° (equivalently, radians, or a half-turn). It has only one line o ...
s.
In the
Beltrami–Klein model of the hyperbolic plane, an ideal triangle is modeled by a Euclidean triangle that is
circumscribed by the boundary circle. Note that in the Beltrami-Klein model, the angles at the vertices of an ideal triangle are not zero, because the Beltrami-Klein model, unlike the Poincaré disk and half-plane models, is not
conformal i.e. it does not preserve angles.
Real ideal triangle group
The real ideal
triangle group
In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle can be an ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolic triangl ...
is the
reflection group In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space. The symmetry group of a regular polytope or of a tiling of the Euclidean space by congruent c ...
generated by reflections of the hyperbolic plane through the sides of an ideal triangle. Algebraically, it is isomorphic to the
free product
In mathematics, specifically group theory, the free product is an operation that takes two groups ''G'' and ''H'' and constructs a new The result contains both ''G'' and ''H'' as subgroups, is generated by the elements of these subgroups, and is ...
of three order-two groups (Schwartz 2001).
References
Bibliography
*
{{polygons
Hyperbolic geometry
Types of triangles