Ideal vertex
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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 In hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane. It consists of three line segments called ''sides'' or ''edges'' and three points called ''angles'' or ''vertices''. Just as in the Euclidean case, three poi ...
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 ''P' ...
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 . F ...
is −1) we also have the following properties: * Any ideal triangle has area π.


Distances in an ideal triangle

* The inscribed circle to an ideal triangle has radius r=\ln\sqrt = \frac \ln 3 = \operatorname\frac = 2 \operatorname(2- \sqrt) = = \operatorname\frac\sqrt = \operatorname\frac\sqrt \approx 0.549 . : 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 d = \ln\left(\frac\right)= 2\ln\varphi\approx 0.962 where \varphi=\frac 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 a = \ln\left(1+ \sqrt 2\right) \approx 0.881, 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 In hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane. It consists of three line segments called ''sides'' or ''edges'' and three points called ''angles'' or ''vertices''. Just as in the Euclidean case, three poi ...
, 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 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, an ideal triangle is modeled by an
arbelos In geometry, an arbelos is a plane region bounded by three semicircles with three apexes such that each corner of each semicircle is shared with one of the others (connected), all on the same side of a straight line (the ''baseline'') that conta ...
, 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 In geometry, the Beltrami–Klein model, also called the projective model, Klein disk model, and the Cayley–Klein model, is a model of hyperbolic geometry in which points are represented by the points in the interior of the unit disk (or ''n'' ...
of the hyperbolic plane, an ideal triangle is modeled by a Euclidean triangle that is
circumscribed In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...
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