Perfect triangle

Three perfect triangles in the Poincare model in a circle

Two perfect triangles in the Poincare model in the upper half-plane

An ideal triangle is a triangle in the Lobachevsky geometry , all three vertices of which are ideal , or infinitely distant, points. Ideal triangles are sometimes called triple asymptotic triangles . Their peaks are sometimes called ideal peaks . All perfect triangles are equal.

Content

Properties

Ideal triangles have the following properties:

All ideal triangles are equal to each other.
All internal angles of an ideal triangle are equal to zero.
A perfect triangle has an infinite perimeter.
An ideal triangle is the largest possible triangle in Lobachevsky geometry.

In the standard Lobachevsky plane (a surface where the Gauss curvature is constant and equal to −1), an ideal triangle also has the following properties:

The area of such a triangle is π. ^[one]

Dimensions related to an ideal triangle and its incircle, depicted in the Beltrami – Klein model (left) and the Poincaré disk model (right)

The radius of the inscribed circle is $r=\ln {\sqrt {3}}={\frac {1}{2}}\ln 3\approx 0.549$ ${\ displaystyle r = \ ln {\ sqrt {3}} = {\ frac {1} {2}} \ ln 3 \ approx 0.549}$ . ^[2]

The distance from any point of the triangle to its nearest side is less than or equal to the above radius, moreover, this equality is true only in the center of the inscribed circle.

The inscribed circle touches the triangle at three points, forming an equilateral triangle with side $d=\ln \left({\frac {{\sqrt {5}}+1}{{\sqrt {5}}-1}}\right)=2\ln \varphi \approx 0.962$ ${\ displaystyle d = \ ln \ left ({\ frac {{\ sqrt {5}} + 1} {{\ sqrt {5}} - 1}} \ right) = 2 \ ln \ varphi \ approx 0.962}$ ^[2] where $\varphi ={\frac {1+{\sqrt {5}}}{2}}$ ${\ displaystyle \ varphi = {\ frac {1 + {\ sqrt {5}}} {2}}}$ there is a golden ratio .

A circle with a radius d around a point inside the triangle will touch or intersect at least two sides of the triangle.

The distance from any point on the side of such a triangle to the other side is less than or equal to $a=\ln \left(1+{\sqrt {2}}\right)\approx 0.881$ ${\ displaystyle a = \ ln \ left (1 + {\ sqrt {2}} \ right) \ approx 0.881}$ , moreover, exactly the equality is fulfilled only for the points of contact described above.

a is also the height of the Schweikart triangle.

If the curvature of the space is -K different from −1, the areas above should be multiplied by $1/K$ ${\ displaystyle 1 / K}$ , and lengths and distances - by $1/{\sqrt {K}}$ ${\ displaystyle 1 / {\ sqrt {K}}}$ .

The position of a δ-thin triangle on a δ-hyperbolic space

Since an ideal triangle is the greatest possible in Lobachevsky geometry, the above values are the greatest possible for triangles in Lobachevsky geometry. This fact is important for studying the Lobachevsky space.

Models

In the Poincaré model in a circle of the Lobachevsky plane, an ideal triangle is formed by three circles intersecting the boundary circle at right angles.

In the Poincare model in the half-plane, the ideal triangle looks like an arbelos - a figure between three contiguous semicircles.

In the projective model, an ideal triangle is a Euclidean triangle inscribed in a boundary circle. Moreover, on the projective model, the angles at the vertices of an ideal triangle are not equal to zero, since this model, unlike the Poincare models, does not preserve angles.

Real group of an ideal triangle

Poincare model paved with perfect triangles
The ideal (∞ ∞ ∞) group of a triangle	Another perfect tiling

A real group of an ideal triangle is a group of transformations generated by reflections of the Lobachevsky plane with respect to the sides of an ideal triangle. As an abstract group, it is isomorphic to the free product of three groups of two elements. As a result of the reflections, the tiling of the Lobachevsky plane by ideal triangles is obtained.

Links

↑ Thurston, Dylan. 274 Curves on Surfaces, Lecture 5 (neopr.) (Fall 2012). Date of treatment July 23, 2013.
↑ ¹ ² What is the radius of the inscribed circle of an ideal triangle (neopr.) . Date of treatment December 9, 2015.

Bibliography

Schwartz, Richard Evan. Ideal triangle groups, dented tori, and numerical analysis (Eng.) // Annals of Mathematics : journal. - 2001. - Vol. 153 , no. 3 . - P. 533-598 . - DOI : 10.2307 / 2661362 . - arXiv : math.DG / 0105264 .

[Thurston_2012-1] Thurston, Dylan. 274 Curves on Surfaces, Lecture 5 (neopr.) (Fall 2012). Date of treatment July 23, 2013.

[MSE1566126-2] ¹ ² What is the radius of the inscribed circle of an ideal triangle (neopr.) . Date of treatment December 9, 2015.