Ideal triangle

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Three ideal triangles in the Poincaré disk model

In hyperbolic geometry an ideal triangle is a hyperbolic triangle whose three vertices all lie on the circle at infinity. In the Poincaré disk model an ideal triangle is bounded by three circles which intersect the boundary circle at right angles. In the hyperbolic metric any two ideal triangles are congruent. Ideal triangles are also sometimes called triply-asymptotic triangles or trebly-asymptotic triangles.

[edit] Properties

The interior angles of an ideal triangle are all zero.
Any ideal triangle has area π and infinite perimeter.
The inscribed circle to an ideal triangle meets the triangle in three points of tangency, forming an equilateral triangle with side length

$4\ln\frac{1+\sqrt 5}{2}\approx 1.925.$ [1]

Any point in the triangle is within constant distance of some two sides of the triangle.

[edit] Real ideal triangle group

The real ideal triangle group is the reflection group generated by reflections of the hyperbolic plane through the sides of an ideal triangle. Algebraically, it is isomorphic to the free product of three order-two groups (Schwarz 2001).

[edit] References

Schwartz, Richard Evan (2001). "Ideal triangle groups, dented tori, and numerical analysis". Ann. of Math., Ser. 2 153 (3): 533–598. arXiv:math.DG/0105264.