Triangle center Since rotations and translations may be regarded as double reflections they too must preserve triangle centers. These invariance properties provide justification for the definition. Some other names for dilation are uniform scaling, isotropic scaling, homothety, and homothecy. The study of triangle centers traditionally is concerned with Euclidean geometry, but triangle centers can also be studied in hyperbolic geometry. Using gyrotrigonometry, expressions for trigonometric barycentric coordinates can be calculated that have the same form for both Euclidean and hyperbolic geometry. In order for the expressions to coincide, the expressions must "not" encapsulate the specification of the angle sum being 180 degrees.
Schwarz triangle function since the case of two cusps is obtained by doubling a triangle with one cusp and non-zero angles /"a" and /2. So consider the geodesic triangle in the upper half plane with angles 0, /"a", /"b" with "a", "b" integers greater than one. The interior of such a triangle can be realised as the region "X" in the upper half plane lying outside the unit disk |"z"| ≤ 1 and between two lines parallel to the imaginary axis through points "u" and "v" on the unit circle. Let Γ be the triangle group generated by the three reflections in the