The triangle group is the finite symmetry group of a tiling of a unit sphere by spherical triangles, or M�bius triangles, whose angles add up to a number greater than ?.
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Its symmetry group is the quotient of the spherical triangle group by the reflection through the origin (-" I " ), which is a central element of order 2.
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If a number of great circles are drawn onto the surface of a sphere then the surface is partioned into a number of shapes, which may be spherical triangles or other spherical polygons.
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In addition to Indian works, Hellenistic methods dealing with spherical triangles were also known, particularly the method of Menelaus of Alexandria, who developed " Menelaus'theorem " to deal with spherical problems.
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It is easy to see how for small spherical triangles, when the radius of the sphere is much greater than the sides of the triangle, this formula becomes the planar formula at the limit, since
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In this area of celestial navigation, Dunthorne has been credited as the first to apply trigonometrical formulae for the general spherical triangle to the reduction of lunar distances and to give auxiliary tables for that purpose.
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Thus, the spherical triangle is defined not only by the three points located at the sun, zenith, and observed point but by both the three interior angles as well as the three angular distances.
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A hyperbolic triangle can be obtained by drawing on a negatively curved surface, such as a saddle surface, and a spherical triangle can be obtained by drawing on a positively curved surface such as a sphere.
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Then the angles of the spherical triangle exceed the corresponding angles of the planar triangle by approximately one third of the spherical excess ( the spherical excess is the amount by which the sum of the three angles exceeds ).
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Spherical triangle groups can be identified with the symmetry groups of regular " n "-gons joined together, or dually hosohedra, which are formed by joining " n " digons together at two vertices.
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