Schwarz triangle function: Difference between revisions

Revision as of 20:08, 15 February 2017

Revision as of 20:20, 15 February 2017

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The construction of a tessellation will first be carried out for the case when ''a'', ''b'' and ''c'' are greater than 2.<ref> {{harvnb|Caratheodory|1954|pages=177–181}}</ref>

The equality above for ''a'', ''b'' and ''c'' implies that if one of the angles is a right angle, say ''a'' = 2, then both ''b'' and ''c'' are greater than 2 and one of them, ''b'' say, must be greater than 3. In this case, reflecting the triangle across the side AB gives an isosceles hyperbolic triangle with angles {{pi}}/''c'', {{pi}}/''c'' and 2{{pi}}/''b''. The construction of the tessellation thatabove isthrough givenincreasing belowconvex appliespolygons equallyadapts wellword infor word to this case except that around the vertex with angle 2{{pi}}/''b'', only ''b''—and not 2''b''—copies of the triangle are required to tile a neighborhood of the vertex. This is possible because the doubled triangle is isosceles.<ref>{{harvnb|Caratheodory|1954|pages=181–182}}</ref>

===Triangles with one or two cusps===