Schwarz triangle function: Difference between revisions

Revision as of 16:41, 16 February 2017

Revision as of 16:43, 16 February 2017

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Before proving (c) and (b), a Möbius transformation can be applied to map the upper half plane to the unit disk and a fixed point in the interior of Δ to the origin.

Then to prove (c), note that the radius joining the origin to a vertex of the polygon ''P''_{''n'' − 1} makes an angle of less than 2{{pi}}/3 with each of the edges of the polygon at that vertex if exactly two triangles of ''P''_{''n'' − 1} meet at the vertex, since each has an angle less than {{pi}}/3 at that vertex. To check this is true when three triangles of ''P''_{''n'' − 1} meet at the vertex, that ''C'' say, suppose that the middle triangle has its base on a side ''AB'' of ''P''_{''n'' − 2}. By induction the radii ''OA'' and ''OB'' makes angles of less than 2{{pi}}/3 with the edge ''AB''. In this case the triangle ''ABC'' lies inside the sector between the radii ''OA'' and ''OB''.

By convexity the new triangle added on an edge of ''P''_{''n'' − 1} lies inside a sector emanating from the origin and going through the endpoints of the edge. Since these open sectors are disjoint, new triangles of this type only intersect as claimed.