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Journal for Geometry and Graphics 20 (2016), No. 1, 041--062
Copyright Heldermann Verlag 2016



Angular Coordinates and Rational Maps

Thomas D. Maienschein
Dept. of Mathematics and Computer Science, Drake University, 2507 University Ave., Des Moines, IA 50311, U.S.A.
thomas.maienschein@drake.edu

Michael Q. Rieck
Dept. of Mathematics and Computer Science, Drake University, 2507 University Ave., Des Moines, IA 50311, U.S.A.
michael.rieck@drake.edu



Associated with a triangle in the real projective plane are three standard transformations: inversion in the circumcircle, isogonal conjugation and antigonal conjugation. These are investigated in terms of angular and related coordinates, and are found to be part of a group of more general transformations. This group can be identified with a group of automorphisms of a real two-torus. The torus is in essence the surface obtained by starting with the projective plane, performing blowups on the three vertices, and then collapsing the triangle's circumcircle and the line at infinity. A conjecture concerning Hofstadter points is proved as an immediate consequence of this viewpoint.

Keywords: Trilinear coordinates, angular coordinates, birational transformation, group action, triangle center.

MSC: 51N20; 14E05, 20B27, 14P25

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