Angular coordinates and rational maps (Q2820330)

The summary of the paper yields the following: ``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.''NEWLINENEWLINE After introducing in Sections 1 and 2 preliminaries and the concepts of angular and atricyclic coordinates, one finds in Section 3, using the authors' ideas, the observation of a result, rather rapidly proven by them, of an older result of \textit{D. M. Bailey} [Math. Mag. 33, 241--259 (1960; Zbl 0196.52801)] and indepently by \textit{J. Van Yzeren} [ibid. 65, No. 5, 339--347 (1992; Zbl 0783.51010)].NEWLINENEWLINE Moreover, in the introduction one finds in respect to Section 4 (loc. cit.): ``\dots points on the circumcircle cannot be distinguished by angular coordinates, and the triangle vertices have ill-defined singular coordinates. -- Algebro-geometric constructions meant to eliminate these issues are carried out in detail. Section 5 introduces the continuous group mentioned above (i.e. in the introduction) as a group of automorphisms of the forms of angular coordinates; it is identified as the set of plane transformations satisfying a property naturally expressed within the framework introduced in Section 2.''