Hyperbolae are the locus of constant angle difference

12/01/2021

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by   Herman Haverkort, et al.

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Given two points A,B in the plane, the locus of all points P for which the angles at A and B in the triangle A,B,P have a constant sum is a circular arc, by Thales' theorem. We show that the difference of these angles is kept a constant by points P on a hyperbola (albeit with foci different from A and B). Whereas hyperbolae are well-known to maintain a constant difference between the distances to their foci, the above angle property seems not to be widely known. The question was motivated by recent work by Alegría et al. and De Berg et al. on Voronoi diagrams of turning rays.