A generalized definition of Apollonius circles is introduced that remains unchanged in Euclidean geometry yet permits a proof that the circle of centers coincides with the equioptic curve of two circles in hyperbolic and spherical geometries.
– Szirmai, J.: Translation-like Apollonius and triangular surfaces in non-constant curvature Thurston geometries.Results in Mathematics,80, 190 (2025)
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Generalized Apollonius Circles As Equioptic Curves Of Circles In Constants Curvature Geometries
A generalized definition of Apollonius circles is introduced that remains unchanged in Euclidean geometry yet permits a proof that the circle of centers coincides with the equioptic curve of two circles in hyperbolic and spherical geometries.