# What is the locus of points from which a given segment AB subtends an angle of alpha in non euclidean geometries?

In euclidean geometry, the locus of points from which a given segment AB subtends a given angle of alpha is made up of two arcs (https://pt.wikipedia.org/wiki/Par_de_arcos_capazes). My question is what are these loci in non euclidean geometries (spherical, taxicab, Poincaré hyperbolic)? In portuguese these loci are called "arco capaz". ChatGPT says that in english it's called apollonius circle. is this correct?

• You haven't mentioned to ChatGPT any interest in non-Euclidean geometries, so why would it answer accordingly? At least as far as Wiki knows, the term "Apollonius circle" is used only in Euclidean geometry (although it does not seem to be what you are seeking even there). Commented Apr 9, 2023 at 21:05

If the angle is different from $$90^\circ$$, then the isoptic of a segment seems to be a curve of degree four in the projective model, see the 2014 article "Isoptic curves of conic sections in constant curvature geometries" by Csima and Szirmai in "Mathematical communications".