Spherical and hyperbolic conics (Q2418620)

This is a survey of interesting results about conics in hyperbolic and spherical geometry, ``to a great extent based'' on \textit{M. Chasles} [``Résumé d'une théorie des coniques sphériques homofocales et des surfaces du second ordre homofocales'', J. Math. Pures Appl. (2) 5, 425--454 (1860)] and \textit{W. E. Story} [Am. J. Math. 5, 358--382 (1883; JFM 15.0458.05)]. After defining and classifying conic sections and polarities in spherical and hyperbolic geometry, the author mentions, with proof, several results about them. Sample theorems for the spherical case: (1) A tangent to a spherical conic cuts from the lune formed by the focal lines a triangle of constant area; (2) Two tangents to a spherical conic intersect the focal lines in four points that are equidistant from the line through the points tangency. Sample theorem for the hyperbolic case: Let $(F_1, F_2)$ be a pair of foci of a hyperbolic conic. Then every light ray originating from $F_1$ reflects from the conic in such a way that it either passes through $F_2$ or contains a ray originating from $F_2$. There is also a result, Ivory's lemma, common to both the spherical and the hyperbolic setting: The diagonals in a quadrilateral formed by four confocal conics have equal lengths. For the entire collection see [Zbl 1412.51001].