The Apollonius Problem and the Caustics of Ellipsoid: Old and New attempts to depict this surface. From A. Cayley to computer graphics

How many normals can be drawn from a point to an ellipse, i.e. How many straight lines can be drawn through a given point A so that they all intersect the conic section at a right angle? The question sounds so natural that it can arise from any enthusiast of geometry who does not suspect that Apollonius of Perga (III-II centuries BC) first set and solved this problem. The answer to this question is given by some astroid (the caustic of the ellipse), outside of which each point has two normals, four inside it, and three on the astroid itself (with the exception of the vertices of the astroid, where there are two normals). We can generalize this problem for space: How many normals can be drawn from a point in space to an ellipsoid? The answer to this question is given by some surface whose history begins with A. Cayley and continues to the present day. At the end of the report, the author will talk about some new results in the article Yagub N. Aliyev, Apollonius Problem and Caustics of an Ellipsoid https://arxiv.org/pdf/2305.06065
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