Apollonius Problem and Caustics of an Ellipsoid
Date Issued
2024
Author(s)
Aliyev, Yagub
Abstract
In the paper we discuss the Apollonius Problem on the number of normals of an ellipse passing
through a given point. It is known that the number is dependent on the position of the given point
with respect to a certain astroida. The intersection points of the astroida and the ellipse are used
to study the case when the given point is on the ellipse. The problem is then generalized for 3-
dimensional space, namely for ellipsoids. The number of concurrent normals in this case is known
to be dependent on the position of the given point with respect to the caustics of the ellipsoid. If
the given point is on the ellipsoid then the number of normals dependends on the position of the
point with respect to the intersections of the ellipsoid with its caustics. The main motivation of
this paper is to find parametrizations and classify all possible cases of these intersections.
through a given point. It is known that the number is dependent on the position of the given point
with respect to a certain astroida. The intersection points of the astroida and the ellipse are used
to study the case when the given point is on the ellipse. The problem is then generalized for 3-
dimensional space, namely for ellipsoids. The number of concurrent normals in this case is known
to be dependent on the position of the given point with respect to the caustics of the ellipsoid. If
the given point is on the ellipsoid then the number of normals dependends on the position of the
point with respect to the intersections of the ellipsoid with its caustics. The main motivation of
this paper is to find parametrizations and classify all possible cases of these intersections.