  In the paper we discuss 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 caustics of the ellipsoid. If the given point is on the ellipsoid
then the number of normals is dependent on 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.
