# Apollonius Problem and Caustics of an Ellipsoid

Yagub N. Aliyev  ADA University, Ahmadbey Aghaoglu str. 61, Baku, 1008,
Azerbaijan<EMAIL_ADDRESS>
###### Abstract.

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.

††footnotetext: MSC2020: Primary 53A05, Secondary: 53A04. Keywords: ellipsoid;
Apollonius; caustics; centro-surface; astroida; ellipse.

1\. Introduction

How many normals can one draw from a point to an ellipse? In the current paper
we will try to solve this problem and its generalization to 3 dimensions,
using the methods of differential and integral calculus and differential
geometry, which were not around when Apollonius of Perga (c. III-II centuries
BC) first asked and answered this question in his famous work Conics [3].
Their number is not the only interesting question about these normals. For
example, theorem proved by Joachimstal in 1843 states that if $AB_{1}$,
$AB_{2}$, $AB_{3}$, and $AB_{4}$ are these normals, then the points $B_{1},\
B_{2},\ B_{3}$, and the point diametrically opposite to $B_{4}$, with respect
to the center $O$, of the ellipse, are concyclic [35] (see also Sect. 17.2 in
[13], [32], [16], [22], [42]). There are more results related to this fact in
[11], Sect. 17.7.3.

Figure 1: Ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ (green), its 4
normals (black), Apollonius hyperbola $y=\frac{xY}{\epsilon X-(\epsilon-1)x}$
(blue), and astroida
$\sqrt[3]{a^{2}X^{2}}+\sqrt[3]{b^{2}Y^{2}}=\sqrt[3]{\left(a^{2}-b^{2}\right)^{2}}$
(red). Created using GeoGebra.

The problem about the number of normals, which Apollonius called as the
shortest and sometimes the longest line segments, appeared in the fifth book
of Apollonius, which survived only in Arabic translation [43]. For the outline
of the solution of Apollonius, one can check [55], Chapter VII, p. 260-261.
There is a lively discussion of this problem also in pages 131-135 of [52],
[33]. The problem was also mentioned by V.I. Arnold in his paper [4], Chapter
IV and related popular lecture [5] available online both as a brochure and as
a YouTube video. The main objective of this paper is to study in detail the
cases when the point $A$ is on the ellipse (1), and generalize these results
to three dimensions for ellipsoids. Some of the results in the current paper
were presented at the Maple Conference 2022 [2], again available as a YouTube
video.

2\. Apollonius problem for plane

Let the ellipse be defined by

$None$ $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,$

where we assume that $a>b>0$. Let us take an arbitrary point $A(X,Y)$ on the
plane of the ellipse. We want to find point $B(x,y)$ on the ellipse such that
$AB$ is perpendicular to the tangent of the ellipse at $B$. The slope of this
tangent line is $y^{\prime}=-\frac{b^{2}x}{a^{2}y}$, and therefore
$\frac{y-Y}{x-X}=\frac{a^{2}y}{b^{2}x}$. From this we obtain the equation of
rectangular hyperbola $y=\frac{xY}{\epsilon X-(\epsilon-1)x},$ where
$\epsilon=\frac{a^{2}}{b^{2}}$. The intersection points $B_{1},\ B_{2},\
B_{3}$, and $B_{4}$ of this hyperbola with the ellipse give us the required
normals $AB_{1}$, $AB_{2}$, $AB_{3}$, and $AB_{4}$. In his solution,
Apollonius also used this hyperbola, which is now known as Apollonius
hyperbola [11], Sect. 17.5.5.6. The asymptotes of the hyperbola are
$x=\frac{a^{2}X}{a^{2}-b^{2}}$ and $y=\frac{b^{2}Y}{b^{2}-a^{2}}$. One of the
branches of this hyperbola passes through the center of the ellipse and
therefore, there are at least 2 intersection points with the ellipse. The
other branch may or may not intersect the ellipse. In the cases when $X=0$ and
$Y=0$, the hyperbola degenerates to a pair of perpendicular lines
$x=0,y=\frac{b^{2}Y}{b^{2}-a^{2}}$ and $x=\frac{a^{2}X}{a^{2}-b^{2}},y=0$,
respectively. Let us denote by $n(A)$ the total number of intersections of the
hyperbola with the ellipse. Since the intersection points are the solutions of
a fourth order equation, $n(A)$ can not exceed 4. Let us find points $A$,
where $n(A)$ jumps from 4 to 2. This happens when Apollonius hyperbola is
tangent to the ellipse i.e. the slopes are equal at the intersection point:
$-\frac{x}{\epsilon y}=\frac{\epsilon XY}{\left(\epsilon
X-(\epsilon-1)x\right)^{2}}$. Using this and the equation of the ellipse, we
obtain

$\frac{x}{a}=\frac{a}{a^{2}-b^{2}}\left(\sqrt[3]{\frac{b^{2}Y^{2}X}{a^{2}}}+X\right),\
\frac{y}{b}=\frac{b}{b^{2}-a^{2}}\left(\sqrt[3]{\frac{a^{2}X^{2}Y}{b^{2}}}+Y\right),$

which when used back in the equation of the ellipse, after some
simplifications gives

$None$
$\sqrt[3]{a^{2}X^{2}}+\sqrt[3]{b^{2}Y^{2}}=\sqrt[3]{\left(a^{2}-b^{2}\right)^{2}}.$

It is the equation of astroida in $X,Y$ coordinates. This curve is of 6th
order and its parametric equations can be written as
$\left(x,y\right)=\left(\frac{a^{2}-b^{2}}{a}\cos^{3}{t},\frac{a^{2}-b^{2}}{b}\sin^{3}{t}\right)$.
In the interior region of this astroida $n(A)=4$. Outside of the astroida
$n(A)=2$. On the astroida itself $n(A)=3$, except the vertices of the astroida
$\left(\pm\frac{a^{2}-b^{2}}{a},0\right)$ and
$\left(0,\pm\frac{a^{2}-b^{2}}{b}\right)$, where again $n(A)=2$. This is
essentially what was done by Apollonius, which is a remarkable achievement,
taking into account the mathematical tools available at the time. In [11],
Sect. 17.7.4 (see also p. 204, [48]) it was mentioned that this astroida is
the evolute of the ellipse and therefore drawing normals to the ellipse can be
done by drawing tangent lines of the astroida.

Let us now suppose that the point $A(X,Y)$ is on the ellipse: $X=x,\ Y=y$.
Since Apollonius hyperbola passes through $A(X,Y)$, one of the points $B_{1}$,
$B_{2}$, $B_{3}$, and $B_{4}$, coincide with $A$. For the points of the
ellipse (1) in the astroida (2), $n(A)=4$. For the points of the ellipse (1)
outside the astroida (2), $n(A)=2$. For the intersection points $N_{1},\
N_{2},\ N_{3},$ and $N_{4}$ of the ellipse (1) and the astroida (2), $n(A)=3$.
The coordinates of these points can be easily determined: $(\pm x_{0},\pm
y_{0})$ and $(\pm x_{0},\mp y_{0})$, where

$x_{0}=\sqrt{\frac{a^{4}(a^{2}-2b^{2})^{3}}{(a^{2}-b^{2})(a^{2}+b^{2})^{3}}},\
y_{0}=\sqrt{\frac{b^{4}(2a^{2}-b^{2})^{3}}{(a^{2}-b^{2})(a^{2}+b^{2})^{3}}}.$

Thus we proved the following

######  Theorem 1.

For the ellipse (1) and the astroida (2), the following cases are possible:

  1. (1)

If $a^{2}>2b^{2}$ then the points $(\pm x_{0},\pm y_{0})$ and $(\pm x_{0},\mp
y_{0})$ separate the ellipse into 4 regions where $n(A)=4$ and $n(A)=2$.

  2. (2)

If $a^{2}\leq 2b^{2}$, then for all the points of the ellipse (1), $n(A)=2$.

Noting this, we can say that Apollonius problem for the number of concurrent
normals of an ellipse is completely solved. There is also a three dimensional
variant of this problem, where one takes point $A(X,Y,Z)$ outside of the plane
of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,\ z=0$ and counts
the number of lines $AB$, such that $B(x,y,0)$ is on the ellipse, and $AB$ is
perpendicular to the tangent of the ellipse at the point $B$. But this variant
is easily reduced to the planar case. Consider the projection
$A^{\prime}(X,Y,0)$ of $A$ onto the plane $z=0$. If $A^{\prime}B$ is a normal
of the ellipse then by The Theorem of the Three Perpendiculars, $AB$ is also
perpendicular to the tangent of the ellipse at the point $B$. Therefore,
$n(A)$ is 2, 3, or 4 dependending on the position of point $A$ with respect
cylindrical surface defined by the same equation for the astroida (2).

Apollonius did not mention any practical uses for his results, except that
these normals corresponding to minimal and maximal distances, are worth
investigating for their own sake and that, in contrast to the tangents (See
Appendix), the normals were not studied much by the earlier mathematicians.
Because of this connection with the extremal distances, there can be
applications in optics, wavefronts, mathematical billiards, etc. One of the
applications of these results in astronomy can be a possible explanation for
the presence of 4 images of a distant quasar, whose light is being bent around
approximately elliptical Einstein Ring formed by two galaxies 3.4 billion
light-years away [28] (see also Figure 1 in [50]).

Figure 2: A possible application of Apollonius Problem. Image credit:
ESA/Hubble & NASA, T. Treu; Acknowledgment: J. Schmidt. Public domain. Used
with permission.

3\. Apollonius problem for space

Let us now consider three dimensional generalization of this problem. How many
concurrent normals of an ellipsoid are there? In this form the problem was
studied through analytic methods in [36], [20] (see also [31], [47] for
geometric considerations) and generalized for higher dimensions in [45]. The
answer to this question will be given in the next section. The literature
about the problem of normals to surfaces of second order is vast and we refer
the reader to Chapter III, Sections E2 and E3 of [46], which contains detailed
discussion of the history and many references for this 3D case and the
previous planar case.

Let an ellipsoid be defined by

$None$ $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1,$

where we assume that $a>b>c>0$. Let us take an arbitrary point $A(X,Y,Z)$ and
find the number $n(A)$ of points $B(x,y,z)$ on the ellipsoid such that $AB$ is
the normal line of the plane tangent to the ellipsoid at $B$. Since the outer
normal vector of the plane tangent to the ellipsoid at $B(x,y,z)$ is
$\textbf{N}=\left(\frac{x}{a^{2}},\frac{y}{b^{2}},\frac{z}{c^{2}}\right)$,

$\frac{x-X}{\frac{x}{a^{2}}}=\frac{y-Y}{\frac{y}{b^{2}}}=\frac{z-Z}{\frac{z}{c^{2}}}=-t,$

where $t$ is a parameter. From this we find parametric representation of cubic
hyperbola (see p. 204, [46])

$\textbf{r}(t)=\left(\frac{a^{2}X}{a^{2}+t},\frac{b^{2}Y}{b^{2}+t},\frac{c^{2}Z}{c^{2}+t}\right),$

whose intersections with the ellipsoid give the base points of the normals
through $A$. The asymptotes of this curve are the lines

$\textbf{r}_{1}(t)=\left(t,\frac{b^{2}Y}{b^{2}-a^{2}},\frac{c^{2}Z}{c^{2}-a^{2}}\right),$
$\textbf{r}_{2}(t)=\left(\frac{a^{2}X}{a^{2}-b^{2}},t,\frac{c^{2}Z}{c^{2}-b^{2}}\right),$
$\textbf{r}_{3}(t)=\left(\frac{a^{2}X}{a^{2}-c^{2}},\frac{b^{2}Y}{b^{2}-c^{2}},t\right).$

If $X=0$, $Y=0$, and $Z=0$ then cubic hyperbola splits into a line, which
served earlier as an asymptote of the cubic hyperbola, and a hyperbola:

$\textbf{r}(t)=\textbf{r}_{1}(t),\
\textbf{r}(t)=\left(0,\frac{b^{2}Y}{b^{2}+t},\frac{c^{2}Z}{c^{2}+t}\right);$
$\textbf{r}(t)=\textbf{r}_{2}(t),\
\textbf{r}(t)=\left(\frac{a^{2}X}{a^{2}+t},0,\frac{c^{2}Z}{c^{2}+t}\right);$
$\textbf{r}(t)=\textbf{r}_{3}(t),\
\textbf{r}(t)=\left(\frac{a^{2}X}{a^{2}+t},\frac{b^{2}Y}{b^{2}+t},0\right),$

respectively. The cubic hyperbola passes through the center of the ellipsoid
when $t=\pm\infty$, and goes to infinity when $t=-a^{2},-b^{2},-c^{2}$.
Therefore, there are at least 2 intersections with the ellipsoid. For example,
one can take the points of the ellipsoid with maximal and minimal distances
from $A$. On the other hand, these intersections are determined by

$None$
$\left(\frac{aX}{a^{2}+t}\right)^{2}+\left(\frac{bY}{b^{2}+t}\right)^{2}+\left(\frac{cZ}{c^{2}+t}\right)^{2}=1,$

which is a sixth order equation with respect to $t$, and therefore can not
have more than 6 real solutions.

Figure 3: Ellipsoid
$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1$ (green), its 6
normals $AB_{1},AB_{2},\ldots,AB_{6}$ (black), cubic hyperbola
$\textbf{r}(t)=\left(\frac{a^{2}X}{a^{2}+t},\frac{b^{2}Y}{b^{2}+t},\frac{c^{2}Z}{c^{2}+t}\right)$
(red). Created using GeoGebra.

As before, let us denote the number of normals through $A$ by $n(A)$. We want
to find points $A$, where $n(A)$ jumps from 2 to 4, or from 4 to 6. This
happens when cubic hyperbola is tangent to the ellipsoid i.e.
$\textbf{r}^{\prime}(t)=\left(-\frac{a^{2}X}{(a^{2}+t)^{2}},-\frac{b^{2}Y}{(b^{2}+t)^{2}},-\frac{c^{2}Z}{(c^{2}+t)^{2}}\right),$
is orthogonal to
$\textbf{N}=\left(\frac{X}{a^{2}+t},\frac{Y}{b^{2}+t},\frac{Z}{c^{2}+t}\right)$.
This can be expressed as $\textbf{r}^{\prime}(t)\cdot\textbf{N}=0$, or as

$None$
$\frac{a^{2}X^{2}}{(a^{2}+t)^{3}}+\frac{b^{2}Y^{2}}{(b^{2}+t)^{3}}+\frac{c^{2}Z^{2}}{(c^{2}+t)^{3}}=0.$

The equations (4) and (5) define the surface known as Caustics of an Ellipsoid
also known as focal surface, surface of centers, evolute of an ellipsoid, or
just Cayley’s astroida [17] (see also p. 165, [56]). Cayley used the name
Centro-surface of an Ellipsoid, and the equations (4) and (5), which appear in
p. 358 of [17], were obtained using the fact that the points of this surface
are the centers of principal curvatures of the ellipsoid (3) (see also p. 218
in [49]). The centers and principal radii of curvature for more general
surfaces were studied by G. Monge in Sect. XXV of [44]. For a modern
definition of the principal curvatures, see, for example, p. 158, [29]. A.
Cayley’s graph of the surface appears in p. 330 of [17] (also shown in p.116,
[30]). One can also find many other images depicting this surface in various
papers, dissertations, and books. See for example pp. 49-53 in [8], p. 154 in
Ch. 7 of [9], [23], [34], p. 218 in [7], p. 257 in [57], p. 49 in [37] (also
shown in p. 356, [12]), [58], p. 61 in [38], p. 10 in [18]. The part of the
surface where the two surfaces corresponding to minimal and maximal curvatures
intersect (named as ”the purse”), was shown and mentioned in pages 37 and 109,
respectively, of [6] (see also p. 218 in [7]). S.K. Lando gave two popular
lectures about the caustics, available online, one with a demonstration of the
surface at the end [41]. Another representation of the surface together with
some applications of it in astronomy and physics appeared in [54] (See also
[53]). According to [54], the idea of using more general caustics in cosmology
is due to Ya. B. Zel’dovich (see [60], [59] and the references therein).

There are many visualizations of this surface as a physical model. Before the
dawn of computer graphics and 3D printers, handmade models and sculptures
represented the best medium for such mathematical objects [29]. In [40], there
is a description of a model made out of gypsum by student H.A. Schwarz in the
Arts Faculty (later Prof. in Univ. Berlin), which is also mentioned in Sect.
197 (p. 282), [25] (see also p. 198, [46]). Stereographic photo of one such
model by unknown artist/maker from the same time period is shown in Figure 4,
[19]. Two more models of this surface together with models of centro-surfaces
of paraboloids and hyperboloids can be found in The Collection of Mathematical
Models and Instruments in The University of Göttingen [21] (See Fig. 5).
Similar models for the centers of curvature of paraboloids and hyperboloids
were described in [51] (see also [15]) and p. 283 in [25] (see also [26]),
respectively (see also p. 264 and p. 34, respectively, in [38]). See also the
website of The TouchGeometry Project [24] for models of caustics of an
elliptical paraboloid and a hyperboloid of one sheet in Geometry Department of
Karazin University in Kharkiv, Ukraine. Another such model is in The National
Museum of American History [14].

Figure 4: Stereograph Card, Unknown artist/maker, Centro-Surface. Ellipsoid.
about 1860. Gift of Weston J. and Mary M. Naef, Getty Museum Collection. Open
Content program. No copyright. Used with permission.

Figure 5: Curvature centre point surface models: 239 and 242. Curvature centre
point surface of the triaxial ellipsoid. Gypsum; curved surfaces are unified.
(Göttinger Sammlung mathematischer Modelle und Instrumente, Georg-August-
Universität Göttingen)

Note that in general, it is not easy to exclude the parameter $t$ from the
equations (4) and (5), to get an explicit equation for the caustics (see [49],
p. 113 in [30]). This surface is of 12th order and its equation can be written
as a $5\times 5$ determinant (see p. 114 in [30]). But if, for example, $c=0$,
then the equations (4) and (5) are transformed to

$\left(\frac{aX}{a^{2}+t}\right)^{2}+\left(\frac{bY}{b^{2}+t}\right)^{2}=1,\
\frac{a^{2}X^{2}}{(a^{2}+t)^{3}}+\frac{b^{2}Y^{2}}{(b^{2}+t)^{3}}=0,$

from which one can easily eliminate parameter $t$, and obtain the equation (2)
for the astroida. This gives us another solution for the planar case
considered in the previous section. Similarly, if $b=c$ then one can introduce
a new variable $Y^{\prime}$, such that $(Y^{\prime})^{2}=Y^{2}+Z^{2}$ and then
the equations (4) and (5) can be written as

$\left(\frac{aX}{a^{2}+t}\right)^{2}+\left(\frac{bY^{\prime}}{b^{2}+t}\right)^{2}=1,\
\frac{a^{2}X^{2}}{(a^{2}+t)^{3}}+\frac{b^{2}(Y^{\prime})^{2}}{(b^{2}+t)^{3}}=0,$

from which again the parameter $t$ is easily eliminated to get

$\sqrt[3]{a^{2}X^{2}}+\sqrt[3]{b^{2}(Y^{\prime})^{2}}=\sqrt[3]{\left(a^{2}-b^{2}\right)^{2}},$

or

$\sqrt[3]{a^{2}X^{2}}+\sqrt[3]{b^{2}(Y^{2}+Z^{2})}=\sqrt[3]{\left(a^{2}-b^{2}\right)^{2}},$

which is a surface of revolution generated by rotating astroida (2) around $x$
axis (see Fig. 6).

Figure 6: The surface of revolution generated by rotating the astroida around
one of its axes. Created using Maple 2022 for the cases $(a,b,c)=(4,4,3)$
(left) and $(4.1,3,3)$ (right).

4\. Caustics of Ellipsoid in GeoGebra and Maple In this section a method of
generating the surface, based on the cartesian coordinates, will be described.
The formulas for Gaussian curvature and mean curvature of an ellipsoid are
given in Corollary 13.41, p. 413 in [1] (see also Chapter 4, [10] for
applications in geodesy):

$K(x,y)=\frac{1}{\left(abc\left(\frac{x^{2}}{a^{4}}+\frac{y^{2}}{b^{4}}+\frac{z^{2}}{c^{4}}\right)\right)^{2}},\
H(x,y)=\frac{|x^{2}+y^{2}+z^{2}-a^{2}-b^{2}-c^{2}|}{2(abc)^{2}\left(\frac{x^{2}}{a^{4}}+\frac{y^{2}}{b^{4}}+\frac{z^{2}}{c^{4}}\right)^{\frac{3}{2}}}.$

The principal curvatures $k_{1}$ and $k_{2}$ are the roots of the quadratic
equation $x^{2}-2Hx+K=0$ (Corollary 13.26, p. 400, in [1]):

$k_{1}=H-\sqrt{H^{2}-K},\ k_{2}=H+\sqrt{H^{2}-K}.$

The corresponding radii of the curvature are $R_{1}=\frac{1}{k_{1}}$ and
$R_{2}=\frac{1}{k_{2}}$, and the respective centers of the curvature
$C_{1}(x_{1},y_{1},z_{1})$ and $C_{2}(x_{1},y_{1},z_{1})$ can be determined
using the formula (see p. 226, [27])

$C_{1}(x_{1},y_{1},z_{1})=(x,y,z)-R_{1}\cdot\frac{\textbf{N}}{|\textbf{N}|},\
C_{2}(x_{2},y_{2},z_{2})=(x,y,z)-R_{2}\cdot\frac{\textbf{N}}{|\textbf{N}|},$

where as before outer normal is
$\textbf{N}=\left(\frac{x}{a^{2}},\frac{y}{b^{2}},\frac{z}{c^{2}}\right)$. The
GeoGebra Activity demonstrating the surface, can be found in
https://www.geogebra.org. The Maple Learn document can be found in
https://learn.maplesoft.com. The images created using GeoGebra and Maple 2022
are shown in Figure 7 and Figure 8, respectively.

Figure 7: The centers corresponding to smaller (left, $R_{2}$) and greater
(right, $R_{1}$) principal radii of curvature. The surfaces intersect
(center). Created using GeoGebra.

Figure 8: The ellipsoid and its caustics with transparency applied. Created
using Maple 2022 (left) and GeoGebra (right).

We can now answer the question asked at the beginning of Sect. 2. The number
of normals outside of the two caustics is 2 ($n(A)=2$). For the points of the
space inside of only one and both of the caustics, $n(A)=4$ and $n(A)=6$,
respectively (see [36], p. 123-124). On the caustics $n(A)=3$ or $n(A)=5$,
with some exceptions on the planes $X=0$, $Y=0$, and $Z=0$ and on the
intersections of the two caustics, where again $n(A)=2$ or $n(A)=4$ (see Fig.
9).

In the case of an ellipsoid of revolution, for example, when $a=b$ or $b=c$,
one of the caustics becomes a surface of revolution, shown in Figure 6, the
other caustic degenerates to a line segment on the axis of symmetry of the
surfaces shown in Figure 6, between the vertices. The number of normals of the
ellipsoid (3) for the points $A$ on this line segment is infinite
($n(A)=\infty$), except the endpoints of this line segment where $n(A)=2$. For
the other points of the space, the situation is identical to the planar case
considered in Sect. 2.

Figure 9: The number of normals in the regions of space separated by the
caustics of the ellipsoid. Half of the caustics is hidden to make the inner
regions visible.

5\. The intersections of an ellipsoid and its caustics with the coordinate
planes

The intersection curves of the ellipsoid (3) and its caustics (see Figure 10)
with the coordinate planes $x=0,\ y=0,\ z=0$ are found in p. 325, [17] (see
also p. 115, [30]).

######  Lemma 2.

The intersections of the ellipsoid (3) and its caustics with the coordinate
planes are the following curves:

  1. (1)

Ellipse $\left(a\cos{t},b\sin{t},0\right)$ (black),

  2. (2)

Ellipse $\left(a\cos{t},0,c\sin{t}\right)$ (yellow),

  3. (3)

Ellipse $\left(0,b\cos{t},c\sin{t}\right)$ (red),

  4. (4)

Astroida
$\left(\frac{a^{2}-b^{2}}{a}\cos^{3}{t},\frac{a^{2}-b^{2}}{b}\sin^{3}{t},0\right)$
(pink),

  5. (5)

Astroida
$\left(\frac{a^{2}-c^{2}}{a}\cos^{3}{t},0,\frac{a^{2}-c^{2}}{c}\sin^{3}{t}\right)$
(light blue),

  6. (6)

Astroida
$\left(0,\frac{b^{2}-c^{2}}{b}\cos^{3}{t},\frac{b^{2}-c^{2}}{c}\sin^{3}{t}\right)$
(purple),

  7. (7)

Ellipse
$\left(\frac{a^{2}-c^{2}}{a}\cos{t},\frac{b^{2}-c^{2}}{b}\sin{t},0\right)$
(green),

  8. (8)

Ellipse
$\left(\frac{a^{2}-b^{2}}{a}\cos{t},0,\frac{b^{2}-c^{2}}{c}\sin{t}\right)$
(dark blue),

  9. (9)

Ellipse
$\left(0,\frac{a^{2}-b^{2}}{b}\cos{t},\frac{a^{2}-c^{2}}{c}\sin{t}\right)$
(orange).

Figure 10: The intersections of a triaxial ellipsoid and its caustics with the
coordinate planes, the nodal curve (grey), and the intersections of the
ellipsoid with its caustics (cyan blue). The other colors are explained in
Lemma 2. The tangency points of the two caustics and some of the intersection
points are also shown. See https://www.geogebra.org/3d/tqjwgxwg for more
details.

Let us now find intersections of Ellipses 1,2,3, Astroidas 4,5,6 and Ellipses
7,8,9, respectively.

######  Lemma 3.

  1. (1)

If $a^{2}\geq 2b^{2}$, then Ellipse 1 and Astroida 4 intersect at $(\pm
x_{0},\pm y_{0},0)$ and $(\pm x_{0},\mp y_{0},0)$, where

$x_{0}=\sqrt{\frac{a^{4}(a^{2}-2b^{2})^{3}}{(a^{2}-b^{2})(a^{2}+b^{2})^{3}}},\
y_{0}=\sqrt{\frac{b^{4}(2a^{2}-b^{2})^{3}}{(a^{2}-b^{2})(a^{2}+b^{2})^{3}}}.$

  2. (2)

If $a^{2}\geq 2c^{2}$, then Ellipse 2 and Astroida 5 intersect at $(\pm
x_{1},0,\pm z_{1})$ and $(\pm x_{1},0,\mp z_{1})$, where

$x_{1}=\sqrt{\frac{a^{4}(a^{2}-2c^{2})^{3}}{(a^{2}-c^{2})(a^{2}+c^{2})^{3}}},\
z_{1}=\sqrt{\frac{c^{4}(2a^{2}-c^{2})^{3}}{(a^{2}-c^{2})(a^{2}+c^{2})^{3}}}.$

  3. (3)

If $b^{2}\geq 2c^{2}$, then Ellipse 3 and Astroida 6 intersect at $(0,\pm
y_{2},\pm z_{2})$ and $(0,\pm y_{2},\mp z_{2})$, where

$y_{2}=\sqrt{\frac{b^{4}(b^{2}-2c^{2})^{3}}{(b^{2}-c^{2})(b^{2}+c^{2})^{3}}},\
z_{2}=\sqrt{\frac{c^{4}(2b^{2}-c^{2})^{3}}{(b^{2}-c^{2})(b^{2}+c^{2})^{3}}}.$

  4. (4)

Ellipse 1 and Ellipse 7 do not have real intersection points and the
coordinates of the non-real intersection points are $(\pm x^{*},\pm y^{*},0)$
and $(\pm x^{*},\mp y^{*},0)$, where

$x^{*}=\sqrt{\frac{a^{2}(a^{2}-c^{2})^{2}(2b^{2}-c^{2})}{(a^{2}-b^{2})(2a^{2}b^{2}-a^{2}c^{2}-b^{2}c^{2})}},$
$y^{*}=\sqrt{\frac{b^{2}(b^{2}-c^{2})^{2}(2a^{2}-c^{2})}{(b^{2}-a^{2})(2a^{2}b^{2}-b^{2}c^{2}-a^{2}c^{2})}}.$

  5. (5)

If $b^{2}\geq 2c^{2}$, then Ellipse 2 and Ellipse 8 intersect at $(\pm
x_{3},0,\pm z_{3})$ and $(\pm x_{3},0,\mp z_{3})$, where

$x_{3}=\sqrt{\frac{a^{2}(a^{2}-b^{2})^{2}(2c^{2}-b^{2})}{(a^{2}-c^{2})(2a^{2}c^{2}-a^{2}b^{2}-b^{2}c^{2})}},$
$z_{3}=\sqrt{\frac{c^{2}(c^{2}-b^{2})^{2}(2a^{2}-b^{2})}{(c^{2}-a^{2})(2a^{2}c^{2}-a^{2}b^{2}-b^{2}c^{2})}},$

and $x_{1}\geq x_{3}$, $z_{1}\leq z_{3}$ with equality cases when
$\frac{1}{a^{2}}+\frac{1}{c^{2}}=\frac{3}{b^{2}}$.

  6. (6)

If $2b^{2}\geq a^{2}\geq 2c^{2}$, then Ellipse 3 and Ellipse 9 intersect at
$(0,\pm y_{4},\pm z_{4})$ and $(0,\pm y_{4},\mp z_{4})$, where

$y_{4}=\sqrt{\frac{b^{2}(b^{2}-a^{2})^{2}(2c^{2}-a^{2})}{(b^{2}-c^{2})(2b^{2}c^{2}-a^{2}b^{2}-a^{2}c^{2})}},$
$z_{4}=\sqrt{\frac{c^{2}(c^{2}-a^{2})^{2}(2b^{2}-a^{2})}{(c^{2}-b^{2})(2b^{2}c^{2}-a^{2}b^{2}-a^{2}c^{2})}}.$

In particular, $y_{2}\geq y_{4}$ and $z_{2}\leq z_{4}$ if and only if
$2b^{4}+2c^{4}-a^{2}b^{2}-a^{2}c^{2}-2b^{2}c^{2}\geq 0$.

  7. (7)

If $a^{2}+c^{2}\geq 2b^{2}$, then Astroida 4 and Ellipse 7 intersect at $(\pm
x_{5},\pm y_{5},0)$ and $(\pm x_{5},\mp y_{5},0)$, where

$x_{5}=\sqrt{\frac{(a^{2}-c^{2})^{3}(2b^{2}-a^{2}-c^{2})^{3}}{a^{2}(b^{2}-a^{2})(a^{2}+b^{2}-2c^{2})^{3}}},\
y_{5}=\sqrt{\frac{(b^{2}-c^{2})^{3}(2a^{2}-b^{2}-c^{2})^{3}}{b^{2}(a^{2}-b^{2})(a^{2}+b^{2}-2c^{2})}}.$

  8. (8)

Astroida 5 and Ellipse 8 are tangent to each other at the points $(\pm
x_{6},0,\pm z_{6})$ and $(\pm x_{6},0,\mp z_{6})$, where

$x_{6}=\sqrt{\frac{(b^{2}-c^{2})^{3}}{c^{2}(a^{2}-c^{2})}},\
z_{6}=\sqrt{\frac{(a^{2}-b^{2})^{3}}{a^{2}(a^{2}-c^{2})}},$

These points also divide Astroida 5 and Ellipse 8 into parts which belong to
different caustics. These points are on, in and outside the ellipsoid (3) if
$\frac{1}{a^{2}}+\frac{1}{c^{2}}=\frac{3}{b^{2}}$, $<\frac{3}{b^{2}}$, and
$>\frac{3}{b^{2}}$, respectively.

  9. (9)

If $a^{2}+c^{2}\leq 2b^{2}$, then Astroida 6 and Ellipse 9 intersect at
$(0,\pm y_{7},\pm z_{7})$ and $(0,\pm y_{7},\mp z_{7})$, where

$y_{7}=\sqrt{\frac{(b^{2}-a^{2})^{3}(2c^{2}-b^{2}-a^{2})^{3}}{b^{2}(c^{2}-b^{2})(b^{2}+c^{2}-2a^{2})^{3}}},$
$z_{7}=\sqrt{\frac{(c^{2}-a^{2})^{3}(2b^{2}-c^{2}-a^{2})^{3}}{c^{2}(b^{2}-c^{2})(b^{2}+c^{2}-2a^{2})^{3}}}.$

These points are on, in and outside the ellipsoid (3) if
$2b^{4}+2c^{4}-a^{2}b^{2}-a^{2}c^{2}-2b^{2}c^{2}=0$, $<0$, and $>0$,
respectively.

Proof.  By direct substitution of the given coordinates in the equations of
the curves, we can prove the claim about the intersections. The inequalities
involving coordinates are also proved by direct substitutions. For tangency of
the curves in part 8, additionally, the slopes of the curves are compared. In
parts 8 and 9, the last claim about the position with respect to the ellipsoid
(3), follows directly from the substitution in (3) of the coordinates
$x_{6},z_{6}$ and $y_{7},z_{7}$, respectively. $\Box$

One can experiment with these intersection points and the curves by moving the
sliders in GeoGebra Activity https://www.geogebra.org/3d/tqjwgxwg.

6\. The number of normals for the points of an ellipsoid As for an ellipse, if
the point $A$ is on the ellipsoid (3), then one of the points
$B_{1},B_{2},\ldots,B_{6}$ coincides with $A$. The description all possible
cases for the regions on an ellipsoid, where $n(A)$ jumps from 2 to 4, or from
4 to 6, is not as trivial as in the planar case. It seems that the problem in
this setting did not attract much attention and remains unstudied. In the
remaining part of the paper we will highlight the 3 cases of intersections of
the ellipsoid and its caustics (see Figure 8). In general, these intersections
are some curves on the ellipsoid (3), and a simple parametrization for these
curves is given at the end of this paper. Using the intersections of these
curves with the coordinate planes, which we found in the previous section, one
can categorize 3 possible cases: (i) none of the caustics intersect the
ellipsoid, (ii) only one of the caustics intersects the ellipsoid, (iii) both
of the caustics intersect the ellipsoid. These intersections have many
different shapes and positions, and the complete categorization of all general
cases is shown in Figure 11.

######  Theorem 4.

For the ellipsoid (3) and its caustics defined by (4) and (5), the following
cases are possible:

  1. (1)

If $a^{2}<2c^{2}$ then there are no intersections of the caustics with the
ellipsoid (3),

  2. (2)

If $b^{2}<2c^{2}\leq a^{2}$ then only one of the caustics intersects the
ellipsoid (3),

  3. (3)

If $b^{2}\geq 2c^{2}$, then both of the caustics intersects the ellipsoid (3).

In all cases, for the points of the ellipsoid (3) lying outside of the two
caustics $n(A)=2$, for the points of the ellipsoid (3) lying in only one of
these caustics $n(A)=4$, for the points of the ellipsoid (3) lying in both of
these caustics $n(A)=6$, and for the intersection points of the ellipsoid (3)
and these caustics $n(A)=3$ or $5$, except some of the points of the ellipsoid
(3), where the caustics intersect each other or these caustics intersect the
coordinate planes.

Proof.  Let us first note that if the ellipsoid (3) and one of its caustics
intersect, then they should also intersect on at least one point of the
coordinate planes. Indeed, suppose on the contrary that the ellipsoid (3) and
one of its caustics intersect but they do not intersect on any of the
coordinate planes. Then one of the intersection curves should be situated
completely in the 1st octant. Denote this curve by $\Gamma$. The part of the
caustic bounded by $\Gamma$ is a smooth surface outside of the ellipsoid (3),
and therefore it has a tangent plane which does not intersect the ellipsoid
(3). This tangent plane contains also the corresponding normal of the
ellipsoid (3) because the principal radii are tangent to the caustics (see p.
312, [39]). This is a contradiction because the normals intersects the
ellipsoid (3). It follows that the problem of existence of intersections of
the ellipsoid (3) and its caustics can be studied just by their cross sections
with the coordinate planes, which was done in Lemma 2 and Lemma 3. The
remaining claims follow directly from the results in [36]. $\Box$

Depending on whether $a^{2}\leq 2b^{2}$ or $a^{2}>2b^{2}$, the caustic
corresponding to the greater principal radius (the red caustics in Figure 11)
is encompassed by the ellipsoid (3) or the ellipsoid (3) is encompassed by
this caustic. Similarly, depending on whether $a^{2}+c^{2}\geq 2b^{2}$ or
$a^{2}+c^{2}<2b^{2}$, the (blue) caustic corresponding to the smaller
principal radius is encompassed by the (red) caustic corresponding to the
greater principal radius or vice versa (cf. p. 326 and p. 363, [17]). It is
obvious that if $a^{2}+c^{2}\leq 2b^{2}$ then $a^{2}<2b^{2}$. Similarly, if
$b^{2}<2c^{2}\leq a^{2}$ then
$\frac{1}{a^{2}}+\frac{1}{c^{2}}<\frac{3}{b^{2}}$.

Detailed classification of the cases of intersection of the ellipsoid and its
caustics is done in Figure 11 based on the sign of the expressions
$\frac{1}{a^{2}}+\frac{1}{c^{2}}-\frac{3}{b^{2}}$ and
$2b^{4}+2c^{4}-a^{2}b^{2}-a^{2}c^{2}-2b^{2}c^{2}$. Note that if $b^{2}\geq
2c^{2}$ and $2b^{4}+2c^{4}-a^{2}b^{2}-a^{2}c^{2}-2b^{2}c^{2}\geq 0$ then
$\frac{1}{a^{2}}+\frac{1}{c^{2}}>\frac{3}{b^{2}}$. Indeed, since
$\frac{1}{a^{2}}+\frac{1}{c^{2}}>\frac{3}{b^{2}}$ can be written as
$b^{2}c^{2}+a^{2}b^{2}-3a^{2}c^{2}>0$, it is sufficient to show that

$b^{2}c^{2}+a^{2}b^{2}-3a^{2}c^{2}>2b^{4}+2c^{4}-a^{2}b^{2}-a^{2}c^{2}-2b^{2}c^{2}.$

This inequality can be written as
$3b^{2}c^{2}+2a^{2}(b^{2}-c^{2})>2b^{4}+2c^{4}$. Since $a>b>c$, it is
sufficient to show that $3b^{2}c^{2}+2b^{2}(b^{2}-c^{2})\geq 2b^{4}+2c^{4}$,
which simplifies to $b^{2}\geq 2c^{2}$. Similarly, if
$2b^{4}+2c^{4}-a^{2}b^{2}-a^{2}c^{2}-2b^{2}c^{2}\geq 0$ then
$a^{2}+c^{2}<2b^{2}$. Indeed, by rewriting the given inequality we obtain
$a^{2}+c^{2}\leq 2b^{2}-\frac{c^{2}(a^{2}+b^{2}-2c^{2})}{b^{2}}<2b^{2}$.

######  Theorem 5.

If $2b^{4}+2c^{4}-a^{2}b^{2}-a^{2}c^{2}-2b^{2}c^{2}\leq 0$ and
$\frac{1}{a^{2}}+\frac{1}{c^{2}}\geq\frac{3}{b^{2}}$ then the two caustics and
the ellipse (3) intersect at a unique point of each octant.

Proof.  The intersection curve of the two caustics (see Figure 10) is called
nodal curve and its parametrization was given in p. 351, [17]:

$(x(t))^{2}=\frac{((\gamma-\alpha)t+\alpha)((\gamma-\alpha)t-2\gamma)^{2}((\beta-\gamma)t+\gamma)^{3}}{-\beta\gamma
a^{2}(\alpha\gamma+\Omega
t)(3t-2)^{2}},(y(t))^{2}=\frac{t^{2}(t-1)((\gamma-\alpha)^{2}t+3\alpha\gamma)^{3}}{-\alpha\gamma
b^{2}(\alpha\gamma+\Omega t)(3t-2)^{2}},$
$(z(t))^{2}=\frac{((\gamma-\alpha)t-\gamma)((\gamma-\alpha)t+2\alpha)^{2}((\alpha-\beta)t-\alpha)^{3}}{-\alpha\beta
c^{2}(\alpha\gamma+\Omega t)(3t-2)^{2}},$

where $\alpha=b^{2}-c^{2}$, $\beta=c^{2}-a^{2}$, $\gamma=a^{2}-b^{2}$,
$\Omega=\alpha^{2}-\beta\gamma$, and $0\leq
t\leq\min\left(\frac{\alpha}{\alpha-\beta},-\frac{\gamma}{\beta-\gamma}\right)$.
Note that $-\frac{\gamma}{\beta-\gamma}\geq\frac{\alpha}{\alpha-\beta}$ iff
$a^{2}+c^{2}\geq 2b^{2}$. Also note that
$\max\left(\frac{\alpha}{\alpha-\beta},-\frac{\gamma}{\beta-\gamma}\right)<\frac{2}{3}$.
Consider the function
$f(t)=\frac{x(t)^{2}}{a^{2}}+\frac{y(t)^{2}}{b^{2}}+\frac{z(t)^{2}}{c^{2}}-1$.
Simplifying using Maple we obtain that
$f(t)=\frac{p(t)(q(t))^{2}}{a^{4}b^{4}c^{4}\left(3t-2\right)^{2}r(t)}$, where

$p(t)=(a^{4}b^{2}+a^{4}c^{2}+a^{2}b^{4}+c^{4}a^{2}+b^{4}c^{2}+c^{4}b^{2}-6a^{2}b^{2}c^{2})(a^{2}-2b^{2}+c^{2})t^{2}$
$+(11a^{4}b^{2}c^{2}+11a^{2}b^{2}c^{4}+3a^{2}b^{6}+3b^{6}c^{2}-b^{2}a^{6}-c^{2}a^{6}-a^{2}c^{6}-b^{2}c^{6}-a^{4}b^{4}-5a^{4}c^{4}-b^{4}c^{4}-17a^{2}b^{4}c^{2})t$
$+(a^{2}-b^{2})(b^{2}-c^{2})(a^{2}b^{2}+b^{2}c^{2}-3a^{2}c^{2}),$
$q(t)=(a^{2}+b^{2}+c^{2})(a^{2}-2b^{2}+c^{2})t^{2}+(4b^{4}+a^{2}b^{2}+b^{2}c^{2}-3a^{2}c^{2})t-2b^{4},$
$r(t)=(a^{4}+b^{4}+c^{4}-a^{2}b^{2}-b^{2}c^{2}-a^{2}c^{2})t+(a^{2}-b^{2})(b^{2}-c^{2}).$

Note that $r(t)>0$ whenever $t\geq 0$. We will consider 3 cases.

Case 1. Suppose that $a^{2}-2b^{2}+c^{2}>0$. Note that

$p(0)=(a^{2}-b^{2})(b^{2}-c^{2})(a^{2}b^{2}+b^{2}c^{2}-3a^{2}c^{2})>0,$
$p\left(\frac{\alpha}{\alpha-\beta}\right)=-\frac{c^{2}(a^{2}-c^{2})(b^{2}-c^{2})(2a^{2}-b^{2}-c^{2})(2a^{4}+2b^{4}-a^{2}c^{2}-b^{2}c^{2}-2a^{2}b^{2})}{(a^{2}+b^{2}-2c^{2})^{2}}<0$
$p\left(1\right)=-b^{4}(a^{2}-c^{2})^{2}<0,\ p(+\infty)=+\infty.$

Therefore, one of the roots of $p(t)$ is in the interval
$\left(0,\frac{\alpha}{\alpha-\beta}\right)$, and the other root is in the
interval $(1,+\infty)$. Similarly,

$q(-\infty)=+\infty,\ q(0)=-2b^{4}<0,$
$q\left(\frac{\alpha}{\alpha-\beta}\right)=-\frac{c^{2}(3a^{2}(b^{2}-c^{2})+c^{2}(a^{2}+c^{2}-2b^{2}))(2a^{2}-b^{2}-c^{2})}{(a^{2}+b^{2}-2c^{2})^{2}}<0,\
q(+\infty)=+\infty,$

Therefore, one of the roots of $q(t)$ is in the interval $(-\infty,0)$, and
the other root is in the interval
$\left(\frac{\alpha}{\alpha-\beta},+\infty\right)$.

Case 2. Now suppose that $a^{2}-2b^{2}+c^{2}<0$.

$p(-\infty)=-\infty,\ p(0)>0,$
$p\left(-\frac{\gamma}{\beta-\gamma}\right)=\frac{a^{2}(a^{2}-c^{2})(a^{2}-b^{2})(a^{2}+b^{2}-2c^{2})(2b^{4}+2c^{4}-a^{2}b^{2}-a^{2}c^{2}-2b^{2}c^{2})}{(2a^{2}-b^{2}-c^{2})^{2}}<0.$

Therefore, one of the roots of $p(t)$ is in the interval $(-\infty,0)$, and
the other root is in the interval
$\left(0,-\frac{\gamma}{\beta-\gamma}\right)$. Similarly,

$q\left(-\frac{\gamma}{\beta-\gamma}\right)=\frac{a^{2}(3c^{2}(b^{2}-a^{2})+a^{2}(a^{2}+c^{2}-2b^{2}))(a^{2}+b^{2}-2c^{2})}{(2a^{2}-b^{2}-c^{2})^{2}}<0,$
$q(1)=a^{4}-a^{2}c^{2}+c^{4}>0,\ q(+\infty)=-\infty.$

Therefore, one of the roots of $q(t)$ is in the interval
$\left(-\frac{\gamma}{\beta-\gamma},1\right)$, and the other root is in the
interval $(1,+\infty)$.

Case 3. Let us now suppose that $a^{2}-2b^{2}+c^{2}=0$. In this case $p(t)$
and $q(t)$ are linear functions and
$-\frac{\gamma}{\beta-\gamma}=\frac{\alpha}{\alpha-\beta}$. As in Case 1,
$p(0)>0,$ $p\left(\frac{\alpha}{\alpha-\beta}\right)<0$, and therefore, the
only root of $p(t)$ is in the interval
$\left(0,\frac{\alpha}{\alpha-\beta}\right)$. As in Case 2,
$q\left(-\frac{\gamma}{\beta-\gamma}\right)<0,$ $q(1)>0$, and therefore, the
only root of $q(t)$ is in the interval
$\left(-\frac{\gamma}{\beta-\gamma},1\right)$.

We proved that in all the cases there is only one zero $t_{0}$ of $f(t)$ in
the interval $0\leq
t\leq\min\left(\frac{\alpha}{\alpha-\beta},-\frac{\gamma}{\beta-\gamma}\right)$
and it is one of the solutions of the quadratic equation $p(t)=0$. The $x,\
y,\ z$ coordinates of the intersection point of the two caustics and the
ellipsoid (3) in the first quadrant are $x(t_{0}),\ y(t_{0}),\ z(t_{0})$,
respectively (See Figure 10, the intersection point of cyan blue curves).

$\Box$

(i) (ii) (iii) (iv)

(v) (vi) (vii) (viii)

(ix) (x) (xi)(xii)

Figure 11: Classification of intersections of ellipsoid (green) and its
caustics (red and blue). Created using Maple 2022 for the following values of
$(a,b,c)$:

(i) $(4.7,4.4,4)$ ($a^{2}<2c^{2}$, $a^{2}+c^{2}<2b^{2}$).

(ii) $(4.9,4.4,4)$ ($a^{2}<2c^{2}$, $a^{2}+c^{2}>2b^{2}$);

(iii) $(4.7,4,3)$ ($b^{2}<2c^{2}<a^{2}$, $a^{2}+c^{2}<2b^{2}$);

(iv) $(5,4,3)$ ($b^{2}<2c^{2}<a^{2}$, $a^{2}+c^{2}>2b^{2}$, $a^{2}<2b^{2}$);

(v) $(5,3.3,2.5)$ ($b^{2}<2c^{2}<a^{2}$, $a^{2}>2b^{2}$);

(vi) $(4,3,2)$ ($b^{2}>2c^{2}$, $a^{2}+c^{2}<2b^{2}$,
$\frac{1}{a^{2}}+\frac{1}{c^{2}}<\frac{3}{b^{2}}$);

(vii) $5,3.7,2.5$ ($b^{2}>2c^{2}$, $a^{2}+c^{2}>2b^{2}$, $a^{2}<2b^{2}$,
$\frac{1}{a^{2}}+\frac{1}{c^{2}}<\frac{3}{b^{2}}$)

(viii) $(5,2.8,1.8)$ ($b^{2}>2c^{2}$, $a^{2}>2b^{2}$,
$\frac{1}{a^{2}}+\frac{1}{c^{2}}<\frac{3}{b^{2}}$);

(ix) $(5,4.4,1.6)$ ($b^{2}>2c^{2}$,
$2b^{4}+2c^{4}-a^{2}b^{2}-a^{2}c^{2}-2b^{2}c^{2}>0$);

(x) $(4.5,3.5,1.4)$ ($b^{2}>2c^{2}$, $a^{2}+c^{2}<2b^{2}$,
$\frac{1}{a^{2}}+\frac{1}{c^{2}}>\frac{3}{b^{2}}$,
$2b^{4}+2c^{4}-a^{2}b^{2}-a^{2}c^{2}-2b^{2}c^{2}<0$);

(xi) $(5,3.7,2)$ ($b^{2}>2c^{2}$, $a^{2}+c^{2}>2b^{2}$, $a^{2}<2b^{2}$,
$\frac{1}{a^{2}}+\frac{1}{c^{2}}>\frac{3}{b^{2}}$)

(xii) $(5,3,1)$ ($b^{2}>2c^{2}$, $a^{2}>2b^{2}$,
$\frac{1}{a^{2}}+\frac{1}{c^{2}}>\frac{3}{b^{2}}$);

In the remaining part of the paper we will find a parametrization for the
intersection curves of each of the caustics with the ellipsoid (3). In p. 324,
[17] (see also p. 218, [49]) parametrization in curvilinear coordinates $\xi$,
$\eta$ of the caustics is given as

$(x(\xi,\eta))^{2}=-\frac{(a^{2}+\xi)^{3}(a^{2}+\eta)}{a^{2}\beta\gamma},(y(\xi,\eta))^{2}=-\frac{(b^{2}+\xi)^{3}(b^{2}+\eta)}{a^{2}\alpha\gamma},(z(\xi,\eta))^{2}=-\frac{(c^{2}+\xi)^{3}(c^{2}+\eta)}{a^{2}\alpha\beta},$

where $-a^{2}\leq\xi,\eta\leq-c^{2}$. By substituting these equalities in (3),
we obtain

$\eta(\xi)=-\frac{1+\frac{(a^{2}+\xi)^{3}}{\beta\gamma
a^{2}}+\frac{(b^{2}+\xi)^{3}}{\alpha\gamma
b^{2}}+\frac{(c^{2}+\xi)^{3}}{\alpha\beta
c^{2}}}{\frac{(a^{2}+\xi)^{3}}{\beta\gamma
a^{4}}+\frac{(b^{2}+\xi)^{3}}{\alpha\gamma
b^{4}}+\frac{(c^{2}+\xi)^{3}}{\alpha\beta c^{4}}}.$

Substituting these equalities in the above parametrization of the caustics and
taking $\xi=t$ gives

$(x(t))^{2}=\frac{a^{2}(a^{2}+t)^{3}((b^{2}+c^{2})t+3b^{2}c^{2})}{(a^{2}-c^{2})(a^{2}-b^{2})((a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2})t+3a^{2}b^{2}c^{2})},$
$(y(t))^{2}=\frac{b^{2}(b^{2}+t)^{3}((a^{2}+c^{2})t+3a^{2}c^{2})}{(b^{2}-a^{2})(b^{2}-c^{2})((a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2})t+3a^{2}b^{2}c^{2})},$
$(z(t))^{2}=\frac{c^{2}(c^{2}+t)^{3}((a^{2}+b^{2})t+3a^{2}b^{2})}{(c^{2}-a^{2})(c^{2}-b^{2})((a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2})t+3a^{2}b^{2}c^{2})},$

where $-a^{2}\leq t\leq-b^{2}$ and $-b^{2}\leq t\leq-c^{2}$ correspond to the
two intersections of the caustics with the ellipsoid (3) (See Figure 10, cyan
blue curves). This parametrization determines the curves on the ellipsoid (3)
which separate the regions of the ellipsoid with different values for $n(A)$,
thus completing the solution of Apollonius problem for triaxial ellipsoid.

7\. Conclusion In the paper Apollonius problems for 2 dimensions (ellipse) and
3 dimensions (ellipsoid) were discussed. The number of concurrent normals of
an ellipse (an ellipsoid) is dependent on the position of the point of
concurrency with respect to caustics of the ellipse (the ellipsoid). The cases
when the point of concurrency is on the ellipse (the ellipsoid), required the
study of several different cases of intersections of the caustics with the
given ellipse (ellipsoid). It would be interesting to generalize the results
to 4 (see [37]) and higher dimensions.

8\. Appendix The tangent lines of an ellipse, and the tangent lines and planes
of an ellipsoid are much easier to study than the normals. For completeness,
the problem on the number of concurrent tangent lines (planes) of an ellipse
(ellipsoid) will be discussed here. Let us take point $A(X,Y)$ outside of the
ellipse (1) and find point $B(x_{1},y_{1})$ and $C(x_{2},y_{2})$ on the
ellipse (1) such that $AB$ and $AC$ are tangent lines of the ellipse (1) at
$B$ and $C$, respectively. Since $AB$ and $AC$ have the same slopes as the
ellipse (1) at the point $B$ and $C$, respectively, we have
$\frac{y-Y}{x-X}=-\frac{b^{2}x}{a^{2}y}$, which can be written as
$\frac{x(x-X)}{a^{2}}+\frac{y(y-Y)}{b^{2}}=0$. This is equation of ellipse
through the points $O$ and $A$, with center at
$\left(\frac{X}{2},\frac{Y}{2}\right)$ and semiaxes parallel to the semiaxes
of the original ellipse. Its intersections with the original ellipse (1) are
on the line $\frac{xX}{a^{2}}+\frac{yY}{b^{2}}=1$, which is obtained from
subtraction of the equations of the ellipses. The coordinates of the points
$B$ and $C$ are then determined by

$x_{1}=a\cdot\frac{\frac{X}{a}-\frac{Y}{b}\sqrt{\left(\frac{X}{a}\right)^{2}+\left(\frac{Y}{b}\right)^{2}-1}}{\left(\frac{X}{a}\right)^{2}+\left(\frac{Y}{b}\right)^{2}},\
y_{1}=b\cdot\frac{\frac{Y}{b}+\frac{X}{a}\sqrt{\left(\frac{X}{a}\right)^{2}+\left(\frac{Y}{b}\right)^{2}-1}}{\left(\frac{X}{a}\right)^{2}+\left(\frac{Y}{b}\right)^{2}},$
$x_{2}=a\cdot\frac{\frac{X}{a}+\frac{Y}{b}\sqrt{\left(\frac{X}{a}\right)^{2}+\left(\frac{Y}{b}\right)^{2}-1}}{\left(\frac{X}{a}\right)^{2}+\left(\frac{Y}{b}\right)^{2}},\
y_{2}=b\cdot\frac{\frac{Y}{b}-\frac{X}{a}\sqrt{\left(\frac{X}{a}\right)^{2}+\left(\frac{Y}{b}\right)^{2}-1}}{\left(\frac{X}{a}\right)^{2}+\left(\frac{Y}{b}\right)^{2}}.$

Figure 12: Ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ (green), its 2
tangents (black), line $\frac{xX}{a^{2}}+\frac{yY}{b^{2}}=1$ (blue), and
ellipse $\frac{x(x-X)}{a^{2}}+\frac{y(y-Y)}{b^{2}}=0$ (red).

If we denote by $t(A)$ the total number of tangent lines of the ellipse
passing through $A$, then $t(A)=2$ outside of the ellipse, $t(A)=0$ inside of
the ellipse, and $t(A)=1$ on the ellipse.

Similarly, for the ellipsoid (3), if $A$ is outside of the ellipsoid (3), then
the points $B$, such that $AB$ is tangent to the ellipsoid (3) can be
determined by intersecting (3) with another ellipsoid
$\frac{x(x-X)}{a^{2}}+\frac{y(y-Y)}{b^{2}}+\frac{z(z-Z)}{c^{2}}=0$. All these
intersection points are on the plane
$\frac{xX}{a^{2}}+\frac{yY}{b^{2}}+\frac{zZ}{c^{2}}=1$. The number of tangent
lines $tl(A)$ and tangent planes $tp(A)$ of the ellipsoid (3), which pass
through $A$ is infinity ($tl(A)=tp(A)=\infty$) for exterior points of the
ellipsoid (3), $tl(A)=tp(A)=0$ for interior points, and $tl(A)=\infty$ and
$tp(A)=1$ for the points of the ellipsoid (3) itself.

Figure 13: Ellipsoid
$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1$ (green), plane
$\frac{xX}{a^{2}}+\frac{yY}{b^{2}}+\frac{zZ}{c^{2}}=1$ (blue), and ellipsoid
$\frac{x(x-X)}{a^{2}}+\frac{y(y-Y)}{b^{2}}+\frac{z(z-Z)}{c^{2}}=0$ (red).

Figure 13: Ellipsoid
$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1$ (green), plane
$\frac{xX}{a^{2}}+\frac{yY}{b^{2}}+\frac{zZ}{c^{2}}=1$ (blue), and ellipsoid
$\frac{x(x-X)}{a^{2}}+\frac{y(y-Y)}{b^{2}}+\frac{z(z-Z)}{c^{2}}=0$ (red).

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