Depolarization dyadics for truncated spheres, spheroids, and ellipsoids

Tom G. Mackay***E–mail<EMAIL_ADDRESS>
School of Mathematics and Maxwell Institute for Mathematical Sciences
University of Edinburgh, Edinburgh EH9 3FD, UK
and
NanoMM — Nanoengineered Metamaterials Group
Department of Engineering Science and Mechanics
The Pennsylvania State University, University Park, PA 16802–6812, USA
Akhlesh Lakhtakia
NanoMM — Nanoengineered Metamaterials Group
Department of Engineering Science and Mechanics
The Pennsylvania State University, University Park, PA 16802–6812, USA

Abstract

Depolarization dyadics play a central role in theoretical studies involving
scattering from small particles and homogenization of particulate composite
materials. Closed-form expressions for depolarization dyadics have been
developed for truncated spheres and truncated spheroids, and the formalism has
been extended to truncated ellipsoids; the evaluation of depolarization
dyadics for this latter case requires numerical integration. The Hölder
continuity condition has been exploited to fix the origin of the coordinate
system for the evaluation of depolarization dyadics. These results will enable
theoretical studies involving scattering from small particles and
homogenization of particulate composite materials to accommodate particles
with a much wider range of shapes than was the case hitherto.

Keywords: Depolarization dyadic, Hölder continuity, singularity, truncated
spheroid, truncated sphere

##  1 Introduction

The calculation of the electric field due to a specified source current
density is a fundamental problem in electromagnetics [1]. The usual approach
involves integration of a suitable product of a dyadic Green function and the
source current density over the source region (i.e., the region occupied by
the source current density) [2, 3]. This process is particularly challenging
if the electric field is sought in the source region, since the singularity of
dyadic Green function must be considered then [4]. The integrated singularity
of the dyadic Green function — known as the _depolarization dyadic_ — is a
central pillar in scattering and homogenization theories [5, 6].

The depolarization dyadic is expressible in terms of a surface integral. The
evaluation of this integral is crucially dependent upon the shape of the
surface. For relatively simple shapes, such as spherical [4], spheroidal [7,
8], ellipsoidal [9, 10, 11, 12], cylindrical [13, 14], cubical [13], and
polyhedral [15], closed-form expressions for depolarization dyadics have been
developed, but for more complex shapes numerical methods must be used for
their evaluation.

In this communication, closed-form expressions are developed for truncated
spheres and spheroids, and these are illustrated numerically. The formalism is
extended to truncated ellipsoids. Hereafter, $\varepsilon_{\scriptscriptstyle
0}$, $\mu_{\scriptscriptstyle 0}$, and $k_{\scriptscriptstyle
0}=\omega\sqrt{\varepsilon_{\scriptscriptstyle 0}\mu_{\scriptscriptstyle 0}}$
are the free-space permittivity, permeability, and wavenumber, respectively,
with $\omega$ being the angular frequency; an $\exp(-i\omega t)$ time-
dependence is implicit; and $\underline{\underline{I}}=\hat{{\bf u}}_{\rm
x}\hat{{\bf u}}_{\rm x}+\hat{{\bf u}}_{\rm y}\hat{{\bf u}}_{\rm y}+\hat{{\bf
u}}_{\rm z}\hat{{\bf u}}_{\rm z}$ is the identity dyadic, with $\hat{{\bf
u}}_{\rm x}$, $\hat{{\bf u}}_{\rm y}$, and $\hat{{\bf u}}_{\rm z}$ being unit
vectors aligned with the Cartesian coordinate axes .

##  2 Depolarization dyadics

Suppose that a time-harmonic source current density ${\bf J}({\bf r})$ exists
inside a finite region $V_{s}$ which is bounded by a closed surface. The
unbounded region outside $V_{s}$ is vacuous. The time-harmonic electric field
both outside and inside $V_{s}$ is given as [1]

${\bf E}({\bf r})=i\omega\mu_{\scriptscriptstyle
0}\int_{V_{s}}\underline{\underline{G}}({\bf r},{\bf r}_{s})\mbox{
\tiny{${}^{\bullet}$} }{\bf J}({\bf r}_{s})\,d^{3}{\bf r}_{s},$ (1)

wherein

$\underline{\underline{G}}({\bf r},{\bf
r}_{s})=\left(\underline{\underline{I}}+\frac{\nabla\nabla}{k_{\scriptscriptstyle
0}^{2}}\right)\frac{\exp\left(ik_{\scriptscriptstyle 0}|{\bf r}-{\bf
r}_{s}|\right)}{4\pi|{\bf r}-{\bf r}_{s}|}$ (2)

is the free-space dyadic Green function [3].

For field points outside the source region, i.e., ${\bf r}\notin V_{s}$, the
integral on the right side of Eq. (1) is well behaved and its evaluation
delivers ${\bf E}({\bf r})$ as an analytic function. For field points inside
the source region, i.e., ${\bf r}\in V_{s}$, the evaluation of the integral on
the right side of Eq. (1) requires care because the dyadic Green function is
singular at ${\bf r}={\bf r}_{s}$ [4]. In particular, the double derivative in
the free-space dyadic Green function (2) gives rise to a $|{\bf r}-{\bf
r}_{s}|^{-3}$ term. This is not integrable unless the source current density
${\bf J}({\bf r})$ satisfies the following Hölder continuity condition: there
exist three positive constants $a_{1}$, $a_{2}$, and $a_{3}$ such that [16]

$|{\bf J}({\bf r})-{\bf J}({\bf r}_{s})|\leq a_{1}|{\bf r}-{\bf
r}_{s}|^{a_{2}}$ (3)

for all ${\bf r}_{s}$ satisfying $|{\bf r}-{\bf r}_{s}|\leq a_{3}$.

Provided that the Hölder continuity condition (3) is satisfied, the electric
field inside the source region $V_{s}$ may be expressed as [17]

${\bf E}({\bf r})=i\omega\mu_{\scriptscriptstyle 0}\lim_{\delta\to
0}\int_{V_{s}-V_{\delta}}\underline{\underline{G}}({\bf r},{\bf r}_{s})\mbox{
\tiny{${}^{\bullet}$} }{\bf J}({\bf r}_{s})\,d^{3}{\bf
r}_{s}+\frac{1}{i\omega\varepsilon_{\scriptscriptstyle
0}}\underline{\underline{L}}\mbox{ \tiny{${}^{\bullet}$} }{\bf J}({\bf
r}),\qquad{\bf r}\in V_{s}.$ (4)

Herein $V_{\delta}\subset V_{s}$ is a convex exclusion region of linear
dimensions quantifiable through a small length $\delta$, and surface
$S_{\delta}$, that contains the singular point ${\bf r}={\bf r}_{s}$; a
schematic representation is displayed in Fig. 1(a). The depolarization dyadic

$\underline{\underline{L}}=\frac{1}{4\pi}\int_{S_{\delta}}\frac{{\bf
u}_{\delta}\,{\bf r}}{|{\bf r}|^{3}}\,d^{2}{\bf r},$ (5)

with ${\bf u}_{\delta}$ being the unit outward normal vector to $S_{\delta}$.

The question arises: Where inside the exclusion region $V_{\delta}$ should the
coordinate origin be taken for the integration that delivers the
depolarization dyadic in Eq. (5)? In the case of the spherical exclusion
region, there is no problem because
$\underline{\underline{L}}=(1/3)\underline{\underline{I}}$ regardless of where
the coordinate origin is located inside $V_{\delta}$ [4]. However,
$\underline{\underline{L}}$ generally depends on the choice of coordinate
origin for less symmetric exclusion regions [17, 18]. The origin of the
coordinate system must therefore be taken as the center of the largest sphere
that can be inscribed inside the exclusion region $V_{\delta}$, so that the
Hölder continuity condition (3) holds over the largest portion of
$V_{\delta}$.

In the following sections, depolarization dyadics are calculated for
exclusions regions shaped as truncated spheres, spheroids, and ellipsoids. For
a truncated ellipsoid whose principal axes are aligned with the axes of the
Cartesian coordinate system, the depolarization dyadic has the diagonal form

$\underline{\underline{L}}=L_{x}\,\hat{{\bf u}}_{\rm x}\hat{{\bf u}}_{\rm
x}+L_{y}\,\hat{{\bf u}}_{\rm y}\hat{{\bf u}}_{\rm y}+L_{z}\,\hat{{\bf u}}_{\rm
z}\hat{{\bf u}}_{\rm z};$ (6)

with $L_{x}=L_{y}\equiv L$ for truncated spheres and truncated spheroids.
Furthermore, the trace of the depolarization dyadic

$\mbox{trace}\left\\{\underline{\underline{L}}\right\\}=\frac{1}{4\pi}\int_{S_{\delta}}\frac{{\bf
u}_{\delta}\mbox{ \tiny{${}^{\bullet}$} }{\bf r}}{|{\bf r}|^{3}}\,d^{2}{\bf
r}$ (7)

represents the normalized solid angle of the surface $S_{\delta}$ as viewed
from ${\bf r}={\bf 0}$ [17]. Hence,
$\mbox{trace}\left\\{\underline{\underline{L}}\right\\}=1$. Accordingly, in
the following presentation of closed-form expressions for depolarization
dyadics for truncated spheres and spheroids, there is no need to provide
explicit expressions for $L_{z}$ because $L_{z}=1-2L$.

##  3 Spherical geometry

###  3.1 Truncated sphere

Suppose that the unit sphere centered at $x=y=0,z=\kappa-1$, i.e.,

$x^{2}+y^{2}+\left(z-\kappa+1\right)^{2}\leq 1,$ (8)

is bifurcated by the plane $z=-\kappa$, where $0<\kappa<1$, as schematically
illustrated in Fig. 1(b). The exclusion region $V_{\delta}$ is the upper part
of the sphere bounded below by the plane $z=-\kappa$, and the largest
inscribed sphere is specified by

$x^{2}+y^{2}+z^{2}\leq\kappa^{2}.$ (9)

After integrating over both the curved and the flat parts of $S_{\delta}$, we
found from Eq. (5) that $\underline{\underline{L}}=L\left(\hat{{\bf u}}_{\rm
x}\hat{{\bf u}}_{\rm x}+\hat{{\bf u}}_{\rm y}\hat{{\bf u}}_{\rm
y}\right)+L_{z}\,\hat{{\bf u}}_{\rm z}\hat{{\bf u}}_{\rm z}$ with

$L=\frac{\kappa}{6\left(1-\kappa\right)^{3}\tau}\,\left\\{6-\tau\left(3-3\kappa+\kappa^{2}\right)-\kappa\left[11-3\kappa\left(3-\kappa\right)\right]\right\\},$
(10)

where

$\tau=\sqrt{\left(4-3\kappa\right)\kappa};$ (11)

and confirmed that $L_{z}=1-2L$. Observe that $L\to 0$ and $L_{z}\to 1$ in the
limit $\kappa\to 0$, while $L\to 1/3$ and $L_{z}\to 1/3$ in the limit
$\kappa\to 1$, in agreement with well-known results [4].

Plots of the depolarization factors $L$ and $L_{z}$ versus $\kappa$ are
provided in Fig. 2. As $\kappa$ increases, the value of $L$ increases
monotonically whereas the value of $L_{z}$ decreases monotonically. For all
$\kappa<1$, $L_{z}>L$. The case of the hemisphere, i.e., $\kappa=1/2$, for
which $L=(19/6\sqrt{5})-7/6=0.24951$ and
$L_{z}=\left(50-19\sqrt{5}\right)/15=0.500981$ is notable: $L_{z}$ is
approximately – but not exactly – equal to $2L$.

###  3.2 Double-truncated sphere

Suppose that the unit sphere centered at the origin of the coordinate system,
i.e.,

$x^{2}+y^{2}+z^{2}\leq 1,$ (12)

is symmetrically trifurcated by the planes $z=-\kappa$ and plane $z=\kappa$,
where $0<\kappa<1$, per the schematic representation in Fig. 1(c). The
exclusion region $V_{\delta}$ is the double-truncated sphere bounded by the
planes $z=\pm\kappa$.

The depolarization dyadic for this $V_{\delta}$ again is of the form
$\underline{\underline{L}}=L\left(\hat{{\bf u}}_{\rm x}\hat{{\bf u}}_{\rm
x}+\hat{{\bf u}}_{\rm y}\hat{{\bf u}}_{\rm y}\right)+L_{z}\,\hat{{\bf u}}_{\rm
z}\hat{{\bf u}}_{\rm z}$. With the largest inscribed sphere specified by Eq.
(9), we determined

$L=\frac{\left(3-\kappa^{2}\right)\kappa}{6}$ (13)

from Eq. (5) and verified that $L_{z}=1-2L$.

In Fig. 3, the depolarization factors $L$ and $L_{z}$ are plotted against
$\kappa$. As is the case for the truncated sphere presented in §3.1, for the
double-truncated sphere $L\to 0$ and $L_{z}\to 1$ in the limit $\kappa\to 0$,
while $L\to 1/3$ and $L_{z}\to 1/3$ in the limit $\kappa\to 1$. Furthermore,
the plots of $L$ and $L_{z}$ for the double-truncated sphere in Fig. 3 are
similar to the corresponding plots in Fig. 2 for the truncated sphere.
Differences are most obvious in the regime of small values of $\kappa$ wherein
$L$ (resp. $L_{z}$) increases (resp. decreases) less rapidly for the double-
truncated sphere as $\kappa$ increases.

##  4 Spheroidal geometry

###  4.1 Hemispheroid

Consider the spheroid

$\frac{x^{2}+y^{2}}{\alpha^{2}}+\left(z+\kappa\right)^{2}\leq 1,$ (14)

with $\alpha>0$ and $0\leq\kappa\leq 1/2$. The spheroid is oblate for
$\alpha>1$ and prolate for $\alpha<1$. Suppose that the spheroid is bifurcated
in two halves by the symmetry plane $z=-\kappa$. Then $V_{\delta}$ is the
hemispheroid bounded below by the plane $z=-\kappa$, as schematically
illustrated in Fig. 1(d). The largest inscribed sphere is specified by Eq. (9)
with radius

$\kappa=\left\\{\begin{array}[]{lcr}1/2&\mbox{for}&\alpha>1/\sqrt{2}\vspace{8pt}\\\
\alpha\sqrt{1-\alpha^{2}}&\mbox{for}&\alpha\leq 1/\sqrt{2}\end{array}\right..$
(15)

Again, $\underline{\underline{L}}=L\left(\hat{{\bf u}}_{\rm x}\hat{{\bf
u}}_{\rm x}+\hat{{\bf u}}_{\rm y}\hat{{\bf u}}_{\rm y}\right)+L_{z}\,\hat{{\bf
u}}_{\rm z}\hat{{\bf u}}_{\rm z}$ with $L_{z}=1-2L$. We found from Eq. (5) for
$\alpha\leq 1/\sqrt{2}$ that

$\displaystyle{L=\frac{1+\nu+\alpha^{2}\left[\alpha\left(\alpha+\sqrt{2-\alpha^{2}}\right)-\nu-3\right]-\left(\alpha\gamma\right)^{2}\sqrt{2-\alpha^{2}}\log\displaystyle{\frac{\left(1-\alpha^{2}+\gamma\right)\left(1-\alpha\right)}{\alpha\left(\nu-\gamma\right)}}}{4\nu\gamma^{4}},}$
(16)

where

$\left.\begin{array}[]{l}\nu=\sqrt{2-3\alpha^{2}+\alpha^{4}}\vspace{6pt}\\\
\gamma=\sqrt{1-\alpha^{2}}\end{array}\right\\};$ (17)

and for $\alpha>1/\sqrt{2}$

$\displaystyle{L=\frac{3\left(1+\sqrt{1+4\alpha^{2}}\right)+\alpha^{2}\left(10-8\sqrt{1+4\alpha^{2}}\right)-8\alpha^{4}}{4\gamma^{2}\left(3+8\alpha^{2}-16\alpha^{4}\right)}+\frac{\alpha^{2}\log\displaystyle{\frac{1-2\alpha^{2}+\gamma}{\sqrt{1+3\alpha^{2}-4\alpha^{4}}-1}}}{4\gamma^{3}}}.$
(18)

Plots of the depolarization factors $L$ and $L_{z}$ versus $\alpha$ are
displayed in Fig. 4. As $\alpha$ increases, the value of $L$ decreases
monotonically whereas the value of $L_{z}$ increases monotonically. For
$\alpha<0.569$, $L>L_{z}$; and for $\alpha>0.569$, $L<L_{z}$. In the limit
$\alpha\to 0$ we find $L=\left(2+\sqrt{2}\right)/8$ and
$L_{z}=\left(2-\sqrt{2}\right)/4$, while in the limit $\alpha\to\infty$ we
find $L=0$ and $L_{z}=1$. For $\alpha=1$ the results for the hemisphere are
recovered.

###  4.2 Double-truncated spheroid

Consider the spheroid

$\frac{x^{2}+y^{2}}{\alpha^{2}}+z^{2}\leq 1,$ (19)

centered at the coordinate origin with $\alpha>0$. The spheroid is oblate for
$\alpha>1$ and prolate for $\alpha<1$. Suppose that the spheroid is
trifurcated symmetrically by the planes $z=\pm\kappa$, where $0<\kappa<1$.
Then $V_{\delta}$ is the double-truncated spheroid bounded by the planes
$z=\pm\kappa$, as shown in Fig. 1(e). The largest inscribed sphere is centered
at the origin of the coordinate system.

Equation (5) yields $\underline{\underline{L}}=L\left(\hat{{\bf u}}_{\rm
x}\hat{{\bf u}}_{\rm x}+\hat{{\bf u}}_{\rm y}\hat{{\bf u}}_{\rm
y}\right)+L_{z}\,\hat{{\bf u}}_{\rm z}\hat{{\bf u}}_{\rm z}$, where

$L=\frac{1-L_{z}}{2}=\frac{-\nu+\alpha^{2}\tan^{-1}\nu}{2\left(\alpha^{2}-1\right)^{3/2}}$
(20)

with

$\nu=\kappa\sqrt{\frac{\alpha^{2}-1}{\alpha^{2}-\kappa^{2}\left(\alpha^{2}-1\right)}};$
(21)

and again $L_{z}=1-2L$.

In Fig. 5, the depolarization factor $L$ is plotted against $\alpha$ and
$\kappa$. As $\alpha$ increases, $L$ decreases monotonically and $L_{z}$
increases monotonically, for all values of $\kappa$. As $\kappa$ increases,
$L$ increases monotonically and $L_{z}$ decreases monotonically for all values
of $\alpha>0$. In the limit $\kappa\to 0$ we find $L=0$ and $L_{z}=1$, while
in the limit $\kappa\to 1$ we find

$L=\frac{1}{2}\left(\frac{1}{1-\alpha^{2}}+\alpha^{2}\frac{\tan^{-1}\sqrt{\alpha^{2}-1}}{\left(\alpha^{2}-1\right)^{3/2}}\right),$
(22)

in agreement with the standard results for spheroids [8]. In the limit
$\alpha\to 0$, $L=1/2$ and $L_{z}=0$; while $L=0$ and $L_{z}=1$ in the limit
$\alpha\to\infty$. And the results for the hemisphere are recovered with
$\alpha=1$.

##  5 Ellipsoidal geometry

In the cases of truncated ellipsoidal geometries, closed-form expressions for
$L_{x}$, $L_{y}$, and $L_{z}$ could not be obtained, but numerical methods
were used to evaluate these depolarization factors.

###  5.1 Hemi-ellipsoid

Consider the ellipsoid

$\frac{x^{2}}{\alpha^{2}}+\frac{y^{2}}{\beta^{2}}+\left(z+\kappa\right)^{2}\leq
1.$ (23)

with $\alpha>0$, $\beta>0$, and $0\leq\kappa\leq 1/2$. Suppose that the
ellipsoid is truncated by the symmetry plane $z=-\kappa$. We chose
$V_{\delta}$ as the upper hemi-ellipsoid bounded below by the plane
$z=-\kappa$. The largest inscribed sphere is specified by Eq. (9) with radius

$\kappa=\left\\{\begin{array}[]{lcr}1/2,&\mbox{for}&\alpha>1/\sqrt{2},\quad\beta>1/\sqrt{2}\vspace{8pt}\\\
\alpha\sqrt{1-\alpha^{2}}&\mbox{for}&\alpha\leq
1/\sqrt{2},\quad\alpha<\beta\vspace{8pt}\\\
\beta\sqrt{1-\beta^{2}}&\mbox{for}&\beta\leq
1/\sqrt{2},\quad\beta<\alpha\end{array}\right..$ (24)

In Fig. 6, plots of the depolarization factors $L_{x}$ and $L_{y}$ versus
$\alpha$ and $\beta$ are displayed; the plot of $L_{z}$ may be inferred from
$L_{z}=1-L_{x}-L_{y}$. All depolarization factors vary smoothly as $\alpha$
and $\beta$ increase. In particular, $L_{x}$ decreases markedly as $\alpha$
increases but is relatively insensitive to changes in $\beta$; $L_{y}$
decreases markedly as $\beta$ increases but is relatively insensitive to
changes in $\alpha$; and $L_{z}$ generally increases as both $\alpha$ and
$\beta$ increase.

###  5.2 Double-truncated ellipsoid

Consider the ellipsoid

$\frac{x^{2}}{\alpha^{2}}+\frac{y^{2}}{\beta}+z^{2}\leq 1,$ (25)

centered at the origin of the coordinate system with $\alpha>0$ and $\beta>0$.
Suppose that the ellipsoid is trifurcated by the planes $z=\pm\kappa$, where
$0<\kappa<1$. Then $V_{\delta}$ is the double-truncated ellipsoid bounded by
the planes $z=\pm\kappa$, and the largest inscribed sphere is centered at the
origin of the coordinate system.

For $\kappa\in\left\\{0.1,0.5,0.9\right\\}$, plots of $L_{x}$ and $L_{y}$
versus $\alpha$ and $\beta$ are provided in Fig. 7; the plot of $L_{z}$ may be
inferred since $L_{z}=1-L_{x}-L_{y}$. The plots in Fig. 7 for the double-
truncated ellipsoid are qualitatively similar to those in Fig. 6 for the hemi-
ellipsoid. The effects of varying $\kappa$ are most appreciable at low values
of $\alpha$ for $L_{x}$, at low values of $\beta$ for $L_{y}$, and at low
values of both $\alpha$ and $\beta$ for $L_{z}$. Generally, the values of the
depolarization factors $L_{x}$, $L_{y}$, and $L_{z}$ are substantially more
sensitive to variations in $\alpha$ and $\beta$ than they are to variations in
$\kappa$.

##  6 Closing remarks

Depolarization dyadics are central to theoretical studies in scattering and
homogenization, but closed-form expressions for these entities have been
available only for a few simple shapes. Closed-form expressions for
depolarization dyadics have been developed herein for truncated spheres and
truncated spheroids, and the formalism has been extended to truncated
ellipsoids; the evaluation of depolarization dyadics for this latter case
requires numerical integration. These theoretical results will enable studies
of scattering from complex-shaped small particles [19] as well as of
homogenization of particulate composite materials containing complex-shaped
particles [6]. These results are particular valuable in view of the ongoing
rapid development of nanocomposite materials for optical applications [20].

Acknowledgments: TGM was supported by EPSRC (grant number EP/V046322/1). AL
was supported by the US National Science Foundation (grant number DMS-1619901)
as well as the Charles Godfrey Binder Endowment at Penn State.

Declaration of competing interest: The authors declare that they have no known
competing financial interests or personal relationships that could have
appeared to influence the work reported in this paper.

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(a) Exclusion region $V_{\delta}$ inside source region $V_{s}$.

(b) Truncated unit sphere with largest inscribed sphere of radius
$\kappa\in(0,1)$.

(c) Double-truncated unit sphere with largest inscribed sphere of radius
$\kappa\in(0,1)$.

(d) Hemispheroid with largest inscribed sphere of radius $\kappa\in(0,1)$ for
$\alpha<1/\sqrt{2}$ (top) and $\alpha>1/\sqrt{2}$ (bottom).

(e) Double-truncated spheroid with largest inscribed sphere of radius $\alpha$
for $\alpha<\kappa$ (top) and of radius $\kappa$ for $\alpha>\kappa$ (bottom).

Figure 1:  Schematic representations of exclusion regions.

Figure 2:  $L$ and $L_{z}$ plotted against $\kappa\in(0,1)$ for the truncated
unit sphere.  Figure 3:  $L$ and $L_{z}$ plotted against $\kappa\in(0,1)$ for
the double-truncated unit sphere.  Figure 4:  $L$ and $L_{z}$ plotted against
$\alpha$ for the hemispheroid.

Figure 5:  $L$ plotted against $\alpha\in(0,4]$ and $\kappa\in(0,1)$ for the
double-truncated spheroid.

Figure 6:  $L_{x}$ and $L_{y}$ plotted against $\alpha\in(0,3]$ and
$\beta\in(0,3]$ for the hemi-ellipsoid.

Figure 7:  $L_{x}$ and $L_{y}$ plotted against $\alpha\in(0,3]$ and
$\beta\in(0,3]$ for the double-truncated ellipsoid with
$\kappa\in\left\\{0.1,0.5,0.9\right\\}$.