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\DOI{10.5802/crphys.231}
\datereceived{2024-05-09}
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\dateposted{2025-02-06}
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\begin{noXML}

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\title{The cat's-eye phenomenon: a mathematical approach}

\alttitle{L'effet \oe{}il-de-chat : une approche math\'{e}matique}

\author{\firstname{Jean-Pierre} \lastname{Gauthier}\IsCorresp}
\address{Centre de Recherches Gemmologiques, 2 rue de la
Houssini\`{e}re, 44322 Nantes, France}
\email[J.-P. Gauthier]{jpk.gauthier@gmail.com}

\author{\firstname{Thanh Nhan} \lastname{Bui}}
\address{Rue du Compas 47/4, 1070 Bruxelles, Belgium}
\email[T. N. Bui]{tnhan93@gmail.com}

\author{\firstname{Jacques} \lastname{Fereire}}
\addressSameAs{1}{Centre de Recherches Gemmologiques, 2 rue de la
Houssini\`{e}re, 44322 Nantes, France}
\email[J. Fereire]{jacques.fereire@hotmail.fr}

\keywords{\kwd{Chatoyancy}
\kwd{Cat's-eye}
\kwd{Asterism}
\kwd{Optical phenomenon}
\kwd{Mathematical approach}}

\altkeywords{\kwd{Chatoyance}
\kwd{{\OE}il-de-chat}
\kwd{Ast\'erisme}
\kwd{Ph\'enom\`ene optique}
\kwd{Approche math\'ematique}}

\begin{abstract}
This paper provides a mathematical basis for the well-known cat's-eye
effect appearing on chatoyant gemstones. After reminding the reader of
the physical origin of this effect and the conditions for observing it,
we introduce all required parameters and some reasonable assumptions in
order to simplify the calculations. In fact, we describe the motion of
a point located at the apex of the cat's-eye line in the ``main''
plane, defined by the directions of the source and the observer and
perpendicular to the set of acicular inclusions responsible for the
optical phenomenon. These calculations confirm the qualitative
cat's-eye behavior commonly observed in various situations, when the
source or the observer moves around the cabochon or when the stone
rotates under them.
\end{abstract}

\begin{altabstract}
Le but de cet article est de donner une base math\'{e}matique \`{a}
l'effet bien connu d'\oe{}il-de-chat apparaissant sur les pierres
pr\'{e}cieuses chatoyantes. Apr\`{e}s avoir rappel\'{e} l'origine
physique de cet effet et les conditions de son observation, nous
introduisons tous les param\`{e}tres n\'{e}cessaires \`{a} cette
\'{e}tude et quelques hypoth\`{e}ses raisonnables afin de simplifier
les calculs. En fait, nous d\'{e}crivons le mouvement d'un point
situ\'{e} \`{a} l'apex de la ligne de l'\oe{}il-de-chat dans le plan
``principal'', d\'{e}fini par les directions de la source et de
l'observateur et perpendiculaire \`{a} l'ensemble des inclusions
aciculaires responsables du ph\'{e}nom\`{e}ne optique. Ces calculs
confirment le comportement qualitatif de l'\oe{}il-de-chat couramment
observ\'{e} dans diverses situations, lorsque la source ou
l'observateur se d\'{e}placent autour du cabochon ou lorsque la pierre
tourne sous eux.
\end{altabstract}

\shortrunauthors

\maketitle

\end{noXML}

\let\rmmu\upmu

\section{Introduction}\label{sec1}

Numerous minerals exhibit chatoyancy or asterism. Chatoyancy is an
optical phenomenon that creates a band of light moving across a
material's surface (or several bands in the case of asterism). The book
by Steinbach~\cite{9} gives a comprehensive account of the most
well-known, spectacular and common to the rarest of these gems showing
these particular effects, and lists over 200 mineral species chatoyant
or asteriated. Cat's-eyes and stars that can be observed on gems cut
\textit{en cabochon} (gem with a curved surface instead of faceted) are
basically mobile depending on different parameters, such as the
position of the source or the observer and the stone orientation. If
the physical origin of this optical phenomenon is well known, its
mathematical description has never been achieved. We propose to follow
the path of a light ray at the origin of the phenomenon in some
particular cases and to carry out a mathematical study for these
various parameters. For clarity, each case considered will be followed
by a short conclusion, in bold.

\section{The cat's-eye phenomenon: physical bases}\label{sec2}

The cat's-eye effect has long been physically described in various
papers~\cite{11,10,6,12}. It arises from the light scattering on
acicular (needle-like) structures, whether they are exsolutions or
tubular cavities present in a 
single-crystal mineral matrix or even
fibrous materials (Figure~\ref{fig1}). From now on, for simplicity, we
will use the term ``needles'' to refer to tubes, needles, and fibers.

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig1}Chatoyancy caused by (a)~hollow tubes in a
Tanzanian cat's-eye opal (2.95~cts); (b) optical fibers in an imitation
cat's-eye (4.56~cts); (c)~magnetite needles in a 4-ray star diopside
(21.70~cts); (d)~rutile needles in a 6-ray synthetic star sapphire
(10.75~cts). Photos: J.~P.~Gauthier.}
\end{figure}

A 3D simulation of the phenomenon was proposed by Yokoi
et~al.~\cite{13} using a model of needles with microfacets, for which
these authors developed an algorithm to describe the distribution of
rays scattered by the inclusions. However, the purpose of this work was
to build a model illustrating the movement of the cat's-eye or the star
during the rotation of the stone, not to mathematically explain the
phenomenon.

The physical basis of light scattering is quite simple to describe when
a single ray of light falls on a thin isolated needle placed in air.
However, the optical effect known as chatoyancy, which results from the
refraction of a light beam on a cabochon followed by scattering on a
set of parallel needles grown along a particular direction within a
mineral matrix (exsolved inclusions) is more challenging to study in a
comprehensive way. We will therefore gradually analyze different
aspects of this ``cat's-eye'' effect.

\section{Scattering of a light beam by a thin needle in the
air}\label{sec3}

The theory developed by the authors referred to in the previous
paragraph tells us that when a light beam hits a very thin linear
scatterer, it is scattered as a conical surface of light centered on
the long axis of the needle, and having as half top angle (i.e.\ angle
generating a cone by rotation around an axis) the angle ${\theta}$
between the incident beam and the needle (Figure~\ref{fig2}). Therefore,
regardless of the direction of the needle, i.e.\ regardless of the
opening of the cone angle, the incident ray is always one of the lines
generating the cone. As the needle tilt increases relative to the
vertical beam, the scattering cone gradually opens, until the cone
becomes flat when the needle is perpendicular to the incident ray.

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig2}A needle placed in the air forming different
angles ${\theta}$  with respect to the incident ray scatters the light
into hollow conical veils. The needle being placed at the center of a
virtual sphere, the scattering cones intercept it in circular rings,
which all share a common point at the lower pole, and widen in diameter
as the needle moves away from the vertical. Two particular cases may be
emphasized: (i)~when the needle is vertical, the scattering cone is
reduced to a line emerging at the lower pole; (ii)~when the needle is
horizontal, the cone becomes flat (vertical disc).}
\end{figure}

To visualize this effect, one can imagine the incident beam entering
through a hole drilled in a translucent sphere acting as a screen. At
the center of the sphere lies a needle that moves in various
directions. The hollow cones of scattered light will then cross the
screen as luminous circles.

\section{Scattering of a light beam by parallel needles exsolved in a
mineral with a flat or curved surface}\label{sec4}

The scattering phenomenon should now be applied not to a single needle
in air, but to a set of parallel needles embedded into a single
crystal. A ray of light hitting the surface is first refracted, then
scattered by the needles and again refracted out of the mineral. For an
incoming parallel beam, two cases arise:
\begin{enumerate}[(i)]
\item The surface of the mineral is flat (Figure~\ref{fig3}a): all the
rays follow similar paths entering the stone and then exiting it. They
therefore remain parallel at the exit. If they are directed toward the
observer, he sees a uniform sheen everywhere, otherwise the surface
remains entirely dark.

\item The surface is curved (Figure~\ref{fig3}b): after entering the
mineral, the rays are no longer parallel, neither in the stone, nor at
their exit. A single ray reaches the observer and its exit point
corresponds to the position of the cat's-eye.
\end{enumerate}

\begin{figure}
\includegraphics{fig03}
\caption{\label{fig3}Illumination of the surface of a chatoyant stone
with a parallel beam. Rays coming from a source~$S$ at infinity are
refracted into the stone at $I_1$ and $I_{2}$, reach the needles at
$N_{1}$  and $N_{2}$ and are scattered as hollow cones. The rays of the
cones located in the working plane emerge at $J_{1}$  and $J_{2}$. The
emergent rays are: (a)~parallel in case of a flat surface: if they are
in the direction of the observer $O$, also at infinity, all the surface
is shining, creating a ``sheen''; otherwise, the whole surface appears
dark; (b)~convergent in case of a curved surface: in this case, only
one ray reaches the observer $O$, and the corresponding exit point
$J_{2}$ indicates the position of the cat's-eye.}
\end{figure}


\smallskip\noindent
\textbf{${\rightarrow}$ This is why the gem has to be cut as a
\textit{cabochon}. In a stone with a curved surface, the observer will
see a bright line, not a bright or dark surface.}


\section{Mathematical approach}\label{sec5}

\subsection{Overview of the problem}\label{sec51}

The general aim of the study is to follow the movement of the bright
line characterizing the cat's-eye phenomenon on the surface of a
cabochon when the stone is rotated or when the observer or the source
moves, and to determine the limits beyond which the phenomenon
disappears.

In the most common situation, the source is fixed (ceiling light or
sun) and the observer sits or stands in front of his work table, and
rotates the cabochon in his hand to look at the cat's-eye
(Figure~\ref{fig4}a). Less frequently, he moves a penlight over the
static cabochon (Figure~\ref{fig4}b). Rarely, he himself moves in front
of the static cabochon illuminated by a ceiling light
(Figure~\ref{fig4}c).

\begin{figure}
{\vspace*{-2pt}}
\includegraphics{fig04}
{\vspace*{-2pt}}
\caption{\label{fig4}The sketch shows the three geometries for
observing a cat's-eye: (a)~the cabochon~$C$ rotates; (b)~the lamp is
rotated (angularly displaced) by the observer~$O$; (c)~the observer moves
in front of the cabochon. N.B.: (a) is the most frequent observation
mode ($S$ and $O$ fixed); (b) often corresponds to a gemmological
examination ($C$ and $O$ fixed). (c) is rarely used as an examination mode
($S$ and $C$ fixed).}
{\vspace*{-2pt}}
\end{figure}

In order to establish a general equation for these different cases, we
define below the necessary parameters. We propose simplifying
assumptions that are not too restrictive in order to facilitate the
understanding and description of the phenomenon, and to develop a clear
mathematical argumentation. We will then examine qualitatively what
happens when these assumptions are not exactly satisfied.

\section{Assumptions and definition of angular parameters
(Figure~\myRef{fig5})}\label{sec6}

\begin{figure}
{\vspace*{-2pt}}
\includegraphics{fig05}
{\vspace*{-2pt}}
\caption{\label{fig5}Angular parameters: Needle ($N$) tilt, ${\alpha}$;
directions of source $S$, $\sigma$ and observer $O$, ${\omega}$; entry
and exit incidence angles $i$ and $i'$; ${\alpha}$, $\sigma$,
${\omega}$, $i$ and $i'$ are counted positively clockwise; refraction
angles at the surface, ${r}$ and $r'$ (same sign as $i$ and $i'$,
respectively).}
{\vspace*{-2pt}}
\end{figure}

\subsection{Shape of the stone}\label{sec61} 
We will first consider a cabochon of spherical shape to facilitate the
mathematical analysis. Later, we'll consider the case of sharp or flat
cabochons.

\subsection{Light source}\label{sec62}

The light source~$S$ is located at infinity, so that the beam reaching
the stone consists of parallel rays. Its angular position is defined by
the angle $\sigma$, whose origin is counted from the vertical. The
incidence angle $i$ of a ray arriving on the sphere is defined from
the direction of the source. This ray is refracted at the surface of
the sphere at the angle ${r}$.

\subsection{Observer}\label{sec63}
The observer~$O$ is also assumed to be at infinity so that he receives
the rays from a specific direction, characterized by the angle
${\omega}$. This is counted from the upper pole of the sphere.

\subsection{Needles}\label{sec64}

Parallel acicular exsolutions are characterized by an azimuthal angle
${\varphi}$ taken equal to zero (i.e.\ the needles are parallel to the
working plane) and an angle ${\alpha}$, counted from the horizontal
plane. The incidence angle of a ray reaching a needle is denoted
$\theta$. This ray gives rise to a scattering cone with a half top
angle $\theta$. The main ray of this cone reaches the surface at a
``refraction'' angle $r'$ and emerges into the air under the
``incidence'' angle $i'$.

N.B.: Angles $i$, $i'$, $\sigma$, ${\omega}$ and ${\alpha}$ are
counted positively clockwise. For ${r}$ and $r'$, which will appear by
their trigonometric functions, it is worth mentioning that, according
to the Snellius--Descartes law, their sign must be the same as that $i$
and $i'$ respectively, in calculations and drawings. 


\subsection{Location of scattering elements}\label{sec65}

Only needles located close to the surface will be considered. This is
certainly true for nearly opaque stones such as cat's-eye black
diopside and star garnet, as well as most synthetic rubies and
sapphires. Light that penetrates deeply into the stone is absorbed and
only the rays scattered near the surface come out. In translucent gems,
the light arriving on deeper needles has already been scattered once or
several times by the needles near the surface~\cite{7}.
The corresponding beams lose their intensity and coherence. Therefore,
they contribute to the milky aspect and significantly less to the
cat's-eye. The proposed simplifying assumption is not unreasonable. It
allows us to consider that the exit point~$J$ of an emerging ray is very
close to the entry point~$I$ of the incident ray. $I$ (${\approx} J$) is
therefore the point at which the observer should aim. We can then
equate the sphere with its tangent plane~$\Sigma$ at
$I\approx J$ (see Figure~\ref{fig7}).

\subsection{Angular restrictions}\label{sec66}

(i) In order to consider all angular positions of the needles with
respect to the light source, the angle ${\alpha}$ should in principle
range from ${-}$90\textdegree~to ${+}$90\textdegree. However, the angular
range over which the cat's-eye can be observed is significantly
reduced, due to the existence of a critical refraction angle.

(ii) The source can only illuminate half of the surface of the sphere
and the observer can only receive rays from another half. If the
directions of the source and the observer do not coincide, the rays
reaching the observer come at most from a spherical section smaller
than half of a sphere. In the plane of Figure~\ref{fig6}, the possible
angular positions of point $I\approx J$ range from
$\omega-90$\textdegree~to $\sigma+90$\textdegree, which means over
$180-(\omega-\sigma)$ degrees.

\begin{figure}
\includegraphics{fig06}
\caption{\label{fig6}Restricted domain of visibility of the cat's-eye
when the directions of the source~$S$ and the observer~$O$ do not
coincide.}
\end{figure}

\smallskip\noindent
\textbf{${\rightarrow}$ The angular range of visibility of the cat's-eye
decreases as the direction of observation moves away from that of the
source. When the source and the observer are oriented in the same
direction, the cat's-eye is seen in the widest angular range.}


\subsection{Refractive index of the material}\label{sec67}

An incident ray at angle $i$ is refracted at angle ${r}$, in accordance
with the Snellius--Descartes law: $\sin i=n{\cdot}\sin r$ (where $n$ is
the refractive index). A ray scattered by a needle emerges in the air
according to the same law, $\sin i'=n{\cdot}\sin r'$, with a limit
value for $i_{\mathrm{lim}}=i'_{\mathrm{lim}}=90\text{\textdegree}$ and a
corresponding critical refraction angle
${r}_{\mathrm{lim}}=r'_{\mathrm{lim}}=\arcsin(1/{n})$. The impact of
the refractive index will be studied later in the case of four
chatoyant minerals.

\section{Needles in the main plane}\label{sec7}

We consider here the great circle of the sphere, coplanar to the
directions of the source and the observer, and we assume that the
needles are parallel to this plane. It constitutes a plane of symmetry
for the scattering cones surrounding the needles it contains. This
plane intersects two specific rays of the cone, one being an extension
of the incident ray on the needle ($IN$). Earlier described as the
\textbf{main ray}, the other ($NJ$) is located in the main plane, even
after refraction in the air at point $J\approx I$ ($I$, $J$, $N$
illustrated on Figure~\ref{fig3}b). Only this point of the cat's-eye
will be mathematically defined, but it will govern the whole movement
of the bright band of chatoyancy.

We will follow the evolution of this point~$J$ when the sphere, and thus
the needles, rotate around an axis perpendicular to the main plane. In
our assumptions, since the needle is just below to the surface,
points~$J$ and $I$ are close to each other on plane $\Sigma$, tangent
to the sphere, as explained earlier. We therefore study the position of
point $J$ (${\approx}I$) at which the emergent ray points toward the
observer. It is characterized by the angle $i$ between the incident ray
and the normal to the plane tangent to the sphere at point~$I$, as a
function of the needle orientation, $\alpha$. This angle~$i$ is the
unknown which corresponds to the angular position $\sigma-i$ of the
cat's-eye apex on the sphere. It must be evaluated as a function of
parameters $\alpha$, $\sigma$, and $\omega$ (Figure~\ref{fig7}).

\begin{figure}
\includegraphics{fig07}
\caption{\label{fig7}Optical path of a ray emanating from the
source~$S$, refracted at point~$I$, scattered by a needle at point~$N$
under angle~${\theta}$. The main ray scattered under the
angle~${\theta}$  is refracted at point~$J$ to emerge in the direction of
the observer~$O$. The respective angular directions~$\sigma$
and~${\omega}$ of the source and the observer are defined with respect
to the vertical, the tilt angle~${\alpha}$ of the needles with respect
to the horizontal, and the angles of incidence and refraction~$i$,
$i'$, ${r}$ and $r'$ with respect to the normal to the tangent
plane~${\Sigma}$ of the sphere at points~$I$ or~$J$. Angles~${\alpha}$,
$\sigma$, ${\omega}$, $i$ and~$i'$ are counted positively clockwise;
${r}$ and $r'$ must be of the same sign as $i$ and $i'$, respectively
(thus, the angle
$NIJ=90\text{\textdegree} -|r'|=90\text{\textdegree}+r'$, where $r'$ is negative
with an absolute value equal to $|r'|$). With the
initial simplifying assumptions, the point $I\approx J$ corresponds to
the top of the bright line representing the cat's-eye at the surface of
the spherical cabochon, at the angular position  $\sigma-i$.
${\Sigma}$ is the trace of the tangent plane to the sphere at point
$I\approx J$.} 
\end{figure}

\section{Determination of the angular position $\sigma-i$ of the
cat's-eye}\label{sec8}

Figure~\ref{fig7} represents the path followed by a ray from the source
to the observer, first refracted at point~$I$, arriving on the needle at
point~$N$ under angle~${\theta}$, then scattered on the needle under the
same angle~${\theta}$, giving rise to the \textbf{main ray}~$NJ$,
refracted and emerging at point $J$. All angles defined previously are
depicted in Figure~\ref{fig7}. With the extension of the directions of
the needle and the trace of the tangent plane~${\Sigma}$ to the
sphere, we can establish angular relations from:

\smallskip\noindent
triangle NIA: $180=(90+r)+\theta+(\sigma-i-\alpha)$.

\smallskip\noindent
triangle NJA: $180=(90+r')+(180-\theta)+(\sigma-i-\alpha)$.


Which express that the sum of the angles in a triangle is equal
to~180\textdegree. Eliminating ${\theta}$  between these two formulae,
we obtain:
{\begin{equation}\label{eq1}
\sigma-i=\alpha-(r+r')/2
\end{equation}}\unskip
or (with $\omega-i'=\sigma-i$ : see angles near point $J$) 
{\begin{equation}\label{eq2}
i'=i+\omega-\sigma.
\end{equation}}\unskip

These simple equations govern the position of the point of emergence
$J\approx I$ and therefore the angular position $\sigma-i$ of the
cat's-eye. The purpose of the following mathematical operations is only
to express $r$, $r'$ and $i'$ as a function of $i$, using the
Snellius--Descartes law, which is written as follows:
{\begin{equation*}
\sin r=\sin i/n  \quad\mbox{or}\quad r=\arcsin(\sin i/n) 
\quad\mbox{and similarly, } r'=\arcsin(\sin i'/n).
\end{equation*}}\unskip

Replacing now $r$, $r'$ and $i'$ versus $i$ in Equations~(\ref{eq1})
and~(\ref{eq2}), we get:
{\begin{equation}\label{eq3}
\sigma-i=\alpha-\{\arcsin[\sin i/n ]+\arcsin
[\sin(i+\omega-\sigma)/n ]\}/2 .
\end{equation}}\unskip

Equation~(\ref{eq3}) is the form of Equation~(\ref{eq1}) which derives
from the simple Snellius--Descartes law applied to the light ray coming
in and out of the spherical sample surface, allowing to operate
practical calculations. Such trigonometric equation (earlier called
``transcendental'') can be solved by considering each of the terms on
either side of the sign equal as a function $y$ or $y'$:
{\begin{equation}\label{eq4}
\begin{array}{c}
y=\sigma-i\\
y'=\alpha-\{\arcsin[\sin i/n ]+\arcsin [\sin(i+\omega-\sigma)/n]\}/2.
\end{array}
\end{equation}}\unskip

The graphical solution of Equations~(\ref{eq4}) is given by the
intersection of the straight-line ${y}$ and the curve $y'$.

Examples are provided in the following paragraphs. The curves are
obtained from tables of values calculated for ${y}$ and $y'$. The range
of application of these curves is limited by the values defined in the
previous paragraph~\ref{sec66} ``angular restrictions''.

\section{Determination of the relative movement of the cat's-eye}\label{sec9}

The pair of Equations~(\ref{eq4}) will be used to follow the
displacement of the cat's-eye as one of the three entities moves
(source, observer or needles), the other two being fixed. To highlight
the direction of motion, we will write Equation~(\ref{eq3}) for two values of
the variable parameter, setting an arbitrary value to the other two
parameters. As example, sillimanite will be considered for practical
calculations, with its refractive index $n=1.66$.

\subsection{Rotation of the stone: $\alpha$ (needles) variable,
$\omega$ and $\sigma$ (source and eye) fixed}\label{sec91}

We take for example ${\sigma}=20$\textdegree~and
$\omega=-30$\textdegree, and solve Equations~(\ref{eq4}) for
$\alpha=-15$ and ${\alpha}=+15$\textdegree. The graphical resolution
is presented in Figure~\ref{fig8}a. The straight line $y$ does not
depend on $\alpha$, and thus is the same for both values of $\alpha$.
The two curves $y'=\alpha-\arcsin[\sin i/n]$ are similar, only offset
vertically by  30\textdegree, when passing from
$\alpha=-15$\textdegree~to $\alpha=+15$\textdegree. The approximate
solutions correspond to $i\sim 47$\textdegree~and
$i\sim-16$\textdegree~(orange and blue vertical arrow, respectively). A
refinement carried out with a step of 0.1\textdegree~on the horizontal
axis gives more accurate values (47.45\textdegree~and
${-}$16.84\textdegree), obtained from the insets in Figure~\ref{fig8}a.
Thus, the angular position of the cat's-eye moves from
${\sigma}-i=-27.45$\textdegree~to ${\sigma}-i=36.84$\textdegree,
corresponding to a displacement of 64.29\textdegree~(orange and blue
horizontal arrows, respectively), while the needle tilt angle $\alpha$
increases as well, from 30\textdegree~(Figure~\ref{fig8}b). The
following can be concluded:

\smallskip\noindent
\textbf{${\rightarrow}$ When the stone rotates in one direction, the observer
and the source being fixed, the cat's-eye moves faster, in the same
direction.}

\begin{figure}
\includegraphics{fig08}
\caption{\label{fig8}(a) Practical example of graphical solution for
the case of sillimanite ($n=1.66$), where Equations~(\ref{eq4}) are
plotted as a function of angle $i$ in order to obtain the angular
positions $\sigma-i$ of the cat's-eye apex with a light source located
in the direction ${\sigma}=20$\textdegree~and an observer at
$\omega=-30$\textdegree, for needle tilt angles 
$\alpha=\pm15$\textdegree. The solutions (orange and blue arrows) lie
at the intersection points between the straight-line $y$ (in black)
and the curves $y'$ (orange for ${\alpha}=-15$\textdegree~and blue for
$\alpha=+15$\textdegree). The line $y$ remains unchanged, while the
curves $y'$ shift by 30\textdegree, corresponding to the change from
$\alpha=-15$\textdegree~to ${\alpha}=+15$\textdegree. (b) Optical
paths for these two situations: When $\alpha$ increases by
30\textdegree, the angular position $\sigma-i$ of the cat's-eye also
increases by 64.29\textdegree~(from the orange point to the blue point
$I\approx J$).}
\end{figure}

\subsection{Movement of the source: $\sigma$ variable, $\alpha$ and
$\omega$ (needle and observer) fixed}\label{sec92}

This time, we assign simple values to the needle tilt and the observer
angle, for instance: $\alpha=0$\textdegree~and $\omega=0$\textdegree.
In these conditions, the range of validity for the point $I\approx J$
extends from ${-}$90\textdegree~to ${+}$90\textdegree.

Let us take two values of the source direction:
$\sigma=0$\textdegree~and $\sigma=30$\textdegree.


For $\sigma=0$\textdegree, Equation~(\ref{eq3}) is easily solved: 
${i}=\arcsin(\sin i/1.66)$, verified if, and only if, $i=0$. Thus, 
$\sigma- i=0$, and point $I\approx J$ stays at the apex of the great
circle.


For $\sigma=30$\textdegree, the Equations~(\ref{eq4}) become:
{\begin{equation*}
y=30\mbox{\textdegree}-i \quad \mbox{and}\quad  y'=\{\arcsin
(\sin i/1.66)+\arcsin[\sin(i-30\mbox{\textdegree})/1.66]\}/2.
\end{equation*}}\unskip

In Figure~\ref{fig9}a, the graphical solution is located at the
intersection of the straight line $y$ (in black) and the curve $y'$ (in
blue). In the inset of the same figure, a refinement of the measure
gives the value: $\sigma-i=-19.36$\textdegree. The whole light path
from the source to the observer is plotted in blue on
Figure~\ref{fig9}b.

\begin{figure}
\includegraphics{fig09}
\caption{\label{fig9}(a) Solving the equations giving the angular
position of the cat's-eye when the source or the observer moves. We fix
the needle tilt equal to 0\textdegree, as well as the initial values of
$\sigma$ and $\omega$, corresponding to the black point on the
horizontal axis. Then, we successively change the value of $\sigma$
($=30$\textdegree) or the value of $\omega$ ($=-30$\textdegree). For
each value of $\sigma$ or $\omega$, curve $y'$ is identical in both
cases (in black). In the former case, the angular position $\sigma-i$
is negative, and positive in the latter. (b) Representation of the
light paths in both cases. The cat's-eye, initially represented by
point $I_{0}$, moves in the opposite direction of the source (from
black point $I_{0}$ to blue point $I$) or of the observer (from black
point $I_{0}$ to red point $I$).}
\end{figure}

Thus, from $\sigma=0$\textdegree~to $\sigma=30$\textdegree, the angle
$\sigma-i$ associated with the point $I$ changes from 0\textdegree~to
${-}$19.36\textdegree~which means the following:

\smallskip\noindent
\textbf{${\rightarrow}$ When the lamp angularly moves away from the observer,
with the latter and the stone fixed, the cat's-eye rotates more slowly,
in the opposite direction.}


\subsection{Movement of the observer: $\omega$ variable,  $\sigma$
and ${\alpha}$ (gem and source) fixed}\label{sec93}

Taking the simple values ${\sigma}=0$\textdegree~and
$\alpha=0$\textdegree, we now choose two values of $\omega$
(0\textdegree~and ${-}$30\textdegree):

For $\omega=0$\textdegree, Equation~(\ref{eq3}) is again
$i=\arcsin(\sin i/1.66)$, verified if, and only if,
$i=0$\textdegree~and the cat's-eye remains at the apex of the spherical
cabochon.

For $\omega=-30$\textdegree, the Equations~(\ref{eq4}) are expressed as:
{\begin{equation*}
y=i \quad\mbox{and}\quad y' =\{\arcsin(\sin i/1.66)+
\arcsin  [\sin(i-30\mbox{\textdegree})/1.66] \}/2.
\end{equation*}}\unskip

Compared to the previous case, $y=\sigma-i=-i$ is a straight line,
parallel to the previous one, but shifted downwards by~30\textdegree,
and $y'$ is formally identical, represented by the same curve (in
black) in Figure~\ref{fig9}a. In Figure~\ref{fig9}b, the light path is
drawn in red. It appears symmetrical with respect to the median
vertical plane, with an inversion of the direction of light
propagation. The position of point $I\approx J$ is also inverted, with
the value of $\sigma-i=19.36$\textdegree, opposite to that in the
previous paragraph.

Thus, when the observer moves from $\omega=0$\textdegree~to
$\omega=-30$\textdegree, point $I\approx J$ slips angularly from
$\sigma-i=0$\textdegree~to $\sigma-i=19.36$\textdegree, which implies
the following:

\smallskip\noindent
\textbf{${\rightarrow}$ When the observer moves in one direction, the stone and
the source being fixed, the cat's-eye moves more slowly, in the
opposite direction.}

\subsection{Angular ranges for observing the cat's-eye, with a source
at vertical and an observer at any angle ${\omega}$}\label{sec94}


By rotating the sphere around an axis perpendicular to the working
plane (which contains the needles regardless of their tilt angle
$\alpha$), in which angular range $\omega$ will an observer
distinguish the cat's-eye? This question is not uninteresting because
the light source is not always located above the gem; it can be placed
at a different angle on a table or a desk, and the observer is
therefore not necessarily in the same direction as the source. And even
if the source is above the gem, the observer being placed next to the
stone can, without moving, raise it towards the lamp, and thus the
source and the observer are no longer in the same direction. To answer
this question, we must express $\alpha$ versus $\sigma-i$ for different
values of $\omega$. Using again the refractive index of sillimanite and
a source at $\sigma=0$\textdegree, Equation~(\ref{eq3}) is inferred as
follows:
{\begin{equation}\label{eq5}
\alpha=-i+\{\arcsin[\sin i/1.66]+\arcsin[\sin(i+\omega)/ 1.66]\}/2.
\end{equation}}\unskip

This function is plotted versus $\sigma-i=-i$ on Figure~\ref{fig10}a
for the following values of $\omega$: 0\textdegree, 20\textdegree,
40\textdegree, 60\textdegree, 80\textdegree, 100\textdegree,
120\textdegree, 140\textdegree, 160\textdegree, 180\textdegree.

\begin{figure}
\includegraphics{fig10}
\caption{\label{fig10}(a) Curves $\alpha$ versus $\sigma-i$ showing the
ranges for which the cat's-eye is visible, for a vertically oriented
source and for an observer looking at different directions $\omega$,
from 0\textdegree~to 180\textdegree. The more this angle increases, the
more the observation of a cat's-eye is reduced, as demonstrated by
increasingly smaller curves. The blue cross corresponds to the most
pleasant situation with a central position of the cat's-eye
($\sigma-i=0$\textdegree~with $\alpha=0$\textdegree,
$\sigma=0$\textdegree, $\omega=0$\textdegree). The horizontal black
arrows indicate an approximate value of the needle inclination
${\alpha}$ (about 28\textdegree) for which the angular variation of the
observation $\Delta \omega$ is the largest (from 0\textdegree~to
100\textdegree), corresponding to an angular variation $\Delta(-i)$
of the cat's-eye (from 60\textdegree~to 10\textdegree), when the stone
is translated upwards. (b) Diagram of the light path for
$\sigma=0$\textdegree~with $\alpha$ and $\omega$ variable.}
\end{figure}

These curves show that for a given value of $\omega$, the angular range
$\sigma-i=-i$ (with $\sigma=0$\textdegree) decreases when the observer
moves angularly away from the source.

\smallskip\noindent
\textbf{$\rightarrow$ When the cabochon rotates under a ceiling lamp, the range
of visibility of the cat's-eye narrows as the observer moves angularly
away from the vertical.}

N.B.: If the observer only moves the cabochon progressively upwards
without rotating the stone, the observation time of the cat's-eye
varies depending on the tilt angle $\alpha$. The widest range of
variation of $\omega$, and therefore of the visibility of the
cat's-eye, indicated by the black arrows, lies between the values
associated with the violet ($\omega=0$\textdegree) and green
($\omega=100$\textdegree) curves. It corresponds to an
$\alpha$ value of about 28\textdegree. The cat's-eye will not be
at the apex of the cabochon, but will evolve between the angular
positions 10\textdegree~and 60\textdegree~approximately (vertical
dotted lines).

\subsection{Influence of the refractive index}\label{sec95}

It is interesting to assess the angular range of ${\alpha}$  for which
the cat's-eye is visible depending on the refractive indices, from low
to high values. We operate in classical and optimal conditions
(${\sigma}=0$\textdegree~and ${\omega}=0$\textdegree), for which
Equation~(\ref{eq3}) becomes:
{$$
\alpha=-i+\arcsin[\sin i/n]=h(i).
$$}\unskip


We consider the cases of four chatoyant minerals: opal, sillimanite,
garnet and the rare cat's-eye rutile~\cite{4} with refractive indices
around 1.43, 1.66, 1.80 and 2.7 respectively. Figure~\ref{fig11} shows
the curves $\alpha$ versus $i$ for these four indices. The maximal
angles $\alpha$ deduced from the graph are about ${\pm}$45\textdegree,
${\pm}$53\textdegree, ${\pm}$56\textdegree~and ${\pm}$68\textdegree.
This means that there is a substantial difference in the overall
observation ranges $\Delta \alpha$ of the cat's-eyes between the
two extreme cases: 90\textdegree~for opal (blue double arrow) and
136\textdegree~for rutile (orange double arrow). In each case, under
the same chosen conditions ($\sigma=0$\textdegree~and
$\omega=0$\textdegree), the incidence angle $i$ changes from
${-}$90\textdegree~to ${+}$90\textdegree~and the position of the cat's-eye
moves from 90\textdegree~to ${-}$90\textdegree. But at the same time, the
angles of rotation of the stone are very different. For the observer,
it is more comfortable to watch the cat's-eye over a larger range of
stone rotation (case of rutile).

\begin{figure}
{\vspace*{4pt}}
\includegraphics{fig11}
{\vspace*{4pt}}
\caption{\label{fig11}$\alpha=h(i)$ curves for different
refractive index values in chatoyant or asteriated gems.
$\Delta\alpha$ values are 90\textdegree~for opal (blue arrow)
and 136\textdegree~for rutile (red arrow).}
\vspace*{4pt}
\end{figure}

\smallskip\noindent
\textbf{$\rightarrow$ The angular range of visibility of the cat's-eye when the
cabochon rotate increases with the refractive index.
In other words, for an identical rotational velocity of the cabochon,
the higher the refractive index is, the longer the observer is able to
watch the cat's-eye.}

\section{Illustration of the three modes of observation: the practical
example of sillimanite}\label{sec10}

To illustrate the results inferred from the mathematical development,
we chose an oval cabochon of sillimanite (approximately
$23.5\times~20.8\times11.2$~mm; mass 46.81~ct) originating from India.
It has a dark greenish-gray color, and its refractive index is 1.66,
measured at the base of the cabochon. It belongs to the collection of
Pascal Entremont. We photographed the stone for five angular values of
a varying parameter $\sigma$, ${\omega}$ or ${\alpha}$, the other two
being fixed at zero: 0\textdegree, 15\textdegree, 30\textdegree,
45\textdegree~and 60\textdegree. Let us consider the three
configurations (note that the case where all parameters are equal to
0\textdegree~belongs to each series of photographs), as illustrated in
Figure~\ref{fig12}.

\begin{figure}
\includegraphics{fig12}
\caption{\label{fig12}Photographs of the cat's-eye positions for three
configurations, with one varying parameter: $\omega$ (top; angle of
the observer away from the normal to the cab), $\sigma$ (middle:
angle of the source with the normal) or $\alpha$ (bottom; needle tilt).
The sharpest line corresponds to the third case, where $\alpha$
varies, $\sigma$ and $\omega$ are set to 0\textdegree. This is also
the most natural and comfortable observation condition. However, the
most pleasant situation occurs when all angles are equal to
0\textdegree~(left). At high values of $\alpha$, $\sigma$ or
$\omega$, the cat's-eye disappears. The stone was illuminated by a
white point light source at a distance of about 30~cm and the needles
producing the cat's-eye are parallel to one of the optic axes of this
sillimanite~\cite{3}. P. Entremont's collection.
Photos by Thanh Nhan Bui.}
\end{figure}

\subsection{Rotation of the stone with $\sigma=\text{0\textdegree}$\ and
${\omega}=\text{0\textdegree}$\ (source and observer directly above the
cab)}\label{sec101}

The cat's-eye rotates in the same direction as the cabochon does
(Figure~\ref{fig12}, bottom series). It should be noted that this type
of observation is the most common and the most convenient, and
definitely the most instinctive. The observer, placed under a lamp
above him (preferably a point light source), holds the stone low enough
against him, and turns it slowly in one direction or another, the
cat's-eye moving around a horizontal rotation axis. The observation
angle ${\omega}$ is low and thus the limitation of the field of view
(${\uppi}$--$\omega$) is reduced to a minimum. The cat's-eye remains
sharper compared to the other cases, even for large angles, but the
most pleasant situation corresponds to a symmetrically positioned
cat's-eye, when all parameters are equal to
0\textdegree~(Figure~\ref{fig12}, center left).

\subsection{Motion of the lamp, with $\alpha=\text{0\textdegree}$\ and
$\omega=\text{0\textdegree}$}\label{sec102}

The cat's-eye rotates in the opposite direction to the lamp. Handling
is easy for the observer, especially with a penlight. But the cat's-eye
quickly becomes diffuse as $\sigma$  increases (Figure~\ref{fig12},
middle series). Since the current sample of sillimanite is not totally
opaque, the ``coffee and cream'' effect appears (see additional
considerations, Section~\ref{sec112}).

\subsection{Motion of the observer with  ${\sigma}=\text{0\textdegree}$\ and 
${\alpha}=\text{0\textdegree}$}\label{sec103}

Actually, the observer does not physically move, but he gradually
raises the stone toward the source without changing its orientation.
This option is the less convenient for him. Therefore, ${\omega}$
varies and the cat's-eye rotates in the opposite direction as the
observer. Moreover, when ${\omega}$ increases, the cat's-eye becomes
diffuse and the ``coffee and cream'' effect also appears
(Figure~\ref{fig12}, top series).

Finally, the experimental cat's-eye behavior, as observed in the case
of sillimanite (Figure~\ref{fig12}), is in full agreement with the
predicted theory when the angular parameters ${\alpha}$, $\sigma$
or $\omega$  vary.

\section{Comments on the preliminary assumptions and additional
considerations}\label{sec11}

\subsection{Source and observer at infinity}\label{sec111}

We assumed that the source is reduced to a point located at infinity.
The sun suits this assumption quite well (apparent diameter:
${\sim}$0.5\textdegree) and allows the observation of a sharp cat's-eye.
A close and large source provides a convergent beam, so that there are
various rays emerging towards the observer, causing the cat's-eye to
widen. With respect to the observer, the apparent pupil diameter of his
eyes (${\sim}$1\textdegree) is not sufficient to cause a noticeable
widening of the cat's-eye.

Conventional desk lamps give a wider and less convincing light strip.
Diffuse ceiling light, ``neon tubes'', multiple LEDs, or cloudy
daylight are too large a source and are not suitable for observing the
cat's-eye phenomenon correctly.

\subsection{Scattering by deep needles within the stone}\label{sec112}

Depending on the translucency of the mineral, deep needles can
contribute to the creation of a cat's-eye. In such cases, the emergence
point $J$ will no longer coincide with the incidence point $I$, and will
depend on the position of each deep needle. This can lead to an
enlargement of the cat's-eye. Moreover, under oblique lighting, an
asymmetrical effect, named ``milk and honey'' in yellow gems such as
chrysoberyl~\cite{2}, Liddicoat~\cite{Liddicoat1972} 
or ``coffee-and-cream'' in brown gems, appears in translucent cabochons
and consists of two distinct areas on both sides of the cat's-eye: a
dark one on the side of the light source and a bright one on the
opposite side. These contrasting areas are due to deep acicular
inclusions within the stone that illuminate one side of the cabochon,
as clearly explained by Killingback~\cite{8}.

Note that when $\sigma=0$\textdegree~and $\omega=0$,
Equation~(\ref{eq2}) is reduced to $i'=i$ and therefore $r'=r$. This
requires the main scattered ray to coincide with the incident refracted
ray. Then, both rays must be perpendicular to the acicular inclusions
(Figure~\ref{fig13}). It follows that, regardless of the inclination
$\alpha$  of the needles, the path of both incident and emergent light
rays is identical. This means that the entry and exit points $I$ and
$J$ are no longer close to each other (as it was the case for needles
close to the surface), but they strictly coincide, whatever their
depth. We can therefore expect a very sharp cat's-eye, without any
``coffee and cream'' effect, even at large $\alpha$ angles
(Figure~\ref{fig12}, bottom series).

\begin{figure}
\includegraphics{fig13}
\caption{\label{fig13}Source and observer in the same vertical
direction ($\sigma=0$\textdegree~and $\omega=0$\textdegree): (a)
The main scattered ray follows the same path as the incident ray, both
perpendicular to the needles and the behavior of the cat's-eye is
symmetrical with respect to the vertical plane passing through the
center of the sphere and perpendicular to the working plane when the
angle ranges from ${-}$90\textdegree~to ${+}$90\textdegree. (b) Details of
angular parameters.}
\end{figure}

\subsection{Shape of the cabochon}\label{sec113}

For the sake of simplified calculations, we considered a spherical
cabochon. In fact, most cabochons display a semi-elliptical or
semi-ovoid cross-section, which is approximately modeled by two
different radii ($R$ and $R'$) of curvatures ($1/R$ and $1/R'$),---one at
the apex, the other at the edges---, leading to two types of shapes:
flat and sharp cabochons.

The cross-section of these two types of cabochon was drawn for a ratio
$R'/R=2.5$ between the central and lateral curvatures, with an
inversion of the radii between Figures~\ref{fig14}a (flat cabochon) and
14b (sharp cabochon). When the stone is rotated through a given angle
(in pink), the cat's-eye travels a distance 2.5 times greater for the
small $1/R'$ curvature than for the larger $1/R$ curvature
(Figure~\ref{fig14}c). In the case of a flat cabochon, the cat's-eye
will move much more spatially around the apex (${\times}$2.5) than in
the case of a sharp cabochon.

\begin{figure}
\includegraphics{fig14}
\caption{\label{fig14}Modeling an ovoid cabochon using a central circle
and two side circles: (a) flat cabochon; (b) sharp cabochon. (c) For a
source and an observer oriented vertically, with a needle ($N$) tilt
varying from 0\textdegree~to ${\alpha}$, the cat's-eye moves away from
the upper pole of the sphere of radius $R$ or $R'$ by a distance
${\alpha}{\cdot}R$ or ${\alpha}{\cdot}R$', respectively. The ratio $R'/R$
is taken as~2.5.}
\end{figure}

On the other hand, while cat's-eyes may widen due to a convergent beam
of light or to some stone transparency, they still remain narrower on
sharp cabochons than on flat cabochons.

\smallskip\noindent
\textbf{$\rightarrow$ When the stone is rotated, the mobility of the cat's-eye
is better experienced on flat cabochons than on sharp cabochons.
However, narrower cat's-eyes are expected on sharp cabochons.}

\subsection{Needles perpendicular to the main plane, with arbitrary
$\sigma$ and $\omega$}\label{sec114}

This case corresponds to the azimuth angle
$\varphi=90$\textdegree, where needles are perpendicular to the
main plane (Figure~\ref{fig15}). Needle portions located in the main
plane containing the source and observer directions scatter light into
flat cones with centrifugal rays. Regardless of the entry points I in
this plane, there is always an emergent ray from the great circle
towards the observer. This is the only case where we can easily
describe the exact position of the entire cat-eye line without further
calculation. Note also that it is no longer necessary to assume that
the needles are close to the surface. On the other hand, needle
portions outside the main plane give rise to scattering cones with a
half top angle below 90\textdegree. The rays thus emerge laterally and
cannot reach the observer.

\begin{figure}
\includegraphics{fig15}
\caption{\label{fig15}As the needles $N$ are perpendicular to the drawing
plane, the scattering cones are flat (half top angle equal to
90\textdegree). Emerging rays, located in the main plane, propagate in
all observation directions belonging to this plane. The dotted lines in
green, tangent to the sphere and parallel to the directions of the
source $S$ and the observer $O$, delimit the cat's-eye (in red)
according to the angular restrictions mentioned in
Section~\ref{sec66}.}
\end{figure}

The cat's-eye is therefore a bright line on the great circle (in red),
in the main plane. However, its extension is limited by the angular
values of $\sigma$ and $\omega$, according to the restrictive
conditions indicated in paragraph~\ref{sec66}.

\smallskip\noindent
\textbf{$\rightarrow$ When needles are perpendicular to the directions of both
the source and the observer, a bright line settles on a great circle in
the main plane. This applies for any position of the sphere rotating
around the needle axis.}


\subsection{Wavy parallel fibers}\label{sec115}

When a bundle of parallel fibers progressively changes direction,
following a wavy pattern in a crystalline matrix with a polished flat
surface, the chatoyancy will not appear everywhere for the same
position of the source and the observer as in Figure~\ref{fig3}a. There
will be bright and dark areas, moving very quickly as the stone is
gradually tilted, giving moir\'{e} patterns. This is the case for
tiger's-eye, hawk's-eye or iron-eye quartz which present an undulating
chatoyancy. Also included are serpentine, chatoyant clinochlore
(seraphinite), as well as charoite.

\subsection{Roughly oriented needles along a mean direction}\label{sec117}

When needles, such as microfibers, are not arranged in a strict
crystallographic orientation in the host crystal, but are slightly
disoriented along an average axis, they do not induce a sharp cat's-eye
on a cabochon, but display a wide band of chatoyancy. This can be
observed for instance in quartz microfibers in chalcedony~\cite{5}.

\subsection{Binocular vision}\label{sec118}

The observer's eyes are two slightly separate receptors, so he should
see two separate cat's-eyes on the stone, but very close to each other.
However, the brain superimposes and merges the two images to create a
single image over the cabochon. This is the stereoscopic effect, as
described by Killingback~\cite{8} and supplemented by Bui
et~al.~\cite{1}, which is beyond the scope of the present paper. 

\section{Conclusion}\label{sec12}

We limited the mathematical description of a cat's-eye to the search
for the position of the light spot generated by a specific ray of a
light beam scattered by parallel needles embedded in a mineral matrix
cut \textit{en cabochon}, which we called ``the main ray''. The study
of the exact location of all points of the cat's-eye line is beyond the
present work. Nonetheless, the angular behavior of the source,
observer, and stone was successfully established through a
straightforward trigonometric formalism, using only the so-called
Snellius--Descartes law, thanks to a number of relevant assumptions.

This well-defined approach allows us to understand and justify the
already experimentally known qualitative behavior of the cat's-eye
moving on the surface of cabochons for three modes of observation.
The motion of the cat's-eye, as intuitively perceived by the observer,
is in full agreement with our mathematical development.

\section*{Declaration of interests}
The authors do not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and have
declared no affiliations other than their research organizations.

\section*{Aknowledgment}
The authors are indebted to Professor Emmanuel Fritsch (University of
Nantes, France) for his invaluable advice and contribution to the
revision of the manuscript.

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