\subsection{The $A$-Viewable Surface}
The $A$-viewable surface consists of locations of azimuthal focus satisfying $\mu = 0$. With
$\phi=\pi/2$, we find ${\hat  {\bf n}}^{\prime} = \left[ \sin \theta\ {\rm \underline{ie^{i \nu}}},\ \cos  \theta  \right]$, and
consequently
 \begin{equ}
 {\hat  {\bf n}}^{\prime} \cdot \hat {\bf t}_i = \frac{1}{h}  \left\{ (1-D \cos \theta) + (\cos \nu -1) \Gamma \right\}, 
 \end{equ}
 where  $\Gamma = \sin\theta (\sin \theta - D \sin 2\theta)$. From \eref{eqn:LawofReflection}
\begin{equ}
\hat {\bf t}_e^{\prime} = -2 ( {\hat  {\bf n}}^{\prime} \cdot \hat {\bf t}_i) \left[ \sin \theta\ {\rm \underline{i e^{i\nu}}},\ \cos \theta \right] + \frac{1}{h} \left[ (\sin \theta - D \sin 2\theta)\ {\rm \underline{i}},\ (\cos \theta -D\cos  2\theta) \right]~.
\end{equ}
The exit ray is ${\bf r}_A(\mu, t) = {\hat  {\bf n}}^{\prime} + s \cdot \hat {\bf t}_e^{\prime}$, and setting $\nu = 0$ the $A$-viewable surface is given by
\begin{equa}
0 = &{\rm d}{\bf r}^{\prime}_A = \left\{ \left[-\sin \theta\ {\rm \underline{1}}, \ 0 \right] -2 s ({\hat  {\bf n}}^{\prime} \cdot \hat {\bf t}_i)\left[ -\sin \theta\ {\rm \underline{1}},\ 0 \right] \right\} {\rm d}\nu\\
                             &\left\{ -2 ({\hat  {\bf n}}^{\prime} \cdot \hat {\bf t}_i) \left[\sin \theta\ {\rm \underline{i}},\ \cos \theta \right] + \frac{1}{h} \left[ (\sin \theta - D \sin 2\theta) \ {\rm \underline{i}},\ (\cos \theta -D\cos  2\theta) \right]
 \right\} {\rm d} s ~.
 \end{equa}
 Projecting in the $y$-direction gives ${\rm d} s=0$, and projecting in the $x$-direction gives
 \begin{equ}
 s_A = -\frac{1}{2} \frac{h}{(D\cos \theta - 1)}~.
 \end{equ}
 The location of the image is
 \begin{equ}
{\bf r}_A = {\hat  {\bf n}} + s_A \cdot \hat {\bf t}_e = \left[ \left\{ \sin \theta + \frac{\sin \theta}{2(D\cos \theta-1)}\right\}\ {\rm \underline{i}}, \cos \theta - \frac{(D-\cos\theta)}{2(D \cos \theta - 1)} \right]~.
\end{equ}
For the $A$-viewable surface $z_A = 1/2 + \mathcal{O}(y_A^2)$ at the symmetry    
  axis, where the quadratic term is a function of $D$ that does not
  vanish at any $D$. 




The $z$-coordinate of the viewable surface moves to $-\infty$ as the
incoming ray reaches grazing incidence at $\theta = {\rm sin^{-1}}
(1/D)$.  \fref{fig:ACaustic2} shows cross sections of the viewable surface
for $D=1.5R$, $D=3.73R$, and $D=20R$. 

\begin{figure}
\C{\includegraphics[width = 0.60\textwidth]{Figure34.png}}
\caption {Cross sections of $A$-viewable surfaces for three values of the
  viewing distance $D$.
  }
\label{fig:ACaustic2} 
\end{figure}


\section {Viewing a Planar Surface from its Center}
\label{sec:ViewingPlane}
\fref{fig:FlatConfig} shows another configuration with non-degenerate viewable surfaces. An image,
  such as a photograph, lies on the yellow
  plane (shown in projection) a distance $D$ from the center of the
  mirror. The observer places the eye at the intersection of the black dashed
  line (looking to the left through a small hole in the center of the
  yellow surface). The reflected image can then be seen in two
  different locations, depending on the astigmatism. In \fref{fig:FlatView},
  the cross-section of the $R$-viewable surface is shown in red, and that of the $A$-viewable surface
 in blue. At the edges, focus
  will be at different depths.



\begin{figure}
\C{\includegraphics[width = 0.60\textwidth]{Figure41.png} }
\caption {Viewing an image  of a plane (yellow) in the witch ball. The image is viewed from $O$, the intersection of the plane with the diameter normal to it. The ray from $P$ reaches $O$ following reflection from the ball.}
\label{fig:FlatConfig} 
\end{figure}


The path of a representative ray, from a point $P=\left[ y_0 {\rm \underline{i}}, \ D \right]$ on the photograph to the observation point $O$ is shown in \fref{fig:FlatConfig}. Note that, since the configuration is axially
symmetric, we have, without loss of generality, chosen $\phi=\pi/2$. The direction of the incoming ray is
\begin{equ}
\hat {\bf t}_i = \frac{1}{h_i} \left[( \sin \theta - y_0)\ {\rm \underline{i}}, \ (\cos \theta - D) \right]~,
\end{equ}
where
\begin{equa}
h_i^2 =\, & |(\sin \theta - y_0)\ {\rm \underline{i}}, (\cos \theta - D)|^2 \\
          =\, & 1 + D^2 + y_0^2 - 2D\cos \theta  - 2 y_0 \sin \theta \\
          =\, & h^2 + y_0^2 -2y_0 \sin \theta~,
\end{equa}
Using the law of reflection; {\it i.e.,\ }
$- \hat {\bf t}_i \cdot {\hat {\bf n}} = {\hat  {\bf n}} \cdot \hat {\bf t}_e$, we find
\begin{equ}
\frac{D \cos \theta + y_o \sin \theta -1}{h_i} = \frac{D \cos \theta - 1}{h}~,
\end{equ}
from which it follows that
\begin{equ}
y_0 = \frac {2D \sin \theta\ (D \cos \theta - 1)}{(D \cos 2\theta - \cos \theta)}~.
\label{eqn:z0}
\end{equ}
The vanishing of the denominator, resulting in $y_0 \rightarrow \infty$, is equivalent to the incoming ray being normal to the symmetry axis.


\subsection {The $R$-Viewable Surface}
The $R$-viewable surface containing points of radial focus are, as in Section~\ref{sec:Infinite}, computed using the one-dimensional set of rays emanating from $P$ that are in the plane of \fref{fig:FlatConfig}. Consider a ray reflecting from ${\hat  {\bf n}}^{\prime} = \left[ \sin (\theta + \varepsilon) \underline{i}, \ \cos (\theta + \varepsilon)\right]$. The direction of the incoming ray is
\begin{equ}
\hat {\bf t}_i^{\prime} = \frac{1}{h_i^{\prime}}\left[ (\sin (\theta + \varepsilon) - y_0)\ {\rm \underline{i}}, \ (\cos (\theta + \varepsilon) - D) \right]~,
\label{eqn:new1_inray}
\end{equ}
with
\begin{equ}
h_i^{\prime 2} = 1 + D^2 + y_0^2 -2D \cos (\theta + \varepsilon) - 2 y_0 \sin(\theta + \varepsilon)~, 
\end{equ}
and hence
\begin{equ}
h_i^{\prime} \ \frac{\partial h_i^{\prime}}{\partial \varepsilon} \Bigg|_{\varepsilon=0} =  D \sin \theta - y_0 \cos \theta ~.
\label{eqn:hi_deriv}
\end{equ}
From \eref{eqn:LawofReflection}, the direction of the reflected ray is
\begin{equa}
\hat {\bf t}_e^{\prime} = \frac{1}{h_i^{\prime}} \bigl[ (& -\sin (\theta + \varepsilon) + D \sin 2(\theta + \varepsilon) - y_0 \cos 2(\theta + \varepsilon))\ {\rm \underline{i}}, \\ & (-\cos (\theta + \varepsilon) + D \cos 2(\theta + \varepsilon) + y_0 \sin 2(\theta + \varepsilon) ) \bigr]~,
\end{equa}
and hence 
\begin{equa}
\frac{\partial \hat {\bf t}_e^{\prime}}{\partial \varepsilon} \Bigg|_{\varepsilon=0} = &\frac{1}{h_i} \left[ (-\cos \theta + 2D\cos 2\theta + 2y_0 \sin 2\theta) \ {\rm \underline{i}}, \ (\sin \theta - 2D \sin 2\theta + 2y_0 \cos 2\theta) \right] \\
 &- \frac{1}{h_i^3}(D\sin \theta - y_0 \cos \theta) \cdot \hat {\bf t}_e^{\prime} \big|_{\varepsilon=0}~.
 \end{equa}
 
 To compute the viewable surface, note, as before that the reflected ray can be parameterized as ${\bf r}_R = {\hat {\bf n}}^{\prime} + s \hat {\bf t}_e^{\prime}$, and that on the viewable surface
 \begin{equ}
 {\rm d}{\bf r}_R = \frac{\partial {\hat {\bf n}}^{\prime}}{\partial \varepsilon}
 \Bigg|_{\varepsilon=0} {\rm d}\varepsilon +  \hat {\bf t}_e^{\prime} \big|_{\varepsilon=0} {\rm d} s+ s \frac{\partial \hat {\bf t}_e^{\prime}}{\partial \varepsilon} \Bigg|_{\varepsilon=0} {\rm d}\varepsilon = 0~,
 \end{equ}
 from which it follows that the location on the surface is
 \begin{equ}
 s_R = \frac {h_i (D\cos \theta - 1) } {(3D \cos \theta - 1 + 2y_0 \sin \theta - 2D^2 \cos 2\theta - 2 y_0 D \sin  2\theta)}~.
 \end{equ}
The coordinates $y_R$ and $z_R$ on the viewable surface are found to be
\begin{equa}
z_R &= \cos \theta + s_R \frac{(D - \cos \theta)}{h}~;\\
y_R &= \sin \theta - s_R \frac{\sin \theta }{h}~.
\label{eqn:RViewSurface}
\end{equa}
 The cross-section of the $R$-viewable surface for a planar
 object placed a distance $2R$ in front of the witch ball is shown in
 red in \fref{fig:FlatView}. Although it is nearly flat, the surface fails to have a vanishing second derivative for any $D$.
 
 
 \begin{figure}
\C{\includegraphics[width = 0.60\textwidth]{Figure42.png}}
\caption {Cross sections of $R$- and $A$-viewable surfaces (red,
  resp.~blue) for a
  planar object placed normal to the axis at  $D = 2R$, and viewed
  through a small hole at its center. }
\label{fig:FlatView} 
\end{figure}
 

\subsection {The $A$-Viewable Surface} 

To compute the $A$-viewable surface, we need the focus of a rays emanating from $P$ normal to the plane of \fref{fig:FlatConfig}, Consider a ray with a differential orientation $\eta$. The incident direction, point of reflection and other relevant quantities are
\begin{equa}
{\hat {\bf n}}^{\prime} &= \left[ \sin \theta\ {\rm \underline{i e^{i\eta}}},\ \cos \theta \right]~, \nonumber\\
{\hat {\bf t}_i}^{\prime} &= \frac{1}{h_i^{\prime}} \left[ \sin \theta \ {\rm \underline{i e^{i\eta} }} - y_0\ {\rm \underline{i}}, \ \cos \theta - D \right]~, \nonumber \\
h_i^{\prime 2} &= D^2 + y_0^2 + 1 - 2D \cos \theta - 2 y_0 \sin \theta \cos \eta~, \nonumber \\
\frac{\partial h_i^{\prime}} {\partial \eta} & \big|_{\eta=0} = 0~, \nonumber \\
\hat {\bf t}_e^{\prime}  \big|_{\eta=0} &= \frac{1}{h_i} \left[ (-\sin \theta -y_0 \cos 2\theta + D \sin 2\theta)\ {\rm \underline{i}},\ (-\cos \theta + y+0 \sin 2\theta + D \cos 2\theta) \right]~,\\
\frac{\partial \hat {\bf t}_e^{\prime}} {\partial \eta} & \big|_{\eta=0} = \frac{1}{h_i} (-\sin \theta - 2 y_0 \sin^2 \theta - D \sin 2\theta) \left[\underline{1},\ 0 \right]~.
\end{equa}
The reflected ray, parameterized, is $r_A = {\hatt {\bf n}}^{\prime} + s {\bf t}_e^{\prime}$, and the location on the viewable surface corresponds to   
\begin{equ}
s_A = -\frac {h_i}{(2D \cos \theta + 2 y_0 \sin \theta -1)}~,
\end{equ}
and consequently,
\begin{equa}
z_A &= \cos \theta + s_A \frac{(D -\cos \theta)}{h}~,\\
y_A &= \sin \theta - s_A \frac{\sin \theta}{h}~.
\label{eqn:AViewSurface}
\end{equa}
The cross-section of the $A$-viewable surface is shown in blue in \fref{fig:FlatView}.

In the next section we experimentally demonstrate the existence of two distinct images along a viewing direction and validate the expressions \eref{eqn:RViewSurface} and \eref{eqn:AViewSurface}.

\section {Experimental Validation}
\label{sec:Expt}

Typical witch balls are not sufficiently smooth to test the calculations outlined in Section~\ref{sec:ViewingPlane}. 
Experiment validation was conducted using the arrangement shown schematically  in \fref{fig:Experiment}.  The mirror is an optical quality convex spherical mirror with a radius of curvature $R=20.0$ cm, a focal length of $f = -10.0 $ cm, a diameter of $50.0$ mm, and a protected aluminum coating. The center of the sphere is $C$. The object at $P$ is a cross with vertical and horizontal line segments of length $6$ cm and width $2$ mm.  Images are recorded using a Scion camera with a $ccd$ chip of $1200\ {\rm pixels} \times 1600\ {\rm pixels}$  with pixel spacing $4.0 $ micrometers.  A Navitar Zoom 7000 18 --108 mm Macro Lens is used to focus the reflected object onto the detector.  Camera positioning was accomplished using translation stages having micrometer drives with a spatial resolution of $10$ micrometers.  The configuration shown lies in a horizontal plane. 

 \begin{figure}
\C{\includegraphics[width = 0.60\textwidth]{Figure51.png}}
\caption {A schematic of the experiment.  The mirror, shown by a solid line is a section of a hypothetical witch ball centered at $C$ is marked by a dashed line. A cross of vertical and horizontal  lines at $P$ is viewed through the camera at $O$. The camera can be moved along $O O^{\prime}$, the direction of the emergent ray.}
\label{fig:Experiment} 
\end{figure}

The mirror and the witch ball from which it is assumed to be cleaved are shown in solid and dashed lines respectively. The mirror-axis is held in place, and each setting is initiated by a pre-specified rotation of the mirror. The center of the sphere,  $C$, is identified using the radius of curvature; the camera is placed on the symmetry axis a distance $D= 65$ mm from $C$. The object $P$ is moved in the plane normal to the symmetry axis until its image is at the focal plane of the mirror. The azimuthal line  ({\it i.e.,\ } vertical line  in our setup) is set in focus, as confirmed using the intensity profile of the cross section, see \fref{fig:TwoFocuses}(c). The best focus is defined by minimizing the half-width of the intensity curve shown in \fref{fig:TwoFocuses}.(c) Typically, this optimal focus remains unchanged when $O$ is moved by $\sim 1-2$ mm;  these distances give an estimate of the measurement error. Generally, the image of the radial (horizontal) line is not in focus at this point as seen in \fref{fig:TwoFocuses}(c).

Refocussing the camera to locate the second image will change the focal plane in a way that is difficult to estimate. Instead, we search for the second image by moving the camera forward, say by a distance $\delta$, along the direction $-{\hat {\bf t}}_e$ of the exit ray while maintaining the same focus. It is found that the horizontal line of the object focusses when the camera has moved to a point $O^{\prime}$;  $\delta = |O O^{\prime}|$. The use of the micrometer permits the estimation of $\delta$ to within 10 microns, if necessary. Generally, the focus of the azimuthal line is lost at this point as seen from \fref{fig:TwoFocuses}(d). The views at $O$ and $O^{\prime}$, shown in \fref{fig:TwoFocuses}, clearly illustrates the existence of two distinct images. 

\begin{figure}
\C{\includegraphics[width = 0.80\textwidth]{Figure52.png}}
\caption {The two images with (a) a sharp focus on the azimuthal (vertical) line, and (b) sharp focus on the radial  (horizontal) line. The variation in intensity for the two images are shown in (c) and (d). The red (resp. blue) line shows the intensity profile for the radial (resp. azimuthal)  lines. Observe that the focussed images have a higher value for the peak intensity gradient. }
\label{fig:TwoFocuses} 
\end{figure} 


Since the camera has moved during the experiment, $\delta$ is not the distance between the $R$ and $A$ viewable surfaces with fixed viewing at $O$. The location of azimuthal focus,  ${\bf r}_R(D, \theta)$ is given by ~\eref{eqn:RViewSurface}, while that at radial focus needs to be calculated for viewing at $O^{\prime}$. To do so, we note first that the angle $\psi$ between the exit ray ${\bf t}_e$ and $CO$ is small is our experiments. In addition, $\delta \ll D$ and hence distance $D^{\prime} \equiv |CO^{\prime}| \approx D-\delta$. The angle subtended by $O O^{\prime}$ at $C$ is, by the law of sines, $\delta \theta = \sin^{-1} (\delta \sin \psi / D^{\prime})$. Hence the angle $\theta^{\prime}$ between $CO^{\prime}$ and $\hat {\bf n}$ is $\theta^{\prime} = \theta  + \mathcal{O}(\psi^2, \psi \delta, \delta^2)$; {\it i.e.,\ } it is unchanged to first order in $\delta$ and $\psi$. The location ${\bf r}_A (D^{\prime}, \theta^{\prime})$ of the horizontal focus with the camera at $O^{\prime}$  can be calculated to first order in perturbation in $\delta$ and $\psi$. The value of $\delta$ can then be obtained by solving
\begin{equ}
\delta = | {\bf r}_R(D, \theta) - {\bf r}_A(D^{\prime}, \theta^{\prime})|.
\end{equ}
The solid line in \fref{fig:Result} shows $\delta$ as a function of $\theta$.  

The experimental values of $\delta$ for $\theta=15^o$, $17.5^o$ and $20^o$ are shown with the error bars originating from our inability to precisely locate the point of best focus.

\begin{figure}
\C{\includegraphics[width = 0.50\textwidth]{Figure53.png}}
\caption {The solid line shows the inclination of the mirror vs. $\delta$, the camera movement to get from the image with vertical focus to that with horizontal focus. The diamonds show the best estimate of $\delta$ and the error bars represent distances over which the with of the intensity remains unchanged.}
\label{fig:Result} 
\end{figure} 


\section{Discussion}
\label{sec:Discussion}
The role of caustics in image formation was highlighted in Chapter 3 of ~\citep{Feigenbaum2050}. Following passage through a general  (non-symmetric) optical system, a wavefront does not appear to emerge from a single point. Rather, it is a Gaussian surface with partial focus along two principal directions; in ~\citep{Feigenbaum2050}, such wavefronts were considered to emerge from ``non-objects." An observer finds two distinct images of a point source and it was conjectured that our visual system preferentially focusses on the one with vertical astigmatism. In viewing an extended object the two sets of images span two surfaces, referred to as viewable surfaces. In general, these viewable surfaces are not sagittal and tangential surfaces discussed in the context of image formation in axisymmetric optical configurations~\citep{AvendanoAlejoHernandez:2019}. The latter are not computed using caustics, but rather through ray diagrams in two orthogonal planes. Anamorphic images and the corresponding viewable surfaces were calculated in~\citep{Feigenbaum2050}. Imaging of underwater objects was discussed in ~\citep{HorvathBucha:2003,Eckmann:2021}.

The goal of our work was to compute images and viewable surfaces in reflections from a spherical mirror. We computed them for three configurations, the first of which was for an incoming parallel beam of light~\citep{Berry:1972,Berry2015}. The two images and the corresponding viewable surface contained points of radial and azimuthal focus. The surface of radial focus was azimuthally symmetric with a cusp singularity on the symmetry axis. The surface of azimuthal focus degenerated to a one-dimensional subset of the symmetry axis. Both viewable surfaces for the next two configurations were non-degenerate. In one we calculated viewable surfaces for an observer at a distance $D$ viewing infinitely far away objects. We found that the surface of radial focus was planar to fourth order when $D = (2+\sqrt{3}) R$. The final example involved viewing of a planar object through an opening at its intersection with a radius of the sphere normal to the plane. Several predictions were experimentally validated. 

The two viewable surfaces in all configurations we study have the following feature: one surface consisted of points of radial focus and the other points of azimuthal focus. Feigenbaum conjectured that the preferential focus for our visual system is for points of horizontal focus or vertical astigmatism. This leads to an interesting question on viewing a plane in the configuration shown in \fref{fig:FlatConfig} . The viewable surfaces, shown in \fref{fig:FlatView}, contain points of radial (red) and azimuthal (blue) focus. The first viewable surface can easily be captured by a camera. What will a subject observe in viewing the image in the mirror? Will they see the surface of radial focus at the upper and lower locations and the surface of azimuthal focus at the sides? Alternatively, will our ability to reconstruct local scenes using global properties~\citep{GreeneOliva:2009} allow a subject to focus on one of the two images?  We expect to conduct a study to address this issue. 

JPE is partially supported by the Fonds National Suisse Swissmap. The work reported here was inspired by studies of Mitchell J. Feigenbaum and insightful discussions with him. We dedicate this work to his memory. 







\bibliographystyle{new4}
