(*Note*: This was orginally posted on New Year's Eve, 2011.)

Let a *body* $B$ be a compact set in $\mathbb{R}^3$ with a piecewise smooth boundary.
Some pieces/patches of the boundary are perfect mirrors; others perfect matte, colored surfaces.
Imagine the view of $B$ from infinity in some direction $u$.
A light ray from the "eye" at infinity travels parallel to $u$ until it hits $B$.
It then reflects from perfect mirror patches and eventually hits a colored patch, or shoots off
(in some direction) to infinity. What is seen in direction $u$ is an array of the colors each ray hits, or
transparency when a ray runs to infinity.

More precisely, let $u$ be along the $z$-direction of a Cartesian coordinate system.
Each ray parallel to $u$ may be identified by $(x,y)$-coordinates. Each point $(x,y)$
is assigned a color, a positive integer corresponding to the color hit by that ray, or 0 if the ray (ultimately) goes to infinity.
The *image* of $B$ in the $z$-direction is the coloring of $\mathbb{R}^2$.
Let $R(u)$ be a region of $\mathbb{R}^2$ large enough to include all the nonzero points
of the image from any direction $u$.

As the directions $u$ vary over $\mathbb{S}^2$, the image $R(u)$ changes, generally continuously (say, under the Hausdorff metric). I am interested in bodies $B$ that change discontinuously:

**Q1**. Does there exist a $B$ whose image changes discontinuously with respect to $u$,
preferably with
rather dramatic differences, and perhaps several or many such changes?

Such a $B$ could be called a *chameleon body*, for its appearance changes
depending on the viewpoint.
Ultimately I would like to control the changes.
For example:

**Q2**. Is there a $B$ whose view changes between these two images? :-)

[This question was inspired by fascinating work by Alexander Plakhov and Vera Roshchina: "Invisibility in billiards" (Nonlinearity

**24**(3)) and "Bodies invisible from one point" (arXiv:1112.6167). They arrange that a in-ray along line $L$, after ricocheting around inside $B$, emerges and continues along $L$. Orchestrating this for all rays from a point makes $B$ invisible from that point.]