Light rays bouncing around inside a sphere in d-dimensions Suppose $S=\mathbb{S}^d$ is a unit sphere in $(d-1)$ dimensional space, with $d=3$ of special interest.
The surface of $S$ is a perfect (internal) mirror.
You stand at point $x$ (not the sphere center $c$) inside $S$ and emit a single laser light ray in direction $u$.
What happens?  I believe that the light ray will remain within the plane containing the three points
$\{ x, x+u, c \}$.
Now suppose instead that from $x$ you shine a flashlight, a cone with angular extent $\pm \epsilon$.
Does this fill the sphere with constant-density energy for any $\epsilon > 0$?  Are there are no dark points within $S$?
A somewhat related question is: What would the flashlight-holder see from $x$?
What would the visual image be, say in a graphics ray-tracing system (in $d$ dimensions!)?  
I've asked enough questions for one MO posting, but ellipsoids in $\mathbb{R}^d$ are the
obvious extension.  Are they integrable or chaotic?
 A: Your first question about the ray remaining in a planar region is trivial considering reflection angles.
In a sphere a ray of light that is at distance $r$ from the center will continue to stay at that distance even after multiple reflections and so for a cone of light to illuminate the sphere one of its rays must pass through the origin. If the cone's distance to the origin is $r$ (assuming that the cone sheds light both ways) then all of the sphere minus a smaller concentric sphere of radius $r$ intersected with the figure you get after rotating a plane on the axis $XC$ so as to cover all of the rays from the flashlight will be illuminated. This is easy to show since you will have a continuum of angles that subtend the chords formed by the rays and so most of them will not be rational multiples of $\pi$ and will not have periodic orbits. 
So basically there are two obstructions, the distance of the cone from the center of the sphere $C$ and the planes determined by the line $XC$ and the rays of light. You can see that both these obstructions disappear if one of the interior rays passes through $C$.
A: Billiards inside an ellipsis all are integrable in dimension $2$, as is beautifully shown in the small book by Tabachnikov. I do not know if this property extends to higher dimensions, but the proof I know certainly doesn't (it relies to the metric bifocal caracterisation of ellipses).
Let me give a more precise statement and mention an important conjecture in this domain. Given an elliptic domain, there is an open neighborhood of its boundary that is foliated by curves (which are in fact confocal ellipses) in such a way that any billiard trajectory starting close enough to the boundary (in position and direction; here the proximity condition is in fact that the first segment does not go between the focal points), has each of its segments tangent to one and the same of these curves. The conjecture is that ellipses are the only convex 2-dimensional billiards having this property.
