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?