ds.dynamical systems - 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?