## Being Inside a Perfectly Reflecting Sphere Or Torus

### Version 1.0 of 1/6/2006-12:00 a.m.

On 31/10/2005, the author posted this problem to sci.optics and sci.physics. The problem is described in more detail below.

Consider the following optical structures:

1. A void sphere S with totally reflecting inner walls.
2. A void torus T with totally reflecting inner walls.
3. A (point or extended) light source L inside S and T.
4. An observer O inside S and T.

Here are some interesting questions for you to ponder about:

1. What would O see when O tries to look at the walls of S or T at various angles, while L is illuminating the inside of S or T?[1]
2. What will O see when O tries to look at the walls of S or T at various angles, when either O or L move relative to the walls of S or T, while O is observing?[1]
3. What happens to a single light ray which emanates from L inside S and T?
4. Will any light ray emanating from L, eventually reach O in S and T?
5. Will O be able to see L's reflection anywhere in S or T? Everywhere? Nowhere?
6. Will O be able to see O's reflection anywhere in S or T? Everywhere? Nowhere?
7. Will O be able to see L's reflection, even when L is hidden from O in T?
8. What will O see in S or T, if L is a strobo flash which lights exactly once?
9. Can S and T's of reasonable sizes (with radii 10-20 m) be built?
10. Do you think they would be "safe" for human observers?
12. How might S and T be related to "cavity resonators"?
13. Assume S or T is built, without O/L in it. What will happen if one opens a little hole on the wall of S or T for some light to get in?
14. Same question as above with O already in S or T.
15. Can you perhaps think of a useful application of S and T in Physics?
16. Would you volunteer to be O in experiments with S and T?
17. Why/Why not?

#### A Short Analysis

The central problem above is the general case of what happens in two dimensions, which was a question posed by Costas Vlachos on sci.math:

Suppose a user shines a beam of light (say a laser) inside a perfectly reflective unit circle. Which points will be visited by the light ray and which will not?

Let us define the problem geometrically: Consider the unit circle and a user A who "shines" a laser at an angle θ from the tangent at A. Clearly we have:

ρ+2*φ=π,