### Reflection from functions

Posted:

**Tue Nov 17, 2009 2:02 am UTC**So I wrote a program a while back that does reflections along mirrors defined by arbitrary functions and I noticed an interesting property of mirrors defined by functions that were power greater than 2. For a cubic mirror, you get something that looks kind of like this:

Now, to me, those two symmetric spaces (filled with the beams going down before they were reflected) look kind of like hyperbolas. I'll give images of some other higher degree mirrors and their patterns if you want, but the gist of my question is this: Is there any way you could prove that those lines, after reflected by a mirror defined by a function, say m(x), are tangent to another function, r(x)?

Now, to me, those two symmetric spaces (filled with the beams going down before they were reflected) look kind of like hyperbolas. I'll give images of some other higher degree mirrors and their patterns if you want, but the gist of my question is this: Is there any way you could prove that those lines, after reflected by a mirror defined by a function, say m(x), are tangent to another function, r(x)?