Assume that you have two parabolic mirrors, one sitting on top of the other, and you sitting at the focal point of the bottom one, looking down:

```\       /
\     /
\___/

X <---- you're here
-|-
\  / \  /
\     /
\___/
```

In a parametric representation, the bottom mirror would consist of the set of points of the form [ x, y, x2 + y2 ], and the top mirror would be those points [ x, y, x2 + y2 ]. To find the focal point of the bottom mirror, you can look at the simpler case with the function f(x) = x2. The focal point is at the same height as the point where the derivative (slope) of the curve is precisely 1. This is because all lines coming stright down that hit the mirror reflect off and hit the mirror's focal point. When your slope is 1, the mirror's surface is at a 45 degree angle to a vertical line, and the reflected ray becomes horizontal, which places it at the same height as the focal point. So, f'(x) = 2x, which means that when x = 1/2, we have this particular point, meaning that the focal point's height is f(x) = (1/2)2 = (1/4). As a result, you're located at the point [0,0,1/4].

Now, also assume you're sitting in a big box, with the bottom floor covered in purple checkers, the top in blue checkers, the right in cyan, the left in green, the front in yellow, and the back in red checkers.

When you look down from your current location, you see basically the same thing that a person sitting at the focal point of the top mirror would see, if they were looking downward as well. To see this, remember that a parabolic mirror focuses vertical rays going down into the mirror get sent to the focal point. Likewise, rays cast from the focal point of the bottom mirror to the mirror surface reflect from the mirror in a vertical ray. Thes rays would travel upwards until they hit the top mirror. It happens that the reflected ray from the bottom of the top mirror is the same as one that is cast from the focal point of the top mirror to the point of intersection. In a really bad ascii-art attempt to describe this reflective behavior:

```\   \   /
\   \ /
\___X
|\
| \
\   \ |  \
\   \|/  \
\___/    \
\
```

Follow the ray from the bottom mirror's focal point towards its surface, to the top mirror, then to the lower right. Then follow the ray from the top mirror's focal point, through its mirror and to the lower right. The lower right section is the same in both cases presuming that both of the rays hit the top mirror at point marked by X.

Now, without any mirrors, you would be looking down on a purple checkered floor. Add in the bottom mirror, and you'll be seeing a non-perspective view of the ceiling (a dark-blue checkered ceiling). Throw the top mirror in, and you'll see generally the purple floor, in a perspective view again (as I said, with the perspective rooted at the focal point of the top mirror).

Now, there is going to be some reflection back into the bottom mirror from the top mirror. This produces a neat bit of chaotic behavior in the system. It's relatively easy to see that the vertical line at the center of the mirrors is going to be a fixed point in a ray-tracing simulation --- Light rays cast from the focal point straight down will bounce straight up from the bottom mirror, straight up to the top mirror, and back down to the bottom mirror, infinitely.

The way that ray-tracers work, the scene is rendered by having the computer cast rays from the camera's eye out into the scene, and the physics of the light rays are simulated, as they bounce from object to object. Typically, the renderer will only follow the rays through a few reflections, after which, if it hasn't hit anything that isn't entirely transparent or reflective, it will just show a black (or other colored) region in the appropriate region of the image. So, in the cases where the rays bounce between the mirrors before escaping them and hitting the walls, you'll see black regions in the ray tracer.

You can look at this ray-tracing behavior much like the iterative complex arithmetic algorithms used to find stable/unstable points in the mandlebrot set. In the mandlebrot set, you start with a point c in the complex plane, and you run it through the formula f(z) = z2 + c, and get another point in the complex plane, and then run that point through the function again, ad infinitum. After a number of iterations, you make the decision as to whether iteration using the point c leads to a stable solution or not, and you color the point a particular color.

Note that the chaotic reflective behavior happens in two places in the system. In the first case, near the center of the image as you look down from the focal point, the rays keep bouncing back and forth between the two mirrors. Depending on the sizes of the mirrors, other such locations can be found as well (ie: if the sides of the mirrors are higher than their focal points).

You can get a movie of this sort of environment here:
parabolics.mpg

The movie starts out straight-forwardly...you can see the generally undistorted purple checkerboard pattern in the background (reflected off of the bottom of the top mirror), and you can make out the silhouette of the bottom mirror in the center of the image. As the top mirror approaches the bottom mirror, the image of the bottom mirror gets larger and larger. After a short while, you start to see a few regions of blackness appearing - these happen in locations where the raytracing depth is too small to converge to a solution, and are much like the "black sections" of a mandlebrot set. Note that after a period of time of seeing these black regions, you enter a stable configuration, where it looks like you slowly fall into a colorful box with a circular pattern in front of you, containing many black regions as you get closer. After a few moments more, the world goes black, as almost all of the points in your view are non-convergant ray-traces (increasing the maximum ray-tracing depth should reveal a bit more here).