# Parabolas and focal points II

Yesterday I wrote about parabolic mirrors. I pointed out that parallel rays hitting a parabola typically do not reflect back to a focal point. That happens only when the rays are parallel to the axis of the parabola.

Then my friend Dan sent me an email encouraging me to look at the envelope of the reflected rays for a given incoming direction (called a catacaustic). He said that in the case of a parabola, these catacaustics are a tear-drop-shaped curve called the Tschirnhausen cubic ($r=a\sec^3(\frac{\theta}{3})$ in polar coordinates or $27ay^{2}=x^{3}+9ax^{2}$ in rectangular coordinates) suitably translated, rotated, and dilated.

So, I made an applet to create such an envelope of reflected rays. Here’s a screen shot for one particular incoming slope.

Furthermore, Dan wrote:

As the angle of the parallel lines change from perpendicular to the axis of the parabola, the cubic rotates and shrinks—with the loop always enclosing the focus of the parabola— until the lines are parallel to the axis, when you get the degenerate case of all the lines meeting at the focus. If you looked at an animation of it, it would look like a loop getting pulled tighter and tighter around the focus until it disappeared.

Very cool. Thanks, Dan!