Posted by: Dave Richeson | June 5, 2009

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.
Picture 1

So, I made an applet to create such an envelope of reflected rays. Here’s a screen shot for one particular incoming slope.
Picture 2

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!

About these ads

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Categories

Follow

Get every new post delivered to your Inbox.

Join 211 other followers

%d bloggers like this: