ASTROID : Caustic Curve

To construct the caustic curve of the astroid relative to the origin O, we simply construct the normal to the astroid at point P of the curve, take the reflection of OP in this normal, and then draw the envelope of these reflected lines.

Click on the diagram below to open the applet, and then click on the ‘Animate’ button to generate the caustic curve of the astroid relative to the origin. Click the ‘Animate’ button again to stop the generation. Do you undertsand what is happening here? Clicking the little red x at bottom right will clear the drawing from the applet window. Dragging the point Q around the circle is less helpful in generating the caustic here, as the lines tend to fill the plane. Can you describe the caustic generated here?

There is a very interesting and unsuspected property of the normal appearing here. What is it?


The caustic curve generated here is a very attractive octagonal figure with inwardly curving edges. It is not a curve we have seen before.

We notice that the normal to the astroid at P passes through the generating point Q on the circle. This is of little interest in terms of the caustic, but might be usefiul in defining the position and attributes of Q.


Bibliography

Wolfram MathWorld : http://mathworld.wolfram.com/AstroidCatacaustic.html