Catacaustics Formed within Parametric Curves

Marc Payne
A catacaustic is the envelope of rays reflected off of a curve or surface. In this demonstration, we can watch as rays reflect off the interior of three different parametric curves. The incoming rays traverse the curves counterclockwise, leaving traces of reflected rays that reveal the corresponding catacaustics. There are two types of "light" source to choose from: radiant point and the point at infinity. If "Radiant Point" is checked, a yellow X becomes visible. This represents a point source of light and can be moved anywhere you wish. If "Point at Infinity" is checked, the yellow X disappears and the incoming rays enter parallel to each other (not shown) in whatever direction you choose. Note that only the reflected rays leave traces. There are three parametric curves to choose from. The catacaustic formed inside the circle from a point source on the circumference is known as a cardioid. This shape can often be seen on the inside of a cup, when light enters at an angle. The other curves produce more complicated catacaustics. This demonstration is meant to be fun, and is not intended to be an accurate model of reality. For a more in-depth analysis, see this Sage worksheet that shows some of the math behind catacaustics. In particular, note the explicit equations for the catacaustic of the tricuspoid, also known as a deltoid. The catacaustic of the epicycloid is much more complicated.