Deltoid Catacaustic

Amazingly, the catacaustic of the deltoid when the rays are parallel in any direction is an astroid. In particular, for a deltoid with parametric equations

			(1)
			(2)

the catacaustic for rays incident from an angle is the offset deltoid given by

			(3)
			(4)

For incident rays making an angle with respect to the -axis, the resulting astroid is titled at an angle and has center offset so that the astroid rolls along the deltoid, completing half a turn as the incident rays rotate around a full turn. This therefore provides a construction for an astroid with a deltoid rotor.

Wolfram Web Resources