Cycloid Catacaustic

The catacaustic of one arch of a cycloid given parametrically as

			(1)
			(2)

is a complicated expression for an arbitrary radiant point. For the case of the radiant point at , however, the catacaustic becomes

			(3)
			(4)

which is, rather amazingly, precisely two arches cycloid with identical beginning and endpoints and twice the period.