Cycloid Catacaustic
CycloidCatacaustic
The catacaustic of one arch of a cycloid given parametrically as
x = t-sint
(1)
y = 1-cost
(2)
is a complicated expression for an arbitrary radiant point. For the case of the radiant point at y=infty, however, the catacaustic becomes
x = 1/2[(1/2t)-sin(1/2t)]
(3)
y = 1/2[1-cos(1/2t)],
(4)
which is, rather amazingly, precisely two arches cycloid with identical beginning and endpoints and twice the period.
See also
Catacaustic, CycloidExplore with Wolfram|Alpha
WolframAlpha
Cite this as:
Weisstein, Eric W. "Cycloid Catacaustic." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CycloidCatacaustic.html