Deltoid Catacaustic
DeltoidCatacaustic
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
x = 2cost+cos(2t)
(1)
y = 2sint-sin(2t),
(2)
the catacaustic for rays incident from an angle theta is the offset deltoid given by
x = 3cost+cos(3t+2theta)-cos(2theta)
(3)
y = 3sint+sin(3t+2theta)+sin(2theta).
(4)
Animation of the catacautic of a deltoid as the angle of incident rays is changed
For incident rays making an angle theta with respect to the x-axis, the resulting astroid is titled at an angle theta/2 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.
See also
Astroid, Catacaustic, Deltoid, RotorExplore with Wolfram|Alpha
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Cite this as:
Weisstein, Eric W. "Deltoid Catacaustic." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DeltoidCatacaustic.html