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
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(1)
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(2)
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the catacaustic for rays incident from an angle 
is the offset deltoid given by
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(3)
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(4)
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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.
