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Astroid Catacaustic

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AstroidCatacaustics

In general, the catacaustics of the astroid are complicated curves. For an astroid with parametric equations

x=cos^3t
(1)
y=sin^3t,
(2)

the catacaustic for a radiant point at (0,0) is

x_r=(cost[8+5cos(2t)+3cos(6t)])/(13+3cos(4t))
(3)
y_r=(4sin^3t[7+6cos(2t)+3cos(4t)])/(13+3cos(4t))
(4)

and for a radiant point at y=infty is

x_r=1/2[5-3cos(2t)]cos^3t
(5)
y_r=1/(16)[12sint-sin(3t)+3sin(5t)].
(6)

Catacaustics for the points (0,0), (1,0), (0,infty), and lim_(r->infty)(r,r) are illustrated above.

See also

Astroid, Catacaustic

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Cite this as:

Weisstein, Eric W. "Astroid Catacaustic." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AstroidCatacaustic.html

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