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Ellipse Catacaustic

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For an ellipse given by the parametric equations

x=acost
(1)
y=bsint,
(2)

the catacaustic is a complicated expression for generic radiant point (x_r,y_r). However, it simplifies for a number of special cases to

x_c=(4a(a-b)(a+b)sin^3t)/(a^2+b^2+(b^2-a^2)cos(2t))
(3)
y_c=(4b(b^2-a^2)cos^3t)/(a^2+b^2+3(b^2-a^2)cos(2t))
(4)

for (0, 0),

x_c=([5a^2-b^2+(a^2+b^2)cos(2t)]sint)/(4a)
(5)
y_c=bcos^3t
(6)

for x=infty, and

x_c=asin^3t
(7)
y_c=-([a^2-5b^2+(a^2+b^2)cos(2t)]cost)/(4b)
(8)

for y=infty.

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