Ellipse Catacaustic

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EllipseCaustic1

EllipseCaustic2

EllipseCaustic3

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 = [画像:(4a(a-b)(a+b)sin^3t)/(a^2+b^2+(b^2-a^2)cos(2t))]

(3)

y_c = [画像:(4b(b^2-a^2)cos^3t)/(a^2+b^2+3(b^2-a^2)cos(2t))]

(4)

for (0, 0),

x_c = [画像:([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 = [画像:-([a^2-5b^2+(a^2+b^2)cos(2t)]cost)/(4b)]

(8)

for y=infty.

See also

Catacaustic, Ellipse

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Cite this as:

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

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