Consider the family of ellipses
(1)
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for . The partial derivative with respect to is
(2)
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(3)
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Combining (1) and (3) gives the set of equations
(4)
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(5)
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(6)
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where the quadratic curve discriminant is
(7)
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so (6) becomes
(8)
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Eliminating then gives
(9)
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which is the equation of the astroid. If the curve is instead represented parametrically, then
(10)
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(11)
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Solving
(12)
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for gives
(13)
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so substituting this back into (10) and (11) gives
(14)
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(15)
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(16)
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(17)
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the parametric equations of the astroid.