To find the minimum distance between a point in the plane and a quadratic plane curve
(1)
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note that the square of the distance is
(2)
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(3)
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Minimizing the distance squared is equivalent to minimizing the distance (since and have minima at the same point), so take
(4)
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Expanding,
(5)
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(6)
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Minimizing the distance to find the closest point therefore requires solution of a cubic equation.