Is there an easy way of finding the shortest distance from a point to a polynomial?

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Simon 2013-7-3

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With access to the point and the coefficients of the polynomial, is there an easy way of getting the perpendicular distance? Currently I'm having to find the distance from the point to every point in the curve and find the shortest manually, and its sloooooow.

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Answer 1

Matt J 2013-7-3

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编辑：Matt J 2013-7-3

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The analytical solution boils down to finding the roots of another polynomial. If your fixed point is (x,y) and your polynomial is p(t), then the distance squared is

f(t) = (t-x)^2 +(p(t)-y)^2

Setting the derivative of f(t) to zero leads to

(t-x)+ (p(t)-y)*dp/dt = 0

which is just a problem of finding the roots of the polynomial on the left hand side. Once you find its roots using the ROOTS command, you then test f(t) at the real roots for the minimum distance.