Point-Quadratic Distance

To find the minimum distance between a point in the plane  and a quadratic plane curve

(1)

note that the square of the distance is

			(2)
			(3)

Minimizing the distance squared is equivalent to minimizing the distance (since  and  have minima at the same point), so take

(4)

Expanding,

	(5)
	(6)

Minimizing the distance to find the closest point  therefore requires solution of a cubic equation.