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.