Point-to-ellipse and point-to-ellipsoid distance equation analysis.

For the problem of distance evaluation from a point X 0 to an ellipse in R 2 and to an ellipsoid in R 3 given by algebraic equation G ( X ) = 0 , we investigate the properties of the distance equation , i.e. an algebraic equation whose zeros coincide with the critical values of the squared distance function. We detail the structure of this equation and an algorithm for finding the point in the quadric nearest to X 0 . We also find analytical formulas for distance approximations using the expansion of the zero of distance equation into power series ∑ j = 1 ∞ ℓ j G j ( X 0 ) . Exact values for the approximation error bounds are obtained via construction of an analogue of the distance equations for the curve-to-curve distance problem. [ABSTRACT FROM AUTHOR]