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

摘要

For the problem of distance evaluation from a point X0 to an ellipse in R2 and to an ellipsoid in R3 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 X0. We also find analytical formulas for distance approximations using the expansion of the zero of distance equation into power series ∑j=1∞ℓjGj(X0) . 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.