A closed-form general solution for the distance of point-to-ellipse in two dimensions

Abstract

This paper resolved a closed-form general solution for the distance of one point and one ellipse in a two dimensional plane. Even such problem is simple as apparently to be understood, the solutions provided from literature are all about iterative algorithms or approximating approaches. None of O(1) methods or general roots solutions are found. The author takes several geometrical transformations to reduce the power of the equation to simplify the problem. Eventually, a closed-form general solution is resolved in this paper. This can be a contribution both for academic and for in practice and should be archived. Based on this general solution, we can develop the algorithm computing the shortest path touring n sequenced ellipses soon. Since the results of this paper is sufficiently fundamental, it can be conducted in a variety of applications: in the fields of computer-aided design and manufacturing, computer graphics, layered manufacturing, prototyping, robot motion (path) planning, wireless sensor networking, multimedia animation, geodesy, astronomy, physics, molecular geometry, electromagnetics, and Fluid mechanics.