In this paper, we present the analytical expressions for computing the minimum distance between a point and an elliptical torus, which is called the orthogonal projection point problem. Firstly, we give the method to find the corresponding orthogonal projection point of the major planar ellipse of the elliptical torus for test point being at eight different planar positions. Secondly, based on the method, we present an approach to find the corresponding orthogonal projection point or point set of the elliptical torus for test point being at nine different spatial positions. Furthermore, in order to improve the accuracy, we use the accurate numerical expression method based on the analytic expression method to compute orthogonal projection point on the elliptical torus.

point projection, elliptical torus, major planar ellipse, minor planar circle, intersection.