Computation of point inversion and ray-surface intersection through tracing along the base surface

This paper addresses new methods to solve point inversion problem for parametric surface. As a by-product a new method for ray-surface intersection is also developed. After further analysis, we reduce finding the corresponding parameters of a given point on a surface to the following steps: (1) construct a line segment with the given point as its one end and an arbitrarily selected point on the surface as its other end point, (2) project the line segment onto the surface orthogonally or along a vector or through a central point, or compute the intersection curve segment of a specially created plane and the surface, (3) trace the parameters along the projected curve or intersection curve with the linear convergence or the second-order convergence. As a matter of fact, we formulate some related systems of first-order or second-order ordinary differential equations met by the corresponding projection curve segment of the line segment or by the intersection curve segment. Using the parameters of the selected point as initial values, we trace the desired parameter on surface along the line segment from its one end to another or along the intersection curve segment. In this method, there is no need to consider the sensitivity to the choice of starting points, iteration convergence and so on, which several existing methods must face. The method is simpler than existing methods for it merely concerns first-order information of the surface, if we only ask for linear convergence, and has better error control mechanism, if we seek for second-order convergence. Implementation examples are also given to demonstrate its validity.