In this paper, a new algorithm with extrapolation process for computing the ray/surface intersection is presented. Also, a ray is defined to be the intersection of two planes, which are non-orthogonal in general, in such a way that the number of multiplication operations is reduced. In the preprocessing step, NURBS surfaces are subdivided adaptively into rational Bezier patches. Parallelepipeds are used to enclose the respective patches as tightly as possible. Therefore, for each ray that hits the enclosure (i.e., parallelepiped) of a patch the intersection points with the parallelepiped's faces can be used to yield a good starting point for the following iteration. The improved Newton iteration with extrapolation process saves CPU time by reducing the number of iteration steps. The intersection scheme is faster than previous methods for which published performance data allow reliable comparison. The method may also be used to speed up tracing the intersection of two parametric surfaces and other operations that need Newton iteration.