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This article presents a method for multipoint inversion and multiray surface intersection problem on the parametric surface. By combining tracing along the surface and classical Newton iteration, it can solve point inversion and ray-surface intersection issues concerning a large number of points or rays in a stable and high-speed way. What is more, the computation result can approximate to exact solutions with arbitrary precision because of the self-correction of Newton-Raphson iteration. The main ideas are adopting a scheme tracing along the surface to obtain a good initial point, which is close to the desired point with any prescribed precision, and conducting Newton iteration process with the point as a start point to compute desired parameters. The new method enhances greatly iterative convergence rate compared with traditional Newton’s iteration related methods. In addition, it has a better performance than traditional methods, especially in dealing with multipoint inversion and multiray surface intersection problems. The result shows that the new method is superior to them in both speed and stability and can be widely applied to industrial and research field related to CAD and CG.

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Purpose This paper aims to provide a series of new methods for projecting a three-dimensional (3D) object onto a free-form surface. The projection algorithms presented can be divided into three types, namely, orthogonal, perspective and parallel projection. Design/methodology/approach For parametric surfaces, the computing strategy of the algorithm is to obtain an approximate solution by using a geometric algorithm, then improve the accuracy of the approximate solution using the Newton–Raphson iteration. For perspective projection and parallel projection on an implicit surface, the strategy replaces Newton–Raphson iteration by multi-segment tracing. The implementation takes two mesh objects as an example of calculating an image projected onto parametric and implicit surfaces. Moreover, a comparison is made for orthogonal projections with Hu’s and Liu’s methods. Findings The results show that the new method can solve the 3D objects projection problem in an effective manner. For orthogonal projection, the time taken by the new method is substantially less than that required for Hu’s method. The new method is also more accurate and faster than Liu’s approach, particularly when the 3D object has a large number of points. Originality/value The algorithms presented in this paper can be applied in many industrial applications such as computer aided design, computer graphics and computer vision.