An Approach to Computing Multipoint Inversion and Multiray Surface Intersection on Parametric Surface

Abstract

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.

New methods for projecting a 3D object onto a free-form surface

“…This article only focuses on three projection types. Therefore, the key technologies between Jingyu et al (2019) and this article have a big difference. More precisely, the problem solved by Jingyu et al (2019) is completely different from the problem stated in this article.…”

“…Therefore, the key technologies between Jingyu et al (2019) and this article have a big difference. More precisely, the problem solved by Jingyu et al (2019) is completely different from the problem stated in this article. Not limited to the specific problems, the proposed method has the potential to solve any non-linear 3D object projection problem.…”

“…Wang et al (2015) provided a series of methods for point inversion and ray-surface intersection by tracing along a curve on the base surface. Jingyu et al (2019) presented some methods for multi-point inversion and multi-ray surface intersection by combining Newton-Raphson iteration and the Runge-Kutta ordinary differential equation (ODE) solver. Xu et al (2012) described a marching method with error adjustment to calculate the projection curve.…”

“…By taking two calculation schemes, including the aforementioned method and multi-segment tracing method, the new method can quickly and steadily complete the calculation of 3D objects projection to implicit surfaces. Unlike the algorithm in Jingyu et al (2019), which is applied to the multi-point inversion and multi-ray surface intersection on the parametric surface, the same problem such as point inversion New methods for projecting a 3D object does not exist for implicit surfaces. Although the multi-ray surface intersection in Jingyu et al (2019) can provide some ideas for perspective projection on the parametric surface.…”

“…Unlike the algorithm in Jingyu et al (2019), which is applied to the multi-point inversion and multi-ray surface intersection on the parametric surface, the same problem such as point inversion New methods for projecting a 3D object does not exist for implicit surfaces. Although the multi-ray surface intersection in Jingyu et al (2019) can provide some ideas for perspective projection on the parametric surface. It cannot meet the technical requirements in a 3D object projection problem.Three-dimensional object projection problem is a vast research field.…”