Research On Algorithms For Orthogonal Projection Of Points On Implicit Curves And Surfaces

Orthogonal projection has attracted much interest in geometric modeling, computer graphics and computer vision. Projecting a point onto a curve or surface in order to find the closest point (footpoint) is important in application of generating a curve on a surface and fitting a curve or surface and it is also a key issues in the ICP (iterative closest point) algorithm for shape registration. Because of the tight relation between orthogonal projection and distance projection, it has been given much recognition in computing the least distance between different geometric bodies. Both the domestic and overseas scholars have done many researches in this aspect and gained pleasing effect.This paper presents algorithms for both projecting a given point onto an implicit curve or an implicit surface based on the aforementioned research work. According to the different forms of definition of planar curve and space curve, we give algorithms of projecting a point onto a two-dimensional implicit curve and a three-dimensional curve respectively. The algorithm of projecting a given point onto an implicit surface is generalized from the algorithm of projecting a given point onto an implicit curve. It mainly involves three parts: tracing the projection point, controlling the tracing step, analyzing and rectifying the error. The part of tracing the projection point is mainly to solve the problem of how to tracing the aimed orthogonal projection point and we construct different manners to trace the projection point for both curves and surfaces respectively.Massive simulation examples are given and experimental results show that the presented algorithms have favorable convergence speed and the sensitivity of it to the initial values is low and it can satisfy arbitrary degree of accuracy. We also apply the algorithms to the areas of distance projection and generation of curve on surface and have gained much approving fruits. And it can also be applied to fitting curves and surfaces.