Research On Orthogonal Projection Of Points Onto Free-From Curves And Surfaces

Orthogonal projection of points onto free-form curves and surfaces is one of the most important topics in CAD/CAGD. It has been widely used in many other fields,such as intersections of curves and surfaces, point cloud registrations, surface fitting, etc.This paper discusses about several main algorithms in orthogonal projection: algorithm for global estimation of orthogonal projection parameters of points onto free-form curves and surfaces, iteration algorithm for orthogonal projection of points onto free-form curves and surfaces, algorithm for multi-point inversions on free-form surfaces.This paper proposes an algorithm for global estimation of orthogonal projection parameters of points onto free-from curves and surfaces, based on pruning on the explicit convex hull of the squared distance function. In each step, clipping circle/sphere criteria are first used to eliminate roughly. If the elimination criteria fail, compute the explicit expression of the squared distance function, and construct the convex hull of the squared distance function incrementally. Then regions that obviously contain no projection points are eliminated from the base curve or surface, by intersecting the current nearest squared distance line with the convex hull. When the user-defined tolerances are satisfied, iteration algorithms are used to get the precise projection points. Experimental results show that compared with algorithms using clipping circle/sphere or clipping square/box, this algorithm possesses higher elimination rate and running speed.This paper proposes a geometric iteration algorithm for orthogonal projection of points onto free-from curves and surfaces, based on local biarc approximation. In each iteration, biarc-splines are used to locally approximate a region on base curve or surface. It has higher approximation precision compared with geometric iteration algorithms based on single-point approximations. As a result, the next projection parameter is closer to the precise projection parameter. The iteration is halted when the user-defined tolerances are satisfied. Experimental results show that compared with geometric iteration algorithms based on single-point approximations, this algorithm converges faster and is less dependent on the choice of the initial projection parameter.This paper proposes an algorithm for multi-point inversions on free-form surfaces,based on 3D uniform grid on free-form surfaces. According to the user-defined grid size,an initial grid is constructed on the base surface to cover the computation region of inversions. Then the uniformity of the grid is optimized on the base surface, and is used to estimate the projection parameter for each test point. The precisions of the estimated parameters are elevated by inserting grid nodes on curved regions of the uniform grid according to the user-defined tolerance. Finally the precise inversion parameters are derived using iteration algorithms. Experimental results show that compared with random points algorithm, point-by-point projection algorithm and CATIA, this algorithm possesses higher estimation precision and running speed.