Hybrid Second-Order Iterative Algorithm for Orthogonal Projection onto a Parametric Surface

Abstract

Abstract:To compute the minimum distance between a point and a parametric surface, three well-known first-order algorithms have been proposed by Hartmann (1999), Hoschek, et al. (1993 and Hu, et al. (2000) (hereafter, the First-Order method). In this paper, we prove the method's first-order convergence and its independence of the initial value. We also give some numerical examples to illustrate its faster convergence than the existing methods. For some special cases where the First-Order method does not converge, we combine it with Newton's second-order iterative method to present the hybrid second-order algorithm. Our method essentially exploits hybrid iteration, thus it performs very well with a second-order convergence, it is faster than the existing methods and it is independent of the initial value. Some numerical examples confirm our conclusion.

Elliptical torus cutter tool path generation based on minimum distance computation

“…The contacting relations between the cutter and the surface is detected by the computation of the minimum distance between the cutter and the workpiece. Generally, it can be regarded as a distance computing issue between two freeform surface and solved with iterative methods 22 , 23 or subdivision methods. 24 , 25 However, with the above methods the characteristics of the elliptical torus are neglected.…”

Hybrid Second Order Method for Orthogonal Projection onto Parametric Curve in n-Dimensional Euclidean Space

“…Based on repeated knot insertion, Mørken et al [31] exploit the relationship between a spline and its control polygon and then present a simple and efficient method to compute zeros of spline functions. Li et al [32] present the hybrid second order algorithm which orthogonally projects onto parametric surface; it actually utilizes the composite technology and hence converges nicely with convergence order being 2. The geometric method can not only solve the problem of point orthogonal projecting onto parametric curve and surface but also compute the minimum distance between parametric curves and parametric surfaces.…”

“…The advantage of these methods is that they can find all solutions, while their disadvantage is that they are computationally expensive and may need many subdivision steps.The third classic methods for orthogonal projection onto parametric curve and surface are geometry methods. They are mainly classified into eight different types of geometry methods: tangent method [22,23], torus patch approximating method [24], circular or spherical clipping method [25,26], culling technique [27], root-finding problem with Bézier clipping [28,29], curvature information method [6,30], repeated knot insertion method [31] and hybrid geometry method [32]. Johnson et al [22] use tangent cones to search for regions with satisfaction of distance extrema conditions and then to solve the minimum distance between a point and a curve, but it is not easy to construct tangent cones at any time.…”

Convergence Analysis on a Second Order Algorithm for Orthogonal Projection onto Curves

“…They essentially formulate ordinary differential equation systems, which are determined by the intersection curve segment or by the projection curve segment. Li et al [27] combined a first order algorithm with Newton's second order algorithm to propose the hybrid second order approach, which goes faster than the current methods and does not depend on the initial value. The adoption of ADMM in [28] contributes to refine the estimate in each iteration because it can incorporate information about the direction of estimates gotten in previous steps.…”