Orthogonal projection of points in CAD/CAM applications: an overview

Ko, Kwang Hee; Sakkalis, Takis

doi:10.7315/jcde.2014.012

Cited by 26 publications

(14 citation statements)

References 45 publications

(60 reference statements)

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“…A representative of this approach is the sphere approximation method which was proposed in Reference [10]. Each step of the sphere approximation method starts with the calculation of the vector s k D kC1 k using the least square projection (9). Next, the normal curvature Ä n is calculated according to formula…”

Section: The Sphere Approximation Methodsmentioning

confidence: 99%

“…The closest point projection of a point to the FFS design also poses a common problem to computer graphics. Computational methods for the orthogonal projection of points in Computer‐Aided Design (CAD)/Computer‐Aided Manufacturing (CAM) applications have recently been presented in a review paper . In order to guarantee robustness of the calculation, it is recommended to initially employ a subdivision‐based global scheme, followed by a Newton‐type iteration method, or the second order algorithms with small sensitivity to the choice of initial estimation such as the sphere approximation method and torus approximation method .…”

Section: Introductionmentioning

confidence: 99%

“…Unfortunately, these properties are not shared by the standard higher‐order finite elements, which are the primary subject matter of this paper. In any case, the solution to the problem of orthogonal projection remains open because the present solution methods are prone to instability and their performance should also be improved .…”

Section: Introductionmentioning

confidence: 99%

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Assessment of methods for computing the closest point projection, penetration, and gap functions in contact searching problems

Kopačka

Gabriel

Plešek

et al. 2015

Int. J. Numer. Meth. Engng

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Summary In computational contact mechanics problems, local searching requires calculation of the closest point projection of a contactor point onto a given target segment. It is generally supposed that the contact boundary is locally described by a convex region. However, because this assumption is not valid for a general curved segment of a three‐dimensional quadratic serendipity element, an iterative numerical procedure may not converge to the nearest local minimum. To this end, several unconstrained optimization methods are tested: the Newton–Raphson method, the least square projection, the sphere and torus approximation method, the steepest descent method, the Broyden method, the Broyden–Fletcher–Goldfarb–Shanno method, and the simplex method. The effectiveness and robustness of these methods are tested by means of a proposed benchmark problem. It is concluded that the Newton–Raphson method in conjunction with the simplex method significantly increases the robustness of the local contact search procedure of pure penalty contact methods, whereas the torus approximation method can be recommended for contact searching algorithms, which employ the Lagrange method or the augmented Lagrangian method. Copyright © 2015 John Wiley & Sons, Ltd.

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Section: The Sphere Approximation Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Assessment of methods for computing the closest point projection, penetration, and gap functions in contact searching problems

Kopačka

Gabriel

Plešek

et al. 2015

Int. J. Numer. Meth. Engng

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“…Song et al [20] constructed a biarc that locally approximates a segment on the original curve starting from the current projective point. Ko [21] reviewed methods for computing orthogonal projection of points onto curves and surfaces, which are given on implicit or parametric form or as points cloud. Through parameterizing the progenitor curve, a novel curve projection scheme is proposed by Xu [22] for computing the orthogonal projection.…”

Section: Introductionmentioning

confidence: 99%

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

Pei

Wang

Zhang

2019

Mathematical Problems in Engineering

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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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“…It uses a second-order tracing method to construct a polyline to approximate the pre-image curve of the orthogonal projection curve in the parametric domain of the base surface. Regarding the projection problem, Kwanghee Ko et al (2014) [28] give a detailed review on literatures before 2014. To sum up, those algorithms employ various techniques such as turning the problem into finding the root of the system of nonlinear equations, geometric methods, subdivision methods and circular clipping algorithms.…”

Section: Introductionmentioning

confidence: 99%

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

Wang

et al. 2017

Symmetry

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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.

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Orthogonal projection of points in CAD/CAM applications: an overview

Cited by 26 publications

References 45 publications

Assessment of methods for computing the closest point projection, penetration, and gap functions in contact searching problems

Assessment of methods for computing the closest point projection, penetration, and gap functions in contact searching problems

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

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

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