Efficient Point Projection to Freeform Curves and Surfaces

Oh, Young‐Taek; Kim, Yong-Joon; Lee, Jieun; Kim, Myung-Soo; Elber, Gershon

doi:10.1007/978-3-642-13411-1_13

Cited by 11 publications

(9 citation statements)

References 13 publications

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“…To some extent, the approach avoids fully and truly the sensitivity to the choice of the initial value in those Newton-Raphson iteration related methods. Young-Taek Oh et al [28] used an efficient culling technique to reduce redundant curves and surfaces and proposed an efficient method to project a point to its nearest point on a set of freedom surfaces and curves. Chen et al [29] revised a rational cubic clipping approach to get two bounding cubes within ( ) time, which could get a faster convergence rate.…”

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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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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“…The obstacles are defined as convex polytopes, parametric curves, or any other compact set to which minimum distances can be computed, Figure 1. A main feature of the proposed algorithms is their ability to provide proximity queries for a large class of parametric curves, more general than ones that have been considered previously [23,10,6,25,5]. Such queries are useful in scenarios when the motion planner's candidate path (i) incurs a distance-based penalty for approaching close to an obstacle, (ii) must keep a safe distance from an obstacle, or (iii) must not intersect with the obstacle's geometry.…”

Section: Introductionmentioning

confidence: 99%

Proximity Queries for Absolutely Continuous Parametric Curves

Lakshmanan¹,

Patterson²,

Cichella³

et al. 2019

Robotics: Science and Systems XV

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In motion planning problems for autonomous robots, such as self-driving cars, the robot must ensure that its planned path is not in close proximity to obstacles in the environment. However, the problem of evaluating the proximity is generally non-convex and serves as a significant computational bottleneck for motion planning algorithms. In this paper, we present methods for a general class of absolutely continuous parametric curves to compute: (i) the minimum separating distance, (ii) tolerance verification, and (iii) collision detection. Our methods efficiently compute bounds on obstacle proximity by bounding the curve in a convex region. This bound is based on an upper bound on the curve arc length that can be expressed in closed form for a useful class of parametric curves including curves with trigonometric or polynomial bases. We demonstrate the computational efficiency and accuracy of our approach through numerical simulations 1 of several proximity problems.

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“…Then, based on clipping sphere strategy, [26] propose a method for computing the minimum distance between a point and a clamped B-spline surface. Being analogous to [25,26], based on an efficient culling technique that eliminates redundant curves and surfaces which obviously contain no projection from the given point, Young-Taek Oh et al (2012) [27] present an efficient algorithm for projecting a given point to its closest point on a family of freeform curves and surfaces. Song et al (2011) [7] propose an algorithm for calculating the orthogonal projection of parametric curves onto B-spline surfaces.…”

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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Efficient Point Projection to Freeform Curves and Surfaces

Cited by 11 publications

References 13 publications

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

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

Proximity Queries for Absolutely Continuous Parametric Curves

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

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