Bezier surface/surface intersection

Aziz, Nadim M.; Bata, Reda M.; Bhat, Shankar

doi:10.1109/38.45810

Cited by 27 publications

(8 citation statements)

References 7 publications

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“…Generally speaking, intersection methods can be classified into the following categories. Tracing methods generate the surface/ surface intersection curve from an initial starting point [17]. However, when using the tracing method, it is easy to lose some branches of the intersection.…”

Section: Surface Intersectionmentioning

confidence: 99%

Affine Arithmetic-Based B-Spline Surface Intersection with GPU Acceleration

Lin

Yang

Liao

et al. 2014

IEEE Trans. Visual. Comput. Graphics

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Because the B-spline surface intersection is a fundamental operation in geometric design software, it is important to make the surface intersection operation robust and efficient. As is well known, affine arithmetic is robust for calculating the surface intersection because it is able to not only find every branch of the intersection, but also deal with some singular cases, such as surface tangency. However, the classical affine arithmetic is defined only for the globally supported polynomials, and its computation is very time consuming, thus hampering its usefulness in practical applications, especially in geometric design. In this paper, we extend affine arithmetic to calculate the range of recursively and locally defined B-spline basis functions, and we accelerate the affine arithmetic-based surface intersection algorithm by using a GPU. Moreover, we develop efficient methods to thin the strip-shaped intersection regions produced by the affine arithmetic-based intersection algorithm, calculate the intersection points, and further improve their accuracy. The many examples presented in this paper demonstrate the robustness and efficiency of this method.

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Section: Surface Intersectionmentioning

confidence: 99%

Affine Arithmetic-Based B-Spline Surface Intersection with GPU Acceleration

Lin

Yang

Liao

et al. 2014

IEEE Trans. Visual. Comput. Graphics

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“…One reason for using optimization-based approaches is that convex corridors guarantee the convexity of the problem, allowing for solving the motion planning problem as a quadratic programming problem [5], which are well-studied and can be implemented using off-the-shelf solvers [6]. Among different parametric curves, piece-wise Bezier curves are more popular as the convex hull property of Bezier curves 1 [7] guarantees the curve segments to be confined within the corresponding safety corridor [8]. As a result, the safety of the generated trajectories are guaranteed.…”

Section: Introductionmentioning

confidence: 99%

A Sufficient Condition for Convex Hull Property in General Convex Spatio-Temporal Corridors

Zhang¹,

Yadmellat²

2021

Preprint

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Motion planning is one of the key modules in autonomous driving systems to generate trajectories for selfdriving vehicles to follow. A common motion planning approach is to generate trajectories within semantic safe corridors. The trajectories are generated by optimizing parametric curves (e.g. Bezier curves) according to an objective function. To guarantee safety, the curves are required to satisfy the convex hull property, and be contained within the safety corridors. The convex hull property however does not necessary hold for timedependent corridors, and depends on the shape of corridors. The existing approaches only support simple shape corridors, which is restrictive in real-world, complex scenarios. In this paper, we provide a sufficient condition for general convex, spatio-temporal corridors with theoretical proof of guaranteed convex hull property. The theorem allows for using more complicated shapes to generate spatio-temporal corridors and minimizing the uncovered search space to O( 1 n 2 ) compared to O(1) of trapezoidal corridors, which can improve the optimality of the solution. Simulation results show that using general convex corridors yields less harsh brakes, hence improving the overall smoothness of the resulting trajectories.

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“…The majority of intersection methods published in the literature deals with parametric entities [1][2][3]8,[11][12][13]15,17,18,27,28,31], subdivision techniques [5,16,30], algebraic methods [14,29], special entity types such as natural quadrics [10,19], evolutionary algorithms [7], or computational geometry techniques [4]. A few honorable attempts at robustness revolve around those of solid modeling [6] or the use of interval arithmetic [9].…”

Section: Introductionmentioning

confidence: 99%

A Knowledge-Guided Approach to Line NURBS Curve Intersection

Rajab

Piegl

2013

Computer-Aided Design and Applications

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This paper presents a robust algorithm to solve the line-curve intersection problem used frequently in design, manufacturing, graphics, art, etc. A global solution is proposed, i.e. all the intersections are found and computed to high accuracy requirements. The emphasis is on robustness, reliability and to handle geometric as well as numerical anomalies. The main thrusts of the method lie in the use of a knowledge-guided NURBS system, a tight biarc decomposition and proper pre-and postprocessing of the entities as well as the intersections. All these contribute to achieve a high level of reliability: the method is immune to such cases as tangential intersections, inflection points, or line-curve overlaps. The intersection points produce various relationships that are recorded in the knowledge-guided system so that all the results are reproducible in the receiving system, should the intersection point be recomputed with a different level of accuracy.

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Bezier surface/surface intersection

Cited by 27 publications

References 7 publications

Affine Arithmetic-Based B-Spline Surface Intersection with GPU Acceleration

Affine Arithmetic-Based B-Spline Surface Intersection with GPU Acceleration

A Sufficient Condition for Convex Hull Property in General Convex Spatio-Temporal Corridors

A Knowledge-Guided Approach to Line NURBS Curve Intersection

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