Conditions for injectivity of toric volumes with arbitrary positive weights

Ji, Yindong; Li, Jinggai; Zhu, Chun‐Gang

doi:10.1016/j.cag.2021.04.026

Cited by 4 publications

(1 citation statement)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Yet the stringent bijectivity conditions set forth in these studies are often too restrictive, leading to inaccuracies when determining the bijectivity of Bézier curves in practical situations. Yu et al [15,16] introduced algorithms to assess self-intersections in toric surfaces and volumes, applicable to Bézier geometries. For more information on self-intersection studies in curves and surfaces, interested readers can refer to [17][18][19].…”

Section: Introductionmentioning

confidence: 99%

On Self-Intersections of Cubic Bézier Curves

Yu,

Li,

2024

Mathematics

Self Cite

View full text Add to dashboard Cite

Cubic Bézier curves are widely used in computer graphics and geometric modeling, favored for their intuitive design and ease of implementation. However, self-intersections within these curves can pose significant challenges in both geometric modeling and analysis. This paper presents a comprehensive approach to detecting and computing self-intersections of cubic Bézier curves. We introduce an efficient algorithm that leverages both the geometric properties of Bézier curves and numerical methods to accurately identify intersection points. The self-intersection problem of cubic Bézier curves is firstly transformed into a quadratic problem by eliminating trivial solutions. Subsequently, this quadratic system is converted into a linear system that may be easily analyzed and solved. Finally, the parameter values corresponding to the self-intersection points are computed through the solution of the linear system. The proposed method is designed to be robust and computationally efficient, making it suitable for real-time applications.

show abstract

Section: Introductionmentioning

confidence: 99%

On Self-Intersections of Cubic Bézier Curves

Yu,

Li,

2024

Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

Penalty function-based volumetric parameterization method for isogeometric analysis

Wang

Pan

et al. 2022

Computer Aided Geometric Design

View full text Add to dashboard Cite

Improved algorithms for determining the injectivity of 2D and 3D rational Bézier curves

Zhao

Wang

et al. 2022

era

View full text Add to dashboard Cite

<abstract><p>Bézier curves and surfaces are important to computer-aided design applications. This paper presents algorithms for checking the injectivity of 2D and 3D Bézier curves. An injective Bézier curve or surface is one that has no self-intersections. The proposed algorithms use recently proposed sufficient and necessary conditions under which Bézier curves are guaranteed to be non-self-intersecting. As well as a rigorous derivation of the proposed algorithms, we present a series of examples and derive the complexity and computation times of the proposed algorithms. We find that the complexity our algorithms is approximately $ O(m) $, representing an improvement over previous injectivity-checking algorithms.</p></abstract>

show abstract

Conditions for injectivity of toric volumes with arbitrary positive weights

Cited by 4 publications

References 23 publications

On Self-Intersections of Cubic Bézier Curves

On Self-Intersections of Cubic Bézier Curves

Penalty function-based volumetric parameterization method for isogeometric analysis

Improved algorithms for determining the injectivity of 2D and 3D rational Bézier curves

Contact Info

Product

Resources

About