Quadrilateral mesh generation I : Metric based method

Chen, Wei; Zheng, Xiaopeng; Ke, Jingyao; Lei, Na; Luo, Zhongxuan; Gu, Xianfeng

doi:10.1016/j.cma.2019.07.023

Cited by 31 publications

(28 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If we subdivide the quad-mesh infinitely many times, we obtain geodesic lamination, each leaf is a closed loop. This is called the finite geodesic lamination condition [6].…”

Section: Quad-mesh Induced Structuresmentioning

confidence: 99%

See 1 more Smart Citation

Quadrilateral mesh generation II: Meromorphic quartic differentials and Abel–Jacobi condition

Lei

Zheng

Luo

et al. 2020

Computer Methods in Applied Mechanics and Engineering

Self Cite

View full text Add to dashboard Cite

This work discovers the equivalence relation between quadrilateral meshes and meromorphic quartic differentials. Each quad-mesh induces a conformal structure of the surface, and a meromorphic quartic differential, where the configuration of singular vertices correspond to the configurations of the poles and zeros (divisor) of the meroromorphic differential. Due to Riemann surface theory, the configuration of singularities of a quad-mesh satisfies the Abel-Jacobi condition. Inversely, if a divisor satisfies the Abel-Jacobi condition, then there exists a meromorphic quartic differential whose divisor equals to the given one. Furthermore, if the meromorphic quadric differential is with finite trajectories, then it also induces a a quad-mesh, the poles and zeros of the meromorphic differential correspond to the singular vertices of the quad-mesh. Besides the theoretic proofs, the computational algorithm for verification of Abel-Jacobi condition is also explained in details. Furthermore, constructive algorithm of meromorphic quartic differential on genus zero surfaces is proposed, which is based on the global algebraic representation of meromorphic differentials. Our experimental results demonstrate the efficiency and efficacy of the algorithm. This opens up a novel direction for quad-mesh generation using algebraic geometric approach.

show abstract

“…If we subdivide the quad-mesh infinitely many times, we obtain geodesic lamination, each leaf is a closed loop. This is called the finite geodesic lamination condition [6].…”

Section: Quad-mesh Induced Structuresmentioning

confidence: 99%

“…However the holonomy condition heavily depends on the singularity configuration. The work in [6] didn't answer when the singularity configurations are appropriate for the holonomy condition.…”

Section: Previous Workmentioning

confidence: 99%

Quadrilateral mesh generation II: Meromorphic quartic differentials and Abel–Jacobi condition

Lei

Zheng

Luo

et al. 2020

Computer Methods in Applied Mechanics and Engineering

Self Cite

View full text Add to dashboard Cite

show abstract

“…This classification is achieved by the remarkable uniformization theorem of Poincaré and Koebe proved in 1907. It states that every simply connected Riemann surface is conformally diffeomorphic to the 2-sphere 2 , the plane 2 or the open unit disc ℍ 2 , as shown in Figure 1. Using covering space theory, the uniformization theorem implies that every connected oriented surface with a Riemannian metric ( , ) is conformally diffeomorphic to one of three canonical models of surfaces: (i) the unit sphere 2 , (ii) a flat torus, or 2 , or 2 − {0} or (iii) a hyperbolic surface ℍ 2 /Γ, where Γ is a discrete torsion-free subgroup of isometries of the hyperbolic plane ℍ 2 .…”

Section: Conformal Maps the Natural Correspondences Betweenmentioning

confidence: 99%

“…In general, the continuity at the singularities will be reduced, and it is highly desirable to allow the users to prescribe the positions of singularities. This can be achieved using the above method based on Abel's condition and Ricci flow, as explained in [2]. 3.5.…”

Section: Applicationsmentioning

confidence: 99%

Computational Conformal Geometry Behind Modern Technologies

Luo

Yau

2020

Notices Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Conformal geometry has deep roots in pure mathematics fields, such as Riemann surface theory, complex analysis, differential geometry, algebraic topology, partial differential equations and others. Historically, conformal geometry has been broadly used in many engineering applications [1], such as electro-magnetics, vibrating membranes, acoustics, elasticity, heat transfer and fluid flow. Most of these applications depend on conformal mappings between planar domains. Recently, with the rapid development of 3D scanning and medical imaging technologies, 3D geometric data has become ubiquitous. Figure 2 shows a human facial surface acquired using a scanning system based on structured light.

show abstract

“…The octree structure is a commonly used data structure for 3D models, which can help improve the efficiency of searching desired elements. Some researchers introduce more metrics to the existing algorithms to improve the mesh quality [30,31]. Soni et al [32] propose an efficient algorithm by applying centroidal voronoi tessellation on the voxelized Surface.…”

Section: Prior Workmentioning

confidence: 99%

Automatic generation of global shell-element meshes for large-scale structural design optimization

Hwang

2020

Aiaa Aviation 2020 Forum

View full text Add to dashboard Cite

Quadrilateral mesh generation I : Metric based method

Cited by 31 publications

References 18 publications

Quadrilateral mesh generation II: Meromorphic quartic differentials and Abel–Jacobi condition

Quadrilateral mesh generation II: Meromorphic quartic differentials and Abel–Jacobi condition

Computational Conformal Geometry Behind Modern Technologies

Automatic generation of global shell-element meshes for large-scale structural design optimization

Contact Info

Product

Resources

About