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Uniform Convergence of Discrete Curvatures from Nets of Curvature Lines

Ulrich Bauer, Konrad Polthier, Max Wardetzky
2010 Discrete & Computational Geometry  
We study discrete curvatures computed from nets of curvature lines on a given smooth surface, and prove their uniform convergence to smooth principal curvatures.  ...  We provide explicit error bounds, with constants depending only on properties of the smooth limit surface and the shape regularity of the discrete net.  ...  Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided  ... 
doi:10.1007/s00454-009-9237-4 fatcat:mzhxhje3tfgtjlivflpednsfwe

A Glimpse into Discrete Differential Geometry

Keenan Crane, Max Wardetzky
2017 Notices of the American Mathematical Society  
For instance, surfaces parameterized by principal curvature lines are replaced by meshes made of circular quadrilaterals (top left), the maximum principle obeyed by harmonic functions is expressed via  ...  Discrete differential geometry reimagines classical ideas from differential geometry without reference to differential calculus.  ...  However, convergence does not always single out a best version: treated appropriately, all four of our discrete curvatures will converge.  ... 
doi:10.1090/noti1578 fatcat:syg2kw3nwbcmbnmpq3vebmisea

Computing general geometric structures on surfaces using Ricci flow

Miao Jin, Feng Luo, Xianfeng David Gu
2007 Computer-Aided Design  
In order to discover the intrinsic geometric structures of general surfaces, we developed a theoretic rigorous and practical efficient method, Discrete Variational Ricci flow.  ...  We solve the problem by introducing a novel method based on discrete variational Ricci flow.  ...  Theorem 6 (Discrete Ricci Flow) The discrete Ricci flows 11 and 12 are convergent to the uniformization metric and the convergence rate is exponential.  ... 
doi:10.1016/j.cad.2007.05.006 fatcat:h2i2rbp62ba7xcqqtleeal6sm4

A Hybrid Inference System for Improved Curvature Estimation in the Level-Set Method Using Machine Learning [article]

Luis Ángel Larios-Cárdenas, Frédéric Gibou
2022 arXiv   pre-print
The core of our hybrid framework is a switching mechanism that relies on well established numerical techniques to gauge curvature.  ...  To reduce data set size and training complexity, we leverage the problem's characteristic symmetry and build our models on just half of the curvature spectrum.  ...  Acknowledgements Use was made of computational facilities purchased with funds from the National Science Foundation (CNS-1725797) and administered by the Center for Scientific Computing (CSC).  ... 
arXiv:2104.02951v4 fatcat:hz52wy2mgfd3tgi47iyv2xoqb4

Topological Equivalence and Curvature Convergence: Bezier Surface Approximation [article]

J. Li
2013 arXiv   pre-print
We also show that the total Gaussian curvature of the triangulated surface converges to the total Gaussian curvature of the Bezier surface.  ...  A set of control points can determine a Bezier surface and a triangulated surface simultaneously.  ...  Peters for stimulating questions that motivate this work, and Anne Berres for her Matlab codes of Bézier surfaces.  ... 
arXiv:1311.5280v1 fatcat:wo5i4qllevagjkkhokp67dspdu

Convergence of Combinatorial Gravity [article]

Christy Kelly, Carlo Trugenberger, Fabio Biancalana
2022 arXiv   pre-print
In particular, we define a discrete Einstein-Hilbert action that converges to its manifold counterpart on sufficiently dense random geometric graphs (more generally on any sequence of graphs that converges  ...  Our construction relies crucially on the Ollivier curvature of optimal transport theory. Our methods also allow us to define an analogous discrete action for Klein-Gordon fields.  ...  such as graphs, regarded as discrete metric spaces; we have begun to pursue this line of thought in [47, 48, 81] , calling the associated Ollivier curvature based model combinatorial quantum gravity  ... 
arXiv:2102.02356v3 fatcat:seaeq4kvznhk7cokaz3iji3ig4

Mem3DG: Modeling Membrane Mechanochemical Dynamics in 3D using Discrete Differential Geometry [article]

C. Zhu, C. T. Lee, P. Rangamani
2021 bioRxiv   pre-print
In this work, we provide a unifying perspective by identifying fundamental geometric invariants of the discrete force derived from a discrete energy.  ...  the terms contributing to both the energy and forces from the discrete and smooth geometric theories remains to be established.  ...  Net force and the benefit of DDG Summing the force terms from each physics, we obtain a net force.  ... 
doi:10.1101/2021.10.30.466618 fatcat:kqaal2algbhv3o3iwi2w4tao64

Page 5601 of Mathematical Reviews Vol. , Issue 2004g [page]

2004 Mathematical Reviews  
Construction of shortest line of restricted curvature in a non-singly-connected polygonal area.  ...  Excerpts from the introduction: “The importance of subdivision to applications in computer aided geometric design became clear with the generalizations of the tensor product spline rules to control nets  ... 

CMC-1 surfaces via osculating Möbius transformations between circle patterns [article]

Wai Yeung Lam
2020 arXiv   pre-print
These correspondences are analogous to the Weierstrass representation for surfaces with constant mean curvature H≡ 1 in hyperbolic space. We further establish convergence on triangular lattices.  ...  Such a function plays the role of an osculating Möbius transformation and induces a realization of the dual graph in hyperbolic space.  ...  resemble curvature line parametrizations.  ... 
arXiv:2007.04253v2 fatcat:xigbxorrsngbfiyhluwr7ghyvy

Dev2PQ: Planar Quadrilateral Strip Remeshing of Developable Surfaces [article]

Floor Verhoeven, Amir Vaxman, Tim Hoffmann, Olga Sorkine-Hornung
2021 arXiv   pre-print
Developable PQ-strip meshes are useful in many areas of shape modeling, thanks to the simplicity of fabrication from flat sheet material.  ...  principal curvatures.  ...  This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 101003104).  ... 
arXiv:2103.00239v2 fatcat:7l45gs5k5bezhoxpoupenfrcfa

Isoradial meshes: Covering elastic gridshells with planar facets

Cyril Douthe, Romain Mesnil, Hugo Orts, Olivier Baverel
2017 Automation in Construction  
Gridshells present some advantages in terms of manufacturing, lightness and time of assembly. Their covering remains  ...  Elastic gridshells are structures made of flat two-way grids which are deformed elastically before they are braced and which afterwards mechanically behave like continuous shells.  ...  The bending energy of the Tchebycheff net measures the discrete curvature of the lines.  ... 
doi:10.1016/j.autcon.2017.08.015 fatcat:to3p5ohesvfotnlvd74akmjo64

Emergence of the Circle in a Statistical Model of Random Cubic Graphs [article]

Christy Kelly, Carlo Trugenberger, Fabio Biancalana
2021 arXiv   pre-print
configurations converges in the sense of Gromov-Hausdorff to S^1_r.  ...  We consider a formal discretisation of Euclidean quantum gravity defined by a statistical model of random 3-regular graphs and making using of the Ollivier curvature, a coarse analogue of the Ricci curvature  ...  of discrete curvature adopted.  ... 
arXiv:2008.11779v3 fatcat:4sopddxir5f45kr7aar33cix3a

Discrete Geodesic Nets for Modeling Developable Surfaces [article]

Michael Rabinovich, Tim Hoffmann, Olga Sorkine-Hornung
2017 arXiv   pre-print
We further present an extension of our model that enables a local definition of discrete isometry.  ...  We demonstrate the effectiveness of our discrete model in a developable surface editing system, as well as computation of an isometric interpolation between isometric discrete developable shapes.  ...  .: StG-2012-306877 (ERC Starting Grant iModel) and by the Deutsche Forschungsgemeinschaft-Collaborative Research Center, TRR 109, "Discretization in Geometry and Dynamics. "  ... 
arXiv:1707.08360v1 fatcat:xzq3xvkc6jho3od452am57v63e

Discrete Willmore Flow [article]

Alexander I. Bobenko, Peter Schröder
2005 Symposium on geometry processing : [proceedings]. Symposium on Geometry Processing  
The Willmore energy of a surface, R(H2 -K)dA, as a function of mean and Gaussian curvature, captures the deviation of a surface from (local) sphericity.  ...  In contrast to traditional approaches it is not based on a finite element discretization, but rather on an ab initio discrete formulation which preserves the Möbius symmetries of the underlying continuous  ...  Another interesting avenue is the use of the Willmore functional to construct curvature line nets.  ... 
doi:10.2312/sgp/sgp05/101-110 fatcat:bx3dew4vrvgzfmk3hfc2x4pyoe

Discrete Willmore flow

Alexander I. Bobenko, Peter Schröder
2005 ACM SIGGRAPH 2005 Courses on - SIGGRAPH '05  
The Willmore energy of a surface, (H 2 − K) dA, as a function of mean and Gaussian curvature, captures the deviation of a surface from (local) sphericity.  ...  In contrast to traditional approaches it is not based on a finite element discretization, but rather on an ab initio discrete formulation which preserves the Möbius symmetries of the underlying continuous  ...  Another interesting avenue is the use of the Willmore functional to construct curvature line nets.  ... 
doi:10.1145/1198555.1198664 dblp:conf/siggraph/BobenkoS05 fatcat:acylfvw7vjbpheivporabauw7e
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