Article

Manifold splines

Authors:

Hong QinAuthors Info & Claims

SPM '05: Proceedings of the 2005 ACM symposium on Solid and physical modeling

Pages 27 - 38

https://doi.org/10.1145/1060244.1060249

Published: 13 June 2005 Publication History

Abstract

Constructing splines whose parametric domain is an arbitrary manifold and effectively computing such splines in real-world applications are of fundamental importance in solid and shape modeling, geometric design, graphics, etc. This paper presents a general theoretical and computational framework, in which spline surfaces defined over planar domains can be systematically extended to manifold domains with arbitrary topology with or without boundaries. We study the affine structure of domain manifolds in depth and prove that the existence of manifold splines is equivalent to the existence of a manifold's affine atlas. Based on our theoretical breakthrough, we also develop a set of practical algorithms to generalize triangular B-spline surfaces from planar domains to manifold domains. We choose triangular B-splines mainly because of its generality and many of its attractive properties. As a result, our new spline surface defined over any manifold is a piecewise polynomial surface with high parametric continuity without the need for any patching and/or trimming operations. Through our experiments, we hope to demonstrate that our novel manifold splines are both powerful and efficient in modeling arbitrarily complicated geometry and representing continuously-varying physical quantities defined over shapes of arbitrary topology.

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Cited By

Takacs T(2025)Approximation properties over self-similar meshes of curved finite elements and applications to subdivision based isogeometric analysisComputer Aided Geometric Design10.1016/j.cagd.2025.102413(102413)Online publication date: Feb-2025
https://doi.org/10.1016/j.cagd.2025.102413
Karčiauskas KPeters J(2024) Quadratic-attraction subdivision with contraction-ratio Computers & Graphics10.1016/j.cag.2024.104001123(104001)Online publication date: Oct-2024
https://doi.org/10.1016/j.cag.2024.104001
Wen ZFaruque MLi XWei XCasquero H(2023)Isogeometric analysis using G-spline surfaces with arbitrary unstructured quadrilateral layoutComputer Methods in Applied Mechanics and Engineering10.1016/j.cma.2023.115965408(115965)Online publication date: Apr-2023
https://doi.org/10.1016/j.cma.2023.115965
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Index Terms

Manifold splines
1. Computing methodologies
  1. Computer graphics
    1. Shape modeling
      1. Parametric curve and surface models
      2. Volumetric models

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cover image ACM Conferences

SPM '05: Proceedings of the 2005 ACM symposium on Solid and physical modeling

June 2005

287 pages

ISBN:1595930159

DOI:10.1145/1060244

Program Chairs:
Leif Kobbelt
RWTH Aachen University, Germany
,
Vadim Shapiro
University of Wisconsin--Madison, USA

Copyright © 2005 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 13 June 2005

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SPM05: 2005 ACM Symposium on Solid and Physical Modeling

June 13 - 15, 2005

Massachusetts, Cambridge

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Cited By

Takacs T(2025)Approximation properties over self-similar meshes of curved finite elements and applications to subdivision based isogeometric analysisComputer Aided Geometric Design10.1016/j.cagd.2025.102413(102413)Online publication date: Feb-2025
https://doi.org/10.1016/j.cagd.2025.102413
Karčiauskas KPeters J(2024) Quadratic-attraction subdivision with contraction-ratio Computers & Graphics10.1016/j.cag.2024.104001123(104001)Online publication date: Oct-2024
https://doi.org/10.1016/j.cag.2024.104001
Wen ZFaruque MLi XWei XCasquero H(2023)Isogeometric analysis using G-spline surfaces with arbitrary unstructured quadrilateral layoutComputer Methods in Applied Mechanics and Engineering10.1016/j.cma.2023.115965408(115965)Online publication date: Apr-2023
https://doi.org/10.1016/j.cma.2023.115965
Karčiauskas KPeters J(2023)Improved Caps for Improved Subdivision SurfacesComputer-Aided Design10.1016/j.cad.2023.103543162(103543)Online publication date: Sep-2023
https://doi.org/10.1016/j.cad.2023.103543
Feng WZhang JZhou YXin S(2022)GDR-Net: A Geometric Detail Recovering Network for 3D Scanned ObjectsIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2021.311065828:12(3959-3973)Online publication date: 1-Dec-2022
https://doi.org/10.1109/TVCG.2021.3110658
Wei XLi XQian KHughes TZhang YCasquero H(2022)Analysis-suitable unstructured T-splines: Multiple extraordinary points per faceComputer Methods in Applied Mechanics and Engineering10.1016/j.cma.2021.114494391(114494)Online publication date: Mar-2022
https://doi.org/10.1016/j.cma.2021.114494
Zhang QNiu DWang XZhao X(2017)A spherical volumetric parameterization approach based on large-scale distortion harmonic energy2017 IEEE 9th International Conference on Communication Software and Networks (ICCSN)10.1109/ICCSN.2017.8230210(742-748)Online publication date: May-2017
https://doi.org/10.1109/ICCSN.2017.8230210
Zhang QZhao XYu S(2016)A volumetric parameterization approach based on bounded-distortion harmonic energy2016 9th International Congress on Image and Signal Processing, BioMedical Engineering and Informatics (CISP-BMEI)10.1109/CISP-BMEI.2016.7852735(352-357)Online publication date: Oct-2016
https://doi.org/10.1109/CISP-BMEI.2016.7852735
Mourrain BVidunas RVillamizar N(2016)Dimension and bases for geometrically continuous splines on surfaces of arbitrary topologyComputer Aided Geometric Design10.1016/j.cagd.2016.03.00345:C(108-133)Online publication date: 1-Jul-2016
https://dl.acm.org/doi/10.1016/j.cagd.2016.03.003
Lai LYuen M(2015)Blending of mesh objects to parametric surfaceComputers and Graphics10.1016/j.cag.2014.09.03046:C(283-293)Online publication date: 1-Feb-2015
https://dl.acm.org/doi/10.1016/j.cag.2014.09.030
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