Article

Second order smoothness over extraordinary vertices

Author:

Charles LoopAuthors Info & Claims

SGP '04: Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on Geometry processing

Pages 165 - 174

https://doi.org/10.1145/1057432.1057454

Published: 08 July 2004 Publication History

Abstract

Catmull & Clark subdivision is now a standard for smooth free-form surface modeling. These surfaces are everywhere curvature continuous except at points corresponding to vertices not incident on four edges. While the surface has a continuous tangent plane at such a point, the lack of curvature continuity presents a severe problem for many applications. Topologically, each n-valent extraordinary vertex of a Catmull & Clark limit surface corresponds to an n-sided hole in the underlying 2-manifold represented by the control mesh. The problem we address here is: How to fill such a hole in a Catmull & Clark surface with exactly n tensor product patches that meet the surrounding bicubic patch network and each other with second order continuity. We convert the problem of filling the hole with n tensor product patches in the spatial domain into the problem of filling the hole in the n frequency modes with a single bidegree 7 tensor product patch.

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Cited By

Wang XMa W(2023)An extended tuned subdivision scheme with optimal convergence for isogeometric analysisComputer-Aided Design10.1016/j.cad.2023.103544162:COnline publication date: 13-Jul-2023
https://dl.acm.org/doi/10.1016/j.cad.2023.103544
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
Oberbichler TBletzinger K(2022)CAD-Integrated Form-Finding of Structural Membranes Using Extended Catmull–Clark Subdivision SurfacesComputer-Aided Design10.1016/j.cad.2022.103360151:COnline publication date: 1-Oct-2022
https://dl.acm.org/doi/10.1016/j.cad.2022.103360
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Index Terms

Second order smoothness over extraordinary vertices
1. Computing methodologies
  1. Computer graphics
    1. Shape modeling
2. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

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cover image ACM Other conferences

SGP '04: Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on Geometry processing

July 2004

259 pages

ISBN:3905673134

DOI:10.1145/1057432

Conference Chairs:
Jean-Daniel Boissonnat
INRIA, France
,
Pierre Alliez
INRIA, France

Copyright © 2004 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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EUROGRAPHICS: The European Association for Computer Graphics

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Association for Computing Machinery

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Published: 08 July 2004

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SGP04: Symposium on Geometry Processing

July 8 - 10, 2004

Nice, France

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Overall Acceptance Rate 64 of 240 submissions, 27%

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Cited By

Wang XMa W(2023)An extended tuned subdivision scheme with optimal convergence for isogeometric analysisComputer-Aided Design10.1016/j.cad.2023.103544162:COnline publication date: 13-Jul-2023
https://dl.acm.org/doi/10.1016/j.cad.2023.103544
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
Oberbichler TBletzinger K(2022)CAD-Integrated Form-Finding of Structural Membranes Using Extended Catmull–Clark Subdivision SurfacesComputer-Aided Design10.1016/j.cad.2022.103360151:COnline publication date: 1-Oct-2022
https://dl.acm.org/doi/10.1016/j.cad.2022.103360
Peters JKarčiauskas K(2022)Subdivision and G-Spline Hybrid Constructions for High-Quality Geometric and Analysis-Suitable SurfacesGeometric Challenges in Isogeometric Analysis10.1007/978-3-030-92313-6_8(171-189)Online publication date: 9-Aug-2022
https://doi.org/10.1007/978-3-030-92313-6_8
唐月(2017)Smooth Connection near Singular Points on Subdivision SurfacesAdvances in Applied Mathematics10.12677/AAM.2017.6914106:09(1163-1173)Online publication date: 2017
https://doi.org/10.12677/AAM.2017.69141
Usumezbas AFabbri RKimia B(2017)The Surfacing of Multiview 3D Drawings via Lofting and Occlusion Reasoning2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)10.1109/CVPR.2017.485(4560-4569)Online publication date: Jul-2017
https://doi.org/10.1109/CVPR.2017.485
Karčiauskas KPeters J(2015)Biquintic G 2 surfaces via functionalsComputer Aided Geometric Design10.1016/j.cagd.2014.11.00333:C(17-29)Online publication date: 1-Feb-2015
https://dl.acm.org/doi/10.1016/j.cagd.2014.11.003
Wu RPeters J(2015)Correct resolution rendering of trimmed spline surfacesComputer-Aided Design10.1016/j.cad.2014.08.01258:C(123-131)Online publication date: 1-Jan-2015
https://dl.acm.org/doi/10.1016/j.cad.2014.08.012
Cashman T(2012)Beyond Catmull–Clark? A Survey of Advances in Subdivision Surface MethodsComputer Graphics Forum10.1111/j.1467-8659.2011.02083.x31:1(42-61)Online publication date: 1-Feb-2012
https://dl.acm.org/doi/10.1111/j.1467-8659.2011.02083.x
Myles APeters J(2011)C 2 splines covering polar configurationsComputer-Aided Design10.1016/j.cad.2011.08.01843:11(1322-1329)Online publication date: 1-Nov-2011
https://dl.acm.org/doi/10.1016/j.cad.2011.08.018
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