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T-splines and T-NURCCs

Authors:

Thomas W. Sederberg,

Jianmin Zheng,

Almaz Bakenov,

Ahmad NasriAuthors Info & Claims

ACM Transactions on Graphics (TOG), Volume 22, Issue 3

Pages 477 - 484

https://doi.org/10.1145/882262.882295

Published: 01 July 2003 Publication History

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Abstract

This paper presents a generalization of non-uniform B-spline surfaces called T-splines. T-spline control grids permit T-junctions, so lines of control points need not traverse the entire control grid. T-splines support many valuable operations within a consistent framework, such as local refinement, and the merging of several B-spline surfaces that have different knot vectors into a single gap-free model. The paper focuses on T-splines of degree three, which are C² (in the absence of multiple knots). T-NURCCs (Non-Uniform Rational Catmull-Clark Surfaces with T-junctions) are a superset of both T-splines and Catmull-Clark surfaces. Thus, a modeling program for T-NURCCs can handle any NURBS or Catmull-Clark model as special cases. T-NURCCs enable true local refinement of a Catmull-Clark-type control grid: individual control points can be inserted only where they are needed to provide additional control, or to create a smoother tessellation, and such insertions do not alter the limit surface. T-NURCCs use stationary refinement rules and are C² except at extraordinary points and features.

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References

[1]

BAKENOV, A. 2001. T-Splines: Tensor Product B-spline Surfaces with T-Junctions. Master's thesis, Brigham Young University.

Google Scholar

[2]

CATMULL, E., AND CLARK, J. 1978. Recursively Generated B-spline Surfaces On Arbitrary Topological Meshes. Computer-Aided Design 10, 350--355.

Crossref

Google Scholar

[3]

FORSEY, D., AND BARTELS, R. H. 1988. Hierarchical B-spline refinement. Computer Graphics 22, 4(August 1988), 205--212.

Digital Library

Google Scholar

[4]

GRINSPUN, E., KRYSL, P., AND SCHRÖDER, P. 2002. Charms: A simple framework for adaptive simulation. ACM Transactions on Graphics 21, 3 (July), 281--290.

Digital Library

Google Scholar

[5]

KOBBELT, L. 2000. √3-subdivision. Computer Graphics, 103--112. SIGGRAPH 2000.

Digital Library

Google Scholar

[6]

KRAFT, R. 1998. Adaptive and linearly independent multilevel B-splines. In Surface Fitting and Multiresolution Methods, A. L. Mehaute, C. Rabut, and L. L. Schumaker, Eds. Vanderbilt University Press, Nashville, 209--218.

Google Scholar

[7]

RAMSHAW, L. 1989. Blossoms are polar forms. Computer Aided Geometric Design 6, 323--358.

Digital Library

Google Scholar

[8]

ROCKWOOD, A. P., HEATON, K., AND DAVIS, T. 1989. Real-time rendering of trimmed surfaces. Proceedings of SIGGRAPH 89, 107--116.

Digital Library

Google Scholar

[9]

SEDERBERG, T. W., ZHENG, J., SEWELL, D., AND SABIN, M. 1998. Non-uniform recursive subdivision surfaces. Proceedings of SIGGRAPH 98 (July), 387--394. ISBN 0-89791-999-8.

Digital Library

Google Scholar

[10]

VELHO, L., AND ZORIN, D. 2001. 4--8 subdivision. Computer Aided Geometric Design 18, 5, 397--428.

Digital Library

Google Scholar

[11]

WELLER, F., AND HAGEN, H. 1995. Tensor product spline spaces with knot segments. In Mathematical Methods for Curves and Surfaces, M. Daehlen, T. Lyche, and L. L. Schumaker, Eds. Vanderbilt University Press, Nashville, 563--572.

Google Scholar

[12]

ZORIN, D., AND SCHRÖDER, P., 2000. Subdivision for modeling and animation, SIGGRAPH'00 course notes.

Google Scholar

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Index Terms

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

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Published In

cover image ACM Transactions on Graphics

ACM Transactions on Graphics Volume 22, Issue 3

July 2003

683 pages

ISSN:0730-0301

EISSN:1557-7368

DOI:10.1145/882262

Issue’s Table of Contents

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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Publication History

Published: 01 July 2003

Published in TOG Volume 22, Issue 3

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Cited By

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Bastl BSlabá K(2025)Adaptive refinement in incompressible fluid flow simulation based on THB-splines-powered isogeometric analysisMathematics and Computers in Simulation10.1016/j.matcom.2024.09.016228(514-533)Online publication date: Feb-2025
https://doi.org/10.1016/j.matcom.2024.09.016
Giannelli CImperatore SMantzaflaris AMokriš D(2025)Efficient alternating and joint distance minimization methods for adaptive spline surface fittingGraphical Models10.1016/j.gmod.2024.101251137(101251)Online publication date: Feb-2025
https://doi.org/10.1016/j.gmod.2024.101251
Du XLei SHuang ZWang WZhao G(2025)Seamless integration of design and analysis for architected shell structures using unstructured T-splinesComputer Methods in Applied Mechanics and Engineering10.1016/j.cma.2024.117619435(117619)Online publication date: Feb-2025
https://doi.org/10.1016/j.cma.2024.117619
Qian KSuarez GNambara TKanekiyo TLiao AWebster-Wood VZhang Y(2025)Neurodevelopmental disorders modeling using isogeometric analysis, dynamic domain expansion and local refinementComputer Methods in Applied Mechanics and Engineering10.1016/j.cma.2024.117534433(117534)Online publication date: Jan-2025
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Grošelj JŠadl Praprotnik A(2025) Rational cubic Powell–Sabin B-splines with application to representation of ruled surfaces Journal of Computational and Applied Mathematics10.1016/j.cam.2024.116292457(116292)Online publication date: Mar-2025
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Qarariyah AYang TDeng F(2024)A Solution-Structure B-Spline-Based Framework for Hybrid Boundary Problems on Implicit DomainsMathematics10.3390/math1224397312:24(3973)Online publication date: 18-Dec-2024
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Harmening CButzer R(2024)Improving the approximation quality of tensor product B-spline surfaces by local parameterizationJournal of Applied Geodesy10.1515/jag-2023-0071Online publication date: 4-Jan-2024
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