The simplest subdivision scheme for smoothing polyhedra

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The simplest subdivision scheme for smoothing polyhedra

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ACM Transactions on Graphics (TOG) archive
Volume 16 , Issue 4 (October 1997) table of contents

Pages: 420 - 431

Year of Publication: 1997

ISSN:0730-0301

Authors

Jörg Peters	Purdue Univ., West Lafayette, IN
Ulrich Reif	Univ. Stuttgart, Stuttgart, Germany

Publisher

ACM New York, NY, USA

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Downloads (6 Weeks): 12, Downloads (12 Months): 79, Citation Count: 30

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abstract references cited by index terms review collaborative colleagues

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ABSTRACT

Given a polyhedron, construct a new polyhedron by connecting every edge-midpoint to its four neighboring edge-midpoints. This refinement rule yields a C1 surface and the surface has a piecewise quadratic parametrozation except at a finite number of isolated points. We analyze and improve the construction.

REFERENCES

Note: OCR errors may be found in this Reference List extracted from the full text article. ACM has opted to expose the complete List rather than only correct and linked references.

	1	Carl de Boor , Klaus Höllig , Sherman Riemenschneider, Box splines, Springer-Verlag New York, Inc., New York, NY, 1993
	2	CATMULL, E. AND CLARK, J. 1978. Recursively generated B-spline surfaces on arbitrary topological meshes. Comput. Aided Des. 10, 350-355.
	3	DOO, D. AND SABIN, M.A. 1978. Behaviour ofrecursive subdivision surfaces near extraordinary points. Comput. Aided Des. 10, 356-360.
	4	LooP, C. 1987. Smooth subdivision for surfaces based on triangles. Master's thesis, University of Utah.
	5	Jörg Peters, Smooth free-form surfaces over irregular meshes generalizing quadratic splines, Computer Aided Geometric Design, v.10 n.3-4, p.347-361, Aug. 1993
	6	Jörg Peters , Ulrich Reif, Analysis of Algorithms Generalizing B-Spline Subdivision, SIAM Journal on Numerical Analysis, v.35 n.2, p.728-748, April 1998 [doi>10.1137/S0036142996304346]
	7	POWELL, M. 1969. Piecewise quadratic surface fitting for contour plotting. In Software for Numerical Mathematics, D. Evans, Ed., Academic Press, Boston, MA.
	8	PRAUTZSCH, g. 1984. Unterteilungsalgorithmen ffir multivariante Splines. Ph.D. thesis, Tech. Universit~t Braunschweig.
	9	Ulrich Reif, A unified approach to subdivision algorithms near extraordinary vertices, Computer Aided Geometric Design, v.12 n.2, p.153-174, March 1995 [doi>10.1016/0167-8396(94)00007-F]
	10	SABIN, M. 1986. Recursive subdivision. In The Mathematics of Surfaces, Clarendon Press, Oxford, England, 269-282.
	11	ZWART, P. 1973. Multivariate splines with non-degenerate partitions. SIAM J. Numer. Anal. 10, 665-673.

CITED BY 30

	Chandrajit L. Bajaj , Valerio Pascucci , Guozhong Zhuang, Progressive compressive and transmission of arbitrary triangular meshes, Proceedings of the conference on Visualization '99: celebrating ten years, p.307-316, October 1999, San Francisco, California, United States
	Carlos Gonzalez-Ochoa , Jorg Peters, Localized-hierarchy surface splines (LeSS), Proceedings of the 1999 symposium on Interactive 3D graphics, p.7-15, April 26-29, 1999, Atlanta, Georgia, United States
	Xuefu Wang , Fuhua (Frank) Cheng , Brian A. Barsky, Blending, smoothing and interpolation of irregular meshes using N-sided Varady patches, Proceedings of the fifth ACM symposium on Solid modeling and applications, p.212-222, June 08-11, 1999, Ann Arbor, Michigan, United States
	Alberto S. Aguado , Eugenia Montiel , Ed Zaluska, Modeling generalized cylinders via Fourier morphing, ACM Transactions on Graphics (TOG), v.18 n.4, p.293-315, Oct. 1999
	Chhandomay Mandal , Hong Qin , Baba C. Vemuri, A novel FEM-based dynamic framework for subdivision surfaces, Proceedings of the fifth ACM symposium on Solid modeling and applications, p.191-202, June 08-11, 1999, Ann Arbor, Michigan, United States
	Leif Kobbelt, √3-subdivision, Proceedings of the 27th annual conference on Computer graphics and interactive techniques, p.103-112, July 2000
	Martin Bertram, Biorthogonal wavelets for subdivision volumes, Proceedings of the seventh ACM symposium on Solid modeling and applications, June 17-21, 2002, Saarbrücken, Germany
	Luiz Velho, Stellar subdivision grammars, Proceedings of the 2003 Eurographics/ACM SIGGRAPH symposium on Geometry processing, June 23-25, 2003, Aachen, Germany
	Hong Qin , Chhandomay Mandal , Baba C. Vemuri, Dynamic Catmull-Clark Subdivision Surfaces, IEEE Transactions on Visualization and Computer Graphics, v.4 n.3, p.215-229, July 1998
	I. P. Ivrissimtzis , N. A. Dodgson , M. A. Sabin, A generative classification of mesh refinement rules with lattice transformations, Computer Aided Geometric Design, v.21 n.1, p.99-109, January 2004
	Chhandomay Mandal , Hong Qin , Baba C. Vemuri, Dynamic Modeling of Butterfly Subdivision Surfaces, IEEE Transactions on Visualization and Computer Graphics, v.6 n.3, p.265-287, July 2000
	Jörg Peters , Le-Jeng Shiue, Combining 4- and 3-direction subdivision, ACM Transactions on Graphics (TOG), v.23 n.4, p.980-1003, October 2004
	Ergun Akleman , Jianer Chen, Regular meshes, Proceedings of the 2005 ACM symposium on Solid and physical modeling, p.213-219, June 13-15, 2005, Cambridge, Massachusetts
	Georg Umlauf, Analysis and tuning of subdivision algorithms, Proceedings of the 21st spring conference on Computer graphics, May 12-14, 2005, Budmerice, Slovakia
	I. P. Ivrissimtzis , H.-P. Seidel, Evolutions of polygons in the study of subdivision surfaces, Computing, v.72 n.1-2, p.93-103, April 2004
	Severinas Zube, Number systems, α-splines and refinement, Journal of Computational and Applied Mathematics, v.172 n.2, p.207-231, 1 December 2004
	Loïc Barthe , Leif Kobbelt, Subdivision scheme tuning around extraordinary vertices, Computer Aided Geometric Design, v.21 n.6, p.561-583, July 2004
	Ioannis P. Ivrissimtzis , Malcolm A. Sabin , Neil A. Dodgson, On the support of recursive subdivision, ACM Transactions on Graphics (TOG), v.23 n.4, p.1043-1060, October 2004
	Chandrajit L. Bajaj , Guoliang Xu, Anisotropic diffusion of surfaces and functions on surfaces, ACM Transactions on Graphics (TOG), v.22 n.1, p.4-32, January 2003
	Yonggang Xue , Thomas P.-Y. Yu , Tom Duchamp, Jet subdivision schemes on the k-regular complex, Computer Aided Geometric Design, v.23 n.4, p.361-396, May 2006
	Charles K. Chui , Qingtang Jiang, Matrix-valued subdivision schemes for generating surfaces with extraordinary vertices, Computer Aided Geometric Design, v.23 n.5, p.419-438, July 2006
	Charles K. Chui , Qingtang Jiang, From extension of Loop's approximation scheme to interpolatory subdivisions, Computer Aided Geometric Design, v.25 n.2, p.96-115, February, 2008
	H. Hussein Karam , F. F. M. Ghaleb , Y. M. Abd El-Latif, Towards fast and smooth subdivision surface reconstruction, International Journal of Computers and Applications, v.28 n.2, p.170-176, April 2006
	Denis Zorin, Modeling with multiresolution subdivision surfaces, ACM SIGGRAPH 2006 Courses, July 30-August 03, 2006, Boston, Massachusetts
	Vinod Srinivasan , Ergun Akleman, Connected & manifold Sierpinsky polyhedra, Proceedings of the ninth ACM symposium on Solid modeling and applications, June 09-11, 2004, Genoa, Italy
	Guiqing Li , Weiyin Ma, Composite 2 subdivision surfaces, Computer Aided Geometric Design, v.24 n.6, p.339-360, August, 2007
	L. Wang , G. H. Wang, Subdivision-based integration of geometric design and finite element mesh generation for sheet metal structures, International Journal of Computer Applications in Technology, v.32 n.1, p.40-46, July 2008
	Elliot English , Robert Bridson, Animating developable surfaces using nonconforming elements, ACM Transactions on Graphics (TOG), v.27 n.3, August 2008
	Marcelo Siqueira , Dianna Xu , Jean Gallier , Luis Gustavo Nonato , Dimas Martínez Morera , Luiz Velho, Technical Section: A new construction of smooth surfaces from triangle meshes using parametric pseudo-manifolds, Computers and Graphics, v.33 n.3, p.331-340, June, 2009
	Ergun Akleman , Jianer Chen , Qing Xing , Jonathan L. Gross, Cyclic plain-weaving on polygonal mesh surfaces with graph rotation systems, ACM Transactions on Graphics (TOG), v.28 n.3, August 2009

INDEX TERMS

Primary Classification:
I. Computing Methodologies
I.3 COMPUTER GRAPHICS
I.3.3 Picture/Image Generation
Subjects: Display algorithms

Additional Classification:
I. Computing Methodologies
I.3 COMPUTER GRAPHICS
I.3.5 Computational Geometry and Object Modeling
Subjects: Geometric algorithms, languages, and systems; Curve, surface, solid, and object representations

General Terms:
Algorithms, Design

Keywords:
arbitrary topology, box-spline, characteristic map, simplest smooth subdivision, surface

REVIEW

"Nickolas S. Sapidis : Reviewer"

This short note deals with the smoothness of the surface produced when the following subdivision scheme is applied to a polyhedron: “connect every edge-midpoint to the four midpoints of the edges sharing a vertex and a face w more...

Collaborative Colleagues:

Jörg Peters: colleagues

Ulrich Reif: colleagues

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