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Parametrizations for triangular Gk spline surfaces of low degree

Published: 01 October 2006 Publication History

Abstract

In this article, we present regularly parametrized Gk free-form spline surfaces that extend box and half-box splines over regular triangular grids. The polynomial degree of these splines is max{4k + 1, ⌈3k/2 + 1⌉r}, where r ∈ ℕ can be chosen arbitrarily and determines the flexibility at extraordinary points. The Gk splines presented in this article depend crucially on low-degree (re-)parametrizations of piecewise polynomial hole fillings. The explicit construction of such parametrizations forms the core of this work and we present two classes of singular and regular parametrizations. Also, we show how to build box and half-box spline surfaces of arbitrarily high smoothness with holes bounded by only n patches, in principle.

References

[1]
Catmull, E. and Clark, J. 1978. Recursive generated b-spline surfaces on arbitrary topological meshes. Comput. Aided Des. 10, 6, 350--355.
[2]
Doo, D. and Sabin, M. 1978. Behaviour of recursive division surfaces near extraordinary points. Comput. Aided Des. 10, 6, 356--360.
[3]
Dyn, N., Gregory, J., and Levin, D. 1990. A butterfly subdivision scheme for surface interpolation with tension control. ACM Trans. Graph. 9, 2, 160--169.
[4]
Hahn, J. 1989. Geometric continuous patch complexes. Comput. Aided Geom. Des. 6, 1, 55--67.
[5]
Karciauskas, K., Peters, J., and Reif, U. 2004. Shape characterization of subdivision surfaces---Case studies. Comput. Aided Geom. Des. 21, 6, 601--614.
[6]
Kobbelt, L. 2000. varun 3 subdivision. In Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques. 103--112.
[7]
Lane, J. and Riesenfeld, R. 1980. A theoretical development for the computer generation and display of piecewise polynomial surfaces. IEEE Trans. Pattern Anal. 2, 1, 35--46.
[8]
Loop, C. 1987. Smooth subdivision surfaces based on triangles. M.S. thesis, Department of Mathematics, University of Utah.
[9]
Peters, J. 2002. C2 free-form surfaces of degree (3,5). Comput. Aided Geom. Des. 19, 113--126.
[10]
Peters, J. and Reif, U. 2004. Shape characterization of subdivision surfaces---Basic principles. Comput. Aided Geom. Des. 21, 6, 585--599.
[11]
Prautzsch, H. 1984. Unterteilungsalgorithmen für multivariate Splines. Ph.D. thesis, Universität Braunschweig.
[12]
Prautzsch, H. 1997. Freeform splines. Comput. Aided Geom. Des. 14, 3, 201--206.
[13]
Prautzsch, H. 1998. Smoothness of subdivision surfces at extraordinary points. Adv. Comput. Math. 9, 377--389.
[14]
Prautzsch, H. and Boehm, W. 2002. Box splines. In Handbook of CAGD G. Farin et al., Eds. Elsevier, 255--282.
[15]
Prautzsch, H., Boehm, W., and Paluszny, M. 2002. Bézier and B-Spline Techniques. Springer-Verlag, New York.
[16]
Prautzsch, H. and Reif, U. 1999. Degree estimates for Ck-piecewise polynomial subdivision surfaces. Adv. Comput. Math. 10, 2, 209--217.
[17]
Prautzsch, H. and Umlauf, G. 2000. Triangular G2-splines. In Curve and Surface Design---Saint-Malo 1999. P.-L. Laurent et al. Eds. Vanderbilt University, Nashville, TN. 335--342.
[18]
Qu, R. 1990. Recursive subdivision algorithms for curve and surface design. Ph.D. thesis, Department of Mathematics and Statistics, Burnel University, Uxbridge, Middlesex, UK.
[19]
Reif, U. 1996. A degree estimate for subdivision surfaces of higher regularity. Proc. Amer. Math. Soc. 124, 7, 2167--2174.
[20]
Reif, U. 1998. TURBS---Topologically unrestricted rational B-splines. Constr. Approx. 14, 1, 57--78.
[21]
Umlauf, G. 2004. A technique for verifying the smoothness of subdivison schemes. In Geometric Modeling and Computing---Seattle 2003. M. Lucian and M. Neamtu, Eds. Nashboro Press, 513--521.
[22]
Zorin, D. and Schröder, P. 2001. A unified framework for primal/dual quadrilateral subdivision schemes. Comput. Aided Geom. Des. 18, 5, 429--454.

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  • (2017)An Effective 3D Shape Descriptor for Object Recognition with RGB-D SensorsSensors10.3390/s1703045117:3(451)Online publication date: 24-Feb-2017
  • (2009)A new construction of smooth surfaces from triangle meshes using parametric pseudo-manifoldsComputers & Graphics10.1016/j.cag.2009.03.01733:3(331-340)Online publication date: Jun-2009
  • (2009)Assembling curvature continuous surfaces from triangular patchesComputers & Graphics10.1016/j.cag.2009.03.01533:3(204-210)Online publication date: Jun-2009

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Published In

cover image ACM Transactions on Graphics
ACM Transactions on Graphics  Volume 25, Issue 4
October 2006
243 pages
ISSN:0730-0301
EISSN:1557-7368
DOI:10.1145/1183287
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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 October 2006
Published in TOG Volume 25, Issue 4

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Author Tags

  1. CAD
  2. Geometric modeling
  3. curves
  4. surfaces

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View all
  • (2017)An Effective 3D Shape Descriptor for Object Recognition with RGB-D SensorsSensors10.3390/s1703045117:3(451)Online publication date: 24-Feb-2017
  • (2009)A new construction of smooth surfaces from triangle meshes using parametric pseudo-manifoldsComputers & Graphics10.1016/j.cag.2009.03.01733:3(331-340)Online publication date: Jun-2009
  • (2009)Assembling curvature continuous surfaces from triangular patchesComputers & Graphics10.1016/j.cag.2009.03.01533:3(204-210)Online publication date: Jun-2009

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