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Spline Surfaces over Arbitrary Topological Meshes: Theoretical Analysis and Application

Part of: Numerical approximation and computational geometry

Published online by Cambridge University Press:  20 July 2016

Chaoyang Liu  and
Xiaoping Zhou
Chaoyang Liu*
Affiliation:
School of Mathematics & Statistics, Zhengzhou University, Zhengzhou 450001, China
Xiaoping Zhou*
Affiliation:
School of Information Engineering, Zhengzhou University, Zhengzhou 450001, China
*
*Corresponding author. Email addresses:[email protected] (C.-Y. Liu), [email protected] (X.-P. Zhou)
*Corresponding author. Email addresses:[email protected] (C.-Y. Liu), [email protected] (X.-P. Zhou)
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Abstract

Based on polyhedral splines, some multivariate splines of different orders with given supports over arbitrary topological meshes are developed. Schemes for choosing suitable families of multivariate splines based on pre-given meshes are discussed. Those multivariate splines with inner knots and boundary knots from the related meshes are used to generate rational spline shapes with related control points. Steps for up to C2-surfaces over the meshes are designed. The relationship among the meshes and their knots, the splines and control points is analyzed. To avoid any unexpected discontinuities and get higher smoothness, a heart-repairing technique to adjust inner knots in the multivariate splines is designed.

With the theory above, bivariate C1-quadratic splines over rectangular meshes are developed. Those bivariate splines are used to generate rational C1-quadratic surfaces over the meshes with related control points and weights. The properties of the surfaces are analyzed. The boundary curves and the corner points and tangent planes, and smooth connecting conditions of different patches are presented. The C1–continuous connection schemes between two patches of the surfaces are presented.

Keywords

Polyhedral splineinner knotmultivariate splinecontrol pointspline shapeCk-surfacearbitrary topological meshrational bivariate quadratic surfacerectangular mesh

MSC classification

Secondary: 65D18: Computer graphics, image analysis, and computational geometry 68U05: Computer graphics; computational geometry
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 9 , Issue 3 , August 2016 , pp. 383 - 415
Copyright
Copyright © Global-Science Press 2016 

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References

[1]Liang, X.-K.. Rational Qudratic B-spline Interpolation of Function Segments. Computation Mathematics, A Jounal of Chinese University (in Chinese), 2012,01:52–60.Google Scholar
[2]Xie, J., Tan, J.-Q. and Li, S.-F.. Quadratic Algebraic Hypobolic B-spline Curves with Weight Parameters. Computation Mathematics, A Jounal of Chinese University (in Chinese), 2011,04:337–346.Google Scholar
[3]Jin, L.-B. and Deng, J.-S.. Conversion Between (3,3,1,1) Hierarchical B-splines and Polynomial splines over Hierarchical T-meshes. Computatation Mathematics, A Jounal of Chinese University (in Chinese), 2009,04:330–342.Google Scholar
[4]Qin, L. and Guan, L.-T.. Minimal Surfaces Design Based on NURBS. Computatation Mathematics, A Jounal of Chinese University (in Chinese), 2005, S1:175–181.Google Scholar
[5]Goodman, T., Polyhedral Splines, in Computation of Curves and Surfaces, Dahmen, W., Gasca, M., and Micchelli, C.A., eds. Kluwer, Dordrecht, 1990, Springer Netherlands. pp: 347–382.Google Scholar
[6]Zhang, Y.-C., Da, F.-P. and Song, W.-Z.. Piecewise C 1 Surfaces Based on Bivariate Quartic Box-Splines for Arbitrary Triangular Meshes. Journal of Software, 2006, 17(10): 2211–2220.Google Scholar
[7]Feng, R.-Z. and Wang, R.-H.. Smooth spline surfaces over arbitrary topological triangular meshes. Journal of Software, 2003, 14(4):830–837.Google Scholar
[8]Zhou, R.-R., Zhang, L.-Y., Pang, X.-F. and Zhou, L.-S.. Smooth Piecewise Spline Surfaces over Arbitrary Control Polyhedron. Chinese Journal of Aeronautics, 2001, 14(1):57–64.Google Scholar
[9]Liu, C.-Y.. Theory and application of Convex Curves and Surfaces in CAGD. ISBN 90-365-1577-7, Press of University of Twente, 2001.Google Scholar
[10]Liu, C.-Y. and Zhou, X.-P.. Research on Algorithm of Precision Modeling Body Surfaces in Electronic Ray-Tracing Technique. College Mathematics (in Chinese), 2014,30(6):34–38.Google Scholar
[11]Traas, C.Practice of Bivariate Quadratic Simplicial Splines, in Computation of Curves and Surfaces, Dahmen, W., Gasca, M., and Micchelli, C.A., eds. Kluwer, Dordrecht, 1990, Springer Netherlands. pp:383–422Google Scholar
[12]Xu, W.-Y.. The Construction and Application of the Bivariate Quadratic Splines on Uniform Rectangular Grids. Zhengzhou University, 2014.Google Scholar
[13]X.-J. Zhuang, . The Construction and Application of the Bivariate Quadratic Spline Function Method In the Bezier Trianglar Mesh. Zhengzhou University, 2014.Google Scholar