Reconstruction of Low Degree B-spline Surfaces with Arbitrary Topology Using Inverse Subdivision Scheme

  • Nga Le-Thi-Thu Quynhon University, Vietnam
  • Khoi Nguyen-Tan Danang University of Science and Technology, Vietnam
  • Thuy Nguyen-Thanh VNU University of Engineering and Technology, Vietnam


Multivariate B-spline surfaces over triangular parametric domain have many interesting properties in the construction of smooth free-form surfaces. This paper introduces a novel approach to reconstruct triangular B-splines from a set of data points using inverse subdivision scheme. Our proposed method consists of two major steps. First, a control polyhedron of the triangular B-spline surface is created by applying the inverse subdivision scheme on an initial triangular mesh. Second, all control points of this B-spline surface, as well as knotclouds of its parametric domain are iteratively adjusted locally by a simple geometric fitting algorithm to increase the accuracy of the obtained B-spline. The reconstructed B-spline having the low degree along with arbitrary topology is interpolative to most of the given data points after some fitting steps without solving any linear system. Some concrete experimental examples are also provided to demonstrate the effectiveness of the proposed method. Results show that this approach is simple, fast, flexible and can be successfully applied to a variety of surface shapes.


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[1] G. Greiner. Geometric modeling. Lecture Notes in Winter Term, 2010.
[2] Denis Zorin, Peter Schroder. Subdivision for Modeling and Animation. SIGGRAPH Course Notes, 2000.
[3] G. Farin. Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide. 5th edn. Morgan Kaufmann, San Mateo, 2002.
[4] M.Botsch, M.Pauly, C.Rossl, S.Bischoff and L.Kobbelt. Geometric Modeling Based on Triangle Meshes. EuroGraphics, 2006.
[5] F.Cheng, F.Fan, S.Lai, C.Huang, J.Wang, J .Yong. Loop subdivision surface based progressive interpolation. Journal of Computer Science and Technology, vol.24, pp.3946, 2009.
[6] M. Eck, H.Hoppe. Automatic reconstruction of B-spline surfaces of arbitrary topological type. In Proceedings of SIGGRAPH96, ACM Press, pp.325334, 1996.
[7] M.Halstead, M.Kass, T.Derose. Efficient, fair interpolation using Catmull-Clark surfaces. In Proceedings of ACM SIGGRAPH 93, pp. 3544, 1993.
[8] C. Deng, W. Ma. Weighted progressive interpolation of Loop subdivision surfaces. Computer-Aided Design, vol.44, pp.42431, 2012.
[9] T.Maekawa, Y.Matsumoto, K.Namiki. Interpolation by geometric algorithm. Computer-Aided Design, vol.39, pp.313323, 2007.
[10] Y.Nishiyama, M.Morioka, T. Maekawa. Loop subdivision surface fitting by geometric algorithms. Poster proceedings of pacific graphics, 2008.
[11] C. Deng, H.Lin. Progressive and iterative approximation for least squares B-spline curve and surface fitting. Computer-Aided Design, vol.47, pp.3244, 2014.
[12] Y.Kineri, M.Wang, H.Lin, T.Maekawa. B-spline surface fitting by iterative geometric interpolation/approximation algorithms. Computer-Aided Design, vol.44(7), pp.697708, 2012.
[13] Y.Xiong, G.Li, A.Mao. Convergence analysis for B-spline geometric interpolation. Computers and Graphics, vol.36, pp.884891, 2012.
[14] Jie Chen, Guo-Jin Wang. Progressive iterative approximation for triangular Bzier surfaces. Computer-Aided Design, vol.43, pp.889895, 2011.
[15] Yu Zhao, Hongwei Lin. The PIA property of low degree nonuniform triangular B-B patches.. In Proceedings of the 12th International Conference on CAD and CG, pp.239-243, 2011.
[16] Christopher K. Ingram. A Geometric B-Spline Over the Triangular Domain.. M.S. Mathematics thesis, 2003.
[17] Dian Pratiwi. The Implementation of Univariate and Bivariate B-Spline Interpolation Method in Continuous. IJCSI International Journal of Computer Science Issues, vol. 10, Issue 2, No 2, March 2013.
[18] Neamtu M. Bivariate simplex B-splines: a new paradigm. In Proceedings of the 17th spring conference on computer graphics, pp.7178, 2001.
[19] C.Loop. Smooth Subdivision Surfaces Based on Triangles. M.S. Mathematics thesis, 1987.
[20] W. Dahmen, C.A. Micchelli, and H.-P. Seidel. Blossoming begets B-spline bases built better by B-patches. Mathematics of Computation,vol 59(199), pp 97-115, 1992.

How to Cite
LE-THI-THU, Nga; NGUYEN-TAN, Khoi; NGUYEN-THANH, Thuy. Reconstruction of Low Degree B-spline Surfaces with Arbitrary Topology Using Inverse Subdivision Scheme. Journal of Science and Technology: Issue on Information and Communications Technology, [S.l.], v. 3, n. 1, p. 82-88, mar. 2017. ISSN 1859-1531. Available at: <>. Date accessed: 24 mar. 2023. doi: