Xueting Han
ABSTRACT
This paper presents an approach to subdivide a Non-Uniform Rational B-Splines (NURBS) curve into several small parts in terms of a given arc length for the small parts. The approach is developed on the De Boor algorithm to perform the subdivision. It first subdivides the NURBS curve into two parts with the original weighted control points and knot vector, and then recursively subdivides the subdivided small parts by evaluating the arc length of the newly-subdivided sub-curves until each sub-curve fits the given arc length. The paper also gives a data structure to store and traverse the subdivided weighted control points and knot vectors. The accuracy and efficiency of the algorithm are also investigated. Numerical experiments show the approach is suitable for treating NURBS of lower degrees in computer graphics, computer images as well as CNC data fitting.
- Piegl, Les, and Wayne Tiller. 1996. The NURBS book. Springer Science & Business Media.Google Scholar
- Piegl, L. 1991. On NURBS: a survey. IEEE Computer Graphics and Applications, 11(01), 55-71.Google ScholarDigital Library
- Di, F. 2014. CNC processing of Nurbs curve based on AutoCAD. In Advanced Materials Research ,Vol. 1016, pp. 167-171. Trans Tech Publications Ltd.Google ScholarCross Ref
- Farin, Gerald E., and Gerald Farin. 2002. Curves and surfaces for CAGD: a practical guide. Morgan Kaufmann.Google Scholar
- Rogers, D. F. 2001. An introduction to NURBS: with historical perspective. Morgan Kaufmann.Google Scholar
- Salomon, D. 2007. Curves and surfaces for computer graphics. Springer Science & Business Media.Google Scholar
- Cashman, T. J., Dodgson, N. A., and Sabin, M. A. 2009. Selective knot insertion for symmetric, non-uniform refine and smooth B-spline subdivision. Computer Aided Geometric Design, 26(4), 472-479.Google ScholarDigital Library
- Beccari, C., Casciola, G., and Romani, L. 2011. Polynomial-based non-uniform interpolatory subdivision with features control. Journal of computational and applied mathematics, 235(16), 4754-4769.Google ScholarDigital Library
- Qu, R., and Gregory, J. A. 1992. A Subdivision Algorithm For Non—Uniform B—Splines. In Approximation Theory, Spline Functions and Applications, pp. 423-436. Springer, Dordrecht.Google Scholar
- Miura, K. T., Sone, J., Yamashita, A., Kaneko, T., Ueda, M., and Osano, M. 2002. Adaptive refinement for B-spline subdivision curves. Journal of Three Dimensional Images, 16(4), 74-78.Google Scholar
- Kosinka, J., Sabin, M., and Dodgson, N. 2014. Creases and boundary conditions for subdivision curves. Graphical Models, 76(5), 240-251.Google ScholarDigital Library
- Goldman, R. N. 1990. Blossoming and knot insertion algorithms for B-spline curves. Computer Aided Geometric Design, 7(1-4), 69-81.Google ScholarDigital Library
- Penner, A. 2019. General Properties of Splines. In Fitting Splines to a Parametric Function, pp. 13-17. Springer, Cham. https://doi.org/10.1007/978-3-030-12551-6_3Google ScholarCross Ref
- Boehm, W. 1980. Inserting new knots into B-spline curves. Computer-Aided Design, 12(4), 199-201.Google ScholarCross Ref
- Cohen, E., Lyche, T., and Riesenfeld, R. 1980. Discrete B-splines and subdivision techniques in computer-aided geometric design and computer graphics. Computer graphics and image processing, 14(2), 87-111.Google Scholar
- Jankauskas K. 2010. Time-efficient NURBS curve evaluation algorithms. Information Technologies 2010: proceedings of the 16th international conference on Information and Software Technologies.Google Scholar
- Raja, S. P. 2020. Bézier and B-spline curves—A study and its application in wavelet decomposition. International Journal of Wavelets, Multiresolution and Information Processing, 18(04), 2050030.Google ScholarCross Ref
- De Boor, C. 1977. Package for calculating with B-splines. SIAM Journal on Numerical Analysis, 14(3), 441-472.Google ScholarDigital Library
- Ershov, S. N. 2019. B-splines and bernstein basis polynomials. Physics of Particles and Nuclei Letters, 16(6), 593-601.Google ScholarCross Ref
- Kang, H., and Li, X. 2019. De Boor-like evaluation algorithm for Analysis-suitable T-splines. Graphical Models, 106, 101042.Google ScholarDigital Library
- Fan, Y. Q., Fu, J. Z., and Gan, W. F. 2010. Research on Look-Ahead Adaptive Interpolation of NURBS Based on De Boor Algorithm. In Key Engineering Materials, Vol. 426, pp. 206-211. Trans Tech Publications Ltd.Google ScholarCross Ref
- Nava-Yazdani, E., and Polthier, K. 2013. De Casteljauʼs algorithm on manifolds. Computer Aided Geometric Design, 30(7), 722-732.Google ScholarDigital Library
- Yeh, S. T. 2002. Using trapezoidal rule for the area under a curve calculation. Proceedings of the 27th Annual SAS® User Group International (SUGI’02), 1-5.Google Scholar
- Hug, S., Schwarzfischer, M., Hasenauer, J., Marr, C., and Theis, F. J. 2016. An adaptive scheduling scheme for calculating Bayes factors with thermodynamic integration using Simpson's rule. Statistics and Computing, 26(3), 663-677.Google ScholarDigital Library
- Kalicharan, N. 2014. Introduction to Binary Trees. In Advanced Topics in Java, pp. 219-257. Apress, Berkeley, CA.Google ScholarCross Ref
- WANG Xing-bo. 2011. Study on non-recursive and stack-free algorithms for preorder traversal of complete binary trees. Computer Engineering and Design, Vol.32, No.9.Google Scholar
- Das, V. V. 2010. A new non-recursive algorithm for reconstructing a binary tree from its traversals. In 2010 International Conference on Advances in Recent Technologies in Communication and Computing, pp. 261-263. IEEE. https://doi.org/10.1109/ARTCom.2010.88Google ScholarDigital Library
- Li, H. 2016. Binary Tree's Recursion Traversal Algorithm and Its Improvement. Journal of Computer and Communications, 4(07), 42.Google ScholarCross Ref
- Hartley, P. J., and Judd, C. J. 1978. Parametrization of Bézier-type B-spline curves and surfaces. Computer-Aided Design, 10(2), 130-134.Google ScholarCross Ref
Index Terms
- De Boor Algorithm Based NURBS Subdivision in Terms of Arc Length
Recommendations
Arc Length Restricted Knot Vector Construction for NURBS Subdivision
ICIIT '23: Proceedings of the 2023 8th International Conference on Intelligent Information TechnologyThis paper presents a method to construct an arc-length restricted knot vector for subdivision of a non-uniform rational B-spline (NURBS) curve. Based on Boehm's knot insertion algorithm of restriction free, the new method calculates the inserted knot ...
Arc-length preserving curve deformation based on subdivision
Special issue: The international symposium on computing and information (ISCI2004)An arc-length preserving deformation for curves is presented by combining subdivision and inverse Kinematic. A curve is discreted into polyline first, then the polyline is deformed with its arc-length preserved in the sense of minimizing energy. ...
Approximating NURBS Curves by Arc Splines
GMP '00: Proceedings of the Geometric Modeling and Processing 2000It is desirable to approximate a smooth curve by arc spline with fewest segments within prescribed tolerance. In this paper, we present an efficient algorithm for fitting planar smooth curve by arc spline. The main idea is that we construct the optimal ...
Comments