Abstract
Non-Uniform Rational B-Spline (NURBS) is the multidisciplinary topic in Mathematics, Computer Science, and Engineering. The NURBS curves together with their geometric properties are widely used in Computer Aided Design, Computer Aided Geometric Design, and Computational Mathematics. When a single weight approaches infinity, the limit of a NURBS curve tends to the corresponding control point. In this paper, a kind of control structure of a NURBS curve, called regular control curve, is defined. We prove that the limit of the NURBS curve is exactly its regular control curve when all of weights approach infinity, where each weight is multiplied by a certain one-parameter function tending to infinity, different for each control point. Moreover, some representative examples are presented to show this property and indicate its application for shape deformation.
Similar content being viewed by others
References
de Boor, C.: On calculating with B-splines. J. Approx. Theory 6(1), 50–62 (1972)
de Boor, C.: A Practical Guide to Splines. Springer, New York (1978)
Davis, P.: Interpolation and Approximation. Dover Publications, New York (1975)
Piegl, L., Tiller, W.: The NURBS Book, 2nd edn. Springer, Berlin, Heidelberg, New York (1997)
Piegl, L.: Modifying the shape of rational B-spline, Part 1: curves. Comput. Aided Des. 21(8), 509–518 (1989)
Piegl, L.: Modifying the shape of rational B-splines, Part 2: surfaces. Comput. Aided Des. 21(9), 538–546 (1989)
Farin, G.: Curve and Surfaces for CAGD: A Practical Guide, 5th edn. Morgan Kaufmann Publishers, San Francisco (2002)
Farin, G., Hoschek, J., Kim, M.S.: Handbook of Computer Aided Geometric Design. Elsevier Science, Amsterdam (2002)
Au, C.K., Yuen, M.M.F.: Unified approach to NURBS curve shape modification. Comput. Aided Des. 27(2), 85–93 (1995)
Sánchez-Reyes, J.: A simple technique for NURBS shape modification. IEEE Comput. Graph. Appl. 17(1), 52–59 (1997)
Juhász, I.: Weight-based shape modification of NURBS curves. Comput. Aided Geom. Design 16(5), 377–383 (1999)
Zhang, G.H., Yang, X.Q., Zhang, C.M.: Weight-based shape modification of NURBS curves. J. Comput. r Aided Design Comput. Graph. 16(10), 1386–1400 (2004)
Sturmfels, B.: Gröbner Bases and Convex Polytopes. American Mathematical Society, Providence (1996)
Krasauskas, R.: Toric surface patches. Adv. Comput. Math. 17(1–2), 89–113 (2002)
García-Puente, L.D., Sottile, F., Zhu, C.G.: Toric degenerations of Bézier patches. ACM Trans. Graph. (TOG) 30(5), 110 (2011)
Zhu, C.G.: Degenerations of toric ideals and toric varieties. J. Math. Anal. Appl. 386(2), 613–618 (2012)
Zhu, C.G., Zhao, X.Y.: Self-intersections of rational Bézier curves. Graph. Models 76(5), 312–320 (2014)
Zhao, X.Y., Zhu, C.G.: Injectivity conditions of rational Bézier surfaces. Comput. Graph. 51, 17–25 (2015)
Boehm, W.: Inserting new knots into B-spline curves. Comput. Aided Des. 12(4), 199–201 (1980)
Acknowledgements
The authors appreciate the comments and valuable suggestions from the anonymous reviewers.Their advice helped to improve the presentation of this paper. This work is partly supported by the National Natural Science Foundation of China (Nos. 11671068, 11271060), Fundamental Research of Civil Aircraft (No. MJ-F-2012-04), and the Fundamental Research Funds for the Central Universities (Nos. DUT16LK38).
Author information
Authors and Affiliations
Corresponding author
Electronic supplementary material
Below is the link to the electronic supplementary material.
About this article
Cite this article
Zhang, Y., Zhu, CG. Degenerations of NURBS curves while all of weights approaching infinity. Japan J. Indust. Appl. Math. 35, 787–816 (2018). https://doi.org/10.1007/s13160-018-0301-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13160-018-0301-4