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A unified approach to non-polynomial B-spline curves based on a novel variant of the polar form

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Abstract

We develop a general, unified theory of splines for a wide collection of spline spaces, including trigonometric splines, hyperbolic splines, and special Müntz spaces of splines by invoking a novel variant of the homogeneous polar form where we alter the diagonal property. Using this polar form, we derive de Boor type recursive algorithms for evaluation and differentiation. We also show that standard knot insertion procedures such as Boehm’s algorithm and the Oslo algorithm readily extend to these general spline spaces. In addition, for these spaces we construct compactly supported B-spline basis functions with simple two term recurrences for evaluation and differentiation, and we show that these B-spline basis functions form a partition of unity, have curvilinear precision, and satisfy a dual functional property and a Marsden identity.

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References

  1. Ait-Haddou, R., Sakane, Y., Nomura, T.: Chebyshev blossoming in Müntz spaces: toward shaping with Young diagrams. J. Comput. Appl. Math. 247, 172–208 (2013). doi:10.1016/j.cam.2013.01.009

    Article  MathSciNet  MATH  Google Scholar 

  2. Alfeld, P., Neamtu, M., Schumaker, L.L.: Circular Bernstein-Bézier polynomials. In: Mathematical methods for curves and surfaces (Ulvik, 1994), pp. 11-20. Vanderbilt Univ. Press, Nashville, TN (1995)

  3. Andrews, G.E.: The theory of partitions. Cambridge Mathematical Library, Cambridge University Press, Cambridge, reprint of the 1976 original (1998)

  4. Barry, P.J.: de Boor-Fix dual functionals and algorithms for Tchebycheffian B-spline curves. Constr. Approx. 12(3), 385–408 (1996). doi:10.1007/s003659900020

    Article  MathSciNet  MATH  Google Scholar 

  5. Boehm, W.: Inserting new knots into B-spline curves. Comput. Aided Des. 12(4), 199–201 (1980). doi:10.1016/0010-4485(80)90154-2

    Article  Google Scholar 

  6. Cohen, E., Lyche, T., Riesenfeld, R.: Discrete \(B\)-splines and subdivision techniques in computer-aided geometric design and computer graphics. Comput. Graph. Image Process. 14(2), 87–111 (1980). doi:10.1016/0146-664X(80)90040-4

    Article  Google Scholar 

  7. Dişibüyük, Ç.: A functional generalization of the interpolation problem. Appl. Math. Comput. 256, 247–251 (2015). doi:10.1016/j.amc.2014.12.152

    MathSciNet  MATH  Google Scholar 

  8. Dişibüyük, Ç., Goldman, R.: A unifying structure for polar forms and for Bernstein Bézier curves. J. Approx. Theory 192, 234-249 (2015). doi:10.1016/j.jat.2014.12.007, http://www.sciencedirect.com/science/article/pii/S0021904514002263

  9. Goldman, R.: Blossoming and knot insertion algorithms for B-spline curves. Comput. Aided Geometr. Des. 7(1–4), 69–81 (1990). doi:10.1016/0167-8396(90)90022-J

    Article  MathSciNet  MATH  Google Scholar 

  10. Goldman, R.: Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling. The Morgan Kaufmann Series in Computer Graphics and Geometric Modeling. Elsevier Science, San Francisco (2003)

  11. Gonsor, D., Neamtu, M.: Non-polynomial polar forms. In: Curves and Surfaces in Geometric Design (Chamonix-Mont-Blanc, 1993), pp. 193-200. A K Peters, Wellesley, MA (1994)

  12. Koch, P.E., Lyche, T.: Exponential B-splines in tension. In: Approximation Theory VI, Vol. II (College Station, TX, 1989). Academic Press, Boston, MA, pp. 361-364 (1989)

  13. Koch, P.E., Lyche, T., Neamtu, M., Schumaker, L.L.: Control curves and knot insertion for trigonometric splines. Adv. Comput. Math. 3(4), 405–424 (1995). doi:10.1007/BF03028369

    Article  MathSciNet  MATH  Google Scholar 

  14. Kulkarni, R., Laurent, P.J., Mazure, M.L.: Nonaffine blossoms and subdivision for \(Q\)-splines. In: Mathematical Methods in Computer Aided Geometric Design, II (Biri, 1991), pp. 367-380. Academic Press, Boston, MA (1992)

  15. Lü, Y., Wang, G., Yang, X.: Uniform hyperbolic polynomial B-spline curves. Comput. Aided Geom Des. 19(6), 379–393 (2002). doi:10.1016/S0167-8396(02)00092-4

    Article  MathSciNet  Google Scholar 

  16. Lyche, T.: Trigonometric splines: a survey with new results. In: Shape Preserving Representations in Computer Aided Geometric Design, pp. 201-227. Nova Science Publishers Inc, New York (1999)

  17. Lyche, T., Mazure, M.L.: Total positivity and the existence of piecewise exponential B-splines. Adv. Comput. Math. 25(1–3), 105–133 (2006). doi:10.1007/s10444-004-7633-0

    Article  MathSciNet  MATH  Google Scholar 

  18. Mainar, E., Peña, J.M.: Corner cutting algorithms associated with optimal shape preserving representations. Comput Aided Geom Des. 16(9), 883–906 (1999). doi:10.1016/S0167-8396(99)00035-7

    Article  MathSciNet  MATH  Google Scholar 

  19. Mazure, M.L.: Blossoming of Chebyshev splines. Mathematical Methods for Curves and Surfaces (Ulvik, 1994), pp. 355–364. Vanderbilt Univ. Press, Nashville, TN (1995)

    Google Scholar 

  20. Mazure, M.L.: Blossoming: a geometrical approach. Constr. Approx 15(1), 33–68 (1999a). doi:10.1007/s003659900096

    Article  MathSciNet  MATH  Google Scholar 

  21. Mazure, M.L.: Chebyshev spaces with polynomial blossoms. Adv. Comput. Math. 10(3–4), 219–238 (1999b). doi:10.1023/A:1018995019439

    Article  MathSciNet  MATH  Google Scholar 

  22. Mazure, M.L.: Chebyshev splines beyond total positivity. Adv. Comput. Math. 14(2), 129–156 (2001). doi:10.1023/A:1016616731472

    Article  MathSciNet  MATH  Google Scholar 

  23. Mazure, M.L., Pottmann, H.: Tchebycheff curves. In: Total positivity and its applications (Jaca, 1994), Math. Appl., vol 359, Kluwer Acad. Publ., Dordrecht, pp 187-218 (1996)

  24. Ramshaw, L.: Blossoms are polar forms. Comput. Aided Geom. Des. 6(4), 323–358 (1989). doi:10.1016/0167-8396(89)90032-0

    Article  MathSciNet  MATH  Google Scholar 

  25. Simeonov, P., Goldman, R.: Quantum B-splines. BIT 53(1), 193–223 (2013). doi:10.1007/s10543-012-0395-z

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Dokuz Eylül University, Fen Fakültesi, Tınaztepe Kampüsü, 35390, Buca, İzmir, Turkey

    Çetin Dişibüyük

  2. Department of Computer Science, 6100 South Main, Rice University, Houston, TX, 77251-1892, USA

    Ron Goldman

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  1. Çetin Dişibüyük
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  2. Ron Goldman
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Dişibüyük, Ç., Goldman, R. A unified approach to non-polynomial B-spline curves based on a novel variant of the polar form. Calcolo 53, 751–781 (2016). https://doi.org/10.1007/s10092-015-0172-x

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  • DOI: https://doi.org/10.1007/s10092-015-0172-x

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