nach oben

Calcolo

Erschienen in:

01.12.2016

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

verfasst von: Çetin Dişibüyük, Ron Goldman

Erschienen in: Calcolo | Ausgabe 4/2016

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

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.

Vorheriger Artikel A Jacobi–Davidson type method for computing real eigenvalues of the quadratic eigenvalue problem

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

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 MathSciNetCrossRefMATH

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)

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

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 MathSciNetCrossRefMATH

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 CrossRef

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 CrossRef

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

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

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 MathSciNetCrossRefMATH

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 MathSciNetCrossRefMATH

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 MathSciNetCrossRef

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 MathSciNetCrossRefMATH

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 MathSciNetCrossRefMATH

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)

20.

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

21.

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

22.

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

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 MathSciNetCrossRefMATH

25.

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

Titel: A unified approach to non-polynomial B-spline curves based on a novel variant of the polar form
verfasst von: Çetin Dişibüyük
Ron Goldman
Publikationsdatum: 01.12.2016
Verlag: Springer Milan
Erschienen in: Calcolo / Ausgabe 4/2016
Print ISSN: 0008-0624
Elektronische ISSN: 1126-5434
DOI: https://doi.org/10.1007/s10092-015-0172-x

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Weitere Artikel der Ausgabe 4/2016

Spectral discretization of an unsteady flow through a porous solid

Error bounds for linear complementarity problems of QN-matrices

An efficient numerical algorithm based on Haar wavelet for solving a class of linear and nonlinear nonlocal boundary-value problems

A Jacobi–Davidson type method for computing real eigenvalues of the quadratic eigenvalue problem

Ball convergence of a sixth order iterative method with one parameter for solving equations under weak conditions

Multi-step Chebyshev spectral collocation method for Volterra integro-differential equations

Premium Partner