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Mixed hyperbolic/trigonometric non-stationary subdivision scheme

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Abstract

This paper proposes two novel families of non-stationary subdivision schemes with a shape parameter of hierarchically and efficiently generating mixed hyperbolic/trigonometric curves of order 3 and 4. An analysis of convergence and smoothness of the proposed schemes is established by using the asymptotic equivalence method. This paper also discusses the bivariate tensor-product subdivision scheme for the surface modeling on the regular meshes. The numerical results they produce are very encouraging. In this context, the performance of our algorithms has been exposed by considering examples, which illustrate how the shape parameter and the control points are assigned to reproduce such analytic curves and surfaces using tensor-product notion.

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References

  1. Brilleaud, M., Mazure, M.-L.: Mixed hyperbolic/trigonometric spaces for design. Comput. Math. Appl. 64, 2459–2477 (2012)

    Article  MathSciNet  Google Scholar 

  2. Cavaretta, A.S., Dahmen, W., Micchelli, C.A.: Stationary subdivision. Memoirs of the American Mathematical Society 93(453), 346–349 (1991)

    Article  MathSciNet  Google Scholar 

  3. Catmull, E., Clark, J.: Recursively generated B-spline surfaces on arbitrary topological meshes. Comput. Aided Des. 10(6), 350–355 (1978)

    Article  Google Scholar 

  4. Chaikin, G.M.: An algorithm for high speed curve generation. Comput. Graph. Image Process. 3(4), 346–349 (1974)

    Article  Google Scholar 

  5. Daniel, S., Shunmugaraj, P.: Geometric Modelling and Imaging 07, 7695–2901 (2007)

    Google Scholar 

  6. Daniel, S., Shunmugaraj, P.: An approximating ${\cal{C}}^2$ non-stationary subdivision scheme. Comput. Aided Geom. Des. 26, 810–821 (2009)

    Article  MathSciNet  Google Scholar 

  7. de Rham, G.: Sur une courbe plane, J. Math. Pures Appl. 9(35), 25–42 (1956)

  8. Dyn, N., Levin, D.: Subdivision schemes in geometric modelling. Acta Numer. 11, 73–144 (2002)

    Article  MathSciNet  Google Scholar 

  9. Dyn, N., Levin, D.: Analysis of asymptotically equivalent binary subdivision schemes. J. Math. Anal. Appl. 193, 594–621 (1995)

    Article  MathSciNet  Google Scholar 

  10. Fang, M.-e, Ma, W., Wang, G.: A generalized surface subdivision scheme of arbitrary order with a tension parameter. Comput. Aided Des. 49, 8–17 (2014)

  11. Fang, M.-e, Ma , W., Wang, G.: A generalized curve subdivision scheme of arbitrary order with a tension parameter. Comput. Aided Geom. Des. 27 , 720–733 (2010)

  12. Fang, M.-e, Jeong, B., Yoon, J.: A family of non-uniform subdivision schemes with variable parameters for curve design. Appl. Math. Comput. 313, 1–11 (2017)

  13. Jena, M.K., Shunmugaraj, P., Das, P.C.: A subdivision algorithm for trigonometric spline curves. Comput. Aided Geom. Des. 19, 71–88 (2002)

    Article  MathSciNet  Google Scholar 

  14. Jena, M.K., Shunmugaraj, P., Das, P.C.: A non-stationary subdivision scheme for curve interpolation, ANZIAM J. 44 (E) (2003) E216-E235

  15. Jeong, B., Lee, Y.J., Yoon, J.: A family of non-stationary subdivision schemes reproducing exponential polynomials. J. Math. Anal. Appl. 402, 207–219 (2013)

    Article  MathSciNet  Google Scholar 

  16. Romani, L.: From approximating subdivision schemes for exponential splines to high-performance interpolating algorithms. J. Comput. Appl. Math. 224, 383–396 (2009)

    Article  MathSciNet  Google Scholar 

  17. Siddiqi, S.S., Younis, M.: Ternary three point non-stationary subdivision scheme. Res. J. Appl. Sci. Eng. Technol. 4(13), 1875–1882 (2012)

    Google Scholar 

  18. Siddiqi, S.S., Younis, M.: A symmetric ${\cal{C}}^3$ non-stationary subdivision scheme. LMS J. Comput. Math 17(1), 259–272 (2014)

    Article  MathSciNet  Google Scholar 

  19. Siddiqi, S.S., Salam, W., Rehan, K.: Binary 3-point and 4-point non-stationary subdivision schemes using hyperbolic function. Appl. Math. Comput 258, 120–129 (2015)

    MathSciNet  MATH  Google Scholar 

  20. Siddiqi, S.S., Salam, W., Kashif, R.: A new non-stationary binary 6-point subdivision scheme. Appl. Math. Comput. 268, 1227–1239 (2015)

    MathSciNet  MATH  Google Scholar 

  21. Zorin, D., Schröder, P.: A unified framework for primal/dual quadrilateral subdivision schemes. Comput. Aided Geom. Des. 18(5), 429–454 (2001)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Laboratory LS3M, Univ. Sultan Moulay Slimane, Khouribga, Morocco

    R. Fakhar

  2. Laboratory MISI, Faculty of Sciences and Technology, Univ. Hassan 1st, Settat, Morocco

    A. Lamnii & M. -Y. Nour

  3. LGIPM, Université Lorraine, Metz, France

    A. Zidna

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  1. R. Fakhar
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  2. A. Lamnii
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  3. M. -Y. Nour
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  4. A. Zidna
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Correspondence to A. Lamnii.

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Fakhar, R., Lamnii, A., Nour, M.Y. et al. Mixed hyperbolic/trigonometric non-stationary subdivision scheme. Math Sci 16, 149–162 (2022). https://doi.org/10.1007/s40096-021-00406-4

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