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A Coons Patch Spanning a Finite Number of Curves Tested for Variationally Minimizing Its Area

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  • Daud Ahmad
  • Bilal Masud

Abstract

In surface modeling a surface frequently encountered is a Coons patch that is defined only for a boundary composed of four analytical curves. In this paper we extend the range of applicability of a Coons patch by telling how to write it for a boundary composed of an arbitrary number of boundary curves. We partition the curves in a clear and natural way into four groups and then join all the curves in each group into one analytic curve by using representations of the unit step function including one that is fully analytic. Having a well‐parameterized surface, we do some calculations on it that are motivated by differential geometry but give a better optimized and possibly more smooth surface. For this, we use an ansatz consisting of the original surface plus a variational parameter multiplying the numerator part of its mean curvature function and minimize with the respect to it the rms mean curvature and decrease the area of the surface we generate. We do a complete numerical implementation for a boundary composed of five straight lines, that can model a string breaking, and get about 0.82 percent decrease of the area. Given the demonstrated ability of our optimization algorithm to reduce area by as much as 23 percent for a spanning surface not close of being a minimal surface, this much smaller fractional decrease suggests that the Coons patch we have been able to write is already close of being a minimal surface.

Suggested Citation

  • Daud Ahmad & Bilal Masud, 2013. "A Coons Patch Spanning a Finite Number of Curves Tested for Variationally Minimizing Its Area," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    • Handle: RePEc:wly:jnlaaa:v:2013:y:2013:i:1:n:645368
    DOI: 10.1155/2013/645368
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    References listed on IDEAS

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    1. Barrera, D. & Fortes, M.A. & González, P. & Pasadas, M., 2008. "Minimal energy Cr-surfaces on uniform Powell-Sabin type meshes," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 77(2), pages 161-169.
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    Cited by:

    1. Sadia Bashir & Daud Ahmad & Ghada Ali, 2024. "Exploring q-Bernstein-Bézier surfaces in Minkowski space: Analysis, modeling, and applications," PLOS ONE, Public Library of Science, vol. 19(5), pages 1-26, May.
    2. Daud Ahmad & Kanwal Hassan & M. Khalid Mahmood & Javaid Ali & Ilyas Khan & M. Fayz-Al-Asad, 2021. "Variationally Improved Bézier Surfaces with Shifted Knots," Advances in Mathematical Physics, John Wiley & Sons, vol. 2021(1).
    3. Daud Ahmad & M. Khalid Mahmood & Qin Xin & Ferdous M. O. Tawfiq & Sadia Bashir & Arsha Khalid, 2022. "A Computational Model for q‐Bernstein Quasi‐Minimal Bézier Surface," Journal of Mathematics, John Wiley & Sons, vol. 2022(1).

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