Research Article
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Year 2018, Volume: 36 Issue: 2, 325 - 340, 01.06.2018

Abstract

References

  • [1] R. Osserman. A survey of Minimal Surfaces. Dover Publications Inc., 1986.
  • [2] J. C. C. Nitsche. Lectures on Minimal Surfaces. Cambridge University Press, 1989.
  • [3] J.A.F Plateau. Statique exprimentale et thorique des liquides soumis aux seules forces molculaires. Gauthier-Villars, Paris, 1873.
  • [4] H.A. Schwarz. Gesammelte Mathematische Abhandlungen. 2 Bande. Springer, 1890.
  • [5] R. Garnier. Le problme de Plateau. Annales Scienti_ques de l'E.N.S., 3(45):53-144, 1928.
  • [6] J. Douglas. Solution of the problem of Plateau. Trans. Amer. Math. Soc., 33(1):263-321, 1931.
  • [7] T. Rad_o. On Plateau's problem. Ann. Of Math., (2)31(3):457-469, 1930.
  • [8] C. Coppin and D. Greenspan. A contribution to the particle modeling of soap films. Applied Mathematics and Computation, 26(4):315-331, 1988.
  • [9] K. Koohestani. Nonlinear force density method for the form-finding of minimal surface membrane structures. Communications in Nonlinear Science and Numerical Simulation, 19(6):2071-2087, 2014.
  • [10] Kenneth A. Brakke. The surface evolver. Experiment. Math., 1(2):141-165, 1992.
  • [11] D. L. Chopp. Computing minimal surfaces via level set curvature ow. Journal of Computational Physics, 106(1):77-91, 1993.
  • [12] Y.Jung, K.T.Chu, and S.Torquato. A variational level set approach for surface area minimization of triply-periodic surfaces. J. Comput. Phys., 223(2):711-730, May 2007.
  • [13] . Trasdahl and E.M. Ronquist. High order numerical approximation of minimal surfaces. Journal of Computational Physics, 230(12):4795-4810, 2011.
  • [14] C. Y. Li, R. H. Wang, and C. G. Zhu. Designing approximation minimal parametric surfaces with geodesics. Applied Mathematical Modelling, 37(9):6415-6424, 2013.
  • [15] O. Kassabov. Transition to canonical principal parameters on minimal surfaces. Computer Aided Geometric Design, 31(78):441-450, 2014.
  • [16] Gang Xu, Yaguang Zhu, Guozhao Wang, Andr_e Galligo, Li Zhang, and Kin-chuen Hui. Explicit form of parametric polynomial minimal surfaces with arbitrary degree. Appl. Math. Comput., 259(C):124-131, May 2015.
  • [17] J. Monterde. Bezier surfaces of minimal area: The Dirichlet approach. Computer Aided Geometric Design, 21:117-136, 2004.
  • [18] X. D. Chen, G. Xua, and Y. Wanga. Approximation methods for the Plateau-Bezier problem. In 2009 11th IEEE International Conference on Computer-Aided Design and Computer Graphics, 2009.
  • [19] Y. X. Hao, R. H. Wang, and C. J. Li. Minimal quasi-Bezier surface. Applied Mathematical Modelling, 36:5751-5757, 2012.
  • [20] G. Xu and G. Wang. Quintic parametric polynomial minimal surfaces and their properties. Differential Geometry and its Applications, 28:697-704, 2010.
  • [21] D. Ahmad and B. Masud. Variational minimization on string-rearrangement surfaces, illustrated by an analysis of the bilinear interpolation. Applied Mathematics and Computation, 233:72-84, 2014.
  • [22] D. Ahmad and B. Masud. A Coons patch spanning a finite number of curves tested for variationally minimizing its area. Abstract and Applied Analysis, 2013, 2013.
  • [23] D. Ahmad and B. Masud. Near-stability of a quasi-minimal surface indicated through a tested curvature algorithm. Computers & Mathematics with Applications, 69(10):1242- 1262, 2015.
  • [24] G. Xu, T. Rabczuk, E. Guler, Q. Wu, K. C. Hui, and G. Wang. Quasi-harmonic Bezier approximation of minimal surfaces for finding forms of structural membranes. Comput. Struct., 161(C):55-63, December 2015.
  • [25] S. S. Chern. An elementary proof of the existence of isothermal parameters on a surface. Proc. Amer. Math. Soc., 60: 771-782,1955.
  • [26] J. Monterde and H. Ugail. On harmonic and biharmonic Bezier surfaces. Computer Aided Geometric Design, 21:697-715, 2004.
  • [27] G. Farin. Triangular Bernstein-Bezier patches. Computer Aided Geometric Design, 3(2):83-127, 1986.
  • [28] J. Warren. Algebraic Geometry and its Applications, chapter A Bound on the Implicit Degree of Polygonal Bezier Surfaces, pages 513-525. Springer New York, 1994.
  • [29] R. Krasauskas. Toric surface patches. Advances in Computational Mathematics, 17(1):89-113, 2002.
  • [30] R. Goldman, R. Krasauskas. Topics in Algebraic Geometry and Geometric Modeling, chapter Toric Bezier patches with depth. American Mathematical Society, 2002.
  • [31] Gang Xu, Tsz-Ho Kwok, and Charlie C.L. Wang. Isogeometric computation reuse method for complex objects with topology-consistent volumetric parameterization. Computer-Aided Design, 91:1 - 13, 2017.
  • [32] L. D. Garcia-Puente, F. Sottile, and C. Zhu. Toric degenerations of Bezier patches. ACM Trans. Graph., 30(5):1-10, 2011.
  • [33] L. Y. Sun and C. G. Zhu. G1 continuity between toric surface patches. Computer Aided Geometric Design, 35-36:255-267, 2015.
  • [34] L.Y. Sun and C.G. Zhu. Approximation of minimal toric Bezier patch. Advances in Mechanical Engineering, 8(6), 2016.
  • [35] J. Monterde and H. Ugail. A general 4th-order PDE method to generate Bezier surfaces from the boundary. Computer Aided Geometric Design, 23:208-225, 2006.
  • [36] G. Farin. Curves and Surfaces for Computer Aided Geometric Design. The Academic Press, USA, 2002.
  • [37] R. Schneider and L. Kobbelt. Geometric fairing of irregular meshes for free-form surface design. Computer Aided Geometric Design, 18(4):359-379, 2001.
  • [38] M.I.G. Bloor and M.J. Wilson. An analytic pseudo-spectral method to generate a regular 4-sided PDE surface patch.

QUASI-HARMONIC CONSTRAINTS FOR TORIC BEZIER SURFACES

Year 2018, Volume: 36 Issue: 2, 325 - 340, 01.06.2018

Abstract

Toric Bezier patches generalize the classical tensor-product triangular and rectangular Bezier surfaces, extensively used in CAGD. The construction of toric Bezier surfaces corresponding to multi-sided convex hulls for known boundary mass-points with integer coordinates (in particular for trapezoidal and hexagonal convex hulls) is given. For these toric Bezier surfaces, we find approximate minimal surfaces obtained by extremizing the quasi-harmonic energy functional. We call these approximate minimal surfaces as the quasi-harmonic toric Bezier surfaces. This is achieved by imposing the vanishing condition of gradient of the quasi-harmonic functional and obtaining a set of linear constraints on the unknown inner mass-points of the toric Bezier patch for the above mentioned convex hull domains, under which they are quasi-harmonic toric Bezier patches. This gives us the solution of the Plateau toric Bezier problem for these illustrative instances for known convex hull domains.

References

  • [1] R. Osserman. A survey of Minimal Surfaces. Dover Publications Inc., 1986.
  • [2] J. C. C. Nitsche. Lectures on Minimal Surfaces. Cambridge University Press, 1989.
  • [3] J.A.F Plateau. Statique exprimentale et thorique des liquides soumis aux seules forces molculaires. Gauthier-Villars, Paris, 1873.
  • [4] H.A. Schwarz. Gesammelte Mathematische Abhandlungen. 2 Bande. Springer, 1890.
  • [5] R. Garnier. Le problme de Plateau. Annales Scienti_ques de l'E.N.S., 3(45):53-144, 1928.
  • [6] J. Douglas. Solution of the problem of Plateau. Trans. Amer. Math. Soc., 33(1):263-321, 1931.
  • [7] T. Rad_o. On Plateau's problem. Ann. Of Math., (2)31(3):457-469, 1930.
  • [8] C. Coppin and D. Greenspan. A contribution to the particle modeling of soap films. Applied Mathematics and Computation, 26(4):315-331, 1988.
  • [9] K. Koohestani. Nonlinear force density method for the form-finding of minimal surface membrane structures. Communications in Nonlinear Science and Numerical Simulation, 19(6):2071-2087, 2014.
  • [10] Kenneth A. Brakke. The surface evolver. Experiment. Math., 1(2):141-165, 1992.
  • [11] D. L. Chopp. Computing minimal surfaces via level set curvature ow. Journal of Computational Physics, 106(1):77-91, 1993.
  • [12] Y.Jung, K.T.Chu, and S.Torquato. A variational level set approach for surface area minimization of triply-periodic surfaces. J. Comput. Phys., 223(2):711-730, May 2007.
  • [13] . Trasdahl and E.M. Ronquist. High order numerical approximation of minimal surfaces. Journal of Computational Physics, 230(12):4795-4810, 2011.
  • [14] C. Y. Li, R. H. Wang, and C. G. Zhu. Designing approximation minimal parametric surfaces with geodesics. Applied Mathematical Modelling, 37(9):6415-6424, 2013.
  • [15] O. Kassabov. Transition to canonical principal parameters on minimal surfaces. Computer Aided Geometric Design, 31(78):441-450, 2014.
  • [16] Gang Xu, Yaguang Zhu, Guozhao Wang, Andr_e Galligo, Li Zhang, and Kin-chuen Hui. Explicit form of parametric polynomial minimal surfaces with arbitrary degree. Appl. Math. Comput., 259(C):124-131, May 2015.
  • [17] J. Monterde. Bezier surfaces of minimal area: The Dirichlet approach. Computer Aided Geometric Design, 21:117-136, 2004.
  • [18] X. D. Chen, G. Xua, and Y. Wanga. Approximation methods for the Plateau-Bezier problem. In 2009 11th IEEE International Conference on Computer-Aided Design and Computer Graphics, 2009.
  • [19] Y. X. Hao, R. H. Wang, and C. J. Li. Minimal quasi-Bezier surface. Applied Mathematical Modelling, 36:5751-5757, 2012.
  • [20] G. Xu and G. Wang. Quintic parametric polynomial minimal surfaces and their properties. Differential Geometry and its Applications, 28:697-704, 2010.
  • [21] D. Ahmad and B. Masud. Variational minimization on string-rearrangement surfaces, illustrated by an analysis of the bilinear interpolation. Applied Mathematics and Computation, 233:72-84, 2014.
  • [22] D. Ahmad and B. Masud. A Coons patch spanning a finite number of curves tested for variationally minimizing its area. Abstract and Applied Analysis, 2013, 2013.
  • [23] D. Ahmad and B. Masud. Near-stability of a quasi-minimal surface indicated through a tested curvature algorithm. Computers & Mathematics with Applications, 69(10):1242- 1262, 2015.
  • [24] G. Xu, T. Rabczuk, E. Guler, Q. Wu, K. C. Hui, and G. Wang. Quasi-harmonic Bezier approximation of minimal surfaces for finding forms of structural membranes. Comput. Struct., 161(C):55-63, December 2015.
  • [25] S. S. Chern. An elementary proof of the existence of isothermal parameters on a surface. Proc. Amer. Math. Soc., 60: 771-782,1955.
  • [26] J. Monterde and H. Ugail. On harmonic and biharmonic Bezier surfaces. Computer Aided Geometric Design, 21:697-715, 2004.
  • [27] G. Farin. Triangular Bernstein-Bezier patches. Computer Aided Geometric Design, 3(2):83-127, 1986.
  • [28] J. Warren. Algebraic Geometry and its Applications, chapter A Bound on the Implicit Degree of Polygonal Bezier Surfaces, pages 513-525. Springer New York, 1994.
  • [29] R. Krasauskas. Toric surface patches. Advances in Computational Mathematics, 17(1):89-113, 2002.
  • [30] R. Goldman, R. Krasauskas. Topics in Algebraic Geometry and Geometric Modeling, chapter Toric Bezier patches with depth. American Mathematical Society, 2002.
  • [31] Gang Xu, Tsz-Ho Kwok, and Charlie C.L. Wang. Isogeometric computation reuse method for complex objects with topology-consistent volumetric parameterization. Computer-Aided Design, 91:1 - 13, 2017.
  • [32] L. D. Garcia-Puente, F. Sottile, and C. Zhu. Toric degenerations of Bezier patches. ACM Trans. Graph., 30(5):1-10, 2011.
  • [33] L. Y. Sun and C. G. Zhu. G1 continuity between toric surface patches. Computer Aided Geometric Design, 35-36:255-267, 2015.
  • [34] L.Y. Sun and C.G. Zhu. Approximation of minimal toric Bezier patch. Advances in Mechanical Engineering, 8(6), 2016.
  • [35] J. Monterde and H. Ugail. A general 4th-order PDE method to generate Bezier surfaces from the boundary. Computer Aided Geometric Design, 23:208-225, 2006.
  • [36] G. Farin. Curves and Surfaces for Computer Aided Geometric Design. The Academic Press, USA, 2002.
  • [37] R. Schneider and L. Kobbelt. Geometric fairing of irregular meshes for free-form surface design. Computer Aided Geometric Design, 18(4):359-379, 2001.
  • [38] M.I.G. Bloor and M.J. Wilson. An analytic pseudo-spectral method to generate a regular 4-sided PDE surface patch.
There are 38 citations in total.

Details

Primary Language English
Journal Section Research Articles
Authors

Daud Ahmad 0000-0003-0966-8237

Saba Naeem This is me 0000-0001-9368-6731

Publication Date June 1, 2018
Submission Date December 31, 2016
Published in Issue Year 2018 Volume: 36 Issue: 2

Cite

Vancouver Ahmad D, Naeem S. QUASI-HARMONIC CONSTRAINTS FOR TORIC BEZIER SURFACES. SIGMA. 2018;36(2):325-40.

IMPORTANT NOTE: JOURNAL SUBMISSION LINK https://eds.yildiz.edu.tr/sigma/