A Computational Approach to a Quasi- Minimal Bezier Surface for Computer Graphics

Authors

  • Daud Ahmad University of the Punjab, Lahore, Pakistan
  • Saba Naeem University of the Punjab, Lahore, Pakistan
  • Abdul Haseeb University of the Punjab, Lahore, Pakistan
  • M.Khalid Mahmood Department of Mathematics, University of the Punjab, Lahore, Pakistan

DOI:

https://doi.org/10.21015/vtse.v9i4.929

Abstract

 

In computer science, the algorithms related to geometry can be exploited in computer aided geometric design, a field in computational geometry. Bézier surfaces are restricted class of surfaces used in computer science, computer graphics and the allied disciplines of science. In this work, a computational approach for finding the Béziersurface related minimal surfaces as the extremal of mean curvature functional is presented. A minimal surface is the surface that locally minimizes its area and has zero mean curvature everywhere. The vanishing mean curvature results in a non-linear partial differential equation for a minimal surface spanned by the boundary of interest and the solution of the partial differential equation does exist for very few special cases whereas, the vanishing condition of the gradient of the area functional is in general not possible as it involves square-root in its integrand. Instead, we find the vanishing condition of the gradient of the mean curvature for a related problem, Plateau-Bézier problem that gives the constraints on the interior control points in terms of boundary control points of the prescribed border. The emerging Bézier surface is the quasi-minimal surface as the extremal of mean curvature.

 

References

J.A.F Plateau. Statique exprimentale et thorique des liquides soumis aux seules forces molculaires. Gauthier-Villars, Paris, 1873.

R. Osserman. A survey of Minimal Surfaces. Dover Publications Inc., 1986.

J. C. C. Nitsche. Lectures on Minimal Surfaces. Cambridge University Press, 1989.

H.A. Schwarz. Gesammelte Mathematische Abhandlungen. 2 B¨ande. Springer, 1890. DOI: https://doi.org/10.1007/978-3-642-50665-9

R. Garnier. Le problme de Plateau. Annales Scientifiques de l’E.N.S., 3(45):53–144, 1928. DOI: https://doi.org/10.24033/asens.783

J. Douglas. Solution of the problem of Plateau. Trans. Amer. Math. Soc., 33(1):263–321, 1931. DOI: https://doi.org/10.1090/S0002-9947-1931-1501590-9

T. Rad´o. On Plateau’s problem. Ann. Of Math., (2)31(3):457–469, 1930. DOI: https://doi.org/10.2307/1968237

A. Ali Shah and Y. D. Khan. Identification of 4-carboxyglutamate residue sites based on position based statistical feature and multiple classification. Scientific Reports, 10(1):16913, Oct 2020.

S. Naseer, W. Hussain, Y. D. Khan, and N. Rasool. iphoss(deep)-pseaac: Identify phosphoserine sites in proteins using deep learning on general pseudo amino acid compositions via modified 5-steps rule. IEEE/ACM Transactions on Computational Biology and Bioinformatics, pages 1–1, 2020.

S. Naseer, W. Hussain, Y. D. Khan, and N. Rasool. Sequence-based identification of arginine amidation sites in proteins using deep representations of proteins and pseaac. Current Bioinformatics, 15(8):937–948, 2020.

S. Naseer, W. Hussain, Y. D. Khan, and N. Rasool. Optimization of serine phosphorylation prediction in proteins by comparing human engineered features and deep representations. Analytical Biochemistry, 615:114069, 2021.

J. Monterde. B´ezier surfaces of minimal area: The Dirichlet approach. Computer Aided Geometric Design, 21:117–136, 2004. DOI: https://doi.org/10.1016/j.cagd.2003.07.009

X. D. Chen, G. Xu, and Y. Wanga. Approximation methods for the Plateau-B´ezier problem. In Computer Aided Design and Computer Graphics, 2009. CAD/Graphics ’09. 11th IEEE International Conference on, pages 588–591, 2009. DOI: https://doi.org/10.1109/CADCG.2009.5246833

Y. X. Hao, R. H. Wang, and C. J. Li. Minimal quasi-B´ezier surface. Applied Mathematical Modelling, 36:5751 – 5757, 2012. DOI: https://doi.org/10.1016/j.apm.2012.01.040

J. Monterde and H. Ugail. A general 4th-order PDE method to generate B´ezier surfaces from the boundary. Computer Aided Geometric Design, 23:208 – 225, 2006. DOI: https://doi.org/10.1016/j.cagd.2005.09.001

G. Farin. Curves and Surfaces for Computer Aided Geometric Design. The Academic Press, USA, 2002.

R. Schneider and L. Kobbelt. Geometric fairing of irregular meshes for free-form surface design. Computer Aided Geometric Design, 18(4):359 – 379, 2001. DOI: https://doi.org/10.1016/S0167-8396(01)00036-X

M.I.G. Bloor and M.J. Wilson. An analytic pseudo-spectral method to generate a regular 4-sided PDE surface patch. Computer Aided Geometric Design, 22(3):203 – 219, 2005. DOI: https://doi.org/10.1016/j.cagd.2004.08.005

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. DOI: https://doi.org/10.1016/j.amc.2014.01.172

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. DOI: https://doi.org/10.1155/2013/645368

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. DOI: https://doi.org/10.1016/j.camwa.2015.03.015

G. Xu, T. Rabczuk, E. G¨uler, Q. Wu, K. Hui, and G. Wang. Quasi-harmonic B´ezier approximation of minimal surfaces for finding forms of structural membranes. Comput. Struct., 161(C):55–63, December 2015. DOI: https://doi.org/10.1016/j.compstruc.2015.09.002

J. Monterde and H. Ugail. On harmonic and biharmonic B´ezier surfaces. Computer Aided Geometric Design, 21(7):697–715, 2004. DOI: https://doi.org/10.1016/j.cagd.2004.07.003

L. S. Velimirovic´, M. S. C´iric´, and M. D. Cvetkovic´. Change of the Willmore energy under infinitesimal bending of membranes. Computers & Mathematics with Applications, 59(12):3679 – 3686, 2010. DOI: https://doi.org/10.1016/j.camwa.2010.03.069

D. Ahmad and S. Naeem. Quasi-harmonic constraints for toric B´ezier surfaces. Sigma Journal of Engineering and Natural Sciences, 36:325–340, 2018.

D. Ahmad, K. Hassan, M. K. Mahmood, Javaid Ali, Ilyas Khan, and M. Fayz-Al-Asad. Variationally improved B´ezier surfaces with shifted knots. Advances in Mathematical Physics, 2021(5):14, 2021.

S. N. Bernstein. Dmonstration du thorme de weierstrass fonde sur le calcul des probabilities. Comm. Soc. Math. Kharkov, 13:1–2, 1912.

M. Do Carmo. Differential Geometry of Curves and Surfaces. Prentice Hall, 1976.

Downloads

Published

2021-12-31

How to Cite

Ahmad, D., Naeem, S., Haseeb, A., & Mahmood, M. (2021). A Computational Approach to a Quasi- Minimal Bezier Surface for Computer Graphics. VFAST Transactions on Software Engineering, 9(4), 150–159. https://doi.org/10.21015/vtse.v9i4.929

Issue

Vol. 9 No. 4 (2021): October-December

Section

Articles

License

Authors who publish with this journal agree to the following terms:

  1. Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License (CC-By) that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal.
  2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
  3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).

                                      

This work is licensed under a Creative Commons Attribution  License CC BY