Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
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ON CONSTRUCTIONS OF MINIMAL SURFACES

  • Yoon, Dae Won (Department of Mathematics Education and RINS Gyeongsang National University)
  • Received : 2020.07.09
  • Accepted : 2020.11.24
  • Published : 2021.02.15
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Abstract

In the recent papers, S'anchez-Reyes [Appl. Math. Model. 40 (2016), 1676-1682] described the method for finding a minimal surface through a geodesic, and Li et al. [Appl. Math. Model. 37 (2013), 6415-6424] studied the approximation of minimal surfaces with a geodesic from Dirichlet function. In the present article, we consider an isoparametric surface generated by Frenet frame of a curve introduced by Wang et al. [Comput. Aided Des. 36 (2004), 447-459], and give the necessary and sufficient condition to satisfy both geodesic of the curve and minimality of the surface. From this, we construct minimal surfaces in terms of constant curvature and torsion of the curve. As a result, we present a new approach for constructions of the minimal surfaces from a prescribed closed geodesic and unclosed geodesic, and show some new examples of minimal surfaces with a circle and a helix as a geodesic. Our approach can be used in design of minimal surfaces from geodesics.

Keywords

Acknowledgement

The author was supported by the Gyeongsang National University Fund for Professors on Sabbatical Leave, 2019.

References

  1. B. Appleton and H. Talbot, Globally optimal geodesic active contours, JMIV., 23 (2005), 67-86. https://doi.org/10.1007/s10851-005-4968-1
  2. Y. Boykov and V. Kolmogorov, Computing Geodesics and Minimal Surfaces via Graph Cuts, Proceedings of International Conference on Computer Vision(ICCV), Nice, France, November (2003), 26-33.
  3. R. J. Haw, An application of geodesic curves to sail design, Comput. Graph. Forum., 4 (1985), 137-139. https://doi.org/10.1111/j.1467-8659.1985.tb00203.x
  4. Y.-X. Hao, R.-H. Wang and C.-J. Li, Minimal quasi-B'ezier surface, Appl. Math. Model., 36 (2012), 5751-5757. https://doi.org/10.1016/j.apm.2012.01.040
  5. E. Kasap and F. T. Akyildiz, Surfaces with common geodesic in Minkowski 3-space, Appl. Math. Comput., 177 (2006), 260-270. https://doi.org/10.1016/j.amc.2005.11.005
  6. E. Kasap, F. T. Akyildiz and K. Orbay, A generalization of surfaces family with common spatial geodesic, Appl. Math. Comput., 201 (2008) 781-789. https://doi.org/10.1016/j.amc.2008.01.016
  7. C.-Y. Li, R.-H. Wang and C.-G. Zhu, Designing approximation minimal parametric surfaces with geodesics Appl. Math. Model., 37 (2013), 6415-6424. https://doi.org/10.1016/j.apm.2013.01.035
  8. F. C. Munchmeyer and R. Haw, Applications of differential geometry to ship design. In D. F. Rogers, B. C. Nehring, and C. Kuo, editors, Proceedings of Computer Applications in the Automation of Shipyard Operation and Ship Design IV, 9, 183-196, Annapolis, Maryland, USA, June 1982.
  9. C. M. C. Riverros and A. M. V. Corro, Geodesics in minimal surfaces, Math. Notes, 101 (2017), 497-514. https://doi.org/10.1134/S0001434617030129
  10. J. S'anchez-Reyes, On the construction of minimal surfaces from geodesics, Appl. Math. Model., 40 (2016), 1676-1682. https://doi.org/10.1016/j.apm.2015.07.011
  11. J. S'anchez-Reyes and R. Dorado, Constrained design of polynomial surfaces from geodesic curves, Comput. Aided Des., 40 (2008), 49-55. https://doi.org/10.1016/j.cad.2007.05.002
  12. M. Tamura, Surfaces which contain helical geodesics, Geod. Dedic., 42 (1992), 311-315. https://doi.org/10.1007/BF02414069
  13. M. Tamura, Surfaces which contain helical geodesics in the 3-sphere, Mem. Fac. Sci. Eng. Shimane Univ. Series B: Math. Sci., 37 (2004), 59-65
  14. G.-J. Wang, K. Tang and C.-L. Tai, Parametric representation of a surface pencil with a common spatial geodesic, Comput. Aided Des., 36 (2004), 447-459. https://doi.org/10.1016/S0010-4485(03)00117-9
  15. G. Xu and G.-Z. Wang, Quintic parametric polynomial minimal surfaces and thier properties, Diff. Geom. Appl., 28 (2010), 697-704. https://doi.org/10.1016/j.difgeo.2010.07.003
  16. Z. K. Yuzbasi and M. Bektas, On the construction of a surface family with common geodesic in Galilean space G3, Open Phys., 14 (2016), 360-363. https://doi.org/10.1515/phys-2016-0041