Research Article
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Construction of Developable Surface with Geodesic or Line of Curvature Coordinates

Year 2021, Issue: 36, 75 - 87, 30.09.2021
https://doi.org/10.53570/jnt.987265

Abstract

In this paper, a developable surface with geodesic or line of curvature coordinates is constructed in the Euclidean 3-space. A developable surface is coordinated by two families of parametric curves, base curves (directrices) and lines (rulings). Since any part of a straight line on a developable surface is geodesic and line of curvature, we only need to show that the directrices curves are geodesics or lines of curvature to ensure that the developable surface is parameterized by geodesic or line of curvature coordinates. The necessary and sufficient conditions for the directrices curves to be geodesics or lines of curvature are studied. The main results of this paper show that the developable surface with geodesic coordinates is a generalized cylinder, and the developable surface with line of curvature coordinates is a tangent surface.

References

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  • C. Y. Li, R. H. Wang, C. G. Zhu, An Approach for Designing a Developable Surface through a Given Line of Curvature, Computer-Aided Design 45(3) (2013) 621-627.
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  • M. I. Shtogrin, Bending of a Piecewise Developable Surface, Proceedings of the Steklov Institute of Mathematics 275(1) (2011) 133-54.
  • N. M. Althibany, Classification of Ruled Surfaces Family with Common Characteristic Curve in Euclidean 3-space, Turkish Journal of Science (2021) In Press.
Year 2021, Issue: 36, 75 - 87, 30.09.2021
https://doi.org/10.53570/jnt.987265

Abstract

References

  • H. Pottmann, A. Asperl, M. Hofer, A. Kilian, Architectural geometry, Bentley Institute Press, 2007.
  • Y. Liu, H. Pottmann, J. Wallner, Y. L. Yang, W. Wang, Geometric Modeling with Conical Meshes and Developable Surfaces, In ACM SIGGRAPH Papers (2006) 681-689.
  • C. Tang, P. Bo, J. Wallner, H. Pottmann, Interactive Design of Developable Surfaces, ACM Transactions on Graphics (TOG) 35(2) (2016) 1-12.
  • W. K. Schief, On the Integrability of Bertrand Curves and Razzaboni Surfaces, Journal of Geometry and Physics 45(1-2) (2003) 130-150.
  • N. Gürbüz, The Motion of Timelike Surfaces in Timelike Geodesic Coordinates, International Journal of Mathematical Analysis 4 (2010) 349-356.
  • Y. Li, C. Chen, The Motion of Surfaces in Geodesic Coordinates and 2+ 1-dimensional Breaking Soliton Equation, Journal of Mathematical Physics 41(4) (2000) 2066-2076.
  • E. Adiels, M. Ander, C. Williams, Brick Patterns on Shells Using Geodesic Coordinates, In Proceedings of IASS Annual Symposia 23 (2017) 1-10 Hamburg, Germany.
  • X. Tellier, C. Douthe, L. Hauswirth, O. Baverel, Surfaces with Planar Curvature Lines: Discretization, Generation and Application to the Rationalization of Curved Architectural Envelopes, Automation in Construction 106 (2019) p.102880.
  • H. Zhao, G. Wang, A New Method for Designing a Developable Surface Utilizing the Surface Pencil through a Given Curve, Progress in Natural Science 18(1) (2008) 105-110.
  • R. A. Al-Ghefaria, A. B. Rashad, An Approach for Designing a Developable Surface with a Common Geodesic Curve, International Journal of Contemporary Mathematical Sciences 8(18) (2013) 875-891.
  • C. Y. Li, R. H. Wang, C. G. Zhu, An Approach for Designing a Developable Surface through a Given Line of Curvature, Computer-Aided Design 45(3) (2013) 621-627.
  • M. D. Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall, New Jersey, 1976.
  • A. N. Pressley, Elementary Differential Geometry, Springer Science & Business Media, 2010.
  • F. Doğan, Y. Yaylı, The Relation between Parameter Curves and Lines of Curvature on Canal Surfaces, Kuwait Journal of Science 44(1) (2017) 29-35.
  • M. I. Shtogrin, Bending of a Piecewise Developable Surface, Proceedings of the Steklov Institute of Mathematics 275(1) (2011) 133-54.
  • N. M. Althibany, Classification of Ruled Surfaces Family with Common Characteristic Curve in Euclidean 3-space, Turkish Journal of Science (2021) In Press.
There are 16 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Nabil Althibany 0000-0001-8057-2938

Publication Date September 30, 2021
Submission Date August 26, 2021
Published in Issue Year 2021 Issue: 36

Cite

APA Althibany, N. (2021). Construction of Developable Surface with Geodesic or Line of Curvature Coordinates. Journal of New Theory(36), 75-87. https://doi.org/10.53570/jnt.987265
AMA Althibany N. Construction of Developable Surface with Geodesic or Line of Curvature Coordinates. JNT. September 2021;(36):75-87. doi:10.53570/jnt.987265
Chicago Althibany, Nabil. “Construction of Developable Surface With Geodesic or Line of Curvature Coordinates”. Journal of New Theory, no. 36 (September 2021): 75-87. https://doi.org/10.53570/jnt.987265.
EndNote Althibany N (September 1, 2021) Construction of Developable Surface with Geodesic or Line of Curvature Coordinates. Journal of New Theory 36 75–87.
IEEE N. Althibany, “Construction of Developable Surface with Geodesic or Line of Curvature Coordinates”, JNT, no. 36, pp. 75–87, September 2021, doi: 10.53570/jnt.987265.
ISNAD Althibany, Nabil. “Construction of Developable Surface With Geodesic or Line of Curvature Coordinates”. Journal of New Theory 36 (September 2021), 75-87. https://doi.org/10.53570/jnt.987265.
JAMA Althibany N. Construction of Developable Surface with Geodesic or Line of Curvature Coordinates. JNT. 2021;:75–87.
MLA Althibany, Nabil. “Construction of Developable Surface With Geodesic or Line of Curvature Coordinates”. Journal of New Theory, no. 36, 2021, pp. 75-87, doi:10.53570/jnt.987265.
Vancouver Althibany N. Construction of Developable Surface with Geodesic or Line of Curvature Coordinates. JNT. 2021(36):75-87.


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