Generalized Cylinder with Geodesic and Line of Curvature Parameterizations

Abstract

Constructing a surface with geodesic or line of curvature parameterization is an important problem in many practical applications. The present paper aims to design a generalized cylinder that is parametrized along the geodesics and lines of curvature curves in Euclidean 3- space. The main results show that the generalized cylinder with geodesic or line of curvature parameterization is a rectifying cylinder or a right cylinder respectively.

Keywords

• Generalized cylinder
• Geodesic
• Line of curvature
• Parameterization

References

1. [1] K. H. Chang, Product Design Modeling using CAD/CAE, The computer aided engineering design series, Academic Press, 2014.
2. [2] R. Goldman, An Integrated Introduction to Computer Graphics and Geometric Modeling, CRC Press, 2009.
3. [3] S. Guha, Computer Graphics Through OpenGL: From Theory to Experiments, Chapman and Hall/CRC, 2018.
4. [4] P. Helmut, A. Andreas, H. Michael, K. Axel, Architectural Geometry, Bentley Institute Press, 2007.
5. [5] H. K. Joo, T. Yazaki, M. Takezawa, T. Maekawa, Differential geometry properties of lines of curvature of parametric surfaces and their visualization, Graph. Models, 76 (2014), 224–238.
6. [6] X. P. Zhang, W. J. Che, J. C. Paul, Computing lines of curvature for implicit surfaces, Comput. Aid. Geom. Des., 26(9)(2009), 923–940.
7. [7] I. Hotz, H. Hagen, Visualizing geodesics, In Proceedings Visualization 2000, VIS 2000 (Cat. No. 00CH37145), IEEE, 311-318.
8. [8] Y. L. Yang, J. Kim, F. Luo, S. M. Hu, X. Gu, Optimal surface parameterization using inverse curvature map, IEEE Transactions on Visualization and Computer Graphics, 14(5)(2008), 1054-1066.

9. [9] A. Sheffer, E. Praun, K. Rose, Mesh parameterization methods and their applications, Foundations and Trends in Computer Graphics and Vision, 2(2)(2006), 105–171.
10. [10] K. Hormann, B. Levy, A. Sheffer, Mesh Parameterization: Theory and Practice, 2007.
11. [11] M. Desbrun, P. Alliez, U. S. C. Inria, M. Meyer, P. Alliez, Intrinsic parameterizations of surface meshes, Comput. Graph. Forum, 21(3)(2002), 209–218.
12. [12] M. S. Floater, K. Hormann, Surface Parameterization: A Tutorial and Survey, In Advances in multiresolution for geometric modelling, Springer, 2005, 157–186.
13. [13] B. H. Jafari, N. Gans, Surface parameterization and trajectory generation on regular surfaces with application in robot-guided deposition printing, IEEE Robotics and Automation Letters, 5(4)(2020), 6113-6120.
14. [14] R. R. Martin, Principal Patches-A new class of surface patch based on differential geometry, Eurographics Proceedings, (1983).
15. [15] L. Garnier, L. Druoton, Constructions of principal patches of Dupin cyclides defined by constraints: four vertices on a given circle and two perpendicular tangents at a vertex, XIV Mathematics of Surfaces (Birmingham, Royaume-Uni, 11-13 September 2013), pp.237-276.
16. [16] M.Takezawa, T. Imai, K. Shida, T. Maekawa, Fabrication of freeform objects by principal strips, ACM T. Graphic., 35(6)(2016), 1-12.
17. [17] N. G¨urb¨uz, The motion of timelike surfaces in timelike geodesic coordinates, Int. J. Math. Anal, 4(2010), 349-356.
18. [18] Y. Li, C. Chen, The motion of surfaces in geodesic coordinates and 2+ 1-dimensional breaking soliton equation, J. Math. Phys., 41(4)(2000), 2066-2076.
19. [19] E. Adiels, M. Ander, C. Williams, Brick patterns on shells using geodesic coordinates, In Proceedings of IASS Annual Symposia, Hamburg, Germany, September 25-28, 23(2017), 1-10.
20. [20] 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), 102880.
21. [21] S. Pillwein, K. Leimer, M. Birsak, P. Musialski, On elastic geodesic grids and their planar to spatial deployment, 2020, arXiv preprint arXiv:2007.00201.
22. [22] H. Wang, D. Pellis, F. Rist, H. Pottmann, C. M¨uller, Discrete geodesic parallel coordinates, ACM T. Graphic., 38(6)(2019), 1-13.
23. [23] M. Rabinovich, T. Hoffmann, O. Sorkine-Hornung, Discrete geodesic nets for modeling developable surfaces, ACM T. Graphic., 37(2)(2018), 1-17.
24. [24] N. M. Althibany, Construction of developable surface with geodesic or line of curvature coordinates, J. New Theory, 36(2021),75-87.
25. [25] M. D. Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, New Jersey, 1976.
26. [26] D. J. Struik, Lectures on Classical Differential Geometry, 2nd Edition, Dover Publications Inc., New York, 1988.
27. [27] A. N. Pressley, Elementary Differential Geometry, Springer Science & Business Media, 2010.
28. [28] F. Dogan, Y. Yayli, The relation between parameter curves and lines of curvature on canal surfaces, Kuwait J. Sci., 44(1)(2017), 29-35.
29. [29] M. I. Shtogrin, Bending of a piecewise developable surface, Proceedings of the Steklov Institute of Mathematics, 275(2011), 133-154.
30. [30] A. Honda, K. Naokawa, K. Saji, M. Umehara, K. Yamada, Curved foldings with common creases and crease patterns, Adv. App. Math., 121(2020), 102083.
31. [31] S. Izumiya, H. Katsumi, T. Yamasaki, The rectifying developable and the spherical Darboux image of a space curve, Banach Center Publ., 50(1999), 137-149.
32. [32] G. H. Georgiev, C. L. Dinkova, Focal curves of geodesics on generalized cylinders, ARPN J. Engineering and Applied Sciences, 14(11)(2019), 2058-2068.