An approach for designing a surface pencil through a given geodesic curve

Abstract

In the present paper, we propose a new method to construct a surface interpolating a given curve as the geodesic curve of it. Also, we analyze the conditions when the resulting surface is a ruled surface. In addition, developablity along the common geodesic of the members of surface family are discussed. Finally, we illustrate this method by presenting some examples.

Keywords

• Geodesic curve
• rotation minimizing frame
• parametric surface

References

1. Bechmann, D., Gerber, D., Arbitrary shaped deformation with dogme, Visual Comput., 19 (2-3) (2003), 175-186.
2. Peng, Q., Jin, X., Feng, J., Arc-length-based axial deformation and length preserving deformation, In Proceedings of Computer Animation., (1997), 86-92.
3. Lazarus, F., Coquillart, S., Jancène, P., Interactive axial deformations, In Modeling in Computer Graphics., Springer, Verlag, 1993.
4. Lazarus, F., Verroust, A., Feature-based shape transformation for polyhedral object, In Proceedings of the 5th Eurographics Workshop on Animation and Simulation., (1994), 1-14.
5. Lazarus, F., Coquillart, S., Jancène, P., Axial deformation: an intuitive technique, Comput. Aid. Des., 26 (8) (1994), 607-613.
6. Llamas, I., Powell, A., Rossignac, J., Shaw, C.D., Bender : A virtual ribbon for deforming 3d shapes in biomedical and styling applications, In Proceedings of Symposium on Solid and Physical Modeling., (2005), 89-99.
7. Bloomenthal,M., Riesenfeld, R.F., Approximation of sweep surfaces by tensor product NURBS, In SPIE Proceedings Curves and Surfaces in Computer Vision and Graphics., 2 (1610) (1991), 132-154.
8. Pottmann, H., Wagner, M., Contributions to motion based surface design, Int. J. Shape Model., 4 (3&4 ) (1998), 183-196.

9. Siltanen, P., Woodward, C., Normal orientation methods for 3D o¤set curves, sweep surfaces, skinning, In Proceedings of Eurographics., (1992), 449-457.
10. Wang,W., Joe, B., Robust computation of rotation minimizing frame for sweep surface modeling, Comput. Aid. Des., 29 (1997), 379-391.
11. Shani, U., Ballard, D.H., Splines as embeddings for generalized cylinders. Comput. Vision Graph. Image Proces., 27 (1984), 129-156.
12. Bloomenthal, J., Modeling the mighty maple, In Proceedings of SIGGRAPH., (1985), 305-311.
13. Bronsvoort, W.F., Klok, F., Ray tracing generalized cylinders, ACM Trans. Graph.,4 (4) (1985) , 291-302.
14. Semwal, S.K., Hallauer, J., Biomedical modeling: implementing line-of-action algorithm for human muscles and bones using generalized cylinders, Comput. Graph., 18 (1) (1994), 105-112.
15. Banks, D.C., Singer, B.A., A predictor-corrector technique for visualizing unsteady flows, IEEE Trans on Visualiz. Comput. Graph., 1 (2) (1995), 151-163.
16. Hanson, A.J., Ma, H., A quaternion approach to streamline visualization, IEEE Trans Visualiz. Comput. Graph., 1 (2) (1995), 164-174.
17. Hanson, A., Constrained optimal framing of curves and surfaces using quaternion gauss map. In Proceedings of Visulization., (1998), 375-382.
18. Barzel, R., Faking dynamics of ropes and springs, IEEE Comput. Graph. Appl., 17 (3) (1997), 31-39.
19. Jüttler, B., Rational approximation of rotation minimizing frames using Pythagoreanhodograph cubics, J. Geom. Graph., 3 (1999), 141-159.
20. Bishop, R. L., There is more than one way to frame a curve, Ame. Math. Mon., 82 (1975), 246-251.
21. O'Neill, B., Elementary Differential Geometry, Academic Press Inc., New York, 1966.
22. Farouki, R.T., Sakkalis, T., Rational rotation-minimizing frames on polynomial space curves of arbitrary degree, J. Symbolic Comput., 45 (2010), 844-856 .
23. Brond, R., Jeulin, D., Gateau, P., Jarrin, J., Serpe, G., Estimation of the transport properties of polymer composites by geodesic propagation, J. Microsc., 176 (1994), 167-177.
24. Bryson, S.,Virtual spacetime: an environment for the visualization of curved spacetimesvia geodesic flows, Technical Report, NASA NAS., Number RNR-92 (1992).
25. Grundig, L., Ekert, L., Moncrieff, E., Geodesic and semi-geodesic line algorithms for cutting pattern generation of architectural textile structures, In: Lan TT, editor. Proceedings of the Asia-Pacific Conference on Shell and Spatial Structures, Beijing, 1996.
26. Haw, R.J., An application of geodesic curves to sail design, Comput. Graphics Forum., 4(2) (1985), 137-139.
27. Haw, R.J., Munchmeyer, R.C., Geodesic curves on patched polynomial surfaces, Comput. Graphics Forum., 2 (4) (1983), 225-232.
28. Wang, G.J., Tang, K., Tai, C.L., Parametric representation of a surface pencil with a common spatial geodesic. Comput. Aided Des., 36 (5) (2004), 447-459.
29. Deng, B., Special Curve Patterns for Freeform Architecture Ph.D. thesis, Eingereicht an der Technischen Universitat Wien, Fakultat für Mathematik und Geoinformation von, 2011.
30. Kasap, E., Akyıldız, F.T., Orbay, K., A generalization of surfaces family with common spatial geodesic, Appl. Math. Comput., 201 (2008), 781-789.
31. Kasap, E., Akyildiz, F.T., Surfaces with common geodesic in Minkowski 3-space, Appl. Math. Comput., 177 (2006), 260-270.
32. Saffak, G., Kasap, E., Family of surface with a common null geodesic, International Journal of Physical Sciences., 4 (8) (2009), 428-433.
33. Atalay, G.¸S., Kasap, E., Surfaces family with common null asymptotic, Appl. Math. Comput., doi: 10.1016/J.amc.2015.03.067.
34. Bayram, E., Güler, F., Kasap, E., Parametric representation of a surface pencil with a common asymptotic curve, Comput. Aided Des., 44 (2012), 637-643.
35. Bayram, E., Bilici, M., Surfaces family with a common involute asymptotic curve, International Journal of Geometric Methods in Modern Physics., 13(5) (2016).
36. Atalay, G.S., Surfaces family with a common Mannheim geodesic curve, Journal of Applied Mathematics and Computation., 2 (4) (2018), 155-165.
37. Atalay, G.S., Surfaces family with a common Mannheim asymptotic curve, Journal of Applied Mathematics and Computation., 2 (4) (2018), 143-154.
38. Ayvacı, K.H., Ortak Mannheim-B ·Isogeodezikli ve ·Isoasimptotikli Yüzey Ailesi, Ondokuz Mayıs Üniversitesi Fen Bilimleri Enstitüsü, Yüksek Lisans Tezi, 2019.
39. Ayvacı, K.H., Atalay, G.S., Surface Family With A Common Bertrand-B Isogeodesic Curve, Journal of the Institute of Science and Technology., 10 (3) (2020), 1975-1983.
40. Do Carmo, M.P., Di¤erential geometry of curves and surfaces, Prentice Hall, Inc., Englewood Cliffs, New Jersey, 1976.
41. Klok, F., Two moving coordinate frames along a 3D trajectory, Comput. Aided Geom. Design., 3 (1986), 217-229.
42. Han, C.Y., Nonexistence of rational rotation-minimizing frames on cubic curves, Comput. Aided Geom. Design., 25 (2008), 298-304.
43. Li ,C.Y., Wang, R.H., Zhu, C.G., An approach for designing a developable surface through a given line of curvature, Comput. Aided Des., 45 (2013), 621-627.