# An approach for designing a surface pencil through a given asymptotic curve

Author：Fatma Güler, Gülnur Şaffak Atalay, Ergin Bayram Date：July 17,2019 Hits：

## Abstract

Surfaces and curves play an important role in geometric design. In recent years, problem of finding a surface passing through a given curve has attracted much interest. In the present paper, we propose a new method to construct a surface interpolating a given curve as the asymptotic curve of it. Also, we analyze the conditions when the resulting surface is a ruled surface. Furthermore, we prove that there exists no developable surface possessing a given curve as an asymptotic curve except plane. Finally, we illustrate this method by presenting some examples.

## References

[1] D. Bechmann, D. Gerber, Arbitrary shaped deformation with dogme, Visual Comput. 19, 2–3 (2003) 175-186.

[2] Q. Peng, X. Jin, J. Feng, Arc-length-based axial deformation and length preserving deformation, In Proceedings of Computer Animation (199) 86-92.

[3] F. Lazarus, S. Coquillart, P. Jancène, Interactive axial deformations, In Modeling in Computer Graphics (1993) 241-254.

[4] F. Lazarus, A. Verroust, Feature-based shape transformation for polyhedral objects, In Proceedings of the 5th Eurographics Workshop on Animation and Simulation (1994) 1-14.

[5] F. Lazarus, S. Coquillart, P. Jancène, Axial deformation: an intuitive technique, Comput. Aided Des. 26, 8 (1994) 607-613.

[6] I. Llamas, A. Powell, J. Rossignac, C.D. Shaw, 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] M. Bloomenthal, R.F. Riesenfeld, Approximation of sweep surfaces by tensor product NURBS, In SPIE Proceedings Curves and Surfaces in Computer Vision and Graphics II, 1610 (1990) 132-154.

[8] H. Pottmann, M. Wagner, Contributions to motion based surface design, Int. J. Shape Model 4, 3&4 (1998) 183-196.

[9] P. Siltanen, C. Woodward, Normal orientation methods for 3D offset curves, sweep surfaces, skinning, In Proceedings of Eurographics (1992) 449-457.

[10] W. Wang, B. Joe, Robust computation of rotation minimizing frame for sweep surface modeling, Comput. Aided Des. 29 (1997) 379-391. [11] U. Shani, D.H. Ballard, Splines as embeddings for generalized cylinders, Comput. Vision Graph. Image Proces. 27 (1984) 129-156.

[12] J. Bloomenthal, Modeling the mighty maple, In Proceedings of SIGGRAPH (1985) 305-311.

[13] W.F. Bronsvoort, F. Klok, Ray tracing generalized cylinders, ACM Trans. Graph. 4, 4 (1985) 291–302.

[14] S.K. Semwal, J. Hallauer, Biomedical modeling: implementing line-of-action algorithm for human muscles and bones using generalized cylinders, Comput. Graph. 18, 1 (1994) 105-112.

[15] D.C. Banks, B.A. Singer, A predictor-corrector technique for visualizing unsteady flows, IEEE Trans. on Visualiz. Comput. Graph. 1, 2 (1995) 151-163.

[16] A.J. Hanson, H. Ma, A quaternion approach to streamline visualization, IEEE Trans. Visualiz. Comput. Graph. 1, 2 (1995) 164-174.

[17] A. Hanson, Constrained optimal framing of curves and surfaces using quaternion gauss map, In Proceedings of Visulization (1998) 375-382.

[18] R. Barzel, Faking dynamics of ropes and springs, IEEE Comput. Graph. Appl. 17 (1997) 31-39.

[19] B. Jüttler, Rational approximation of rotation minimizing frames using Pythagorean-hodograph cubics, J. Geom. Graph. 3 (1999) 141-159. [20] R.L. Bishop, There is more than one way to frame a curve, Am. Math. Mon. 82 (1975) 246-251.

[21] B. O’Neill, Elementary Differential Geometry, Academic Press Inc., New York, 1966.

[22] R.T. Farouki, T. Sakkalis, Rational rotation-minimizing frames on polynomial space curves of arbitrary degree, J. Symbolic Comput. 45 (2010) 844-856.

[23] G. Contopoulos, Asymptotic curves and escapes in Hamiltonian systems, Astron. Astrophys. 231 (1990) 41-55.

[24] C. Efthymiopoulos, G. Contopoulos, N. Voglis, Cantori, islands and asymptotic curves in the stickiness region, Celestial Mech. Dynam. Astronom. 73 (1999) 221–230.

[25] S. Flöry, H. Pottmann, Ruled surfaces for rationalization and design in architecture, In: Proceedings of the conference of the association for computer aided design in architecture (ACADIA) (2010).

[26] M.P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall Inc., Englewood Cliffs (New Jersey), 1976.

[27] G.J. Wang, K. Tang, C.L. Tai, Parametric representation of a surface pencil with a common spatial geodesic, Comput. Aided Des. 36(5) (2004) 447–59.

[28] E. Kasap, F.T. Akyıldız, K. Orbay, A generalization of surfaces family with common spatial geodesic, Appl. Math. Comput. 201 (2008) 781–789.

[29] C.Y. Li, R.H. Wang, C.G. Zhu, Parametric representation of a surface pencil with a common line of curvature, Comput. Aided Des. 43(9) (2011) 1110–1117. [30] E. Bayram, F. Güler, E. Kasap, Parametric representation of a surface pencil with a common asymptotic curve, Comput. Aided Des. 44 (2012) 637-643.

[31] G. Şaffak Atalay, E. Kasap, Surfaces family with common null asymptotic, Appl. Math. Comput. 260 (2015) 135-139.

[32] F. Klok, Two moving coordinate frames along a 3D trajectory, Comput. Aided Geom. Design 3 (1986) 217-229.

[33] C.Y. Han, Nonexistence of rational rotation-minimizing frames on cubic curves, Comput. Aided Geom. Design 25 (2008) 298-304.

[34] C.Y. Li, R.H. Wang, C.G. Zhu, An approach for designing a developable surface through a given line of curvature, Comput. Aided Des. 45 (2013) 621-627.

## Full-Text HTML

**An approach for designing a surface pencil through a given asymptotic curve**

Fatma Güler, Gülnur Şaffak Atalay, Ergin Bayram

Emin Kasap Ondokuz Mayıs University, Arts and Science Faculty, Department of Mathematics, 55139, Samsun, Turkey

***Corresponding author**:Fatma Güler. Ondokuz Mayıs University, Arts and Science Faculty, Department of Mathematics, 55139, Samsun, Turky.

Email: f.guler@omu.edu.tr

**How to cite this paper**: Güler F, Atalay G, Ş, Bayram E, Kasap E. (2019). An approach for designing a surface pencil through a given asymptotic curve*. Journal of Applied Mathematics and Computation, 3(4), 606-615.

DOI: 10.26855/jamc.2019.06.002

>> No Data