Embankment Surfaces in Euclidean 3-Space and Their Visualizations

Authors

  • Ahmet Kazan Department of Computer Technologies, Dogan¸sehir Vahap Küçük Vocational School of Higher Education, Malatya Turgut í–zal University, Malatya
  • H. Bayram Karadağ Department of Mathematics, Faculty of Arts and Sciences, Inönü University, Malatya

DOI:

https://doi.org/10.26713/cma.v10i3.916

Keywords:

Cone, Directrix, Embankment Surface, Gaussian Curvature, Mean curvature

Abstract

In the present paper, we obtain the parametric representation of an embankment surface and give an example for it. We define the notions of embankmentlike surfaces and tubembankmentlike surfaces. Furthermore, we create some embankmentlike and tubembankmentlike surface examples with the aid of different directrix and draw these directrix and surfaces. Also, we find the Gaussian, mean and second Gaussian curvatures of these surfaces and draw the Gaussian, mean and second Gaussian curvature functions' graphics and the variations of Gaussian, mean and second Gaussian curvatures on related surfaces with the aid of Mathematica.

Downloads

Download data is not yet available.

References

S. Aslan and Y. Yaylı, Canal surfaces with quaternions, Advances in Applied Clifford Algebras 26 (2016), 31 – 38, DOI: 10.1007/s00006-015-0602-5.

M. Babaarslan and Y. Yaylı, A new approach to constant slope surfaces with quaternions, International Scholarly Research Network ISRN Geometry 2012 (2012), Article ID 126358, 8 pages, DOI: 10.5402/2012/126358.

C. Baikoussis and D. E. Blair, On the Gauss map of ruled surfaces, Glasgow Mathematical Journal 34 (1992), 355 – 359, DOI: 10.1017/S0017089500008946.

F. Dillen and W. Kühnel, Ruled Weingarten surfaces in Minkowski 3-space, Manuscripta Mathematica 98 (1999), 307 – 320, DOI: 10.1007/s002290050142.

M. P. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall, Englewood Cliffs, NJ (1976).

O. J. Garay, On a certain class of finite type surfaces of revolution, Kodai Mathematical Journal 11(1) (1988), 25 – 31, DOI: 10.2996/kmj/1138038815.

R. Garcia, J. Llibre and J. Sotomayor, Lines of principal curvature on canal surfaces in R3, Annals of the Brazilian Academy of Sciences 78(3) (2006), 405 – 415, DOI: 10.1590/S0001-37652006000300002.

Geotechnical Engineering Manual: Guidelines for Embankment Construction, GEM-12, State of New York, Department of Transportation, Geotechnical Engineering Bureau (2015).

S. Gorjanc, Some examples of using Mathematica and webMathematica in teaching geometry, Journal for Geometry and Graphics 8(2) (2004), 243 – 253.

W. R. Hamilton, On quaternions, or on a new system of imaginaries in algebra, The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science 25(3) (1844), 489 – 495.

E. Hartman, Geometry and Algorithms for Computer Aided Design, Dept. of Math., Darmstadt Univ. of Technology (2003).

A. Kazan and H. B. Karada˘g, A classification of surfaces of revolution in Lorentz-Minkowski space, International Journal of Contemporary Mathematical Sciences 39(6) (2011), 1915 – 1928.

Y.H. Kim, H. Liu and J. Qian, Some characterizations of canal surfaces, Bulletin of the Korean Mathematical Society 53(2) (2016), 461 – 477, DOI: 10.4134/BKMS.2016.53.2.461.

S. N. Krivoshapko and C. A. Bock Hyeng, Classification of cyclic surfaces and geometrical research of canal surfaces, International Journal of Research and Reviews in Applied Sciences 12(3) (2012), 360 – 374.

T. Maekawa, M. N. Patrikalakis, T. Sakkalis and G. Yu, Analysis and applications of pipe surfaces, Computer Aided Geometric Design 15(5) (1998), 437 – 458, DOI: 10.1016/S0167-8396(97)00042-3.

M. Peternell and H. Pottmann, Computing rational parametrizations of canal surfaces, Journal of Symbolic Computation 23(2-3) (1997), 255 – 266, DOI: 10.1006/jsco.1996.0087.

J. S. Ro and D. W. Yoon, Tubes of Weingarten types in a Euclidean 3-space, Journal of the Chungcheong Mathematical Society 22(3) (2009), 360 – 366.

A. Uçum and K. Ë™Ilarslan, New types of canal surfaces in Minkowski 3-space, Advances in Applied Clifford Algebras 26 (2016), 449 – 468, DOI: 10.1007/s00006-015-0556-7.

D. W. Yoon, On the second Gaussian curvature of ruled surfaces in Euclidean 3-space, Tamkang Journal of Mathematics 37(3) (2006), 221 – 226, DOI: 10.5556/j.tkjm.37.2006.167.

D. W. Yoon, Ruled surfaces whose mean curvature vector is an eigenvector of the Laplacian of the second fundamental forms, International Mathematical Forum 36(1) (2006), 1783 – 1788.

Downloads

Published

30-09-2019
CITATION

How to Cite

Kazan, A., & Karadağ, H. B. (2019). Embankment Surfaces in Euclidean 3-Space and Their Visualizations. Communications in Mathematics and Applications, 10(3), 617–636. https://doi.org/10.26713/cma.v10i3.916

Issue

Section

Research Article