On Chebsyshev Solution of Curves by Using Gaussian Curvature

Year 2017, Volume: 13 Issue: 3, 745 - 746, 30.09.2017

Sibel Paşalı Atmaca

https://doi.org/10.18466/cbayarfbe.339350

Abstract

Gaussian curvature is commonly seen in the study of   differential geometry. Gaussian curvature of
a surface at a point is the product of the principal curvatures. They measure
how the surface bends by different amounts 
in different directions at the point. Also, Gaussian curvature is given
as the determinant of shape operator. In pure mathematics, differential
equations are studied from different viewpoints. There are a lot of methods for
solving differential equations in mathematics.  From the differential equations viewpoint,
Gaussian curvature solves the differential equation to find the main curve. One
of them is Chebsyshev expansion method by using Chebsyshev polynomials. Also,
they are important study in approximation theory.  Chebyshev polynomials are a sequence of
orthogonal polynomials and compose a polynomial sequence.The series solution is
also used in surface of revolution.  A
surface of revolution is a surface generated by 
rotating a two-dimensional curve. In this study, our aim is to find the
main curve by using Gaussian curvature. We substitute solution into the
differential equation to find a relation for coeeficients of system. So, we use
Chebsyshev polynomials for solutions to determine the curve and demonstrate our
results on some well-known surfaces such as sphere, catenoid and torus.

Keywords

Chebsyshev Polynomials, Gaussian Curvature, Differential Equations, Shape Operator, Surface of Revolution

References

• 1. Adomian, G., Convergent series solution of nonlinear equa-tions, Journal of Computational and Applied Mathematics, 1984, 11(1), 225-230.
• 2. Carmo, M. P. DiferansiyelGeometri: EğrilerveYüzeyler, Anka-ra, 2012.
• 3. Meek, D. S., Walton, D. J., On surface normal and Gaussian curvature approximations given data sampled from a smooth sur-face, Computer Aided Geometric Design, 2000, 17(6), 521-543.
• 4. Han, Z., Prescribing Gaussian curvature on S2, Duke Mathe-matical Journal, 1990, 61(3), 679.
• 5. Stewart, J., Calculus, Fourth edition, United States, 1999.
• 6. Hacısalihoglu , H. H., Diferansiyel Geometri, Ankara,1998.
• 7. Schoen, R., Zhang, D., Prescribed scalar curvature on the n-sphere, Calculus of Variations and Partial Differential Equations, 1996, 4(1), 1-25.
• 8. Hua Hau, Z., Hypersurfacesin a sphere with constant mean curvature, Proceedings of the American Mathematical Society, 1997, 125, 1193-1196.
• 9. Carmo, M., Dajczer, M., Rotation hypersurfaces in spaces of constant curvature, Transactions of the American Mathematical Society, 1983, 277, 685-709.