On Chebsyshev Solution of Curves by Using Gaussian Curvature

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.