On natural darboux mate of a curve in 3-dimensional euclidean space

In this thesis, a k-type natural Darboux mate definition for k = 1,2,3 is given using the Darboux frame of a curve lying on a surface in 3-dimensional Euclidean space E^3. The expression of the curvature and torsion of the k-type natural Darboux mate curve in terms of geodesic curvature kg, normal curvature k_n, and geodesic torsion τg of the curve lying on the surface is obtained. Next, the necessary and sufficient conditions are obtained for the k-type natural Darboux mate curve to be a helix and a slant helix. Also, the expression of the axes of these helices is defined.