Tubular surface and characterisations according to Sabban frame in Euclidean 3- space
Abstract
Sabban frame is closely related to the study of the curvature of a surface. It provides a way to describe the normal and tangent directions of the surface, which are essential when studying the second fundamental form, Gaussian curvature, and mean curvature of a surface. The frame helps in expressing the curvature of the surface in terms of a local coordinate system and enables a more efficient calculation of these geometrical quantities. A tubular surface is a surface that is a neighbour of a curve and can be thought of as a ‘tube’ around the curve. Tubular surfaces are widely used to study surfaces deformed or displaced from a given curve in space. Since the Sabban framework has important applications here as well, in this work obtained the tubular surface and its characterisations defined according to the Sabban frame of the curve given on the unit sphere. Singularity, Gaussian curvature, mean curvature and basic forms of the tubular surface given according to this frame were calculated. In addition, necessary and sufficient conditions were given for the parameter curves of the tubular surface to be geodesic, asymptotic and line of curvature. Finally, the study was supported with various examples.
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