An approach for designing a surface pencil through a given geodesic curve

Abstract

In the present paper, we propose a new method to construct a surface interpolating a given curve as the geodesic curve of it. Also, we analyze the conditions when the resulting surface is a ruled surface. In addition, developablity along the common geodesic of the members of surface family are discussed. Finally, we illustrate this method by presenting some examples.

Surface family interpolating a common spherical indicatrix curve

“…The study of curves and surfaces has wide application areas such as architectural design, computer aided design, astronomy, astrophysics and genetics [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. We encounter curves and surfaces in every differential geometry book.…”

Hypersurfaces With a Common Geodesic Curve in 4D Euclidean space E4

“…Nowadays, numerous research results have concentrated on surfaces family having a common geodesic curve in a diversity of applications, such as the tent manufacturing, designing industry of shoes, cutting and painting path. In general, the goal of mainly works on geodesics is to define a family of surfaces with a given geodesic curve and express it as a linear combination of the Serret-Frenet frame (See for example [1,2,4,5,11,12,14,16]).…”