Family of Surfaces with a Common Special Involute and Evolute Curves

Abstract

In this paper, we define the necessary and sufficient conditions for a parametric surface on which both the involute and evolute of any given curve lie to be geodesic, asymptotic and curvature line. Then, the first and second fundamental forms of these surfaces are calculated. By using the Gaussian and mean curvatures, the developability and minimality assumptions are drawn, as well. Moreover we extended the idea to the ruled surfaces. Finally, we provide a set of examples to illustrate the corresponding surfaces.

On Ruled Surfaces Constructed by the Evolution of a Polynomial Space Curve

“…D and 1 D be tangent, principal normal-like and binormal-like vectors at point () s  of a polynomial space curve  , respectively, then the Frenet like curve frame is given by matrix form 12 , dd and 3 d are the curvatures of the polynomial curve  with the arc-length s (see for more details [7][8][9]), respectively. Let s X and u X be tangent vectors of a surface   , X s u , then the normal vector field of the surface   , X s u can be defined by…”

“…To solve this problem, Dede defined the Flc frame for moving polynomial curves [7,8]. The Flc frame [9][10][11][12][13] and ruled surfaces on different frames [14][15][16][17][18][19][20][21][22][23] have been investigated by many researchers. Inspired by these studies, we conducted this research to create a new resource on the subject of surfaces and to form a basis for future studies.…”

The Transformation of the Evolute Curves Using Lifts on R^3 to Tangent Space Tr^3

“…manifold) to extended spaces (resp. extended manifolds) using the lift function [1][2][3][4][5][6]. Thus, it may be extended the following theorem given on space to its tangent space (see extended frames and curves [7][8][9][10][11]).…”