Some integral curves with a new frame

Abstract

In this paper, some new integral curves are defined in three-dimensional Euclidean space by using a new frame of a polynomial spatial curve. The Frenet vectors, curvature and torsion of these curves are obtained by means of new frame and curvatures. We give the characterizations and properties of these integral curves under which conditions they are general helix. Also, the relationships between these curves in terms of being some kinds of associated curves are introduced. Finally, an example is illustrated.

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.…”