The new study of some characterization of canal surfaces with Weingarten and linear Weingarten types according to Bishop frame

Abstract

In this paper, we have a tendency to investigate a particular Weingarten and linear Weingarten varieties of canal surfaces according to Bishop frame in Euclidean 3-space E 3 satisfying some fascinating and necessary equations in terms of the Gaussian curvature, the mean curvature, and therefore the second Gaussian curvature. On the premise of those equations, some canal surfaces are introduced.

Timelike Sweeping Surfaces According to Type-2 Bishop Frame in Minkowski 3–space

“…Consequently, the surface spacelike normal U(s 0 , u) is identical with the spacelike principal normal N ζ (u), i.e., the curve ζ(u) is a geodesic planar spacelike line of curvature on X(u, s 0 ). Surfaces whose parametric curves are lines of curvature have various applications in geometric design [6][7][8]. In the case of sweeping surfaces, one has to compute the offset surfaces X f (u, s) = X(u, s) + f U(s, u) of a given surface X(u, s) at a certain distance f .…”

“…In CAGD, conditions that guarantee the convexity or curves which output parabolic points of a surface are required in different applications [6][7][8][9]. However, for the timelike sweeping surface M the convexity can be controlled with the help of the Gaussian curvature as:…”

“…If the spine curve is a circular arc, the resulting sweeping surface is a surface of revolution. Consequently, both cylindrical and rotation surfaces can be considered as special sets of sweeping surface [1][2][3][4][5][6][7][8]. A sweeping surface has the ownership that the cross section is a circle and the normal of the circle plane is mostly parallel to that of the cross section.…”

“…For such a kind of surfaces, some interesting and important relations about the Gaussian curvature, the mean curvature and the second Gaussian curvature are derived. Soliman et al [7] investigated a particular Weingarten and linear Weingarten varieties of canal surfaces according to Bishop frame in Euclidean 3-space satisfying some fascinating and necessary equations in terms of the Gaussian curvature, the mean curvature. A significant fact about sweeping surfaces is that they can be developable ruled surfaces [8] .…”

Sweeping surfaces with Natural mate curve of a spatial curve in Euclidean 3-Space

“…This concept is a generalization ofthe classical notion of a mate of a planar curve [1-8. ] Sweeping surfaces play an essential rolein Computer Aided Geometric Design (CAGD),such as construction of robotic path planning,blending surfaces, transition surfaces betweenpipes, manufacturing of sculptured surfaces [8,9].One of the important fact about sweeping surfaceis that the sweeping surface can be a developablesurface [11][12][13]. Developable surfaces havea very important place in mathematics and engineeringsuch as motion analysis or designing carsand ships.…”