Dual curves associated with the Bonnet ruled surfaces

Abstract

An interest problem arises to determine the surfaces in the Euclidean three space, which admit at least one nontrivial isometry that preserves the principal curvatures. This leads to a class of surface known as a Bonnet surface. The intention of this study is to examine a Bonnet ruled surface in dual space and to calculate the dual geodesic trihedron of the dual curve associated with the Bonnet ruled surface and derivative equations of this trihedron by the dual geodesic curvature. Also, we find that the dual curvature, the dual torsion for the dual curves associated with the Bonnet ruled surface which are different from any dual curves. Moreover, some examples are obtained about the Bonnet ruled surface.

One-Parameter Hyperbolic Spatial Locomotions and Invariants of the Axode

“…The axodes can be employed to characterize the trajectory, velocity, and acceleration of an item as it is mobile through space, providing perceptions into its physical behavior. Furthermore, they can be utilized to form mathematical samples of mobile frameworks, which can be utilized to resolve and optimize complex engineering frameworks [4][5][6][7][8][9]. Moreover, the axodes can be utilized to characterize the geometry of ruled surfaces, such as their curvature functions.…”

“…Moreover, the axodes can be utilized to characterize the geometry of ruled surfaces, such as their curvature functions. This association among the axodes and ruled surfaces has significant implementations in subjects such as computer graphics, architecture mechanism designs, and robot kinematics [7][8][9][10].…”

Kinematic Differential Geometry of a Line Trajectory in Spatial Movement

“…It authorizes a perfect expansion of mathematical statement into spherical point geometry into spatial line geometry through dual numbers expansion, that is, replacing all ordinary quantities by the congruent dual numbers quantities. There exists a significant body of work on E. Study maps, including various monographs [10][11][12][13][14][15][16][17][18].…”

Time-like Ruled Surface in One-Parameter Hyperbolic Dual Spherical Motions

“…The E. Study map states that "The set of all oriented lines in Euclidean 3-space E 3 is directly linked to the set of points on the dual unit sphere in the dual 3-space D 3 ." Thus, the differential geometry of ruled surfaces based on the E. Study map has rederived the curvature theory of a line trajectory and exposed the fundamental curvature functions that describe the shape of ruled surface (see, for example, [4][5][6][7][8][9][10][11]).…”

“…Similarly, the space-like (resp., time-like) curve on S 2 1 represents a time-like (resp., space-like) ruled surface at E 3 1 . In view of its relationships with engineering and physical sciences in Minkowski space, many geometers and engineers have studied and gained many ownerships of the ruled surfaces (see [8][9][10][11][12][13]).…”