Object of study in the present paper are slant and Legendre null curves 3-dimensional Sasaki-like almost contact B-metric manifolds. For examined we express general Frenet frame for which original parameter is distinguished, as well corresponding curvatures, terms structure on manifold. We prove that curvatures a framed curve constants if only specific function considered manifolds constant. fi...

In this research we consider the problem of path following control for a small fixedwing unmanned aerial vehicle (UAV). Assuming the UAV is equipped with an autopilot for low level control, we adopt a kinematic error model with respect to the moving Serret-Frenet frame attached to a path for tracking controller design. A kinematic path following control law that commands heading rate is present...

This paper studies the problem of designing a rational Bézier developable surface pencil with a common isogeodesic, and provides an algorithm for the representation of complicated geometric models in industrial applications which need to satisfy that the shape surface can be developed and a given curve is geodesic. By employing the local Frenet orthonormal frame, the explicit expression of the ...

In this paper smooth trajectories are computed in the Lie group SO(2, 1) as a motion planning problem by assigning a Frenet frame to the rigid body system to optimize the cost function of the elastic energy which is spent to track a timelike curve in Minkowski space. A method is proposed to solve a motion planning problem that minimize the integral of the square norm of Darboux vector of a time...

In this paper, we define the curve at a constant distance from edge of regression on r(s) with arc length parameter s in Galilean 3-space. Here, d is non-isotropic or isotropic vector defined as tightly fastened to Frenet trihedron 3-dimensional space. We build frame constructed respect two types and indicate properties related curvatures . Also, for , give conditions be circular helix. Further...

The geometry of a smooth line is characterized locally by its curvature and torsion, or globally by its writhe. In many situations of physical interest the line is, however, not smooth so that the classical Frenet description of the geometry breaks down everywhere. One example is a thermalized stiff polymer such as DNA, where the shape of the molecule is the integral of a Brownian process. In s...

We consider the motion of a particle described by an action that is a functional of the Frenet-Serret [FS] curvatures associated with the embedding of its worldline in Minkowski space. We develop a theory of deformations tailored to the FS frame. Both the Euler-Lagrange equations and the physical invariants of the motion associated with the Poincaré symmetry of Minkowski space, the mass and the...

In this paper we describe Poisson structures defined on the space of Serret-Frenet equations of curves in a flat homogeneous space G/H where G is semisimple. These structures are defined via Poisson reduction from Poisson brackets on Lg∗, the space of Loops in g∗. We also give conditions on invariant geometric evolution of curves in G/H which guarantee that the evolution induced on the differen...

For a two-dimensional surface M in the four-dimensional Euclidean space E we introduce an invariant linear map of Weingarten type in the tangent space of the surface, which generates two invariants k and κ. The condition k = κ = 0 characterizes the surfaces consisting of flat points. The minimal surfaces are characterized by the equality κ − k = 0. The class of the surfaces with flat normal con...

A new three-dimensional nonlinear path-following guidance law is proposed using differential geometry. The guidance law is designed based on the look-ahead angle and the radially shifted distance which gives an additional degree of freedom to generate acceleration command for precise path-following. A modified Frenet-Serret frame is utilized to describe the desired path, and the closest project...