A New Moving Frame for Trajectories with Non-Vanishing Angular Momentum

Abstract

The theory of curves has a very long history. Moving frames defined on curves are important parts of this theory. They have never lost their importance. A point particle of constant mass moving along a trajectory in space may be seen as a point of the trajectory. Therefore, there is a very close relationship between the differential geometry of the trajectory and the kinematics of the particle moving on it. One of the most important elements of the particle kinematics is the jerk vector of the moving particle. Recently, a new resolution of the jerk vector, along the tangential direction and two special radial directions, has been presented byÖzen et al. (JTAM 57(2)( 2019)). By means of these two special radial directions, we introduce a new moving frame for the trajectory of a moving particle with non vanishing angular momentum in this study. Then, according to this frame, some characterizations for the trajectory to be a rectifying curve, an osculating curve, a normal curve, a planar curve and a general helix are given. Also, slant helical trajectories are defined with respect to this frame. Afterwards, the necessary and sufficient conditions for the trajectory to be a slant helical trajectory (according to this frame) are obtained and some special cases of these trajectories are investigated. Moreover, we provide an illustrative numerical example to explain how this frame is constructed. This frame is a new contribution to the field and it may be useful in some specific applications of differential geometry, kinematics and robotics in the future.

Bertrand partner P-trajectories in the Euclidean 3-space $E^3$

“…form the Positional Adapted Frame {T(s), M(s), Y(s)} (see [15] for more details on PAF). The relation between the Serret-Frenet frame and PAF is as in the following:…”

“…where Ω(s) is the angle between the vector B(s) and the vector Y(s) which is positively oriented from the vector B(s) to vector Y(s). On the other hand, the derivative formulas of PAF are presented as follows [15]:…”

“…Theorem 1. [15] Let α = α(s) be the unit speed parameterization of the trajectory. Then, α is an osculating curve if and only if k 1 = 0.…”

“…Then, α is an osculating curve if and only if k 1 = 0. More details can be found in the studies [11,15,16,19] for Positional Adapted Frame (PAF).…”

“…One of these moving frames called as Positional Adapted Frame (PAF) was introduced by Özen and Tosun in 2021. The authors defined this moving frame for the trajectories having non-vanishing angular momentum in Euclidean 3space [15]. There can be found some other studies [11,16,19] which are performed by considering this frame.…”