Siacci’s resolution of the acceleration vector for a space curve (original) (raw)

A Note on the Acceleration and Jerk in Motion Along a Space Curve

Analele Universitatii "Ovidius" Constanta - Seria Matematica

The resolution of the acceleration vector of a particle moving along a space curve is well known thanks to Siacci [1]. This resolution comprises two special oblique components which lie in the osculating plane of the curve. The jerk is the time derivative of acceleration vector. For the jerk vector of the aforementioned particle, a similar resolution is presented as a new contribution to field [2]. It comprises three special oblique components which lie in the osculating and rectifying planes. In this paper, we have studied the Siacci’s resolution of the acceleration vector and aforementioned resolution of the jerk vector for the space curves which are equipped with the modified orthogonal frame. Moreover, we have given some illustrative examples to show how the our theorems work.

On the instantaneous acceleration of points in a rigid body

Mechanism and Machine Theory, 2011

This work reexamines some classical results in the kinematics of points in space using modern vector-matrix methods. In particular, some very simple Lie theory allows the velocities and accelerations of points to be found in terms of the instantaneous twist of the motion and its derivative. From these results many of the classical results follow rather simply. Although most of the results are well known, some new material is presented. In particular, the discriminant curve that separates cases with one or three real acceleration axes is found and plotted. Another new result concerns the chords to the cubic of inflection points. It is shown that for points on such a chord the osculating planes of the point's trajectories are parallel. Also a new result is found which distinguishes between cases where the Bresse hyperboloid of points whose velocities and accelerations are perpendicular, has one or two sheets.

Kinematics of point motion along curves of the second order

2024

In the scientific literature, the differential equation of an ellipse is derived through dynamic quantities and laws. Kepler's laws are then derived. However, Kepler's laws are kinematic. This article examines the kinematic equation of an ellipse. The equation is derived through the oscillations of a parametric pendulum. It is shown that Kepler's laws are properties of kinematic equations of motion of a point along second-order curves. The motion of a material point along second-order curves is represented by kinematic equation (1.10). The kinematics of second-order curves is studied on an ellipse. Formulas for the dependence of acceleration and radius, speed and radius are derived. The direction of the velocity and acceleration vectors is determined. The conditions for the conservation of Kepler's laws when a material point moves along an ellipse are shown. The article "Kinematics of point motion along curves of the second order" https://www.academia.edu/116906587/ describes the kinematics of uniform, uniformly accelerated, elliptical (Keplerian) motion along an ellipse.

Differential Geometry: An Introduction to the Theory of Curves

Differential geometry is a discipline of mathematics that uses the techniques of calculus and linear algebra to study problems in geometry. The theory of plane, curves and surfaces in the Euclidean space formed the basis for development of differential geometry during the 18th and the 19th century. The core idea of both differential geometry and modern geometrical dynamics lies under the concept of manifold. A manifold is an abstract mathematical space, which locally resembles the spaces described by Euclidean geometry, but which globally may have a more complicated structure. The purpose of this paper is to give an elaborate introduction to the theory of curves, and those are, in general, curved. Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and in the Euclidean space by applying the concept of differential and integral calculus. The curves are represented in parametrized form and then their geometric properties and various quantities associated with them, such as curvature and arc length expressed via derivatives and integrals using the idea of vector calculus.

Motion along a Space Curve with a Quasi-Frame in Euclidean 3-Space: Acceleration and Jerk

Symmetry

The resolution of the acceleration and jerk vectors of a particle moving on a space curve in the Euclidean 3-space is considered. By applying this resolution and Siacci’s theorem, alternative resolutions of acceleration and jerk vectors are derived based on the quasi-frame. In the osculating plane, the acceleration vector is resolved as the sum of its tangential and radial components. In addition, in the osculating and rectifying planes, the jerk vector is resolved along the tangential direction and two special radial directions. The maximum permissible speed on a space curve at all trajectory points is established via the jerk vector formula. Finally, some examples are presented to illustrate how the results work.

Motion of Parallel Curves and Surfaces in Euclidean 3-SPACE

2020

The main goal of this paper is to investigate motion of parallel curves and surfaces in Euclidean 3-space R3. The characteristic properties for such objects are given. The geometric quantities are described. Finally, the evolution equations of the curvatures and the intrinsic geometric formulas are derived. keywords: Curvature, Evolution, Motion, Parallel curves, Parallel surfaces.