Physics Oriented Mathematical Perspective for Creating Trochoids and Ellipses through the Combination Rolling and Sliding Motions of a Circle Along Another One in Forward and Backward Sliding Modes

TL;DR

This paper introduces a novel mathematical approach to generate trochoids and ellipses by combining rolling and sliding motions of a circle, challenging the traditional pure rolling paradigm.

Contribution

It proposes a new concept for creating trochoids and ellipses through combined rolling and sliding motions, expanding the mathematical and physical understanding of these curves.

Findings

Trochoids can be generated without pure rolling motion.

Two methods for simulating combined rolling and sliding motions are presented.

Ellipses are formed by two co-polarized rotational motions with different frequencies.

Abstract

Usual mathematical method for creating trochoids is based on a solid rule that requires a pure rolling motion of a circle along another one. In this vision a trochoid defined as a traced path by an attached point (a non-conceptive issue) to a pure rolling circle! Except for using the restriction of pure rolling motion for a circle beside using the attached points to it, authors of this article have not found other conceptive solutions for this issue in the references of mathematics and physics. This article provides a novel conceptive solution for creating trochoids and ellipses based on combination of rolling and sliding motions of a circle along another one! Therefore, we have not to define a trochoid as a path that is swept by an attached point to a pure rolling circle along another circle! Instead, a trochoid can be defined as a path is swept by a point on the circumference of a circle that is doing rolling and sliding motions uniformly along another one! Also, this article presents two different methods to implement a mathematical simulation for a moving circle does uniform simultaneous rolling and sliding motions along another one! With the help of this innovative solution, it is possible to define ellipses and trochoids as closed plane curves that can be generated through the combination of rolling and sliding motions [ellipse is created through the combination of two co-polarized rotational motions with different commensurable angular frequencies (in two different modes!)]. This article presents a novel idea titled by Virtual Rotating Circles Technique (VRCT) that can be implemented by Mathematical Simulator Machine.