Creating Centered Trochoids and Co-Centered Ellipses through the Combinations of Uniform Rolling and Sliding Operations by Using Virtual Rotating Circles Technique (VRCT)

In this article we present an innovative mental vision for creating uniform rolling and sliding motions for a circle along another circle. Also, we define two different methods for combining rolling and sliding motions through VRCT in order to introduce a simple practical method for creating centered trochoids and co-centered ellipses. In this article traditional mathematical perspective for creating centered trochoids (through the pure rolling process for a circle along another circle) is changed to a novel mathematical perspective which is based on the combination of uniform rolling and sliding motions of a circle along another one. In this novel vision we have not to define a centered trochoid as a swept path by an attached point to a pure rolling circle along another circle. Instead, a centered trochoid can be defined as a traced path by a certain point on the circumference of a rolling and sliding circle along another one. Also, through this new vision an ellipse can be visualized as a closed plane curve that is generated by uniform combination of rolling and sliding motions (due to superposition of two co-polarized rotational motions) with different commensurable angular frequencies! Detailed points in the process of plotting centered trochoids and ellipses through the combination of rolling and sliding operations are observable directly by application an innovative instrument that we have named it mechanical oscilloscope. The function of our device is independent from any other electronic devices such as computer and does not require programming to plot centered trochoids and ellipses.