Representations of Epitrochoids and Hypotrochoids

Abstract

Representations of a given curve may consist of implicit or parametric equations, along with any envelopes that produce that curve. We will describe the different methods of passing from one of these representations to another, then apply these methods with regards to epitrochoids and hypotrochoids. These are the families of curves that are produced by tracing the path of a point affixed to a circle as it rolls around the inside or outside of a stationary circle. Epicycloids and hypocycloids are produced when the point affixed to the moving circle is on the circumference. We will provide several conjectures and results on the representations of epitrochoids and hypotrochoids, with emphasis on epicycloids and hypocycloids, including their implicit representations and their construction as envelopes.