Discrete and Smooth Bicycle “Unicycle” Paths

This paper explores the properties of bicycle paths in which the front wheel path and the back wheel path coincide. First, we extend previous work by establishing an increase in the number of points of inflection when the path is an infinitely smooth curve. Second, we consider a discrete model when the path consists of line segments and circular arcs; in this context, we prove conjectures on the complexity of these ``unicycle" paths including exponential growth of both the path length and total absolute curvature.