In this chapter we will gain some understanding of the nature of a general planar motion µ via two very important curves associated to µ. The first real illumination arises by asking, at any given instant t, for those tracing points w with the property that t is irregular for the trajectory under w. In general, the answer is that there is a unique tracing point w with this property, the ‘instantaneous centre’ of rotation at that instant. That leads naturally to the (fixed and moving) centrodes associated to general motions, and to the classical result of Chasles, that such motions arise as the roulettes associated to these two curves. That provides the content of Section 14.3. In this way the concept of a roulette finally sheds its mantle as an amusing construct for special curves, and assumes its central role as a significant geometric idea in planar kinematics. This basic result provides more than just an insight into the nature of planar motions: it allows one to deduce useful properties of the motion from the geometry of the centrodes.

Generic Parameters

Recall that for a planar motion µ the trajectory generated by the tracing point w can be written in the complex form µ(t)(w) = ρ(t)w + τ(t) where ρ(t), τ(t) are complex numbers and ρ(t) has unit modulus. Given a parameter t it is natural to ask for which tracing points w the parameter t is irregular for the trajectory.