In this chapter we will study how exceptional geometric features (irregular points, inflexions and vertices) appear on the trajectories of a general planar motion. To put matters into perspective it helps to make some preliminary comments on the exceptional features which can appear on a general curve z. The first thing to notice is that irregular points are basically different from inflexions and vertices in that they represent the common zeros of two smooth functions in one variable (the derivatives of the components x, y) whilst inflexions and vertices appear as the zeros of a single smooth function in one variable (the curvature k in the case of inflexions, and its derivative k′ in the case of vertices). We expect a general smooth function in one variable to have a discrete set of zeros, hence that inflexions or vertices will arise from a discrete set of parameters. On the other hand we do not expect two general smooth functions in one variable to have a common zero, so do not expect to have irregular parameters on a general curve z. Thus irregular parameters should be viewed as unstable features of a general curve z (expected to disappear under small changes in z) whereas inflexions and vertices should be viewed as stable features (expected to persist under small changes).

The picture alters fundamentally when we move from the study of a single curve z to the two parameter family of trajectories resulting from a motion µ, one for each tracing point w.