The central object in the previous chapter is the evolute of a parametrized curve, the locus of centres of the circles of curvature. Recall that the circle of curvature has at least three point contact with the curve. In this chapter we will pursue these ideas to study the exceptional points on a curve where the circle of curvature actually has at least four point contact. Our first result is that such exceptional points correspond to stationary values of the curvature, the ‘vertices’ of the curve, enabling us to determine them in explicit examples. One of their virtues is that they tend to appear as highly visible points on a tracing of the evolute, whereas they may be effectively invisible on a tracing of the original curve. That emphasizes the point that the evolute picks up very subtle geometric information about a curve: indeed two visually similar curves may have quite dissimilar evolutes. It is for that reason that evolutes provide sensitive methods for distinguishing one curve from another, a matter of practical importance in some physical disciplines.

The Concept of a Vertex

Before proceeding to formalities it might be profitable to look at an explicit example in some detail.

Example 9.1 Consider the parabola z with components x(t) = at2, y(t) = 2at where a > 0. In Example 8.2 we showed that the circle of curvature at the parameter t = 0 has exactly four point contact with the parabola.