In the previous chapter we were able to gain some insight into the local behaviour of a curve z by studying its contact with lines. In this chapter we extend that philosophy by studying its contact with circles, building on the fruitful concept of ‘multiplicity’ introduced in the previous chapter. Section 8.1 achieves the extension, via explicitly defined contact functions. There are very clear analogies. For instance, just as we have a unique line having at least two point contact with z at a regular parameter t, so we have a unique circle (the ‘circle of curvature’ at t) having at least three point contact with z at t, at least provided t is not inflexional. The locus of centres for the ‘circles of curvature’ gives rise to a new curve known as the ‘evolute’ of z, providing the subject matter for Section 8.2: it will turn out that the evolute construction can be reversed via the notion of an ‘involute’. Evolutes play an important role in understanding the geometry of curves, and we will devote time to alternative descriptions. Thus in Section 8.3 evolutes are described dynamically, as the locus of irregular points for the curves ‘parallel’ to z. Later (Chapter 10) we will meet a third description of the evolute as the ‘envelope’ of the normal lines, providing the key to the crucial role played by evolutes (Chapter 12) in understanding the idea of a ‘caustic’.