ABSTRACT
In this chapter we discuss the 'closeness' of two curves passing through a given point and having a common tangent there. This closeness is measured by contact order. Contact order has several applications. Taking one of the curves to be a line, we show how inflexions and undulations of the second curve can be determined using the method of contact order. In Chapter 6, again using the method of contact order, we show how tangent lines at cusps and other non-regular points of parametric curves can be determined, and how cusps can be classified by their order. Inflexions, undulations, and cusps are 'singularies' of parametrised curves. Contact order can be used to specify the osculating circle or circle of curvature at a point of a parametric curve (see Chapter 9). Also we use contact order in Chapter 15 to give a partial classification of singular points of algebraic curves and to determine the tangent lines at such singular points where appropriate.
