ABSTRACT
In Chapters 1 and 3, we gave parametrisations of some standard curves. We consider in this chapter the formal definitions of parametrised curve and differentiability of a curve. We also give some examples of parametri sations having properties which the reader may not expect or may even find bizarre. Examples of non-differentiable curves include space-filling curves which can fill up a whole square in the plane. In the rest of the book we gen erally consider only curves which are differentiable and whose derivatives have certain 'nice' properties. We show how the tangent and normal at a point of a differentiable parametric curve may be defined by differentiating with respect to the parameter. Next we show how algebraic curves can be parametrised locally. The theoretical existence of such a local parametrisation enables us to determine the tangent and normals of an algebraic curve simply using the partial derivatives of the defining polynomial of the curve. We do not need to determine a local parametrisation in order to do this. Next we show how the length of a parametric curve can be determined. In §4.7 we include a number of results in calculus and analysis which we shall refer to in this chapter and elsewhere.
