ABSTRACT
In this chapter we give a further application of contact order. We deter mine the tangent lines, if any, at singular points of algebraic curves. These tangent lines are tangents to 'branches' of the curve at the singular point. We show that in certain cases these branches can be given smooth local parametrisations. We give conditions in terms of the defining polynomial of the algebraic curve at the singular point that the singular point is a node, acnode, or ordinary cusp. We also give further examples of curves having other types of behaviour at singular points. Our results enable us to classify cubic curves in terms of their singular point behaviour. We show that non-degenerate cubic curves can have at most one singular point and that such a point must be a node, an acnode, or an ordinary cusp. We also show that a non-degenerate cubic curve having a singular point has a rational parametrisation. Finally, in this chapter, we give a formula for de termining the curvature of certain branches at singular points. Some of the proofs given in this chapter are technical. It is suggested that the reader, for whom this a first university course in geometry, simply concentrates on solving specific examples, and understands the examples illustrating the behaviour at more general singular points.
