  We present an approach to finding the implicit equation of a planar rational
parametric cubic curve, by defining a new basis for the representation. The
basis, which contains only four cubic bivariate polynomials, is defined in
terms of the B\'ezier control points of the curve. An explicit formula for the
coefficients of the implicit curve is given. Moreover, these coefficients lead
to simple expressions which describe aspects of the geometric behaviour of the
curve. In particular, we present an explicit barycentric formula for the
position of the double point, in terms of the B\'ezier control points of the
curve. We also give conditions for when an unwanted singularity occurs in the
region of interest. Special cases in which the method fails, such as when three
of the control points are collinear, or when two points coincide, will be
discussed separately.
