Using duality to implicitize and find cusps and inflection points of Bézier curves Original Research Article

A planar algebraic curve C has an implicit equation and a tangential equation. The tangential equation defines a dual curve to C. Starting with a parametrization of C, we find a parametrization of the dual curve, and the tangential equation and implicit equation of C in a novel way. We also find equations whose roots are the parameter values of the cusps and inflection points of C. Methods include polar reciprocation and the theory of envelopes.