The singular point of an algebraic cubic (Q5957861)

The problem of determining the singularity of an algebraic curve with rational coefficients is treated in the special case of cubics. The cubic is written in terms of barycentric coordinates with respect to a triangle. Formulas are given for the coordinates of its singular point in terms of the coefficients of the curve implicit form in barycentric coordinates (rational functions of degree at most five in the coefficients). The implicit expression of the cubic is constructed in terms of its weights and Bézier points, leading to an implicit formula for the singular points in terms of Bézier points and weights.