Trident curve

In mathematics, a trident curve (also trident of Newton or parabola of Descartes) is any member of the family of curves that have the formula:


 * $$xy+ax^3+bx^2+cx=d$$



Trident curves are cubic plane curves with an ordinary double point in the real projective plane at x = 0, y = 1, z = 0; if we substitute x = $x⁄z$ and y = $1⁄z$ into the equation of the trident curve, we get


 * $$ax^3+bx^2z+cxz^2+xz = dz^3,$$



which has an ordinary double point at the origin. Trident curves are therefore rational plane algebraic curves of genus zero.