Trident curve

Trident curve

In mathematics, Trident curve (also trident of Newton or parabola of Descartes) is the name for a 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=∞, 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 one.

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