Trident curve explained

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 =  and y =  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.

References

• Book: Lawrence, J. Dennis . A Catalog of Special Plane Curves . Dover Publications . 1972 . 0-486-60288-5 . 110 . registration .