# 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:

${\displaystyle xy+ax^{3}+bx^{2}+cx=d\,}$
trident curve with a = b = c = d = 1

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

${\displaystyle ax^{3}+bx^{2}z+cxz^{2}+xz=dz^{3},\,}$
trident curve at y = ∞ with a = b = c = d = 1

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

## References

• J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. p. 110. ISBN 0-486-60288-5.