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:

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

Image:Newton_trident.svg

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 = {{sfrac|x|z}} and y = {{sfrac|1|z}} into the equation of the trident curve, we get

:$ax^3+bx^2z+cxz^2+xz\; =\; dz^3,$

Image:Newton trident y=inf.svg

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

Solving for y, we get

$y=\backslash frac\{d\}\{x\}-ax^2-bx-c$

Solving for x, we get

$x=\backslash frac\{d-ax^3-bx^2-cx\}\{y\}$