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:

$xy+a{x}^3+b{x}^2+cx=d$

File:Newton trident.svg

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 = Template:Sfrac and y = Template:Sfrac into the equation of the trident curve, we get

$a{x}^3+b{x}^{2}z+cx{z}^2+xz=d{z}^3,$

File:Newton trident y=inf.svg

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.

Solving for y, we get

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

Solving for x, we get

$x=\frac{d-a{x}^3-b{x}^2-cx}{y}$

External links

Template:Algebraic-geometry-stub