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:

$xy+ax^{3}+bx^{2}+cx=d$.

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

$ax^{3}+bx^{2}z+cxz^{2}+xz=dz^{3},$

Trident curve at $y=\infty$ with $a=b=c=d=1$. This curve partially resembles the folium of Descartes.

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

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

Solving for $x$, we get

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

References

• Lawrence, J. Dennis (1972). A Catalog of Special Plane Curves. New York: Dover. p. 110. ISBN 0-486-60288-5.

External links

• O'Connor, John J.; Robertson, Edmund F., "Trident of Newton", MacTutor History of Mathematics Archive, University of St Andrews