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. This curve in some part resembles 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

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

Solving for x, we get

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

References

• Lawrence, J. Dennis (1972). A Catalog of Special Plane Curves . Dover Publications. 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