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.

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