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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

\( ax^3+bx^2z+cxz^2+xz = dz^3,\, \)

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.


J. Dennis Lawrence (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.

Mathematics Encyclopedia

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