Trident curve

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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:

<math>xy+ax^3+bx^2+cx=d,</math>




Trident curves are cubic plane curves with an ordinary double point in the real projective plane at x&nbsp;=&nbsp;0, y&nbsp;=&nbsp;1, z&nbsp;=&nbsp;0; if we substitute x&nbsp;=&nbsp;x/z and y&nbsp;=&nbsp;1/z into the equation of the trident curve, we get

<math>ax^3+bx^2z+cxz^2+xz = dz^3,,</math>




which has an ordinary double point at the origin. Trident curves are therefore rational plane algebraic curves of genus zero.

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