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SEMICUBICAL PARABOLA
Curve studied by Neile in 1657, Leibniz in 1697 and Newton in 1701.
Other name: Neile parabola. 

Cartesian equation: .
Cartesian parametrization:. Polar equation: (compare to the right cissoid) Curvilinear abscissa: (first algebraic curve to be parametrized by the curvilinear abscissa). Polynomial cubic with a cuspidal point. 
The semicubical parabola is a divergent parabola in the case where the polynomial P has a triple root.
It is the evolute of the parabola, and the pedal of the cissoid of Diocles.
Contrary to the parabola, the semicubical parabola can be parametrized by the curvilinear abscissa using rational functions.
The semicubical parabola possesses the important property of being an isochronous curve: see isochronous curve of Leibniz.
The surface generated by revolution of the curve around its axis is the neiloid.
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© Robert FERRÉOL, Jacques MANDONNET 2017