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MIXED CUBIC
Curve studied by de Longchamps in 1890, who named it.

 
Cartesian equation: 
or   (b > 0).
Cartesian parametrization: .
Rational cubic with isolated point (O).
Polar equation: 
(written  when a = b).

Given a parabola (P) (here with equation  ) and a line (D) perpendicular to the axis of (P) (here, with equation x = a), the associated mixed cubic is the locus of the point M on a variable line (D) passing by O, cutting (P) in P and cutting (D) in Q such that . In other words, the mixed cubic is the cissoid of the parabola (P) and the straight line (D); it is a special case of Zahradnik cissoid.


The mixed cubic is the cissoid of the line and the parabola traced in full lines (and thus the median of the line and the parabola traced in dotted lines).

The name mixed cubic comes from the fact that this curve has a linear asymptote () as well as a parabolic asymptote ().
 
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© Robert FERRÉOL, Jacques MANDONNET 2017