Semicubical parabola

In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve that has an implicit equation of the form

${\displaystyle y^{2}-a^{2}x^{3}=0}$

(with a ≠ 0) in some Cartesian coordinate system.

Solving for y leads to the explicit form

${\displaystyle y=\pm ax^{\frac {3}{2}},}$

which imply that every real point satisfies x ≥ 0. The exponent explains the term semicubical parabola. (A parabola can be described by the equation y = ax².)

Solving the implicit equation for x yields a second explicit form

${\displaystyle x=\left({\frac {y}{a}}\right)^{\frac {2}{3}}.}$

The parametric equation

${\displaystyle \quad x=t^{2},\quad y=at^{3}}$

can also be deduced from the implicit equation by putting ${\textstyle t={\frac {y}{ax}}.}$ ^[1]

The semicubical parabolas have a cuspidal singularity; hence the name of cuspidal cubic.

The arc length of the curve was calculated by the English mathematician William Neile and published in 1657 (see section History).^[2]