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Semicubical parabola
Algebraic plane curve of the form y² – a²x³ = 0 / From Wikipedia, the free encyclopedia
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In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve that has an implicit equation of the form
![Thumb image](http://upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Neil-parab-3.svg/320px-Neil-parab-3.svg.png)
(with a ≠ 0) in some Cartesian coordinate system.
Solving for y leads to the explicit form
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
can also be deduced from the implicit equation by putting [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]