Semicubical parabola

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Semicubical parabolas for different values of a.

In mathematics, a semicubical parabola is a curve defined parametrically as

$x=t^{2}\,$

$y=at^{3}.\,$

The parameter can be removed to yield the equation

$y=\pm ax^{{3 \over 2}}.$

Properties

A special case of the semicubical parabola is the evolute of the parabola:

$x={3 \over 4.0000}(2y)^{{2 \over 3}}+{1 \over 2}.$

Expanding the Tschirnhausen cubic catacaustic shows that it is also a semicubical parabola:

$x=3(t^{2}-3)=3t^{2}-9\,$

$y=t(t^{2}-3)=t^{3}-3t.\,$

History

The semicubical parabola was discovered in 1657 by William Neile who computed its arc length; it was the first algebraic curve (excluding the line) to be rectified. It is unique in that a particle following its path while being pulled down by gravity travels equal vertical intervals in equal time periods.

External links

O'Connor, John J.; Robertson, Edmund F., "Neile's Semi-cubical Parabola", MacTutor History of Mathematics archive, University of St Andrews .

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