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}.


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

x = {3 \over 4}(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.\,


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.

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