Cartesian equation: y
= a x
Click below to see one of the Associated curves.
Definitions of the Associated curves Evolute Involute 1 Involute 2 Inverse curve wrt origin Inverse wrt another circle Pedal curve wrt origin Pedal wrt another point Negative pedal curve wrt origin Negative pedal wrt another point Caustic wrt horizontal rays Caustic curve wrt another point
William Neile was born at Bishopsthrope in 1637. He was a pupil of Wallis and showed great promise. Neile's parabola was the first algebraic curve to have its arc length calculated; only the arc lengths of transcendental curves such as the cycloid and the logarithmic spiral had been calculated before this. Unfortunately Neile died at a young age in 1670 before he had achieved many further results.
In 1687 Leibniz asked for the curve along which a particle may descend under gravity so that it moves equal vertical distances in equal times. Huygens showed that the semi-cubical parabola x = ay satisfied this property. Because of this it is an isochronous curve.
The semi-cubical parabola is the evolute of a parabola.
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