semicubical parabola

The evolute of a parabola is a semicubical parabola.

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

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

As an example of his method, he determined the arc length of a semicubical parabola, which required finding the area under a parabola.

The surface contains a semicubical parabola ("Neile's parabola") and can be derived from solving the corresponding Björling problem.

His major mathematical work, the rectification of the semicubical parabola, was carried out when he was aged nineteen, and was published by John Wallis.

Applying the method to a semicubical parabola produces the Henneberg surface, and to a circle (with a suitably twisted normal field) a minimal Möbius strip.

In 1659, Wallis credited William Neile's discovery of the first rectification of a nontrivial algebraic curve, the semicubical parabola.

Curves with closed-form solution for arc length include the catenary, circle, cycloid, logarithmic spiral, parabola, semicubical parabola and (mathematically, a curve) straight line.

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.

In the same year he gave his exact rectification of the semicubical parabola and communicated his discovery to William Brouncker, Christopher Wren and others connected with Gresham College.