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 can be used to define 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:

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

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

History

The semicubical parabola was discovered in 1657 by William Neil. It is unique in that a particle following its path while being pulled down by gravity travels equal vertical intervals in equal time periods. Aside from the linear equation, it was the first curve to have its arc length computed.