Semicubical parabola
Encyclopedia
In mathematics
, a semicubical parabola is a curve
defined parametric
ally as
The parameter can be removed to yield the equation
of the parabola
:
Expanding the Tschirnhausen cubic catacaustic shows that it is also a semicubical parabola:
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a semicubical parabola is a curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
defined parametric
Parametric
Parametric may refer to:*Parametric equation*Parametric statistics*Parametric derivative*Parametric plot*Parametric model*Parametric oscillator *Parametric contract*Parametric insurance*Parametric feature based modeler...
ally as
The parameter can be removed to yield the equation
Properties
A special case of the semicubical parabola is the evoluteEvolute
In the differential geometry of curves, the evolute of a curve is the locus of all its centers of curvature. Equivalently, it is the envelope of the normals to a curve....
of the parabola
Parabola
In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...
:
Expanding the Tschirnhausen cubic catacaustic shows that it is also a semicubical parabola:
History
The semicubical parabola was discovered in 1657 by William NeileWilliam Neile
William Neile was an English mathematician and founder member of the Royal Society. His major mathematical work, the rectification of the semicubical parabola, was carried out when he was aged nineteen, and was published by John Wallis...
who computed its arc length
Arc length
Determining the length of an irregular arc segment is also called rectification of a curve. Historically, many methods were used for specific curves...
; 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.