Semicubical parabola
In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve that has an implicit equation of the form (with a ≠ 0) in some Cartesian coordinate system.
Solving for y leads to the explicit form which imply that every real point satisfies x ≥ 0.
The exponent explains the term semicubical parabola.
Solving the implicit equation for x yields a second explicit form The parametric equation can also be deduced from the implicit equation by putting
The arc length of the curve was calculated by the English mathematician William Neile and published in 1657 (see section History).
is similar to the semicubical unit parabola
(uniform scaling) maps the semicubical parabola
the curve has a singularity (cusp).
The proof follows from the tangent vector
Differentiating the semicubical unit parabola
of the upper branch the equation of the tangent: This tangent intersects the lower branch at exactly one further point with coordinates [3] (Proving this statement one should use the fact, that the tangent meets the curve at
Determining the arclength of a curve
one gets (The integral can be solved by the substitution
Example: For a = 1 (semicubical unit parabola) and b = 2, which means the length of the arc between the origin and point (4,8), one gets the arc length 9.073.
is a semicubical parabola shifted by 1/2 along the x-axis:
In order to get the representation of the semicubical parabola
in polar coordinates, one determines the intersection point of the line
( see List of identities) one gets [4] Mapping the semicubical parabola
(involutory perspectivity with axis
The cusp (origin) of the semicubical parabola is exchanged with the point at infinity of the y-axis.
This property can be derived, too, if one represents the semicubical parabola by homogeneous coordinates: In equation (A) the replacement
(the line at infinity has equation
One gets the equation of the curve Choosing line
as line at infinity and introducing
yields the (affine) curve
An additional defining property of the semicubical parabola is that it is an isochrone curve, meaning that a particle following its path while being pulled down by gravity travels equal vertical intervals in equal time periods.
In this way it is related to the tautochrone curve, for which particles at different starting points always take equal time to reach the bottom, and the brachistochrone curve, the curve that minimizes the time it takes for a falling particle to travel from its start to its end.
The semicubical parabola was discovered in 1657 by William Neile who computed its arc length.
Although the lengths of some other non-algebraic curves including the logarithmic spiral and cycloid had already been computed (that is, those curves had been rectified), the semicubical parabola was the first algebraic curve (excluding the line and circle) to be rectified.
[1][disputed (for: It appears that parabola and other conic sections have been rectified a long time before) – discuss]
