Semicubical parabola


In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve defined
by an equation of the form
Solving for leads to the explicit form
which is the cause for the term semicubical parabola.

Solving ' for yields the second explicit form
Equation ' shows, that
is a parametric representation of the curve.
The arc length of the curve was calculated by the English mathematician William Neile and published in 1657..

Properties of semicubical parabolas

Similarity

Proof: The similarity maps the semicubical parabola onto the curve with.

Singularity

The proof follows from the tangent vector. Only for this vector has zero length.

Tangents

Differentiating the semicubical unit parabola one gets at point of the upper branch the equation of the tangent:
This tangent intersects the lower branch at exactly one further point with coordinates

Arclength

Determining the arclength of a curve one has to solve the integral. For the semicubical parabola one gets
Example: For and , which means the length of the arc between the origin and point, one gets the arc length

Evolute of the unit parabola

In order to get the representation of the semicubical parabola in polar coordinates, one determines the intersection point of the line with the curve. For there is one point different from the origin:. This point has distance from the origin. With and one gets

Relation between a semicubical parabola and a cubic function

Mapping the semicubical parabola by the projective map yields, hence the cubic function. The cusp 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 the replacement and the multiplication by is performed. One gets the equation of the curve
Choosing line as line at infinity and introducing yields the curve

Isochrone 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.

History

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, the semicubical parabola was the first algebraic curve to be rectified.