Semicubical parabola
In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve that has an implicit equation of the form
in some Cartesian coordinate system.
Solving for leads to the explicit form
which imply that every real point satisfies. The exponent explains the term semicubical parabola.
Solving the implicit equation for yields a second explicit form
The parametric equation
can also be deduced from the implicit equation by putting
The semicubical parabolas have a cuspidal singularity; hence the name of cuspidal cubic.
The arc length of the curve was calculated by the English mathematician William Neile and published in 1657.
Properties of semicubical parabolas
Similarity
Any semicubical parabola is similar to the semicubical unit parabolaProof: The similarity maps the semicubical parabola onto the curve with
Singularity
The parametric representation is regular except at point At point the curve has a 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 getsand which means the length of the arc between the origin and point, one gets the arc length 9.073.
Evolute of the unit parabola
The evolute of the parabola is a semicubical parabola shifted by 1/2 along the x-axis:Polar coordinates
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 getsRelation between a semicubical parabola and a cubic function
Mapping the semicubical parabola by the projective map 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 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.The length of the semicubical parabola was computed by van Heuraet in 1659.