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Wikipedia / Semicubical parabola

In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve that has an implicit equation (with ) in some Cartesian coordinate system.

Solving for leads to the explicit form which implies that every real point satisfies . The exponent explains the term semicubical parabola. (A parabola can be described by the equation .)

Solving the implicit equation for

The parametric equation

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 (see section History).

Properties of semicubical parabolas

Similarity

is similar to the semicubical unit parabola

Proof: with

Singularity

is regular except at point the curve has a singularity (cusp). The proof follows from the tangent vector this vector has zero length.

Tangents

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

twice.)

Arclength

Determining the arclength one gets

(The integral can be solved by the substitution

Example: For (semicubical unit parabola) and which means the length of the arc between the origin and point (4,8), 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

Polar coordinates

( see List of identities) one gets

Relation between a semicubical parabola and a cubic function

by the projective map (involutory perspectivity and center hence the cubic function 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 line at infinity has equation is performed. One gets the equation of the curve

  • in homogeneous coordinates

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 (that is, those curves had been rectified), the semicubical parabola was the first algebraic curve (excluding the line and circle) to be rectified.

The length of the semicubical parabola was computed by van Heuraet in 1659.

References