Semicubical parabola

Summary

In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve that has an implicit equation of the form

Semicubical parabola for various a.

(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. (A parabola can be described by the equation y = ax2.)

Solving the implicit equation for x yields a second explicit form

The parametric equation

can also be deduced from the implicit equation by putting [1]

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).[2]

Properties of semicubical parabolas

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Similarity

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Any semicubical parabola   is similar to the semicubical unit parabola  .

Proof: The similarity   (uniform scaling) maps the semicubical parabola   onto the curve   with  .

Singularity

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

 
Tangent at a semicubical parabola

Tangents

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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 [3]

 

(Proving this statement one should use the fact, that the tangent meets the curve at   twice.)

Arclength

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Determining the arclength of a curve   one has to solve the integral   For the semicubical parabola   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.

Evolute of the unit parabola

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The evolute of the parabola   is a semicubical parabola shifted by 1/2 along the x-axis:  

Polar coordinates

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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   ( see List of identities) one gets [4]

 
 
Relation between a semicubical parabola and a cubic function (green)

Relation between a semicubical parabola and a cubic function

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Mapping the semicubical parabola   by the projective map   (involutory perspectivity with axis   and center  ) yields   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 replacement   (the line at infinity has equation  .) and the multiplication by   is performed. One gets the equation of the curve

  • in homogeneous coordinates:  

Choosing line   as line at infinity and introducing   yields the (affine) curve  

Isochrone curve

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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

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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]

References

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  1. ^ a b Pickover, Clifford A. (2009), "The Length of Neile's Semicubical Parabola", The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics, Sterling Publishing Company, Inc., p. 148, ISBN 9781402757969.
  2. ^ August Pein: Die semicubische oder Neil'sche Parabel, ihre Sekanten und Tangenten , p.2
  3. ^ August Pein: Die semicubische oder Neil'sche Parabel, ihre Sekanten und Tangenten , p.26
  4. ^ August Pein: Die semicubische oder Neil'sche Parabel, ihre Sekanten und Tangenten , p. 10
  • August Pein: Die semicubische oder Neil'sche Parabel, ihre Sekanten und Tangenten , 1875, Dissertation
  • Clifford A. Pickover: The Length of Neile's Semicubical Parabola
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