Semicubical Parabola

$\begin{figure}\begin{center}\BoxedEPSF{SemicubicalParabola.epsf scaled 700}\end{center}\end{figure}$

A Parabola-like curve with Cartesian equation

$\begin{displaymath} y=ax^{3/2}, \end{displaymath}$

(1)

parametric equations

$\displaystyle x$	$\textstyle =$	$\displaystyle t^2$	(2)
$\displaystyle y$	$\textstyle =$	$\displaystyle at^3,$	(3)

and Polar Coordinates,

$\begin{displaymath} r={\tan^2\theta\sec\theta\over a}. \end{displaymath}$

(4)

The semicubical parabola is the curve along which a particle descending under gravity describes equal vertical spacings within equal times, making it an Isochronous Curve. The problem of finding the curve having this property was posed by Leibniz

in 1687 and solved by Huygens

(MacTutor Archive).

The Arc Length, Curvature, and Tangential Angle are

$\displaystyle s(t)$	$\textstyle =$	$\displaystyle {\textstyle{1\over 27}}(4+9t^2)^{3/2}-{\textstyle{8\over 27}}$	(5)
$\displaystyle \kappa(t)$	$\textstyle =$	$\displaystyle {6\over t(4+9t^2)^{3/2}}$	(6)
$\displaystyle \phi(t)$	$\textstyle =$	$\displaystyle \tan^{-1}({\textstyle{3\over 2}}t).$	(7)

See also Neile's Parabola, Parabola Involute

References

Gray, A. ``The Semicubical Parabola.'' §1.7 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 15-16, 1993.

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 85-87, 1972.

Lee, X. ``Semicubic Parabola.'' http://www.best.com/~xah/SpecialPlaneCurves_dir/SemicubicParabola_dir/semicubicParabola.html

MacTutor History of Mathematics Archive. ``Neile's Parabola.'' http://www-groups.dcs.st-and.ac.uk/~history/Curves/Neiles.html.

Yates, R. C. ``Semi-Cubic Parabola.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 186-187, 1952.