Piriform

$\begin{figure}\begin{center}\BoxedEPSF{piriform.epsf scaled 800}\end{center}\end{figure}$

A plane curve also called the Peg Top and given by the Cartesian equation

$\begin{displaymath} a^4 y^2=b^2 x^3(2a-x) \end{displaymath}$

(1)

and the parametric curves

$\displaystyle x$	$\textstyle =$	$\displaystyle a(1+\sin t)$	(2)
$\displaystyle y$	$\textstyle =$	$\displaystyle b\cos t(1+\sin t)$	(3)

for $t\in [-\pi/2,\pi/2]$ . It was studied by G. de Longchamps in 1886. The generalization to a Quartic 3-D surface

$\begin{displaymath} (x^4-x^3)+y^2+z^2=0, \end{displaymath}$

(4)

is shown below (Nordstrand).

$\begin{figure}\begin{center}\BoxedEPSF{PiriformSurface.epsf}\end{center}\end{figure}$

References

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub. p. 71, 1989.

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 148-150, 1972.