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\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

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
\end{displaymath} (4)

is shown below (Nordstrand).


See also Butterfly Curve, Dumbbell Curve, Eight Curve, Heart Surface, Pear Curve


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.

Nordstrand, T. ``Surfaces.''

© 1996-9 Eric W. Weisstein