A quartic algebraic curve also called the peg top and given by the Cartesian equation

(1)

and the parametric curves

			(2)
			(3)

for . It was studied by G. de Longchamps in 1886.

The area of the piriform is

(4)

which is exactly the same as the ellipse with semiaxes and .

The curvature of the piriform is given by

(5)

A generalization to a quartic three-dimensional surface is the quartic surface of revolution

(6)

illustrated above. With , this surface is termed the "zeck" surface by Hauser. It has volume

(7)

geometric centroid

			(8)
			(9)
			(10)

and inertia tensor

(11)

for constant density and mass .

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

Hauser, H. "Algebraic Surfaces." http://www.freigeist.cc/gallery.html.

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

Nordstrand, T. "Surfaces." http://jalape.no/math/surfaces.

CITE THIS AS:

Weisstein, Eric W. "Piriform." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Piriform.html