Piriform quartic

PIRIFORM QUARTIC
Poire ou goutte d'eau ?

Curve studied by Wallis in 1685 and Bonnet in 1844.
From the Latin Pirum "pear".
Other names: drop of water, peg-top.

Cartesian equation:

, or

.
For b = a / 2, the equation can be written:

.
Cartesian parametrization:

with

.
Rational quartic.
Area:

The 3D drop (revolution of the quartic around its axis put on Oz) has the equation : .
Volume of this drop : .

Given a point P describing the circle (C) with diameter [OA] (where A is the point with coordinates (a, 0)), let Q be the point on the line x = b with same ordinate as P. The piriform quartic is the locus of the point M on the line (OQ) with same abscissa as P.

In other words, the piriform quartics are the antihyperbolisms of the circle with respect to a point O on this circle and a line perpendicular to the diameter one end of which is O.

Note that the piriform quartics for any value of b are images of the curve obtained when a = b by a scaling along Oy.
They are special cases of tear curves.

Up to scaling, the piriform quartic is a plane projection of the pancake curve x=cos(u),y=sin(u),z=sin(2u).

A tube with a bore shaped like a piriform quartic is a representation of the Klein bottle.

See also the double drop of water, the kieroids, and the spherical cycloids.

See also here the "true" profile of the drop of water.

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