The **piriform curve** (also called the *peg-top curve* or *pear-shaped quartic*) is a teardrop-shaped family of curves given by the Cartesian equation

- a
^{4}y^{2}= b^{2}x^{3}(2a – x)

where the curve’s midline lies on the x-axis and

- y = (o, ±b) for x = a.

The area of a pirifrom is the same as the area of an ellipse with semiaxes a and b [1].

It can also be described by the algebraic equation

- x
^{4}– 2ab^{2}x^{3}+ a^{4}y^{2}= 0

or the parametric representation

- x = a (1 + sin
*t*) and y = b cos*t*(1 + sin*t*).

This famous curve, which is symmetric about the x-axis, is named after the Latin *pirum *“pear” and appears in many places in nature including some birds eggs, seeds, and part of the human nasal passage.

The piriform was first studied by French mathematician Gaston Albert Gohierre de Longchamps in 1866 [2].

## Construction of the Piriform Curve

The piriform curve can be constructed as follows: Given a point A on the circumference of a circle and a line L

_{1}perpendicular to the diameter through point A:

- Draw a line through point A that crosses L
_{1}in point B.

Draw a second line, L_{2}, perpendicular to L_{1}, intersecting the circle at point C. - Draw a third line L
_{3}, perpendicular to L_{2}through point C and intersecting line L in point P.

The piriform curve is the locus of P for all possible lines L_{n} [2].

## Piriform Curve Derivatives

for the parametric representation x = a (1 + sin*t*) and y = b cos*t* (1 + sin*t*), the derivatives are:

## References

Top image: Mike Williams,

[1] Unger, J. How Wasanka Did Integration: The Case of the Japanese Wedge. Sangaku Journal of Mathematics (SJM). Volume 5 (2021), pp. 43–55.

[2] Cole, D. Playing with Dynamic Geometry.