h06pirif.gif (1506 bytes)

This curve has a form resembling a pear 1), why it is also called pear-shaped quartic or pear-shaped curve.
The first to study the curve was the French mathematician G. de Longchamps (1886).

The curve is defined in a rather complex way, in the following steps:

  • let there be a circle C with a point O on it
  • let there be a line l, perpendicular to the diameter of the circle through O
  • now, draw an arbitrary line m through O, crossing l in P
  • draw a line n perpendicular to l through P, crossing C in points Q
  • then, draw a line o parallel to l through Q
  • now the curve is formed by the points R that are intersection of the lines o and the arbitrary line m

Could you follow me? If not, take a look at the picture.

In fact, the given formula is a (linear) generalization of the curve. The construction is only valid for the curve when the constant 1 is replaced by the 2.

The piriform has as parameter equation:

The curve is also called the pegtop, it is easy to see why.

In your throat there is a place where pieces of food can be stuck, e.g. a fish bone. This pit is called the piriform fossa 2).

The curve can be generalized to the teardrop curve.


1) Pirum (Lat.)  = pear

2) Fossa (Lat.) = moat, pit.