Bifolium

For other uses, see Bifolium (disambiguation).

Bifolium with a = 1

A bifolium is a quartic plane curve with equation in Cartesian coordinates:

${\displaystyle (x^{2}+y^{2})^{2}=ax^{2}y.}$

Construction and equations

Construction of the bifolium

Given a circle C through a point O, and line L tangent to the circle at point O: for each point Q on C, define the point P such that PQ is parallel to the tangent line L, and PQ = OQ. The collection of points P forms the bifolium. ^[1]

In polar coordinates, the bifolium's equation is

${\displaystyle \rho =a\sin \theta \cdot \cos ^{2}\theta ,}$

while (first eqn.)

${\displaystyle \rho ^{2\cdot 2}=a\cdot x^{2}y,\,\,\rho ^{2}=\pm x\cdot (ay)^{1/2}.}$

For a = 1, the total included area is approximately 0.10.

See also

• Folium of Descartes
• Trifolium curve

Reference

1. Kokoska, Stephen. "Fifty Famous Curves, Lots of Calculus Questions, And a Few Answers" (PDF). facstaff.bloomu.edu. Retrieved 6 January 2018.

External links

• Bifolium at MathWorld by Wolfram