Bifolium (original) (raw)

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Quartic plane curve

Bifolium with a = 1

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

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

Construction and equations

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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

ρ = a sin ⁡ θ cos 2 ⁡ θ . {\displaystyle \rho =a\sin \theta \cos ^{2}\theta .} {\displaystyle \rho =a\sin \theta \cos ^{2}\theta .}

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

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