Bifolium

[[File:Bifolium01.svg|thumb|500px|Bifolium with {{math|1=''a'' = 1}}]] A '''bifolium''' is a [[quartic plane curve]] with equation in [[Cartesian coordinate]]s:

:<math>(x^2 + y^2)^2 = ax^2y.</math>

== Construction and equations == [[File:Bifolium.gif|thumb|Construction of the bifolium|right|upright=2.0]] Given a [[circle]] C through a [[point (geometry)|point]] O, and [[line (geometry)|line]] L [[tangent line|tangent]] to the circle at point O: for each point Q on C, define the point P such that PQ is [[Parallel (geometry)|parallel]] to the tangent line L, and PQ = OQ. The collection of points P forms the bifolium.<ref>{{Cite web |last=Kokoska |first=Stephen |date= |title=Fifty Famous Curves, Lots of Calculus Questions, And a Few Answers |url=https://facstaff.bloomu.edu/skokoska/curves.pdf |access-date=6 January 2018 |website=facstaff.bloomu.edu |publisher=}}</ref>

In [[polar coordinates]], the bifolium's equation is :<math>\rho=a\sin\theta\cdot\cos^2\theta,</math> :while (first eqn.) :<math>\rho^{2\cdot2}=a\cdot x^2y,\,\,\rho^2=\pm x\cdot(ay)^{1/2}.</math>

For ''a'' = 1, the total included [[area (geometry)|area]] is approximately 0.10.

== See also ==

* [[Folium of Descartes]] * [[Trifolium curve]]

==References== {{Reflist}}

==External links== *[https://mathworld.wolfram.com/Bifolium.html Bifolium at MathWorld] by Wolfram

[[Category:Plane curves]] [[Category:Algebraic curves]]