# Bifolium

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Bifolium
> Markdown URL: https://mediated.wiki/source/Bifolium.md
> Source: https://en.wikipedia.org/wiki/Bifolium
> Source revision: 1322507986
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

Quartic plane curve

For other uses, see [Bifolium (disambiguation)](/source/Bifolium_(disambiguation)).

Bifolium with *a* = 1

A **bifolium** is a [quartic plane curve](/source/Quartic_plane_curve) with equation in [Cartesian coordinates](/source/Cartesian_coordinate):

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

## Construction and equations

Construction of the bifolium

Given a [circle](/source/Circle) C through a [point](/source/Point_(geometry)) O, and [line](/source/Line_(geometry)) L [tangent](/source/Tangent_line) to the circle at point O: for each point Q on C, define the point P such that PQ is [parallel](/source/Parallel_(geometry)) to the tangent line L, and PQ = OQ. The collection of points P forms the bifolium.[1]

In [polar coordinates](/source/Polar_coordinates), the bifolium's equation is

- ρ = a sin ⁡ θ ⋅ cos 2 ⁡ θ , {\displaystyle \rho =a\sin \theta \cdot \cos ^{2}\theta ,}

- while (first eqn.)

- ρ 2 ⋅ 2 = a ⋅ x 2 y , ρ 2 = ± x ⋅ ( a y ) 1 / 2 . {\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](/source/Area_(geometry)) is approximately 0.10.

## See also

- [Folium of Descartes](/source/Folium_of_Descartes)

- [Trifolium curve](/source/Trifolium_curve)

## References

1. **[^](#cite_ref-1)** Kokoska, Stephen. ["Fifty Famous Curves, Lots of Calculus Questions, And a Few Answers"](https://facstaff.bloomu.edu/skokoska/curves.pdf) (PDF). *facstaff.bloomu.edu*. Retrieved 6 January 2018.

## External links

- [Bifolium at MathWorld](https://mathworld.wolfram.com/Bifolium.html) by Wolfram

---
Adapted from the Wikipedia article [Bifolium](https://en.wikipedia.org/wiki/Bifolium) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Bifolium?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.