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A bifolium is a
quartic plane curve In algebraic geometry, a quartic plane curve is a plane algebraic curve of the fourth degree. It can be defined by a bivariate quartic equation: :Ax^4+By^4+Cx^3y+Dx^2y^2+Exy^3+Fx^3+Gy^3+Hx^2y+Ixy^2+Jx^2+Ky^2+Lxy+Mx+Ny+P=0, with at least one of ...
with equation in
Cartesian coordinate In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
s: :(x^2 + y^2)^2 = ax^2y.


Construction and equations

Given a
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
C through a point O, and line L
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
to the circle at point O: for each point Q on C, define the point P such that PQ is
parallel Parallel may refer to: Mathematics * Parallel (geometry), two lines in the Euclidean plane which never intersect * Parallel (operator), mathematical operation named after the composition of electrical resistance in parallel circuits Science a ...
to the tangent line L, and PQ = OQ. The collection of points P forms the bifolium. In
polar coordinates In mathematics, the polar coordinate system specifies a given point (mathematics), point in a plane (mathematics), plane by using a distance and an angle as its two coordinate system, coordinates. These are *the point's distance from a reference ...
, the bifolium's equation is :\rho=a\sin\theta\cdot\cos^2\theta, :while (first eqn.) :\rho^=a\cdot x^2y,\,\,\rho^2=\pm x\cdot(ay)^. For ''a'' = 1, the total included
area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
is approximately 0.10.


See also

*
Folium of Descartes In geometry, the folium of Descartes (; named for René Descartes) is an algebraic curve defined by the implicit equation x^3+y^3-3a\cdot xy=0. dy/dx=(x^2-ay)/(ax-y^2), \,dx/dy=(ax-y^2)/(x^2-ay). History The curve was first proposed and studi ...
* Trifolium curve


Reference

{{Reflist


External link


Bifolium at MathWorld
by Wolfram Plane curves Algebraic curves