# folium

## quartic 1)

The folium 2 has three forms:

•  a >= 1: single folium or simple folium

Some authors confine the simple folium to a = 1.
In this case the equation of the curve can be rewritten as r=cos3φ.
The curve is the inverse of Tschirnhausen's cubic.
• a = 0: regular bifolium (or regular double folium)
The curve is sometimes called the bifolium, but I see the curve as a special case of this bifolium.
Alternative names for the curve are: right bifolium, right double folium, bifoliate, and rabbit-ear3.  The Cartesian equation of the curve can be written as y = ± √x ± √x(1-x).

The curve can be constructed with a given circle C through O as follows: draw for each point Q on C points P, so that PQ = OQ. Then the collection of the points P forms the regular bifolium.
Another construction is as the mediane curve of a parabola and an ellipse.

The curve can be generalized to the generalized regular bifolium.
• 0 < a < 1: trifolium
For a=1/2 the curve is called the trefoil4, torpedo curve5 or siluroid (see the website of Dario de Judicibus).
Its equation can be written as r = cos2φ cosφ or as r = sin4φ sinφ.  A Cartesian form of its equation is (x2+y2)2 = x(x2-y2).

For a=1/4 we see the regular trifolium, which is in fact a rosette.

Each of the three folia is a (different) pedal of the deltoid.

It was Johann Kepler (1609) who was the first to describe the curve.
Therefore the curve is also known as Kepler's folium. notes

1) In Cartesian coordinates: (x2 + y2)2 + a x3 = (1-a) x y

2) Folium (Lat.) = leaf.

3) In French: oreilles de lapin.

5) In French: torpille.