folium

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=cos³f.
  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 or rabbit-ear ³.
  The curve can be generalized to the generalized regular bifolium.
  
  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.
  
  The Cartesian equation of the curve can be written as y = ± x ± Öx(1-x), the regular bifolium can also be constructed as the mediane curve of a parabola and an ellipse.

• 0 < a < 1: trifolium
  For a=1/2 the curve is called the torpedo curve ⁴.
  Its equation can be written as r = cos2f cosf or as r = sin4f /sinf.
  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 y²  

2) Folium (Lat.) = leaf.

3) In French: oreilles de lapin.

4) In French: torpille.