Kepler's folium is a folium curve explored by Kepler in 1609 (Lawrence 1972, p. 151; Gray et al. 2006, p. 85). When used without qualification, the term "folium" sometimes refers to Kepler's folium (e.g., Lawrence 1972, pp. 152-153; MacTutor).
Kepler's folium has polar equation
 |
(1)
|
and Cartesian equation is
![(x^2+y^2)[x(x+b)+y^2]=4axy^2.](/images/equations/KeplersFolium/NumberedEquation2.svg) |
(2)
|
If 
, it is a single folium. If 
, it is a bifolium.
If 
, it is a three-lobed curve
sometimes called a trifolium. A modification of the case 
, 
is sometimes called the trefoil
curve (Cundy and Rollett 1989, p. 72).
The area of Kepler's folium is
 |
(3)
|
The plots above show families of Kepler's folium for 
between 0 and 4.
The single folium is the pedal curve of the deltoid where the pedal point is one of the cusps.
