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
and Cartesian equation is
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
The plots above show families of Kepler's folium for between 0 and 4.