Kepler's Folium

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

(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.