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# 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.

Bifolium, Folium, Fish Curve, Folium of Descartes, Quadrifolium, Rose Curve, Trefoil Curve, Trifolium

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## References

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 72, 1989.Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 85, 2006.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 152-153, 1972.MacTutor History of Mathematics Archive. "Folium." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Folium.html.

## Cite this as:

Weisstein, Eric W. "Kepler's Folium." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KeplersFolium.html