دانشنامه ریاضی و کامپیوتر

Folium of Descartes

A plane curve proposed by Descartes to challenge Fermat's extremum-finding techniques. In parametric form,

			(1)
			(2)

The curve has a discontinuity at . The left wing is generated as runs from to 0, the loop as runs from 0 to , and the right wing as runs from to .

In Cartesian coordinates,

(3)

(MacTutor Archive). The equation of the asymptote is

(4)

The curvature and tangential angle of the folium of Descartes are

			(5)
			(6)

where is the Heaviside theta function.

Converting the parametric equation s to polar coordinates gives

			(7)
			(8)

so the polar equation is

(9)

The area enclosed by the curve is

			(10)
			(11)
			(12)

The arc length of the loop is given by

			(13)
			(14)

SEE ALSO:Folium

REFERENCES:

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 218, 1987.

Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 77-82, 1997.

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 106-109, 1972.

MacTutor History of Mathematics Archive. "Folium of Descartes." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Foliumd.html.

Smith, D. E. History of Mathematics, Vol. 2: Special Topics of Elementary Mathematics. New York: Dover, p. 328, 1958.

Stroeker, R. J. "Brocard points, Circulant Matrices, and Descartes' Folium." Math. Mag. 61, 172-187, 1988.

Yates, R. C. "Folium of Descartes." In A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 98-99, 1952.