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Fermat's Spiral


An Archimedean Spiral with $m=2$ having polar equation

r = a\theta^{1/2},

discussed by Fermat in 1636 (MacTutor Archive). It is also known as the Parabolic Spiral. For any given Positive value of $\theta$, there are two corresponding values of $r$ of opposite signs. The resulting spiral is therefore symmetrical about the origin. The Curvature is

\kappa(\theta)={{3a^2\over 4\theta}+a^2\theta\over\left({{a^2\over 4\theta}+a^2\theta}\right)^{3/2}}.


Dixon, R. Mathographics. New York: Dover, p. 121, 1991.

Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 69-70, 1993.

Lee, X. ``Equiangular Spiral.''

Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, p. 175, 1967.

MacTutor History of Mathematics Archive. ``Fermat's Spiral.''

Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. Middlesex, England: Penguin Books, 1991.

© 1996-9 Eric W. Weisstein