Archimedean Spiral

A Spiral with Polar equation

$$r=a\theta^{1/n},$$

where $r$ is the radial distance, $\theta$ is the polar angle, and $n$ is a constant which determines how tightly the spiral is ``wrapped.'' The Curvature of an Archimedean spiral is given by

$$\kappa={n\theta^{1-1/n}(1+n+n^2\theta^2)\over a(1+n^2\theta^2)^{3/2}}.$$

Various special cases are given in the following table.

Name	$n$
Lituus	$-2$
Hyperbolic Spiral	$-1$
Archimedes' Spiral	1
Fermat's Spiral	2

See also Archimedes' Spiral, Daisy, Fermat's Spiral, Hyperbolic Spiral, Lituus, Spiral

References

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

Lauwerier, H. Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, pp. 59-60, 1991.

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 186 and 189, 1972.

Lee, X. ``Archimedean Spiral.'' http://www.best.com/~xah/SpecialPlaneCurves_dir/ArchimedeanSpiral_dir/archimedeanSpiral.html

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

MacTutor History of Mathematics Archive. ``Spiral of Archimedes.'' http://www-groups.dcs.st-and.ac.uk/~history/Curves/Spiral.html.

Pappas, T. ``The Spiral of Archimedes.'' The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 149, 1989.