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Archimedean Spiral

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An Archimedean spiral is a spiral with polar equation

r=atheta^(1/n),
(1)

where r is the radial distance, theta is the polar angle, and n is a constant which determines how tightly the spiral is "wrapped."

ArchimedeanSpiral

Values of n corresponding to particular special named spirals are summarized in the following table, together with the colors with which they are depicted in the plot above.

spiralcolorn
lituusred-2
hyperbolic spiralorange-1
Archimedes' spiralgreen1
Fermat's spiralblue2

The curvature of an Archimedean spiral is given by

kappa=(|n|theta^(1-1/n)(1+n+n^2theta^2))/(a(1+n^2theta^2)^(3/2)),
(2)

and the arc length for n>0 by

s=atheta^(1/n)_2F_1(-1/2,1/(2n);1+1/(2n);-n^2theta^2),
(3)

where _2F_1(a,b;c;x) is a hypergeometric function.

If a fly crawls radially outward along a uniformly spinning disk, the curve it traces with respect to a reference frame in which the disk is at rest is an Archimedean spiral (Steinhaus 1999, p. 137). Furthermore, a heart-shaped frame composed of two arcs of an Archimedean spiral which is fixed to a rotating disk converts uniform rotational motion to uniform back-and-forth motion (Steinhaus 1999, pp. 136-137).

See also

Archimedean Spiral Inverse Curve, Archimedes' Spiral, Daisy, Fermat's Spiral, Hyperbolic Spiral, Lituus

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References

Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 90-92, 1997.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.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.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 136-137, 1999.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 8-9, 1991.

Cite this as:

Weisstein, Eric W. "Archimedean Spiral." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ArchimedeanSpiral.html

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