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Fermat's Spiral

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FermatsSpiral

Fermat's spiral, also known as the parabolic spiral, is an Archimedean spiral with m=2 having polar equation

r^2=a^2theta.
(1)

This curve was discussed by Fermat in 1636 (MacTutor Archive). For any given positive value of theta, there are two corresponding values of r of opposite signs. The left plot above shows

r=atheta^(1/2)
(2)

only, while the right plot shows equation (1) in red and

r=-atheta^(1/2)
(3)

in blue. Taking both signs, the resulting spiral is symmetrical about the origin.

The curvature and arc length of the positive branch of Fermat's spiral are

kappa=(2sqrt(theta)(3+4theta^2))/(a(1+4theta^2)^(3/2))
(4)
s=asqrt(theta)_2F_1(-1/2,1/4;5/4;-4theta^2)
(5)
=1/8(1-i)aB(-4theta^2;1/4,3/2),
(6)

where _2F_1(a,b;c;z) is a hypergeometric function and B(z;a,b) is an incomplete beta function.

See also

Archimedean Spiral, Fermat's Spiral Inverse Curve

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References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 225, 1987.Dixon, R. "The Mathematics and Computer Graphics of Spirals in Plants." Leonardo 16, 86-90, 1983.Dixon, R. Mathographics. New York: Dover, p. 121, 1991.Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 90 and 96, 1997.Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, p. 175, 1967.MacTutor History of Mathematics Archive. "Fermat's Spiral." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Fermats.html.Smith, D. E. History of Mathematics, Vol. 2: Special Topics of Elementary Mathematics. New York: Dover, p. 330, 1958.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. Middlesex, England: Penguin Books, pp. 74-75, 1991.

Cite this as:

Weisstein, Eric W. "Fermat's Spiral." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FermatsSpiral.html

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