Fermat's spiral, also known as the parabolic spiral, is an Archimedean spiral with
having polar equation
(1)
This curve was discussed by Fermat in 1636 (MacTutor Archive). For any given positive value of ,
there are two corresponding values of of opposite signs. The left plot above shows
(2)
only, while the right plot shows equation (1) in red and
(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
where
is a hypergeometric function and 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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