Archimedes' Spiral
Archimedes' spiral is an Archimedean spiral with polar equation
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(1)
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This spiral was studied by Conon, and later by Archimedes in On Spirals about 225 BC. Archimedes was able to work out the lengths of various tangents to the spiral.
The curvature of Archimedes' spiral is
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(2)
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and the arc length is
 |  |  |
(3)
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 |  | ![1/2a[thetasqrt(1+theta^2)+ln(theta+sqrt(1+theta^2))].](/images/equations/ArchimedesSpiral/Inline6.gif) |
(4)
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This has the series expansion
 |  | ![a{theta+1/2sum_(k=3)^(infty)[P_(n-3)(0)+(n+1)/nP_(n-1)(0)]theta^k}](/images/equations/ArchimedesSpiral/Inline9.gif) |
(5)
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 |  |  |
(6)
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(OEIS A091154 and A002595), where 
is a Legendre
polynomial.
Archimedes' spiral can be used for compass and straightedge division of an angle into 
parts (including
angle trisection) and can also be used for circle squaring. In addition, the curve can be used
as a cam to convert uniform circular motion into uniform linear motion (Brown 1923;
Steinhaus 1999, p. 137). The cam consists of one arch of the spiral above the
x-axis together with its reflection in the x-axis.
Rotating this with uniform angular velocity about its center will result in uniform
linear motion of the point where it crosses the y-axis.
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archimedes' spiral
