The Spiral of Archimedes
Introduction:
The spiral plotted above is one branch of a curve known as the spiral of Archimedes. In fact, it is part of a family of spirals, many of which have different names. Some examples are given below.
Definition:
,
where a is the radius of the spiral, and n is some real number. In the case n = 1, plotted at the top of the page, the spiral is called the spiral of Archimedes. The family of spirals also has the polar parameterization
Several values of n other than 1 are also given specific names:
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| Spiral of Archimedes (n = 1) | Fermat's Spiral (n = 2) |
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| Hyperbolic Spiral (n = -1) | Lituus (n = -2) |
Properties:
Velocity and Acceleration:
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The Tangent Vector:
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Arc Length: |
Curvature:
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