How can I tell when it is going to be a Archimedean Spiral. What is a general fo

Question

How can I tell when it is going to be a Archimedean Spiral. What is a general formula? Only one I know is r=theta. What else would I look for?

Explanation / Answer

The Archimedean spiral has the property that any ray from the origin intersects successive turnings of the spiral in points with a constant separation distance (equal to 2?b if ? is measured in radians), hence the name "arithmetic spiral". In contrast to this, in a logarithmic spiral these distances, as well as the distances of the intersection points measured from the origin, form a geometric progression. The Archimedean spiral has two arms, one for ? > 0 and one for ? < 0. The two arms are smoothly connected at the origin. Only one arm is shown on the accompanying graph. Taking the mirror image of this arm across the y-axis will yield the other arm. the term Archimedean spiral is used for the more general group of spirals r= a+ b(theta)^1/x where a,b are real numbers The normal Archimedean spiral occurs when x = 1. Other spirals falling into this group include the hyperbolic spiral, Fermat's spiral, and the lituus. Virtually all static spirals appearing in nature are logarithmic spirals, not Archimedean ones. Many dynamic spirals (such as the Parker spiral of the solar wind, or the pattern made by a Catherine's wheel) are Archimedean.

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