Archimedean spiral

An Archimedean spiral is a curve which in polar coordinates (r, θ) can be described by the equation

<math>r=a+b\theta</math>

with real numbers a and b. Changing the parameter a will turn the spiral, while b controls the distance between the arms.

This Archimedean spiral is distinguished from the logarithmic spiral by the fact that successive arms have a fixed distance (equal to 2πb if θ is measured in radians), while in a logarithmic spiral these distances form a geometric progression.

Note that the Archimedean spiral has two arms, one for θ > 0 and one for θ < 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 at the y axis will yield the other arm.

Sometimes the term Archimedean spiral is used for the more general group of spirals

<math>r=a+b\theta^{1\!/\!x}</math>

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 spirals appearing in nature are logarithmic spirals, not Archimedean ones.