# Archimedean spiral

Archimedean spiral.

An Archimedean spiral is traced by a point that travels around a center, varying in distance from it in proportion to the angle it moves through. Similarly a helix is the curve of a point that moves around a cylinder, tracing along it in proportion to its total angular rotation. Both are created automatically by a lathe whose cutting tool traverses a rotating work piece. Many cylindrical objects show their machine finishing as a fine helical path.

An Archimedean spiral is a spiral, like that of the
groove in a phonograph record, in which the distance between adjacent coils,
measured radially out from the center, is constant. It was first studied
by Archimedes and was the main subject of
his treatise *On Spirals*. The Archimedean spiral has a very simple
equation in polar coordinates (*r*, *θ*):

*r*=

*a*+

*b*

*θ*

where *a* and *b* can be any real numbers. Changing the parameter *a* turns the spiral, while *b* controls the distance between
the arms.

The Archimedean spiral is distinguished from the logarithmic
spiral by the fact that successive arms have a fixed distance (equal
to 2π*b* if *θ* is measured in radians),
whereas in a logarithmic spiral these distances form a geometric
sequence.

Note that the Archimedean spiral has two possible arms that coil in opposite
directions, one for *θ* > 0 and the other for *θ* < 0. Many examples of spirals in the man-made world, such as a watch spring
or the end of a rolled carpet, are either Archimedean spirals or another
curve that is very much like it – the circle
involute.