Equation        (polar)

Safari Guide  radius increases at constant rate as spiral turns; gives an evenly spaced spiral  the spacing between arcs is constant

In a DLP projector, a colour wheel rotates in front of the light source, so that the three primary colours are produced in sequence, one after the other. For some (though, curiously, not all) viewers, this creates colour shadows in a rainbow effect. However, if the colour wheel is replaced by an Archimedean spiral, the primary colours are displayed simultaneously in  bands that move down the screen. This eliminates the fraction-of-a-second gap between colours that causes the rainbow effect.

DLP projectors use Archimedean spirals as an anti-rainbow technology

A Closer Look

Technically,  is a spiral of Archimedes. The more general class of Archimedean spirals have polar equation ; the larger n is, the tighter the spiral. For a spiral of Archimedes, ; this is the only type with constant branch spacing.

Archimedean spirals for various n

Notice that for negative n, the spirals curve inward in an anticlockwise sense. That is, they get tighter as θ increases from zero, having started from very small θ and very large r.

Field Study

What exactly is the curvature of an Archimedean spiral? Using the polar formula for the
curvature ,

For specific values of n, this boils down to

The branch spacing for Archimedean spirals is , which for  simplifies to 2πk. So the spacing of a spiral of Archimedes is constant. This is reflected in the asymptotic curvature of 1/θ; as θ and r increase, the curvature decreases to compensate.