The Arenberg Phase Wheel

Creating a perceptually smooth color wheel is in general a difficult task, and comes with inherent compromises. This document serves to list the design decisions taken for the phase wheel, which we call Arenberg, used in our implementation of domain coloring. We carefully selected the following sweep through CIE L*a*b* space:

\[\begin{aligned} L^* &= 67 - 12 \cos(3\theta),\\ a^* &= 46 \cos(\theta + .4) - 3,\quad\text{and}\\ b^* &= 46 \sin(\theta + .4) - 16. \end{aligned}\]

This is implemented by the internal function DomainColoring.arenberg, giving the following phase wheel.

DomainColoring.arenberg.(0:.01:2π)

The main issue of the typically used HSV map that we try to prevent here is the overall variability of the lightness which creates false detail, making certain parts of the phase look longer than others, etc. Unlike the HSV map our peaks and troughs are equispaced and of equal height/depth. Furthermore, the lightness is smooth everywhere.

Lightnesswise the entire range is separated into six equal parts. For data analysis it would be better to minimise the number of oscillations, however, for our purposes the turning points serve as visual anchors dividing the phase range. Note that some lightness variation is wanted, as our eyes mainly rely on lightness to discern higher frequency information^[1].

The target color space is sRGB, including dips in lightness near its red, green, and blue primaries buys us more range in additional lightness variation to show magnitude in more complicated plots. This way we have only slight clipping in sRGB space when adding the lightness variations used to show magnitude changes in domaincolor.

Of course, different contexts require different compromises, hence why domaincolor provides the color keyword argument to replace Arenberg by a color scheme of your choice.