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 analytical 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 problem in the usually used HSV map is the erratic nature of everything, creating false detail, making certain parts of the phase look longer than others, etc. Unlike, HSV our peaks and troughs are equispaced, and smooth.

Lightnesswise the entire range is separated into six equal parts. For usual 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, so adding 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.