# Another color scheme for 3D visualizations of complex functions

I am looking for visualizations of complex functions $$f(z) = r(z)e^{i\varphi(z)}$$ which plot the magnitude $$r(z)$$ as height and display the argument $$\varphi(z)$$ on this graph.

On the cover of Abramowitz and Stegun's Handbook of Mathematical Functions

the argument is displayed as labelled lines perpendicular to the lines of equal height ("iso-phase lines").

Displaying the phase by color wouldn't be a big deal, especially by this HSL color function:

$$H(\varphi)$$ $$= \begin{cases} 0 &\text{ if } 0 \leq \varphi < \pi \ \ \ \ \text{(red)}\\ 2\pi/3 &\text{ if } \pi \leq \varphi < 2\pi \ \ \text{(green)} \end{cases}$$

$$L(\varphi)$$ $$= |\varphi - \pi|/\pi$$

$$S(\varphi)$$ $$= 1$$

These colors would indicate how far the phase is away from $$0$$ (white for $$\varphi = 0$$, black for $$\varphi = \pi$$ and if it is less or greater than $$\pi$$ (shades of red for $$\varphi < \pi$$, shades of green for $$\varphi > \pi$$):

Full red means $$\varphi = \pi/2$$, full green means $$\varphi = 3\pi/2$$.

But you may also plot the magnitude as height and display the phase by simply

$$H(\varphi) = \varphi$$

$$L(\varphi) = 1$$

$$S(\varphi) = 1$$

which seems to be some kind of standard, right?

To emphasize the difference between these two coloring schemes consider the function $$f(z) = z^3 + z^2 + z$$ and the images of the circle with radius $$r = 1.3$$:

It's probably not only a matter of taste which of these two colorings one prefers.

My questions are:

• Is this color mapping already established? Does it have a name (to search for)?

• Can you give me links to specific examples of visualized complex functions plotting magnitude as height and making use of this color mapping?

Ideally, you would be able (as unfortunately I am not) to produce such plots - e.g. with Mathematica - and upload them in an answer. (If so: please without any contur lines, if possible.)

• Are you asking about the general idea of mapping magnitude to height and argument to colour, or specifically your particular choice of colour map? If the former, see e.g. functions.wolfram.com/ElementaryFunctions/Csc/visualizations/5 (scroll down to "Absolute value part over the complex plane"). – Rahul Jan 15 at 12:42
• @Rahul: I am asking specifically about my particular choice of color map (which makes use of only two base colors, which may be interpreted somehow intuitively). In the examples on the Wolfram page we always see many colors which a) looks somehow psychedelic and (to me) unpleasant, and b) is confusing and the colors are much harder to interpret. But that's - admittedly - opinion-based, so I didn't mention it in my question. – Hans-Peter Stricker Jan 15 at 12:50
• @Rahul: The standard approach seems to be $H(\varphi) = \varphi$ - just like in the case of domain-coloring - and $L(\varphi) = 1$. Should I mention this in my question? – Hans-Peter Stricker Jan 15 at 12:51
• @Rahul: Would you mind having a look at my edited question? – Hans-Peter Stricker Jan 15 at 14:03
• I don't know specifically about visualizing complex functions. But colour map choices have received a fair amount of attention in the scientific visualization community, although they usually work with colour maps on an interval rather than on a circle. In that context, they call colour maps similar to the one you've devised diverging colour maps. See Moreland, "Diverging Color Maps for Scientific Visualization" (2009) for a detailed discussion. – Rahul Jan 15 at 16:32