Set a fixed colour scale with DensityPlot

Question

I am trying to plot some Zernike polynomials, but I want all of them to be plotted by using the same colour scale from -1 to +1 (i.e. the range of these polynomials). I am using the following code:

DensityPlot[
  ZernikeR[n, m, Norm[{x, y}]] Cos[m ArcTan[x, y]], {x, -1, 1}, {y, -1,1}
, PlotRange -> {{-1, 1}, {-1, 1}, {-1, 1}}
, ColorFunction -> (Hue[2 (1 - #1)/3] &)
, PlotPoints -> 200
, PlotLegends -> Automatic
, RegionFunction -> Function[{x, y, z}, -1 < x^2 + y^2 < 1]
]

I get what I want with some polynomials, e.g. when m=0 and n=2:

and not with others, e.g. when m=0 and n=4:

where the colour range goes down to -0.5 and not to -1. I want all polynomials to be coloured with a scale from -1 to 1, from blue to red. I haven't found a way to fix this. What am I missing? Thanks.

Answer 1

(This answer will be very similar to halirutan's answer (+1). The difference is that I preserved the color function from the OP, and that I made a fixed bar legend.)

We can do it like in this answer, which is to say, we can turn off the color function scaling and scale the values ourselves in a way that is the same for all plots. We can make our own bar legend that matches this scaling. We only need to change three options:

ColorFunction -> (Hue[2 (1 - Rescale[#1, {-1, 1}])/3] &),
ColorFunctionScaling -> False,
PlotLegends -> BarLegend[{Hue[2 (1 - Rescale[#1, {-1, 1}])/3] &, {-1, 1}}]

Now we get plots like this:

Answer 2

You have currently two mistakes. The first one is your color-function itself. It needs to give Hue[0] for a value of -1 and Hue[1] for a value of 1. The transformation is

col[v_] := Hue[Rescale[v, {-1, 1}]]

Secondly, you need to turn off that Mathematica rescales the values for each plot which can be done with the ColorFunctionScaling -> False option.

plot[m_, n_] := 
 DensityPlot[
  ZernikeR[n, m, Norm[{x, y}]] Cos[m ArcTan[x, y]], {x, -1, 
   1}, {y, -1, 1}, PlotRange -> {{-1, 1}, {-1, 1}, {-1, 1}}, 
  ColorFunction -> col, PlotPoints -> 200, PlotLegends -> Automatic,
  ColorFunctionScaling -> False,
  RegionFunction -> Function[{x, y, z}, -1 < x^2 + y^2 < 1]]

And then, plot[0,2] and plot[0,4] will use the same scaling for the colors:

If you don't want a circular color scheme like Hue is (begins and ends in red), then use one of the other schemes.