# ColorFunctionScaling and OpacityFunctionScaling with values other than 0 and 1

I have a 3D density plot that I was able to get to look okay with setting ColorFunctionScaling and OpacityFunctionScaling to True. I know that there are other ways to view this data, but I checked the documentation and both these functions treat the minimum value as 0 and the maximum as 1. With the 3D plot, this obscures the middle. Is there a way to have it so that for example, the whole plot is scaled with opacity between 0 and 0.7?

Here is the example I am using. If possible I would also like the color function reduced to between 0 and 1/14.

prob1 = (E^(-2 Im[ArcTan[x, y]] - Re[Sqrt[x^2 + y^2 + z^2]]) Abs[x^2 + y^2])/(64 \[Pi]);

P1 =
DensityPlot3D[prob1, {x, -20, 20}, {y, -20, 20}, {z, -20, 20},
AxesLabel -> {"x", "y", "z"}, ColorFunction -> Hue,
OpacityFunction -> Function[a, a^1.2],
ColorFunctionScaling -> True, OpacityFunctionScaling -> True,
FaceGrids -> {{-1, 0, 0}, {0, 1, 0}, {0, 0, -1}},
Ticks -> Table[{{-20,"-20a0"}, {-10,"-10a0"}, {0, "0"}, {10,"10a0"}, {20, "20a0"}}, {i, 3}],
PlotLegends -> Automatic]


This yields the above:

I'm looking for a way to lower the opacity of the inside of the doughnut in that same kind of exponential function I have, and I want the maximum opacity to be 0.7. The prob1 function just takes to long to run if I add a Maximize. Any suggestions?

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Set

OpacityFunctionScaling -> False


and adjust the OpacityFunction to perform the scaling itself. Since the minimum value of prob1 is 0 and from the legend (produced by DensityPlot3D but not shown in the plot in the question) the maximum is about .0027, the function should read

OpacityFunction -> Function[a, (a/.0027)^1.2]


which indeed reproduces the plot in the question. To reduce the maximum opacity to 0.7 instead of 1, use

OpacityFunction -> Function[a, .7 (a/.0027)^1.2]


This choice does not change the plot much, but reducing the factor from .7 to, for instance .3 produces a very noticeable change.

How much is enough is in the eye of the beholder. Incidentally, the scale factor, .0027, also can be obtained from

prmax = FindMaximum[prob1, {x, y, z}] // First
(* 0.00269241 *)