ColorFunction ranges for SphericalPlot3D

Question

I'm not sure why

SphericalPlot3D[1, {θ, 0, π}, {ϕ, 0, 2 π}, 
 ColorFunction -> 
  Function[{θ, ϕ}, 
   ColorData["Rainbow"][
    Re[SphericalHarmonicY[5, 2, θ, ϕ]]]], 
 ColorFunctionScaling -> False]

looks different to

SphericalPlot3D[1, {θ, 0, π}, {ϕ, 0, 2 π}, 
 ColorFunction -> (ColorData["Rainbow"][
     Re[SphericalHarmonicY[5, 2, #4, #5]]] &), 
 ColorFunctionScaling -> False]

(I don't understand how the # symbols work here.) Which one is correct for plotting this spherical harmonic? But my main question is why do the plots not show the full Rainbow color palette? I would like to plot this spherical harmonic using the full Rainbow color palette from min to max value.

Any help would be greatly appreciated!

UPDATE:

The first question has now been answered. Thank you to all who contributed :-) Here is a simple example to expose my confusion about the color palette (my unanswered question):

SphericalPlot3D[1, {\[Theta], 0, \[Pi]}, {\[Phi], 0, 2 \[Pi]}, 
 ColorFunction -> (ColorData["Rainbow"][
     Re[SphericalHarmonicY[5, 2, #4, #5]]] &), 
 ColorFunctionScaling -> False]
BarLegend[{ColorData[
    "Rainbow"], {MinValue[{ComplexExpand[
       Re[SphericalHarmonicY[5, 
         2, \[Theta], \[Phi]]]], {0 <= \[Theta] < \[Pi], 
       0 <= \[Phi] < 2 \[Pi]}}, {\[Theta], \[Phi]}], 
    MaxValue[{ComplexExpand[
       Re[SphericalHarmonicY[5, 
         2, \[Theta], \[Phi]]]], {0 <= \[Theta] < \[Pi], 
       0 <= \[Phi] < 2 \[Pi]}}, {\[Theta], \[Phi]}] }}, 
  LegendLayout -> "Row"] // Quiet
BarLegend[{"Rainbow", {MinValue[{ComplexExpand[
       Re[SphericalHarmonicY[5, 
         2, \[Theta], \[Phi]]]], {0 <= \[Theta] < \[Pi], 
       0 <= \[Phi] < 2 \[Pi]}}, {\[Theta], \[Phi]}], 
    MaxValue[{ComplexExpand[
       Re[SphericalHarmonicY[5, 
         2, \[Theta], \[Phi]]]], {0 <= \[Theta] < \[Pi], 
       0 <= \[Phi] < 2 \[Pi]}}, {\[Theta], \[Phi]}] }}, 
  LegendLayout -> "Row"] // Quiet

In this example, I suspect that the color function is scaling from 0 to 1 (the first legend), whereas I want the color function to scale from min to max of whatever I'm plotting (the second legend). How do I achieve this?

Answer 1

The ColorFunction of a SphericalPlot3D has six arguments, the first three being the $x$, $y$ , $z$ coodinates in $\mathbb{R}^3$. The fourth and fifth argument are the actual parameterization parameters of the surface and the last argument is the distance from the origin (the radius).

# (Slot) and & (Function) together allow to define anonymous function. #4 and #5 refer to the fourth and fifth argument. Here is (essentially) equivalent rewrite with Function in long form:

SphericalPlot3D[1, {θ, 0, π}, {ϕ, 0, 2 π}, 
 ColorFunction -> 
  Function[
   {x, y, z, u, v}, 
   ColorData["Rainbow"][Re[SphericalHarmonicY[5, 2, u, v]]]
   ], 
 ColorFunctionScaling -> False
 ]

Here is the example from the documentation:

GraphicsGrid[
 Partition[#, 3] &@
  Table[SphericalPlot3D[
    1 + Sin[5 ϕ]/10, {θ, 0, Pi}, {ϕ, 0, 2 Pi},
    PlotPoints -> 100,
    ColorFunction -> 
     Function[{x, y, z, θ, ϕ, r}, Evaluate[f]], 
    PlotLabel -> f, 
    Axes -> None], {f, {Hue[x], Hue[y], Hue[z], Hue[θ], 
     Hue[ϕ], Hue[r]}}],

 ImageSize -> Full
 ]