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plotting - ColorFunction and/or ColorScaling issue with ParametricPlot3D


My goal is to use ColorFunction to color a cylinder based on the $z$ value of a separate Plot3D. I'm not sure if I am making a mathematical error somewhere or if I am just messing up some snippet of the following code.


a = Pi;
b = 2;
f[x_, y_] = y;
u[x_, y_, t_] = -((8 E^(-((π^2 t)/20)) (2 - π) Cos[(π y)/4])/π^2) - 
   (8 E^(-((π^2 t)/5)) (2 + 3 π) Cos[(3 π y)/4])/(9 π^2) - 
   (8 E^(-((9 π^2 t)/20)) (2 - 5 π) Cos[(5 π y)/4])/(25 π^2);

Table[ContourPlot[u[x, y, t], {x, -a, a}, {y, 0, b}, 

   ColorFunction -> ColorData["TemperatureMap"], 
   Contours -> 8], {t, {0, .1, .3, .5, 1, 2, 3, 4}}]

enter image description here


Now, I want to wrap each contour plot onto a cylinder by gluing the $x=-a$ edge to the $x=a$ edge; the bottom of the cylinder is then at $y=0$ and the top at $y=b$.


 Table[ParametricPlot3D[{Cos[theta], Sin[theta], rho}, {theta, -Pi, Pi}, {rho, 0, 2},  
    AxesLabel -> {x, y, z}, ColorFunctionScaling -> False, 
    ColorFunction -> Function[{x, y, z, theta, rho}, 
    ColorData["TemperatureMap"][u[x, rho, t]]],
    Mesh -> 8, MeshFunctions -> {Function[{x, y, z, theta, rho}, 

    f[Cos[theta], rho]]}, ViewPoint -> {-2.3, 0.77, -2}, 
    ViewVertical -> {-0.08, 1, -0.06}], 
    {t, {0, .1, .3, .5, 1, 2, 3, 4}}]

enter image description here


This is clearly not correct based on what I am after. Where am I going wrong?


I tried turning ColorFunctionScaling off and on with no luck. I also tried replacing that middle snippet dealing with the coloring with


ColorFunctionScaling -> False, 
ColorFunction -> Function[{x, y, z, theta, rho}, 
ColorData["TemperatureMap"][Rescale[u[Cos[theta], rho, t], {0,2}]]]


where a Plot3D shows $u(x,y,t)$ ranges over (about) $[0,2]$. I suspect the solution lies in some tweaking of this part.



Answer



You can get the colors from the 2D contour plots and use them as the setting for MeshShading:


 tt = {0, .1, .3, .5, 1, 2, 3, 4}; 
 cps = Table[ContourPlot[u[x, y, t], {x, -a, a}, {y, 0, b}, 
   ColorFunction -> ColorData["TemperatureMap"], Contours -> 8],  {t, tt}];
 colors = Cases[cps[[#]], _RGBColor, Infinity] & /@ Range[8];

Then



Grid[Partition[Table[ParametricPlot3D[{Cos[theta], Sin[theta], rho}, 
       {theta, -Pi,  Pi}, {rho, 0, 2},
   Mesh -> 8,
   MeshFunctions -> {Function[{x, y, z, theta, rho}, u[x, z, tt[[i]]]]},
   MeshShading -> colors[[i]],
   ColorFunction -> Function[{x, y, z, theta, rho},
      ColorData["TemperatureMap"][u[x, z, tt[[i]]]]],
   AxesLabel -> {x, y, z}], {i, Range[8]}], 4]]

gives



enter image description here


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