Mathematica Plot parametricplot3d like this example

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The discussion centers on creating visual representations of vibrating membranes using mathematical functions in a 3D parametric plot. The initial example provided demonstrates a complex mathematical expression for generating a plot, with a focus on varying color based on height. The user seeks assistance in replicating a specific brown plot from a reference link, expressing confusion about the origin of the formula involving ArcTan. A successful solution is later shared, showcasing a method to adjust color gradients in relation to height, enhancing the visual output of the plot. The conversation emphasizes the intricacies of mathematical modeling and visualization techniques in computational graphics.
member 428835
Hi PF!

Here looking at the first answer are two awesome examples of a vibrating membrane plotted from a top view. I can create the first example via
Code: 
fXYZ =
{Cos[\[Theta]] Csc[\[Pi]/180] Sin[s Sin[\[Pi]/180]] -
  0.001 Cos[\[Theta]] Cos[2 \[Theta]] Sin[
    s Sin[\[Pi]/
      180]] (10.7721 (0. -
        Sqrt[1 -
          Cos[s Sin[\[Pi]/180]]^2] (0. +
           1.52712 (BesselJ[1,
               175.004 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] -
              BesselJ[3,
               175.004 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]]) Cosh[
             3.05424 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])]) +
        Cos[s Sin[\[Pi]/180]] (0. +
           3.05424 BesselJ[2,
             175.004 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] Sinh[
             3.05424 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])])) +
     0.0939376 (0. -
        Sqrt[1 -
          Cos[s Sin[\[Pi]/180]]^2] (0. +
           3.35307 (BesselJ[1,
               384.253 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] -
              BesselJ[3,
               384.253 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]]) Cosh[
             6.70613 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])]) +
        Cos[s Sin[\[Pi]/180]] (0. +
           6.70613 BesselJ[2,
             384.253 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] Sinh[
             6.70613 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])])) -
     0.000899129 (0. -
        Sqrt[1 -
          Cos[s Sin[\[Pi]/180]]^2] (0. +
           4.98473 (BesselJ[1,
               571.237 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] -
              BesselJ[3,
               571.237 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]]) Cosh[
             9.96947 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])]) +
        Cos[s Sin[\[Pi]/180]] (0. +
           9.96947 BesselJ[2,
             571.237 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] Sinh[
             9.96947 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])])) +
     0.0000163397 (0. -
        Sqrt[1 -
          Cos[s Sin[\[Pi]/180]]^2] (0. +
           6.58519 (BesselJ[1,
               754.645 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] -
              BesselJ[3,
               754.645 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]]) Cosh[
             13.1704 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])]) +
        Cos[s Sin[\[Pi]/180]] (0. +
           13.1704 BesselJ[2,
             754.645 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] Sinh[
             13.1704 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])])) -
     3.74518*10^-7 (0. -
        Sqrt[1 -
          Cos[s Sin[\[Pi]/180]]^2] (0. +
           8.17376 (BesselJ[1,
               936.692 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] -
           
              BesselJ[3,
               936.692 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]]) Cosh[
             16.3475 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])]) +
        Cos[s Sin[\[Pi]/180]] (0. +
           16.3475 BesselJ[2,
             936.692 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] Sinh[
             16.3475 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])])) +
     9.80625*10^-9 (0. -
        Sqrt[1 -
          Cos[s Sin[\[Pi]/180]]^2] (0. +
           9.75646 (BesselJ[1,
               1118.06 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] -
              BesselJ[3,
               1118.06 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]]) Cosh[
             19.5129 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])]) +
        Cos[s Sin[\[Pi]/180]] (0. +
           19.5129 BesselJ[2,
             1118.06 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] Sinh[
             19.5129 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])])) -
     2.94642*10^-10 (0. -
        Sqrt[1 -
          Cos[s Sin[\[Pi]/180]]^2] (0. +
           11.3358 (BesselJ[1,
               1299.05 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] -
              BesselJ[3,
               1299.05 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]]) Cosh[
             22.6716 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])]) +
        Cos[s Sin[\[Pi]/180]] (0. +
           22.6716 BesselJ[2,
             1299.05 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] Sinh[
             22.6716 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])]))),
 Csc[\[Pi]/180] Sin[\[Theta]] Sin[s Sin[\[Pi]/180]] -
  0.001 Cos[2 \[Theta]] Sin[\[Theta]] Sin[
    s Sin[\[Pi]/
      180]] (10.7721 (0. -
        Sqrt[1 -
          Cos[s Sin[\[Pi]/180]]^2] (0. +
           1.52712 (BesselJ[1,
               175.004 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] -
              BesselJ[3,
               175.004 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]]) Cosh[
             3.05424 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])]) +
        Cos[s Sin[\[Pi]/180]] (0. +
           3.05424 BesselJ[2,
             175.004 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] Sinh[
             3.05424 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])])) +
     0.0939376 (0. -
        Sqrt[1 -
          Cos[s Sin[\[Pi]/180]]^2] (0. +
           3.35307 (BesselJ[1,
               384.253 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] -
              BesselJ[3,
               384.253 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]]) Cosh[
             6.70613 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])]) +
        Cos[s Sin[\[Pi]/180]] (0. +
           6.70613 BesselJ[2,
             384.253 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] Sinh[
             6.70613 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])])) -
     0.000899129 (0. -
        Sqrt[1 -
          Cos[s Sin[\[Pi]/180]]^2] (0. +
           4.98473 (BesselJ[1,
               571.237 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] -
              BesselJ[3,
               571.237 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]]) Cosh[
             9.96947 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])]) +
        Cos[s Sin[\[Pi]/180]] (0. +
           9.96947 BesselJ[2,
             571.237 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] Sinh[
             9.96947 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])])) +
     0.0000163397 (0. -
        Sqrt[1 -
          Cos[s Sin[\[Pi]/180]]^2] (0. +
           6.58519 (BesselJ[1,
               754.645 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] -
              BesselJ[3,
               754.645 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]]) Cosh[
             13.1704 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])]) +
        Cos[s Sin[\[Pi]/180]] (0. +
           13.1704 BesselJ[2,
             754.645 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] Sinh[
          
             13.1704 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])])) -
     3.74518*10^-7 (0. -
        Sqrt[1 -
          Cos[s Sin[\[Pi]/180]]^2] (0. +
           8.17376 (BesselJ[1,
               936.692 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] -
              BesselJ[3,
               936.692 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]]) Cosh[
             16.3475 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])]) +
        Cos[s Sin[\[Pi]/180]] (0. +
           16.3475 BesselJ[2,
             936.692 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] Sinh[
             16.3475 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])])) +
     9.80625*10^-9 (0. -
        Sqrt[1 -
          Cos[s Sin[\[Pi]/180]]^2] (0. +
           9.75646 (BesselJ[1,
               1118.06 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] -
              BesselJ[3,
               1118.06 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]]) Cosh[
             19.5129 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])]) +
        Cos[s Sin[\[Pi]/180]] (0. +
           19.5129 BesselJ[2,
             1118.06 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] Sinh[
             19.5129 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])])) -
     2.94642*10^-10 (0. -
        Sqrt[
         1 - Cos[s Sin[\[Pi]/180]]^2] (0. +
           11.3358 (BesselJ[1,
               1299.05 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] -
              BesselJ[3,
               1299.05 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]]) Cosh[
             22.6716 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])]) +
        Cos[s Sin[\[Pi]/180]] (0. +
           22.6716 BesselJ[2,
             1299.05 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] Sinh[
             22.6716 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])]))), (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
    180] + 0.001 Cos[2 \[Theta]] Cos[
    s Sin[\[Pi]/
      180]] (10.7721 (0. -
        Sqrt[1 -
          Cos[s Sin[\[Pi]/180]]^2] (0. +
           1.52712 (BesselJ[1,
               175.004 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] -
              BesselJ[3,
               175.004 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]]) Cosh[
             3.05424 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])]) +
        Cos[s Sin[\[Pi]/180]] (0. +
           3.05424 BesselJ[2,
             175.004 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] Sinh[
             3.05424 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])])) +
     0.0939376 (0. -
        Sqrt[1 -
       
          Cos[s Sin[\[Pi]/180]]^2] (0. +
           3.35307 (BesselJ[1,
               384.253 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] -
              BesselJ[3,
               384.253 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]]) Cosh[
             6.70613 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])]) +
        Cos[s Sin[\[Pi]/180]] (0. +
           6.70613 BesselJ[2,
             384.253 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] Sinh[
             6.70613 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])])) -
     0.000899129 (0. -
        Sqrt[1 -
          Cos[s Sin[\[Pi]/180]]^2] (0. +
           4.98473 (BesselJ[1,
               571.237 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] -
              BesselJ[3,
               571.237 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]]) Cosh[
             9.96947 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])]) +
        Cos[s Sin[\[Pi]/180]] (0. +
           9.96947 BesselJ[2,
             571.237 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] Sinh[
             9.96947 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])])) +
     0.0000163397 (0. -
        Sqrt[1 -
          Cos[s Sin[\[Pi]/180]]^2] (0. +
           6.58519 (BesselJ[1,
               754.645 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] -
           
              BesselJ[3,
               754.645 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]]) Cosh[
             13.1704 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])]) +
        Cos[s Sin[\[Pi]/180]] (0. +
           13.1704 BesselJ[2,
             754.645 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] Sinh[
             13.1704 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])])) -
     3.74518*10^-7 (0. -
        Sqrt[1 -
          Cos[s Sin[\[Pi]/180]]^2] (0. +
           8.17376 (BesselJ[1,
               936.692 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] -
              BesselJ[3,
               936.692 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]]) Cosh[
             16.3475 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])]) +
        Cos[s Sin[\[Pi]/180]] (0. +
           16.3475 BesselJ[2,
             936.692 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] Sinh[
             16.3475 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])])) +
     9.80625*10^-9 (0. -
        Sqrt[1 -
          Cos[s Sin[\[Pi]/180]]^2] (0. +
           9.75646 (BesselJ[1,
               1118.06 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] -
              BesselJ[3,
               1118.06 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]]) Cosh[
             19.5129 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])]) +
        Cos[s Sin[\[Pi]/180]] (0. +
           19.5129 BesselJ[2,
             1118.06 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] Sinh[
             19.5129 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])])) -
     2.94642*10^-10 (0. -
        Sqrt[1 -
          Cos[s Sin[\[Pi]/180]]^2] (0. +
           11.3358 (BesselJ[1,
               1299.05 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] -
              BesselJ[3,
               1299.05 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]]) Cosh[
             22.6716 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])]) +
        Cos[s Sin[\[Pi]/180]] (0. +
           22.6716 BesselJ[2,
             1299.05 Sqrt[1 - Cos[s Sin[\[Pi]/180]]^2]] Sinh[
             22.6716 (1 + (1 - Cos[s Sin[\[Pi]/180]]) Csc[\[Pi]/
                  180])])))};

ParametricPlot3D[
 Evaluate[fXYZ], {s,
   0,1/180 \[Pi] Csc[\[Pi]/180]}, {\[Theta], 0, 2 \[Pi]}, Boxed -> False,
 ViewPoint -> {0, 0, Infinity}, Axes -> False,
 ColorFunction ->
  Function[{x, y, z}, Glow[ColorData["GrayTones", z]]], Mesh -> None,
 Lighting -> None]
However, I can't figure out how to create that brown plot they do (their second plot). Any suggestions (obviously my plot is a parametric 3D plot, so the form is different, hence what's killing me).
 
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joshmccraney said:
Hi PF!

Here looking at the first answer are two awesome examples of a vibrating membrane plotted from a top view.
Does anyone have any idea where the
Code: 
2 ArcTan[10 x]/Pi + .5
comes from at the link? I'm clueless, but the magic seems to be here.
 
For future regard, this ended up working out very nicely (figured out how to vary the color proportional to height):
Code: 
ParametricPlot3D[
 Evaluate[fXYZ], {s, 0, 1/180 \[Pi] Csc[\[Pi]/180]}, {\[Theta], 0,
  2 \[Pi]}, PlotRange -> All,
 ColorFunction ->
  Function[{x, y, z}, Blend[{Black, White, White}, Abs[ 10 z]]],
 ColorFunctionScaling -> False, ViewPoint -> {0, 0, Infinity},
 Axes -> False, Mesh -> None, PlotPoints -> 300, MaxRecursion -> 0]
 

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