Plot parametricplot3d like this example

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  • Thread starter member 428835
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  • #1

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).
 

Answers and Replies

  • #2
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.
 
  • #3
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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