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Combining a parametric plot with a Graphics3D plot

  • To: mathgroup at smc.vnet.net
  • Subject: [mg79510] Combining a parametric plot with a Graphics3D plot
  • From: chuck009 <dmilioto at comcast.com>
  • Date: Sat, 28 Jul 2007 05:32:02 -0400 (EDT)

Hello guys,

I do a lot of work plotting complex contours over the real and imaginary sheets of complex functions.  Often though, when I combine both the contour and the sheet, the contour is shown broken-up.  The code below shows this problem.  It combines the real component of the function:

f[z]=(1+2z}/(1+Exp[z]Sqrt[z])

shown in blue with the path of a contour z=6 Exp[it] shown in red.  Note how the red contour is broken-up.  Sure would be nice if I could have a plot with the red-contour cleanly trajecting over the surface.  I know I can "raise" the contour slightly to reduce the break-up but this often still does not work and also the precision of the plot is reduced.  Can anybody suggest a way to integrate both graphics into a single graphics object so that the plot is nice and clean? 

I have ver. 5.2.  Does ver 6 eliminate this problem?

tophalf = ParametricPlot3D[{Re[z] /. z -> r*Exp[I*t], 
    Im[z] /. z ->r*Exp[I*t], 
    Re[(1 + 2*z)/(1 + Exp[z]*Exp[(1/2)*(Log[Abs[z]] + 
            I*Arg[z])]) /. z ->r*Exp[I*t]], Blue}, 
   {r, 0.3, 6.5}, {t, 0, Pi}, PlotPoints -> {30, 50}]
   
bothalf = ParametricPlot3D[{Re[z] /. z ->r*Exp[I*t], 
    Im[z] /. z ->r*Exp[I*t], 
    Re[(1 + 2*z)/(1 + Exp[z]*Exp[(1/2)*(Log[Abs[z]] + 
            I*(Arg[z] + 2*Pi))]) /. z ->r*Exp[I*t]], 
    Blue}, {r, 0.3, 6.5}, {t, Pi, 2*Pi}, PlotPoints -> {30, 50}]

cfunction = {6 Cos[t],6 Sin[t], 
    Re[(1 + 2*z)/(1 + Exp[z]*Exp[(1/2)*(Log[6] + I*t)]) /. 
       z ->6 Exp[I*t]] + 0.01}; 
       
path1 = ParametricPlot3D[Evaluate[Append[cfunction, 
     {Red, Thickness[0.01]}]], {t, 0, 2*Pi}]
     
Show[{tophalf, bothalf, path1}, Lighting -> False]


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