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MathGroup Archive 2007

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A question about RegionFunction in ver. 6.0

  • To: mathgroup at smc.vnet.net
  • Subject: [mg77765] A question about RegionFunction in ver. 6.0
  • From: chuck009 <dmilioto at comcast.com>
  • Date: Sat, 16 Jun 2007 03:29:59 -0400 (EDT)

I understand RegionFunction allows a specific region of a surface to be plotted.  I've checked the documentation and think that's still not going to be enough to provide a nice interface to plot "clean" sheets of multi-functions.  Consider the principal sheet of the imaginary component for the function w=(z^2-z^3)^(-1/3): 

p1 = Plot3D[Evaluate[Im[(z^2 - z^3)^(-1/3) /. z -> x + I 
                  y]], {x, -3, 3}, {y, -3, 3}, 
            PlotPoints -> 80, AspectRatio -> 1, 
      PlotRange -> {{-3, 3}, {-3, 3}, {-3, 3}}, BoxRatios -> {1, 1, 1}];

Note how Plot3D produces a jagged boundary at the branch cuts.  The following code shows the branch cuts in the plane:

sol = Solve[z^2 - z^3 == v, z];
Show[(ParametricPlot[Evaluate[({
        Re[#1], Im[#1]} &) /@ (z /. sol)], #1, DisplayFunction -> 
      Identity] &) /@ {{v, -8, -2}, {v, -2, 0}},
           DisplayFunction -> $DisplayFunction, 
PlotRange -> All, Frame -> True, Axes -> False, AspectRatio -> Automatic];

Does anyone feel that the RegionFunction could be adapted to plot the three sectors of this sheet cleanly without the jagged edges?  

I can plot a clean sheet in ver 5.2, but must create the polygons directly using the data from the branch cuts.  It's messy though and lots of code.


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