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Displaying contours over normal Riemann surfaces

  • To: mathgroup at smc.vnet.net
  • Subject: [mg126293] Displaying contours over normal Riemann surfaces
  • From: James <icorone at hotmail.com>
  • Date: Sun, 29 Apr 2012 02:09:47 -0400 (EDT)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com

Hi,
The code below draws a simple figure-8 contour in the complex z-plane, then by steriographic projection, maps it to a Riemann sphere as a yellow contour over the sphere.  I have a red contour drawn from the points -1 to 1 on the sphere to represent a branch cut.  I believe it's possible to "open up" the sphere along the red contour, and deform it into a half-sphere with the red contour along the hemisphere edge.  Could someone help me write that code?

Thank you,
James

mySphereX[x_, y_] := (2*x)/(1 + x^2 + y^2); 
mySphereY[x_, y_] := (2*y)/(1 + x^2 + y^2); 
mySphereZ[x_, y_] := (x^2 + y^2 - 1)/(1 + x^2 + y^2); 
myx[t_] := 2*Sin[t]; 
myy[t_] := 2*Cos[t]*Sin[t]; 
pp1 = ParametricPlot3D[{mySphereX[myx[t], myy[t]], mySphereY[myx[t], myy[t]], mySphereZ[myx[t], myy[t]]}, {t, 0, 2*Pi}, 
   PlotStyle -> {Thickness[0.005], Yellow}]
pp3 = ParametricPlot3D[{mySphereX[x, 0], mySphereY[x, 0], mySphereZ[x, 0]}, {x, -1, 1}, 
   PlotStyle -> {Thickness[0.005], Red}]
myplane = Graphics3D[{Opacity[0.5], Polygon[{{2, 2, 0}, {-2, 2, 0}, {-2, -2, 0}, {2, -2, 0}}]}]; 
Show[{myplane, pp1, pp3, SphericalPlot3D[1, {\[Theta], 0, Pi}, {\[Phi], 0, 2*Pi}, PlotStyle -> Opacity[0.5]]}]



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