MathGroup Archive 2007

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Changing Sqrt{(z-1)(z-2)(z-3)} to coffee mug

I sure would like to see animation of this over the Riemann sphere:  Take two Riemann spheres corresponing to the two branches of sqrt showing the two branch cuts on each.  Now:  open up the branch cuts, stretch them into necks, join the necks, form the double flask, double neck, transform this into a donut, indent the donut into a coffee mug.  Would make a nice addition to the Wolfram demonstration project.  

I can only get as far as using ContourPlot3D with the two holes in the sphere:

r2 = ContourPlot3D[x^2 + y^2 + z^2 == 2, {x, -2, 2}, 
     {y, -2, 2}, {z, -2, 2}, RegionFunction -> 
       Function[{x, y, z}, (x - 0.2)^2 + (y - 0.2)^2 > 0.05 && 
           (x - 0.7)^2 + (y - 0.7)^2 > 0.05], 
     BoundaryStyle -> Directive[Red, Thick], Boxed -> False, 
     Axes -> None]

Anyone know of this being done or how one might do this?


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