21
Jan
09

Punctured Helicoid

Here is a helicoidwith holes in it. The following Mathematica code uses some complicated functions that were adapted from Matthias Weber’sMathematica notebook:
(* runtime: 4 seconds *)
<< Graphics`Shapes`;
tau0 = Exp[1.23409 I]; b0 = 0.629065; theta[z_] := EllipticTheta[1, Pi z, Exp[I Pi tau0]];
r1[z_] := theta[z + 0.5 (b0 - 2) (tau0 + 1)]/theta[z + 0.5 (b0 - 1) (tau0 + 1)]; r2[z_] := theta[z - 0.5 b0 (tau0 + 1)]/theta[z - 0.5 (b0 + 1) (tau0 + 1)];
omega3[z_] := r1[z] r2[z]/(0.386191 - 0.169839 I); G[z_] := (108.37 - 62.8417 I) Exp[I Pi (b0 - 2 z + 2 tau0 + b0 tau0)]r1[z]/r2[z];
f[z0_] := Re[NIntegrate[Evaluate[{-(G[z] omega3[z] - omega3[z]/G[z] )/2, I(G[z] omega3[z] + omega3[z]/G[z] )/2, omega3[z]}], {z, tau0/2, z0}]] + {0.434156, 0, -1};
a0 = -0.409956; r0 = 2.43051; g[z_] := (EllipticF[ArcSin[(a0 + r0 E^z)/(1 - a0 E^z)], 1/r0^2]/(2EllipticF[Pi/2, 1/r0^2]) + 0.5)(1 + tau0)/2;
surface = ParametricPlot3D[f[g[x + I y]], {x, -2.5, 2.5 - 0.8881}, {y, 0.001,0.999Pi}, PlotPoints -> {15, 10}, Compiled -> False, DisplayFunction -> Identity][[1]];
surface = {surface, RotateShape[surface, 0, 0, Pi]};
Show[Graphics3D[{surface, TranslateShape[surface, {0, 0, 2}]}, ViewPoint -> {1, 6, 3}]];

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