10
Feb
09

Scherk-Collins Surface

This surface can be formed by twisting and warping a singly-periodic Scherk’s minimal surface. This idea was originally attributed to Brent Collins. Technically, the surface is no longer considered exactly “minimal” after twisting but it still looks minimal (it is actually very difficult to find the exact shape for most minimal surfaces). Click here to download some POV-Ray code.

Here is some Mathematica code:
(* runtime: 0.3 second *)
<< Graphics`Master`; n = 5; r = 0.75n;
Twist[{x_, y_, z_}, theta_] := {x Cos[theta] - y Sin[theta], x Sin[theta] + y Cos[theta], z};
Warp[{x_, y_, z_}, theta_] := {(x + r) Cos[theta], (x + r) Sin[theta], y};
f[z_] := Module[{t1 = Sqrt[2Cot[z]], t2 = Cot[z] + 1}, Warp[Twist[Re[{0.5xsign(Log[t1 - t2] - Log[t1 + t2])/Sqrt[2], ysign I(ArcTan[1 - t1] - ArcTan[1 + t1])/Sqrt[2], z}], 2Re[z]/n], 2Re[z]/n]];
DisplayTogether[Table[ParametricPlot3D[f[x + I y], {x, 0, n Pi}, {y, 0.001, 0.75}, PlotPoints -> {8n + 1, 5}, Compiled -> False], {xsign, -1, 1, 2}, {ysign, -1, 1, 2}]]

The following Mathematica code can be used to increase the number of edges (or “branches”). This code uses some complicated functions that were adapted from Matthias Weber’s Mathematica notebook:
(* runtime: 1.2 seconds *)
<< Graphics`Shapes`; k = 4; phi = Pi(0.6/k - 0.5)/(1 - k);
f[z_] := Re[NIntegrate[Evaluate[{0.5 (w^(1 - k) - w^(k - 1)), 0.5 I (w^(1 - k) + w^(k - 1)), 1}/(w^(k + 1) + w^(1 - k) - 2w Cos[k phi])], {w, 0, z}]];
alpha = Pi/k; zbeta = Exp[I Pi(phi/alpha - 0.5)];
surface = ParametricPlot3D[Re[f[Exp[I alpha/2]((1 + I zbeta Exp[r + I theta])/(I Exp[r + I theta] -zbeta))^(alpha/Pi)]], {r, 0, 4}, {theta, 0, Pi}, PlotPoints -> 10, Compiled -> False, DisplayFunction -> Identity][[1]];
z0 = f[1][[3]]; surface = {surface, AffineShape[TranslateShape[surface, {0, 0, -2z0}], {1, 1, -1}]};
surface = {surface, AffineShape[surface, {1, -1, 1}]}; surface = Table[RotateShape[surface, 2Pi i/k, 0, 0], {i, 1, k}];
dz = Pi Csc[k phi]/k; Show[Graphics3D[Table[TranslateShape[surface, {0, 0, i dz}], {i, 0, 1}]]]

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2 Responses to “Scherk-Collins Surface”


  1. 1 metodykomputerowe
    March 31, 2009 at 9:36 pm

    Amazing

  2. April 7, 2009 at 3:57 am

    How are your animations coming out so smooth?

    I’m using povray on linux to make the graphics now. They certainly don’t look anything like yours with respect to quality. What is different?

    Also what software are you using to compile the animation? I’ve tried mencoder and ffmpeg (both on linux) and of course the compression makes the animation looks to be a complete disservice to the math.


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