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MathGroup Archive 2012

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torus connections in Riemannian sheet trajectories?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg124383] torus connections in Riemannian sheet trajectories?
  • From: Roger Bagula <roger.bagula at gmail.com>
  • Date: Wed, 18 Jan 2012 05:56:43 -0500 (EST)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com

This curve is a Beauville-Weierstrass solution:x^5+y^5+z^5=0;
{1 + t^5, -1 + t^5, -(2 + 20 t^10 + 10 t^20)^(1/5)}
which was Integrated by hand and a substitution used:
 t->r*Exp[i*t]
with r =1 giving the real part as Cos[n*t].
What I want to do is see all 5 Riemannian sheets
instead of just two.
Mathematica:
Clear[x, y, z, w, r, q0, p0]
TubePlotFrenet[curve_List, {var_, min_, max_}, radius_, opts___] :=
 Module[{tangent, unitTangent, normal, unitNormal, biNormal},
  tangent = D[curve, t];
  unitTangent = tangent/Sqrt[tangent.tangent];
  normal = D[unitTangent, t];
  unitNormal = normal/Sqrt[normal.normal];
  biNormal = Cross[unitTangent, unitNormal];
  ParametricPlot3D[
   curve + radius Cos[s] unitNormal + radius Sin[s] biNormal //
    Evaluate, {var, min, max}, {s, 0, 2 \[Pi]}, opts]]

x = Cos[t] + Cos[6*t]/6;
y = Cos[t] - Cos[6*t]/6;
z = -(2*Cos[t] + (10/21)*Cos[21*t] + (20/11)*Cos[11*t])^(1/5);
w = {x, y, z};
w1 = {x, y, -z};
h = TubePlotFrenet[w, {t, -Pi, Pi}, 1/10, Axes -> None,
  Boxed -> False, ViewPoint -> {10, 0, 0}, PlotPoints -> {100, 16},
  Mesh -> False, ColorFunction -> "Rainbow", PlotRange -> All]
g = TubePlotFrenet[w1, {t, -Pi, Pi}, 1/10, Axes -> None,
  Boxed -> False, ViewPoint -> {10, 0, 0}, PlotPoints -> {64, 16},
  Mesh -> False, ColorFunction -> "Rainbow", PlotRange -> All]
gw = Show[{h, g}, PlotRange -> All]



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