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MathGroup Archive 2003

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RE: Fields or forms on manifolds

  • To: mathgroup at smc.vnet.net
  • Subject: [mg43859] RE: [mg43834] Fields or forms on manifolds
  • From: "David Park" <djmp at earthlink.net>
  • Date: Thu, 9 Oct 2003 01:54:32 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Christopher,

A very interesting question. It would help if you had a specific example in
mind.

There are two packages at my web site that might help with the problem,
Tensorial 3.0 and DrawGraphics. DrawGraphics makes it easy to combine
various 3D graphics objects in one plot and has an Arrow3D routine. I have
made some plots showing vector fields, or sets of basis vectors on a curved
surface.

The problem is to keep the graphic from being cluttered. Sometimes you might
want to show selected parts of some field. Or you could have multiple plots,
or an animation where each frame showed a selected part, say the field along
a curve. How to best graphically show the features of a field would depend
upon the nature of the field itself and what particular characteristic you
might want to emphasize. What would work in one case might not work in
another case.

Another possibility is to show a two panel graphic. The left hand panel
would show a plot of the manifold and mark a point on the manifold. The
right hand panel would show the field item plotted in its tangent space. One
could then animate this and show how the field item varied as the point
moved on a curve in the manifold.

Here is a simple example of a Frenet frame sliding on a twisted curve,
alpha. (It's the curve in Chapter II, Section 4, Exercise 3 of O'Neill,
Elementary Differential Geometry, 1966 edition.)

Needs["DrawGraphics`DrawingMaster`"]

\[Alpha][t_] := {t Cos[t], t Sin[t], t}

The following is a parametrization of the Frenet frame using the standard
formulas of differential geometry.

{tt[t_], nn[t_], bb[t_]} = {{(Cos[t] - t*Sin[t])/Sqrt[2 + t^2],
     (t*Cos[t] + Sin[t])/Sqrt[2 + t^2], 1/Sqrt[2 + t^2]},
    {((-t)*(3 + t^2)*Cos[t] - (4 + t^2)*Sin[t])/
      Sqrt[16 + 18*t^2 + 7*t^4 + t^6],
     ((4 + t^2)*Cos[t] - t*(3 + t^2)*Sin[t])/Sqrt[16 + 18*t^2 + 7*t^4 +
t^6],
     -(t/Sqrt[16 + 18*t^2 + 7*t^4 + t^6])},
    {(-2*Cos[t] + t*Sin[t])/Sqrt[8 + 5*t^2 + t^4],
     -((t*Cos[t] + 2*Sin[t])/Sqrt[8 + 5*t^2 + t^4]),
     (2 + t^2)/Sqrt[8 + 5*t^2 + t^4]}};

The following produces one frame of the animation...

slideplot[t_] := Show[Graphics3D[
        {Blue,
          ParametricDraw3D[Evaluate[\[Alpha][s]], {s, -2 \[Pi], 2 \[Pi]}],
          Black,
          (Arrow3D[\[Alpha][t], \[Alpha][t] + 2 #1, HeadLength3D -> 0.3,
                  HeadWidth3D -> 0.5] &) /@ {tt[t], nn[t], bb[t]},
          OrangeRed, AbsolutePointSize[4], Point[\[Alpha][t]]}],
      AspectRatio -> Automatic,
      PlotRange -> {{-8, 8}, {-5, 5}, {-8, 8}},
      BoxStyle -> Gray,
      Background -> Linen,
      ImageSize -> 400];

And the following produces the animation.

Animate[slideplot[t], {t, -2\[Pi], 2\[Pi], 4\[Pi]/40}]; SelectionMove[
  EvaluationNotebook[], All, GeneratedCell]
FrontEndTokenExecute["OpenCloseGroup"]; Pause[0.5];
FrontEndExecute[{FrontEnd`SelectionAnimate[200, AnimationDisplayTime -> 0.1,
      AnimationDirection -> ForwardBackward]}]

David Park
djmp at earthlink.net
http://home.earthlink.net/~djmp/



From: Christopher Arthur [mailto:caa2460 at dcccd.edu]
To: mathgroup at smc.vnet.net

Is there a mathematica add on that provides a means for visualizing forms or
fields on 2 or 3 manifolds? Perhaps there is a way to use the plot vector
field restricted to an arbitrary manifold as domain, or just coloring a
polygon to indicate the value in that area....

Christopher Arthur




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