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Re: 3D Animations

  • To: mathgroup at smc.vnet.net
  • Subject: [mg105889] Re: 3D Animations
  • From: dh <dh at metrohm.com>
  • Date: Wed, 23 Dec 2009 02:42:38 -0500 (EST)
  • References: <hgq21f$g51$1@smc.vnet.net>


Hi Arthur,

you are asking for a stereographic projection between a sphere and a plane.

We may identify the points on a sphere by 2 angles (spherical 

coordinates: elevation and azimuth): theta and phi. We may the give 

transformations between cartesian coordinates: x,y on the plane and 

corresponding spherical coordinates on the sphere. If we assume the 

center of the sphere to be at: p0={p0x,p0y,p0z}:



toPlane[{theta_, phi_}, p0_] :=

   p0[[{1, 2}]] + Tan[theta] p0[[3]] {Sin[phi], Cos[phi]};

toSphere[{x_, y_},

   p0_] := {ArcTan[p0[[3]], Norm[{x, y} - p0[[{1, 2}]]]], ArcTan[x, y]}



we may now e.g. create a grid and project it from the sphere to the 

plane or vice versa:



grid = Flatten[

    Table[{{theta, phi}, k + {theta, phi}}, {theta, 0,

      1, .1}, {phi, -1, 1, .2}, {k, {{.1, 0}, {0, .2}}}], 2];



Show@Graphics[Line /@ grid, Axes -> True]

s2 = Map[toSphere[#, p0] &, grid, {2}];

p2 = Map[toPlane[#, p0] &, grid, {2}];



Show@Graphics[(Line /@ p2), Axes -> True]

Show@Graphics[(Line /@ s2), Axes -> True]



If you project a figure from the plane to the sphere, the move and/or 

rotate the sphere, you can generate all the pictures in the video.

Daniel



Artur wrote:

> Dear Mathematica Gurus,

> I would like to ask that in version up 6 are available such 3D 

> animations like follwing (I'm mean about second part of this video):

> http://www.youtube.com/watch?v=JX3VmDgiFnY&feature=related

> Merry Christmas

> Artur

> 

> 




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