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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