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Old   November 27, 1998, 23:44
Default Mapping a sphere
  #1
Afshin Azari
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Hello everybody. I've heard that it is possible to map a grid about a sphere onto a two-dimensional rectangle. Is this true? If yes, how is it done?

Thanks for your time.
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Old   December 1, 1998, 14:51
Default Re: Mapping a sphere
  #2
jay
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Wow! This is new! I'd really like to see this one, if it can be done at all.

A few quick Q's: Are we talking "...grid about a sphere..," meaning outside of it, or the region inside, or simply the surface only? If it is only the surface, it is simple: map the sphere to a cube (this can be done); open out the cube; fill the empty regions with "dead zones" grid to get one 2D rectangle. There is no volume region to map, so nothing practical (as regards doing computational analyses) is really accomplished from doing this mapping.

I'd wager that the other two exercises (mapping volume region outside, or volume region inside to 2D) are not possible. If it were, any 3D geometry could be similary mapped into 2D. All 3D analyses, be it CFD, or stress analysis, or ANY type of computational technique for 3D problems would become a triviality. In fact, 3D as we understand it would not exist! Comments anyone?

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Old   December 2, 1998, 00:23
Default Re: Mapping a sphere
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Glenn Horrocks
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Mapping a sphere to a rectangle - it's a bit of a give away isn't it?

If you want to see many different methods of mapping a spherical(ish) surface to a rectangle just look at the hundreds of map (ie cartography) projections there are out there. How else can you map a spherical Earth to a rectangular bit of paper (ie a map)?

If you mean map a 3D sphere to a 2D rectangle, it can't be done. The 2D rectangle should be a 3D box - that is possible.

Glenn
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Old   December 2, 1998, 03:03
Default Re: Mapping a sphere
  #4
Krishnakumar
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Mapping a three-d sphere to a rectangle is not possible, basically because a point in the rectangle is specified by two co-ordinates, and even if you do manage it, it might not help reduce computational time. But I have heard of stacked grids; Two-d grids stacked up to make up a three-d grid. A cylinder can consist of stacked up circles. Maybe this is what you call a sphere mapped to a rectangle( in this case only part of the sphere is mapped to the rectangle)
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Old   December 2, 1998, 03:31
Default Re: Mapping a sphere
  #5
Afshin Azari
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Alright everyone. There is a need for the definition of the word "sphere". Here it is (together with that of a "ball"):

A ball: (x - h)(squared) + (y - k)(squared) + (z - l)(squared) <= r(squared)

A sphere: (x - h)(squared) + (y - k) (squared) + (z - l)(squared) = r(squared)

(See "Calculus, One and Several Variables", by Salas).
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Old   December 2, 1998, 10:14
Default Re: Mapping a sphere
  #6
Krishnakumar
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Oh.. Then the problem becomes very simple indeed. Just take the two angles phi,teta which you use to represent points in a spherical co-ordinate and let them be the parameters of the grid. You have thus mapped a sphere to a rectangle... since r is a constant.
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Old   December 2, 1998, 18:55
Default Re: Mapping a sphere
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jay
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Sure. Altitude and Azimuth are sufficient to map a "sphere" as defined by Afshin. But, the issue that arises from his original words:"...map the grid *about* a sphere..." seems to be still unresolved. Does *about* mean some bounded space around the sphere, or am I splitting hairs.... ?

And, is this merely an exercise in mental callisthenics? Or is there any practical value in these argmnts?

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