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3 to 4 dimensional inverse mapping

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I have been trying to find the equation that will let me project the interior values of a 2-sphere boundary onto the "hypersurface" of a 3-sphere. This will be done by an inverse azimuthal equidistant projection. I have found plenty of information on 3-spheres, but none of it dealing specifically with this particular projection. I have also found the inverse projection equation from the interior of a one-sphere (a circular disc) onto a two-sphere on Mathworld [http://mathworld.wol...rojection.html], but it didn't get into higher dimensions.

In other words, if you take the temperature readings of each point inside of the earth as your "3 dimensional map," I want to project these values onto the "surface" of a hypersphere, so that the temperature readings can exist as spatial elements of varying distances from the origin of the 4-dimensional polar coordinate system. Because of the projection, the surface of the earth will become a single point that resides at some arbitrary location on the surface of the hypershere.

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