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Suppose I have a sphere. Inside the sphere I have an inscribed cube. What I am interested in is finding out what is the latitude and longitude (or coordinates) of a point on the sphere which will be projected on a cube's face given the coodinate of a point on one of the cube's faces.

Does anyone have any equations for this?

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Is this the same as this earlier question? –  Henning Makholm Nov 28 '11 at 21:36

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Here I assume, since it is not specified in the question, that the projection is taken along the radius ray.

Suppose $(x,y,z)$ denote coordinate of a point on a cube with unit-edge length and position so that its centroid is exactly at the origin. The radius of the circumscribed sphere is $\frac{\sqrt{3}}{2} $. The coodinates of point's projection onto the sphere are: $$ (x^\prime,y^\prime,z^\prime) = (x,y,z) \frac{1}{2} \sqrt{\frac{3}{x^2+y^2+z^2}} $$ You can now work on mapping these into spherical coordinates as needed.

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isn't the radius for the sphere $ \sqrt{3} $?/2? Equal to the diameter of the cube/2? As the sphere is circumscribed to the cube... –  Ryan Nov 29 '11 at 23:24
    
@Ryan Coordinates of cube's vertexes are $\left( \pm\frac{1}{2}, \pm\frac{1}{2}, \pm\frac{1}{2} \right)$. The square of the distance from any of it to the origin is $\frac{3}{4}$, so yes, you are correct! My bad, sorry. –  Sasha Nov 29 '11 at 23:29

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