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If (a,b) is a point in Affine space, then its projective coordinates are [a:b:1]. To select the representative on the unit sphere we can choose (a/r,b/r,1/r) where r=sqrt(a^2+b^2+1). Revering the transformation, given a point on the unit sphere (x,y,z) we can recover the affine point [a:b:1] as [x/z:y/z:1]. Using this idea try: ContourPlot3D[x^2 + y^2 + z^2 ...

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