In hyperbolic geometry one can use the Möbius addition $$ x\oplus_K y := \frac{ (1-2K\langle x,y \rangle-K||y||_2^2)x + (1+K||x||_2^2) y }{ 1-2K\langle x,y\rangle +K^2||x||_2^2||y||_2^2 } $$ for elements $x,y$ on the Poincaré disk of sectional curvature $K<0$.

Note that for $K=0$ one just recovers the Euclidean vector space addition.

This allows to perform additions as follows:

Is there some corresponding addition that is defined for the stereographic projection of the sphere? Can one maybe just use the $\oplus_K$ of above with $K>0$?



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