More generally, stereographic projection may be applied to the unit -sphere  in ()-dimensional Euclidean space .  If  is a point of  and  a hyperplane in , then the stereographic projection of a point  is the point  of intersection of the line  with . In Cartesian coordinates (,  from 0 to ) on  and (,  from 1 to n) on , the projection from  is given by
Defining
the inverse is given by
Still more generally, suppose that  is a (nonsingular) quadric hypersurface in the projective space . In other words,  is the locus of zeros of a non-singular quadratic form  in the homogeneous coordinates . Fix any point  on  and a hyperplane  in  not containing .  Then the stereographic projection of a point  in  is the unique point of intersection of  with .  As before, the stereographic projection is conformal and invertible on a non-empty Zariski open set. The stereographic projection presents the quadric hypersurface as a rational hypersurface.  This construction plays a role in algebraic geometry and conformal geometry.