Affine projection identities in spherical and hyperbolic geometry

Establish whether an analogue of the Euclidean affine projection identity for right simplices exists in spherical or hyperbolic geometry, where orthogonal projection is no longer a linear operation.

Background

The paper derives an exact affine projection identity for a Euclidean right simplex and a possibly nonincident affine projection subspace. The identity includes correction terms involving the distance from the distinguished vertex to the affine subspace, the altitude of the simplex, and the angle between the displacement vector and a normal direction.

The concluding remarks identify spherical and hyperbolic analogues as unresolved because orthogonal projection in those geometries is not linear. The problem is therefore to determine whether a comparable identity, with an appropriate geometric interpretation and correction terms, can be formulated in either non-Euclidean setting.