Mutually orthogoval projective and affine spaces

A recent paper showed how to find sets of finite affine or projective planes constructed on a common set of points, so that lines of one plane meet lines of a different plane in at most two points. In this paper, those results are generalized in two different ways to spaces of higher dimension. The simpler of the two generalizations admits many solutions, both affine and projective. For the stronger definition, where a line of one space must be an arc in the other, we show the existence of pairs of projective spaces of dimension one less than a prime.