Biplanar space

A real $(2n+1)$-dimensional projective space, with two non-intersecting $n$-dimensional subspaces which are real (biplanar spaces of hyperbolic type) or complex conjugate (biplanar spaces of elliptic type), the fundamental group of which consists of the projective transformations that map each of these subspaces into itself. These two $n$-dimensional subspaces are called the absolute planes. The linear congruence of real straight lines intersecting both absolute planes is said to be an absolute congruence. This congruence serves as a real model of an $n$-dimensional projective space over the algebra of double or complex numbers. If $n=1$, a biplanar space is said to be a bi-axial space. The study of properties of geometrical figures in biplanar spaces which are preserved under the operation of the fundamental group is the subject of biplanar geometry. The bi-axial geometry in which one studies the theory of curves, surfaces and complexes of straight lines, has been investigated in detail.