Clifford parallel

In elliptic geometry, two lines are Clifford parallel or paratactic lines if the perpendicular distance between them is constant from point to point. The concept was first studied by William Kingdon Clifford in elliptic space and appears only in spaces of at least three dimensions. Since parallel lines have the property of equidistance, the term "parallel" was appropriated from Euclidean geometry, although the "lines" of elliptic geometry are geodesic curves and, unlike the lines of Euclidean geometry, are of finite length. The algebra of quaternions provides a descriptive geometry of elliptic space in which Clifford parallelism is made explicit. Introduction The lines on 1 in elliptic space are described by versors with a fixed axis r: For an arbitrary point u in elliptic space, two Clifford parallels to this line pass through u. The right Clifford parallel is