Elliptic Geometry - Elliptic Space

Elliptic Space

The three-dimensional elliptic geometry makes use of the 3-sphere S3, and these points are well-accessed with the versors in the theory of quaternions. A versor is a quaternion of norm one, which must necessarily have the form

The origin corresponds to a = 0 and is the identity of the topological group consisting of versors. With r fixed, the versors

form an elliptic line. The distance from to 1 is a . For an arbitrary versor u, the distance will be that θ for which cos θ = (u + u*)/2 since this is the formula for the scalar part of any quaternion.

An elliptic motion is described by the quaternion mapping

where u and v are fixed versors.

Distances between points are the same as between image points of an elliptic motion. In the case that u and v are quaternion conjugates of one another, the motion is a spatial rotation, and their vector part is the axis of rotation. In the case u = 1 the elliptic motion is called a right Clifford translation, or a parataxy. The case v = 1 corresponds to left Clifford translation.

Elliptic lines through versor u may be of the form

or for a fixed r .

They are the right and left Clifford translations of u along an elliptic line through 1. The elliptic space is formed by identifying antipodal points on S3. Elliptic space has special structures called Clifford parallels and Clifford surfaces.