Split-quaternion - Counter-sphere Geometry

Counter-sphere Geometry

The quadratic form q q* is positive definite on the planes C_v and N. Consider the counter-sphere {q: q q* = −1}.

Take where . Fix theta (θ) and suppose

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Since points on the counter-sphere must line on the conjugate of the unit hyperbola in some plane D_p ⊂ P, m can be written, for some p ∈ J

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Let φ be the angle between the hyperbolas from r to p and m. This angle can be viewed, in the plane tangent to the counter-sphere at r, by projection:

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Now as in the expression of angle of parallelism in the hyperbolic plane H2 . The parameter θ determining the meridian varies over the unit circle S1. Thus the counter-sphere appears as the manifold S1 × H2.