Cross ratio on sphere Project from the north pole to have an identification of the sphere in real space $\mathbb{R}^3$ and the complex projective line. Given $4$ (say, different) points on the sphere, I can project and then compute their cross-ratio. 
Where can I find a formula that directly computes this value from the four vectors in $\mathbb{R}^3$ in terms of cross product, scalar product etc?

Identify the sphere $S^2 \subset \mathbb{R}^3$ with the space of imaginary quaternions of norm $1$. Then an explicit expression for the cross-ratio can be found in the arXiv preprint On a Quaternionic Analogue of the Cross-Ratio, see in particular Section 3.
