Distance in Hopf coordinates

Question

I have been reading about Hopf coordinates (https://en.wikipedia.org/wiki/3-sphere#Hopf_coordinates) which parameterize points on $\mathbb{S}^3$ as follows: \begin{align} x_0 &= \cos \xi_1 \sin\eta \\ x_1 &= \sin \xi_1\sin\eta \\ x_2 &= \cos\xi_2 \cos\eta \\ x_3 &= \sin\xi_2\cos\eta \end{align} where $(\xi_1,\xi_2,\eta)\in [0,2\pi)\times[0,2\pi)\times [0,\pi/2)$. Is it known if there are any formulas for relating geodesic distances on $\mathbb{S}^3$ to expressions in these coordinates?

Answer 1

$\newcommand{Vec}[1]{\mathbf{#1}}$If $\Vec{u}$ and $\Vec{u}'$ are points on the unit sphere in Euclidean $n$-space, they may be identified with unit vectors, and the geodesic distance between them is the angle they subtend at the origin: $$d(\Vec{u}, \Vec{u}') = \arccos(\Vec{u} \cdot \Vec{u}').$$

In your situation, taking $\Vec{u} = (x_{0}, x_{1}, x_{2}, x_{3})$ and $\Vec{u}' = (x_{0}', x_{1}', x_{2}', x_{3}')$ with \begin{align*} x_{0} &= \cos\xi_{1} \sin\eta, & x_{0}' &= \cos\xi_{1}' \sin\eta', \\ x_{1} &= \sin\xi_{1} \sin\eta, & x_{1}' &= \sin\xi_{1}' \sin\eta', \\ x_{2} &= \cos\xi_{2} \cos\eta, & x_{2}' &= \cos\xi_{2}' \cos\eta', \\ x_{3} &= \sin\xi_{2} \cos\eta, & x_{3}' &= \sin\xi_{2}' \cos\eta', \end{align*} the geodesic distance is $$d(\Vec{u}, \Vec{u}') = \arccos(x_{0}x_{0}' + x_{1}x_{1}' + x_{2}x_{2}' + x_{3}x_{3}').$$