  Each non-zero point in $\mathbb{R}^d$ identifies a closest point $x$ on the
unit sphere $\mathbb{S}^{d-1}$. We are interested in computing an
$\epsilon$-approximation $y \in \mathbb{Q}^d$ for $x$, that is exactly on
$\mathbb{S}^{d-1}$ and has low bit size. We revise lower bounds on rational
approximations and provide explicit, spherical instances.
  We prove that floating-point numbers can only provide trivial solutions to
the sphere equation in $\mathbb{R}^2$ and $\mathbb{R}^3$. Moreover, we show how
to construct a rational point with denominators of at most
$10(d-1)/\varepsilon^2$ for any given $\epsilon \in \left(0,\tfrac 1 8\right]$,
improving on a previous result. The method further benefits from algorithms for
simultaneous Diophantine approximation.
  Our open-source implementation and experiments demonstrate the practicality
of our approach in the context of massive data sets Geo-referenced by latitude
and longitude values.
