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Each non-zero point in $\mathbb{R}^d$ identifies a closest point $x$ on theunit 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 rationalapproximations and provide explicit, spherical instances.  We prove that floating-point numbers can only provide trivial solutions tothe sphere equation in $\mathbb{R}^2$ and $\mathbb{R}^3$. Moreover, we show howto 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 forsimultaneous Diophantine approximation.  Our open-source implementation and experiments demonstrate the practicalityof our approach in the context of massive data sets Geo-referenced by latitudeand longitude values.

Rational Points on the Unit Sphere: Approximation Complexity and Practical Constructions