Each non-zero point in R^d identifies one closest point x on the unit sphere S^d-1. We are interested in computing an ε-approximation y ∈ Q^d for x, that is exactly on 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 R² and R³. Moreover, we show how to construct a rational point with denominators of at most 32(d-1)²/ε² for any given ε, 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.