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

Daniel Bahrdt, Martin P. Seybold

Abstract: Each non-zero point in Rd identifies one closest point x on the unit sphere Sd-1. We are interested in computing an ε-approximation y ∈ Qd for x, that is exactly on Sd-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.