Rational Points on the Unit Sphere: Approximation Complexity and Practical Constructions
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Each non-zero point in R^d identifies a 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 R^2 and R^3. Moreover, we show how to construct a rational point with denominators of at most 10(d-1)/ε^2 for any given ϵ∈(0, 1 8], 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.
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