Papers › Rational Points on the Unit Sphere: Approximation Complexity and Practical Constructions
Rational Points on the Unit Sphere: Approximation Complexity and Practical Constructions
Daniel Bahrdt, Martin P. Seybold
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Each non-zero point in ℝᵈ identifies a 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 10(d-1)/ε² 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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