{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/rational-points-on-the-unit-sphere","title":"Rational Points on the Unit Sphere: Approximation Complexity and Practical Constructions","arxiv_id":"1707.08549","date":"2017-07-26","proceeding":null,"authors":["Daniel Bahrdt","Martin P. Seybold"],"abstract":"Each non-zero point in $\\mathbb{R}^d$ identifies a closest point $x$ on the unit 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 rational approximations and provide explicit, spherical instances. We prove that floating-point numbers can only provide trivial solutions to the sphere equation in $\\mathbb{R}^2$ and $\\mathbb{R}^3$. Moreover, we show how to 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 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.","url_abs":"http://arxiv.org/abs/1707.08549v1","url_pdf":"http://arxiv.org/pdf/1707.08549v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"rational-points-on-the-unit-sphere","repo_url":"https://github.com/fmi-alg/libratss","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}