{"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/cycles-cross-ratio-an-invitation","title":"Cycles Cross Ratio: an Invitation","arxiv_id":"2105.05634","date":"2021-05-12","proceeding":null,"authors":["Vladimir V. Kisil"],"abstract":"The paper introduces cycles cross ratio, which extends the classic cross ratio of four points to various settings: conformal geometry, Lie spheres geometry, etc. Just like its classic counterpart cycles cross ratio is a measure of anharmonicity between spheres with respect to inversion. It also provides a M\\\"obius invariant distance between spheres. Many further properties of cycles cross ratio awaiting their exploration. In abstract framework the new invariant can be considered in any projective space with a bilinear pairing.","url_abs":"https://arxiv.org/abs/2105.05634v6","url_pdf":"https://arxiv.org/pdf/2105.05634v6.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":"cycles-cross-ratio-an-invitation","repo_url":"https://github.com/vvkisil/Cycles-cross-ratio-Invitation","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}