{"paper":{"title":"On the zone of a circle in an arrangement of lines","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Gabriel Nivasch","submitted_at":"2015-03-11T19:36:59Z","abstract_excerpt":"Let $\\mathcal L$ be a set of $n$ lines in the plane, and let $C$ be a convex curve in the plane, like a circle or a parabola. The \"zone\" of $C$ in $\\mathcal L$, denoted $\\mathcal Z(C,\\mathcal L)$, is defined as the set of all cells in the arrangement $\\mathcal A(\\mathcal L)$ that are intersected by $C$. Edelsbrunner et al. (1992) showed that the complexity (total number of edges or vertices) of $\\mathcal Z(C,\\mathcal L)$ is at most $O(n\\alpha(n))$, where $\\alpha$ is the inverse Ackermann function. They did this by translating the sequence of edges of $\\mathcal Z(C,\\mathcal L)$ into a sequence "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.03462","kind":"arxiv","version":6},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}