{"paper":{"title":"A canonical parameterization of paths in $\\mathbb{R}^n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GN","authors_text":"E. D. Tymchatyn, L. C. Hoehn, L. G. Oversteegen","submitted_at":"2013-01-25T15:41:36Z","abstract_excerpt":"For sufficiently tame paths in $\\mathbb{R}^n$, Euclidean length provides a canonical parametrization of a path by length. In this paper we provide such a parametrization for all continuous paths. This parametrization is based on an alternative notion of path length, which we call $\\mathsf{len}$. Like Euclidean path length, $\\mathsf{len}$ is invariant under isometries of $\\mathbb{R}^n$, is monotone with respect to sub-paths, and for any two points in $\\mathbb{R}^n$ the straight line segment between them has minimal $\\mathsf{len}$ length.\n  Unlike Euclidean path length, the $\\mathsf{len}$ length"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.6070","kind":"arxiv","version":2},"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"}