{"paper":{"title":"$G 1$-smooth splines on quad meshes with 4-split macro-patch elements","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Ahmed Blidia (AROMATH), Bernard Mourrain (AROMATH), Nelly Villamizar","submitted_at":"2017-03-20T12:47:58Z","abstract_excerpt":"We analyze the space of differentiable functions on a quad-mesh  $\\cM$, which are composed of 4-split spline macro-patch elements on  each quadrangular face.  We describe explicit transition maps across  shared edges, that satisfy conditions which ensure that the space of  differentiable functions is ample on a quad-mesh of arbitrary  topology.  These transition maps define a finite dimensional vector  space of $G^{1}$ spline functions of bi-degree $\\le (k,k)$ on each  quadrangular face of $\\cM$.  We determine the dimension of this space  of $G^{1}$ spline functions for $k$ big enough and prov"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.06717","kind":"arxiv","version":1},"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"}