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In this paper, we interpret these schemes as defining bases for\n            <jats:italic>discrete differential 0- resp. 2-forms<\/jats:italic>\n            , and complete the picture by introducing\n            <jats:italic>edge-based<\/jats:italic>\n            subdivision schemes to construct the missing bases for discrete differential 1-forms. Such subdivision schemes map scalar coefficients on edges from the coarse to the refined mesh and are\n            <jats:italic>intrinsic<\/jats:italic>\n            to the surface. Our construction is based on treating vertex-, edge-, and face-based subdivision schemes as a\n            <jats:italic>joint triple<\/jats:italic>\n            and enforcing that subdivision commutes with the topological exterior derivative. We demonstrate our construction for the case of arbitrary topology triangle meshes. Using Loop's scheme for 0-forms and generalized half-box splines for 2-forms results in a\n            <jats:italic>unique<\/jats:italic>\n            generalized spline scheme for 1-forms, easily incorporated into standard subdivision surface codes. We also provide corresponding boundary stencils. Once a metric is supplied, the scalar 1-form coefficients define a smooth tangent vector field on the underlying subdivision surface. Design of tangent vector fields is made particularly easy with this machinery as we demonstrate.\n          <\/jats:p>","DOI":"10.1145\/1141911.1141991","type":"journal-article","created":{"date-parts":[[2006,7,25]],"date-time":"2006-07-25T14:14:26Z","timestamp":1153836866000},"page":"1041-1048","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":42,"title":["Edge subdivision schemes and the construction of smooth vector fields"],"prefix":"10.1145","volume":"25","author":[{"given":"Ke","family":"Wang","sequence":"first","affiliation":[{"name":"Caltech"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"family":"Weiwei","sequence":"additional","affiliation":[{"name":"Caltech"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yiying","family":"Tong","sequence":"additional","affiliation":[{"name":"Caltech"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mathieu","family":"Desbrun","sequence":"additional","affiliation":[{"name":"Caltech"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Peter","family":"Schr\u00f6der","sequence":"additional","affiliation":[{"name":"Caltech"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2006,7]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"crossref","unstructured":"Arnold D. 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