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As a result, most spline constructions for such domains typically focus on regular (or semi-regular) knot geometries. In the planar harmonic case, we show that the discrete Laplacian plays a role similar to that of the divided differences and can be used to define well-behaved harmonic B-splines. In our main contribution, we then construct an analogous discrete bi-Laplacian for both planar and curved domains and show that its corresponding biharmonic B-splines are also well-behaved. Finally, we derive a fully irregular, discrete refinement scheme for these splines that generalizes knot insertion for univariate B-splines.<\/jats:p>","DOI":"10.1145\/2185520.2185611","type":"journal-article","created":{"date-parts":[[2012,8,6]],"date-time":"2012-08-06T18:11:37Z","timestamp":1344276697000},"page":"1-11","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":17,"title":["Discrete bi-Laplacians and biharmonic b-splines"],"prefix":"10.1145","volume":"31","author":[{"given":"Powei","family":"Feng","sequence":"first","affiliation":[{"name":"Rice University"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Joe","family":"Warren","sequence":"additional","affiliation":[{"name":"Rice University"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2012,7]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/17.3.343"},{"key":"e_1_2_2_2_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0925-7721(01)00018-9"},{"key":"e_1_2_2_3_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01203417"},{"key":"e_1_2_2_4_1","doi-asserted-by":"crossref","unstructured":"Buhmann M. 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