A shift-invariant C 1 2 -subdivision algorithm which rotates the lattice
In order to study differential properties of a subdivision surface at a markpoint, it is necessary to parametrise it over a so-called characteristic map defined as the infinite union of $C^k$-parametrised rings. Construction of this map is known when a single step of the subdivision scheme does not...
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| Published in | Computer aided geometric design Vol. 118; p. 102430 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier
01.05.2025
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0167-8396 |
| DOI | 10.1016/j.cagd.2025.102430 |
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| Abstract | In order to study differential properties of a subdivision surface at a markpoint, it is necessary to parametrise it over a so-called characteristic map defined as the infinite union of $C^k$-parametrised rings. Construction of this map is known when a single step of the subdivision scheme does not rotate a regular lattice. Otherwise, two steps are considered as they realign the lattice and its subdivided version. We present a new subdivision scheme which rotates the lattice and nevertheless allows a direct construction of the characteristic map. It is eigenanalysed with techniques introduced in a companion article and proved to define a $C_1^2$-algorithm around a face-centre. This scheme generalises Loop's scheme, allowing the designer to choose between extraordinary vertices or faces in regard to the shape of the mesh,the location of the extraordinary elements, and the aimed limit shape. |
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| AbstractList | In order to study differential properties of a subdivision surface at a markpoint, it is necessary to parametrise it over a so-called characteristic map defined as the infinite union of $C^k$-parametrised rings. Construction of this map is known when a single step of the subdivision scheme does not rotate a regular lattice. Otherwise, two steps are considered as they realign the lattice and its subdivided version. We present a new subdivision scheme which rotates the lattice and nevertheless allows a direct construction of the characteristic map. It is eigenanalysed with techniques introduced in a companion article and proved to define a $C_1^2$-algorithm around a face-centre. This scheme generalises Loop's scheme, allowing the designer to choose between extraordinary vertices or faces in regard to the shape of the mesh,the location of the extraordinary elements, and the aimed limit shape. |
| ArticleNumber | 102430 |
| Author | Sabin, Malcolm A. Gérot, Cédric |
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| Cites_doi | 10.1016/S0167-8396(01)00039-5 10.1007/s11155-005-6891-y 10.1145/263834.263851 10.1016/j.cagd.2014.06.002 10.1016/0010-4485(78)90111-2 10.1007/s003659910006 10.1007/s00607-006-0211-1 10.1016/0010-4485(78)90110-0 10.1016/0167-8396(94)00007-F 10.1016/j.cagd.2004.04.003 10.1145/1027411.1027415 10.1016/j.cagd.2004.04.006 10.1111/j.1467-8659.2006.00945.x 10.1137/S0036142996304346 10.1007/978-1-4757-2244-4 10.1016/j.cagd.2024.102391 10.1111/j.1467-8659.2008.01163.x 10.1016/j.cag.2021.10.008 |
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| Keywords | Differential analysis Subdivision Extraordinary face Characteristic map |
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| Title | A shift-invariant C 1 2 -subdivision algorithm which rotates the lattice |
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