Abstract
In this paper we construct an edge based, or 1-form, subdivision scheme consistent with \(\sqrt{3}\) subdivision. It produces smooth differential 1-forms in the limit. These can be identified with tangent vector fields, or viewed as edge elements in the sense of finite elements. In this construction, primal (0-form) and dual (2-form) subdivision schemes for surfaces are related through the exterior derivative with an edge (1-form) based subdivision scheme, amounting to a generalization of the well known formulé de commutation.
Starting with the classic \(\sqrt{3}\) subdivision scheme as a 0-form subdivision scheme, we derive conditions for appropriate 1- and 2-form subdivision schemes without fixing the dual (2-form) subdivision scheme a priori. The resulting degrees of freedom are resolved through spectrum considerations and a conservation condition analogous to the usual moment condition for primal subdivision schemes.
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Huang, J., Schröder, P. (2012). \(\sqrt{3}\)-Based 1-Form Subdivision. In: Boissonnat, JD., et al. Curves and Surfaces. Curves and Surfaces 2010. Lecture Notes in Computer Science, vol 6920. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27413-8_22
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DOI: https://doi.org/10.1007/978-3-642-27413-8_22
Publisher Name: Springer, Berlin, Heidelberg
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