Journal
COMPUTER AIDED GEOMETRIC DESIGN
Volume 30, Issue 1, Pages 20-34Publisher
ELSEVIER SCIENCE BV
DOI: 10.1016/j.cagd.2012.09.002
Keywords
Dimension formula; Spline space; T-mesh; Homology
Funding
- 973 Program [2011CB302400]
- NSF of China [60873109, 11031007, 61073108]
- Program for New Century Excellent Talents in University [NCET-08-0514]
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This paper discusses the dimension of spline spaces S(m, n, m - 1, n - 1, I) over a certain type of hierarchical T-mesh. The key step is to establish a bijection between the spline space S(m, n, m - 1, n - 1, I) and a univariate spline space defined in terms of the l-edges of the extended T-mesh with respect to bi-degree (m, n). We decompose the univariate spline space into an isomorphic direct sum using the theory of short exact sequences from homological algebra. Using this decomposition, we can obtain a formula for the dimension of the spline space S(m, n, m - 1, n - 1, I) over the required type of hierarchical T-mesh. We also construct a set of basis functions for the spline space. (C) 2012 Elsevier B.V. All rights reserved.
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