Journal
ACM TRANSACTIONS ON GRAPHICS
Volume 35, Issue 3, Pages -Publisher
ASSOC COMPUTING MACHINERY
DOI: 10.1145/2870629
Keywords
Vector field design; covariant derivative; discrete connection; discrete differential geometry
Categories
Funding
- NSF [CCF-1011944, IIS-0953096, CMMI-1250261, III-1302285]
- Pixar Animations Studios
- Disney Animation Studios
- CAD/CG State Key Lab at Zhejiang University
- Directorate For Engineering
- Div Of Civil, Mechanical, & Manufact Inn [1250261] Funding Source: National Science Foundation
- Division of Computing and Communication Foundations
- Direct For Computer & Info Scie & Enginr [1011944] Funding Source: National Science Foundation
- Div Of Information & Intelligent Systems
- Direct For Computer & Info Scie & Enginr [0953096, 1302285] Funding Source: National Science Foundation
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In this article, we introduce a discrete definition of connection on simplicial manifolds, involving closed-form continuous expressions within simplices and finite rotations across simplices. The finite-dimensional parameters of this connection are optimally computed by minimizing a quadratic measure of the deviation to the (discontinuous) Levi-Civita connection induced by the embedding of the input triangle mesh, or to any metric connection with arbitrary cone singularities at vertices. From this discrete connection, a covariant derivative is constructed through exact differentiation, leading to explicit expressions for local integrals of first-order derivatives (such as divergence, curl, and the Cauchy-Riemann operator) and for L-2-based energies (such as the Dirichlet energy). We finally demonstrate the utility, flexibility, and accuracy of our discrete formulations for the design and analysis of vector, n-vector, and n-direction fields.
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