A Caltech Library Service

Discrete Connection and Covariant Derivative for Vector Field Analysis and Design

Liu, Beibei and Tong, Yiying and de Goes, Fernando and Desbrun, Mathieu (2016) Discrete Connection and Covariant Derivative for Vector Field Analysis and Design. ACM Transactions on Graphics, 35 (3). Art. No. 23. ISSN 0730-0301. doi:10.1145/2870629.

Full text is not posted in this repository. Consult Related URLs below.

Use this Persistent URL to link to this item:


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.

Item Type:Article
Related URLs:
URLURL TypeDescription
Desbrun, Mathieu0000-0003-3424-6079
Additional Information:© 2016 ACM. Received August 2014; revised November 2015; accepted December 2015. This work was completed in January 2014, and we are grateful to Patrick Mullen for proofreading the first version of this article, Max Budninskiy for comments, and Santiago Lombeyda for his timely help with figures. Visualization of vector fields was done using the code from Palacios and Zhang [2007]. The authors also acknowledge funding from NSF grants CCF-1011944, IIS-0953096, CMMI-1250261, and III-1302285, and the support of Pixar Animations Studios, Disney Animation Studios, and Google. MD gratefully thanks the Inria International Chair program and all the members of the TITANE team for support. YT thanks the CAD/CG State Key Lab at Zhejiang University for support.
Funding AgencyGrant Number
Pixar Animation StudiosUNSPECIFIED
Disney Animation StudiosUNSPECIFIED
Subject Keywords:Vector field design, covariant derivative, discrete connection, discrete differential geometry
Issue or Number:3
Record Number:CaltechAUTHORS:20160316-072837925
Persistent URL:
Official Citation:Beibei Liu, Yiying Tong, Fernando de Goes, and Mathieu Desbrun. 2016. Discrete connection and covariant derivative for vector field analysis and design. ACM Trans. Graph. 35, 3, Article 23 (March 2016), 17 pages. DOI:
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:65381
Deposited By: Tony Diaz
Deposited On:16 Mar 2016 20:29
Last Modified:10 Nov 2021 23:44

Repository Staff Only: item control page