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Shape from metric

Chern, Albert and Knöppel, Felix and Pinkall, Ulrich and Schröder, Peter (2018) Shape from metric. ACM Transactions on Graphics, 37 (4). Art. No. 63. ISSN 0730-0301. doi:10.1145/3197517.3201276.

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We study the isometric immersion problem for orientable surface triangle meshes endowed with only a metric: given the combinatorics of the mesh together with edge lengths, approximate an isometric immersion into R^3. To address this challenge we develop a discrete theory for surface immersions into R^3. It precisely characterizes a discrete immersion, up to subdivision and small perturbations. In particular our discrete theory correctly represents the topology of the space of immersions, i.e., the regular homotopy classes which represent its connected components. Our approach relies on unit quaternions to represent triangle orientations and to encode, in their parallel transport, the topology of the immersion. In unison with this theory we develop a computational apparatus based on a variational principle. Minimizing a non-linear Dirichlet energy optimally finds extrinsic geometry for the given intrinsic geometry and ensures low metric approximation error. We demonstrate our algorithm with a number of applications from mathematical visualization and art directed isometric shape deformation, which mimics the behavior of thin materials with high membrane stiffness.

Item Type:Article
Related URLs:
URLURL TypeDescription
Schröder, Peter0000-0002-0323-7674
Additional Information:© 2018 Association for Computing Machinery. This work was supported in part by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics.” Additional support was provided by SideFX software. The Bunny model is courtesy Stanford Computer Graphics laboratory and the Duck model courtesy Keenan Crane.
Funding AgencyGrant Number
Deutsche Forschungsgemeinschaft (DFG)TRR 109
Subject Keywords:Differential geometry, geometry processing, isometric immersions, spin structures, quaternions
Issue or Number:4
Record Number:CaltechAUTHORS:20180731-140714307
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Official Citation:Albert Chern, Felix Knöppel, Ulrich Pinkall, and Peter Schröder. 2018. Shape from Metric. ACM Trans. Graph. 37, 4, Article 63 (August 2018), 17 pages.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:88391
Deposited By: Tony Diaz
Deposited On:31 Jul 2018 22:05
Last Modified:16 Nov 2021 00:26

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