Linear Surface Reconstruction from Discrete Fundamental Forms on Triangle Meshes
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Date
2012
Journal Title
Journal ISSN
Volume Title
Publisher
The Eurographics Association and Blackwell Publishing Ltd.
Abstract
We present a linear algorithm to reconstruct the vertex coordinates for a surface mesh given its edge lengths and dihedral angles, unique up to rotation and translation. A local integrability condition for the existence of an immersion of the mesh in 3D Euclidean space is provided, mirroring the fundamental theorem of surfaces in the continuous setting (i.e. Gauss's equation and the Mainardi–Codazzi equations) if we regard edge lengths as the discrete first fundamental form and dihedral angles as the discrete second fundamental form. The resulting sparse linear system to solve for the immersion is derived from the convex optimization of a quadratic energy based on a lift from the immersion in the 3D Euclidean space to the 6D rigid motion space. This discrete representation and linear reconstruction can benefit a wide range of geometry processing tasks such as surface deformation and shape analysis. A rotation‐invariant surface deformation through point and orientation constraints is demonstrated as well.We present a linear algorithm to reconstruct the vertex coordinates for a surface mesh given its edge lengths and dihedral angles, unique up to rotation and translation. A local integrability condition for the existence of an immersion of the mesh in 3D Euclidean space is provided, mirroring the fundamental theorem of surfaces in the continuous setting (i.e., Gauss's equation and the Mainardi‐Codazzi equations) if we regard edge lengths as the discrete first fundamental form and dihedral angles as the discrete second fundamental form.
Description
@article{10.1111:j.1467-8659.2012.03153.x,
journal = {Computer Graphics Forum},
title = {{Linear Surface Reconstruction from Discrete Fundamental Forms on Triangle Meshes}},
author = {Wang, Y. and Liu, B. and Tong, Y.},
year = {2012},
publisher = {The Eurographics Association and Blackwell Publishing Ltd.},
ISSN = {1467-8659},
DOI = {10.1111/j.1467-8659.2012.03153.x}
}