Linear Surface Reconstruction from Discrete Fundamental Forms on Triangle Meshes
Article first published online: 28 JUN 2012
© 2012 The Authors Computer Graphics Forum © 2012 The Eurographics Association and Blackwell Publishing Ltd.
Computer Graphics Forum
Volume 31, Issue 8, pages 2277–2287, December 2012
How to Cite
Wang, Y., Liu, B. and Tong, Y. (2012), Linear Surface Reconstruction from Discrete Fundamental Forms on Triangle Meshes. Computer Graphics Forum, 31: 2277–2287. doi: 10.1111/j.1467-8659.2012.03153.x
- Issue published online: 26 OCT 2012
- Article first published online: 28 JUN 2012
- differential coordinates;
- first fundamental forms;
- second fundamental forms;
- surface deformation
- I.3.5 [Computer Graphics]: Computational Geometry and Object Modelling—Curve;
- solid and object representations
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.