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.

Based on a new spectral vector field analysis on triangle meshes, we construct a compact representation for near conformal mesh surface correspondences. Generalizing the functional map representation, our representation uses the map between the low‐frequency tangent vector fields induced by the correspondence. While our representation is as efficient, it is also capable of handling a more generic class of correspondence inference. We also formulate the vector field preservation constraints and regularization terms for correspondence inference, with function preservation treated as a special case. A number of important vector field–related constraints can be implicitly enforced in our representation, including the commutativity of the mapping with the usual gradient, curl, divergence operators or angle preservation under near conformal correspondence. For function transfer between shapes, the preservation of function values on landmarks can be strictly enforced through our gradient domain representation, enabling transfer across different topologies. With the vector field map representation, a novel class of constraints can be specified for the alignment of designed or computed vector field pairs. We demonstrate the advantages of the vector field map representation in tests on conformal datasets and near‐isometric datasets.

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The thermal diffusivity of elastomers (i.e., rubber-like materials) can change substantially with elastic finite deformation. Initially isotropic elastomers may be thermally anisotropic when deformed. Data from several experimental studies demonstrate significant changes in the thermal conductivity or diffusivity tensor with finite deformation. Formulating the thermal diffusivity tensor and deformation in terms of the reference configuration may aid in the development of constitutive relations by use of material symmetry. Illustrated here is a relationship between the diffusivity and deformation of representative materials during uniaxial and equibiaxial deformation. Each component of the diffusivity tensor appears to be related to the deformation in the direction of the component only.