Mesh Parameterization
Parameterization corresponds to finding a mapping from a discrete surface patch to an isomorphic planar mesh. Each mesh node in the surface will be assigned to coordinates (u,v) in the planar region. Such a one-to-one mapping provides a flat parametric space, allowing one to perform any complex operation directly on the flat domain rather than on the curved surface. This allows for mesh processing like texture mapping, remeshing, etc. in the flat parametric space.
Discrete Conformal Parameterization
We look for a smooth parameterization of triangulated surface patches that minimize distortion. Distortion is measured for different intrinsic measures of the original mesh, hence leading to Intrinsic parameterization. One such measure is the angle in the triangulation, and conformal mapping is angle-preserving. Such a parameterization can be computed using a simple, sparse, linear system. Here, additionally, a boundary constraint is needed to make the system uniquely solvable.
​
Given a surface mesh with boundary:
1. Detect surface boundary loops and fix the boundary vertices in the planar region while preserving length between boundary points.
2. Solve the linear system to find the location of internal vertices.
3. Visualize the flattened mesh using the normal of the original 3D mesh
​
Refer here for the technical details
Figure 2. Some more examples of surfaces and their parameterizations
Texture mapping
This is a typical application of parameterization. Here, we generate a texture board pattern in flattened mesh thus coloring triangles. The corresponding faces in the original mesh are visualized using the same colors. Thus, mapping textures onto surface patches is as simple as mapping texture to the planar region. Texture mapping additionally allows evaluating the robustness of parameterization.
​
​
Figure 3. Surface patches are mapped with textures using their isomorphic parameterizations
Full Report
Source code
References
Desbrun, Mathieu et al. “Intrinsic Parameterizations of Surface Meshes.” Comput. Graph. Forum 21 (2002): 209-218.