There are many mesh processing methods that become much more tractable when we’re provided with a mapping of a complex surface onto a simple domain. The mapping of surfaces into the plane is also of obvious importance in texture mapping.
Our first topic will focus on conformal mappings of a triangulated surface into the plane.
M. Desbrun, M. Meyer, and P. Alliez. Intrinsic parameterizations of surface meshes. In Eurographics 2002 Conference Proceedings. [PDF]
B. Lévy, S. Petitjean, N. Ray, and J. Maillot. Least squares conformal maps for automatic texture atlas generation In Proceedings of SIGGRAPH 2002. [PDF]
Plus, you’ll want to read why LSCM=DNCP (from David Cohen-Steiner and Mathieu Desbrun).
Next, we’ll be focusing on a slightly different problem. Namely: how can we construct a parameterization of an input mesh onto a simpler base polyhedron? One way of doing this is provided by the MAPS algorithm.
I highly recommend the fine surveys written by Michael Floater and Kai Hormann. In the context of this course, you’ll probably find the second one to be the most helpful.
M. Floater and K. Hormann. Parameterization of triangulations and unorganized points, in Tutorials on Multiresolution in Geometric Modelling, A. Iske, E. Quak, and M. S. Floater (eds.), Springer-Verlag, Heidelberg (2002), 287–315. [PS.gz]
M. Floater and K. Hormann. Surface parameterization: A tutorial and survey. [PDF]
Here’s a later improvement of the MAPS approach that produces a smoother parameterization of the mesh:
Another good choice of edge weighting scheme in linear parameterization methods is Floater’s mean value coordinate construction:
Wei-Wen Feng presented the Desbrun et al. paper on intrinsic parameterizations.