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=DCP (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 algorithm described here:
Finally, here’s an even more general problem. How do we construct smooth mappings of one surface onto another? Mapping onto a simple base polyhedron is now just a special case.
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]
The GSP paper is an improvement of the earlier MAPS approach. Reading about the MAPS algorithm might make it easier to understand the GSP algorithm:
To fully appreciate Inter-surface mapping by Schreiner et al., I recommend the following papers:
V. Kraevoy and A. Sheffer. Cross-parameterization and compatible remeshing of 3D models. In Proceedings of SIGGRAPH 2004. [PDF]
P. V. Sander, J. Snyder, S. J. Gortler, and H. Hoppe. Texture mapping progressive meshes. In Proceedings of SIGGRAPH 2001. [PDF]
Another good choice of edge weighting scheme in linear parameterization methods is Floater’s mean value coordinate construction: