Surface parameterization is all about finding a suitable one-to-one mapping from a given surface in 3D to the plane. Kai Hormann and I wrote two survey papers on this topic: (1) survey1 for the Munich summer school on multiresolution in geometric modelling, August 2001, and (2) survey2 for the Cambridge workshop on multiresolution in geometric modelling, September 2003.
There is a C++ parameterization library for parameterizing triangular meshes and point clouds, available as free software under the GPL licence from SINTEF. Look for the `GoTools parametrization module' on the SINTEF software page.
Usually the surface is represented by or approximated by a triangular mesh. There are many applications, such as smooth surface fitting and texture mapping. I developed a method (CAGD 1997) which involves solving a sparse linear system of equations arising from convex combinations. When the boundary is fixed and convex, the mapping is guaranteed to be one-to-one, i.e. the mapped triangular mesh never folds over. In contrast, the standard discretization of a harmonic map (Eck et al 1996) is not always one-to-one, and I constructed a simple example of this using a mesh with only three triangles in (IJSM 1998). In the same paper I pointed out that, unlike discrete harmonic maps, the convex combination method also applies to arbitrary polygonal meshes (`tilings').
A closely related `convex combination' method was developed jointly with Craig Gotsman (JCAM 1999) for morphing pairs of compatible planar triangular meshes, i.e., continuously deforming one mesh into the other without foldover. More recently, Martin Reimers and I (CAGD 2001) proposed a more general convex combination method for parameterizing point clouds, which we called meshless parameterization. By triangulating the parameter points, we build a triangulation of the original 3D points, often referred to as surface reconstruction. More numerical examples can be found in (Math. of Surfaces 2000).
Between 2000 and 2004, I worked with several visiting researchers, Craig Gotsman, Vitaly Surazhsky, Kai Hormann, and Valerie Pham-Trong, within the MINGLE project on these and related topics.
Kai, Martin, and I (Approximation Theory X 2002) developed a method for parameterizing manifold triangular meshes.
In 2003 I worked out a new proof (Math. Comp. 2003) that convex combination maps over triangular meshes are one-to-one when the domain boundary is mapped to a convex polygon. The proof is much simpler than Tutte's original, and uses no graph theory. The basic idea for the proof is based on Kneser's proof of the Rado-Kneser-Choquet theorem that harmonic maps are one-to-one when they map the domain boundary to a convex curve. For a summary and some open problems consult (Algorithms for Approximation IV, 2002).
Valerie Pham-Trong and I extended the simple proof of Tutte's theorem to general planar graphs ("tilings") and also showed that convex combination maps over tetrahedral meshes (in 3D) are NOT necessarily one-to-one (AiCM 2004).
We also know that meshless parameterization `works' for curves, i.e., if the point sample from a given curve is dense enough and the curve is tangent continuous and has no self-intersections, then the parameterized points will have the same ordering as the sampled points (NA 2003). We so far have no theoretical results on the surface case, and this is an interesting topic for future research.
In (CAGD 2003), I found a simpler way of choosing `good' convex combinations for parameterization. These mean value coordinates come from forcing a piecewise linear function over a triangulation to satisfy the mean value theorem (for harmonic functions) at every interior vertex.
The discrete harmonic map was proposed in