What are the parameters in mesh parameterization?

Question

I recently heard about mesh parameterization as a way of parameterizing the geometry of an arbitrary mesh by mapping it onto a surface, like a sphere. I am not a mathematician, so I don't know how to formally define this problem, but can someone help me understand (in simple terms) what exactly is meant by parameterizing a mesh? Is this parameterization a function, and if so, is there a closed-form equation for it? Or, are the parameters a fixed set of coefficients?

Thanks so much.

Answer 1

One sensible interpretation of "mesh parameterization" is that it describes the process of fitting a parametric surface to a triangular mesh.

A parametric surface in three dimensions is a function $f(u,v) = (x,y,z) : R2 \rightarrow R3$ and a domain for the parameters $u,v$

What are the parameters?

"parameters" refers to $u,v$, the inputs to the parametric equation that yields a surface which resembles your triangular mesh. It does not refer to the parameters in some function that yields a parameterization.

The best fitting parametric surface and method of finding it will entirely depend on how you define "best".

Is this parameterization a function, and if so, is there a closed-form equation for it?

The thing you are describing, "mesh parameterization", is an extremely general idea. Depending on the goals you could approach it many different ways and end up with very different parameterizations. It does not refer to any specific closed-form algorithm.

Here is a paper that surveys different approaches to mesh parameterization, their definition is even more general than just surface fitting.

[1] Mesh Parameterization Methods and Their Applications, 2006 https://www.cs.ubc.ca/~sheffa/papers/CGV011-journal.pdf

Given any two surfaces with similar topology, it is possible to compute a one-to-one and onto mapping between them. If one of these surfaces is represented by a triangular mesh, the problem of computing such a mapping is referred to as mesh parameterization [7, 35]. The surface that the mesh is mapped to is typically referred to as the parameter domain. ...

The purpose of mesh parameterization is to obtain a map between such a mesh and a triangulation of a domain. The map is piecewise linear, associating each triangle of the original mesh with a triangle in the domain. An important goal of parameterization is to obtain bijective (invertible) maps, where each point on the domain corresponds to exactly one point of the mesh. ...

One of the earlier applications of mesh parameterization is surface fitting. Many applications in geometry processing require a smooth analytical surface to be constructed from an input mesh.