Parametrization and smooth approximation of surface triangulations, Michael Floater, CAGD 1997. This paper is a classic for mapping a disk of triangles to the plane, using a linear least-squares solution. The boundary of the mesh is mapped to the boundary of the desired parameterization (usually a square). Each interior vertex tries to map to the centroid of its neighbors. This mapping is guaranteed not to fold, if the boundary is convex. Variations:

• Basic approach: Map the boundary to a convex, closed curve. (Usually a circle or square.) For each boundary vertex this creates a linear constraint vertex -> u,v. For each interior vertex, create a linear constraint that places the vertex at the centroid of its neighbors (1/n sum vj - vi = 0), vj is the star of the vertex, i.e., all of vi's neighbors. Solve the resulting system of equations.
• Variation: Instead of using 1/n, which equally weights all the neighboring vertices, choose any set of weights wi such that sum wi = 1 and wi are non-negative.
• Chord length variation: choose the weights to make the relative areas of the triangles be similar. Approach 1) use linear constraints on the interior edges, forcing them to be particular relative lengths by weighting them by lij. wi = lij / sum lij for all edges adjacent to vi.
• Chord length variation 2: For each vertex, take the star of the vertex and place it in the plane so that the interior edge lengths are the same, and the ratio of interior angles is preserved. Make wi be the area of the resulting triangles, if there are only three adjacent vertices. Otherwise, try to do the equivalent area ratio with the polygon defined by the star.