Least Squares Conformal Maps for Automatic Texture Atlas Generation, Bruno Levy, Sylvain Petitjean, Nicolas Ray, Jerome Maillot, Siggraph 2002. A technique for producing a conformal mapping of a set of polygons. No guarantees about folding. Very similar to Desbrun's technique for conformal mapping, but more expensive to compute. Contains methods to split up the mesh into sections and arrange them in the texture space.

• Segmentation: Splitting up the mesh.
  • Mark the high-curvature features/edges.
  • Seed with a small number of charts.
  • Grown the charts until they bump into features.
  • Merge charts that are small.
  • Grow charts in parallel, always restarting at centroid.
• Parameterization: Flattening the mesh.
  • Minimizes angle deformations and non-uniform scalings.
  • Linear solver, therefore no local minimum.
  • No need to fix the boundary.
  • No flipping, but may overlap.
  • Conformality: Put the local coordinate frame so that (x,y) is in the plane of the triangle, z is up. Now the map from the triangle to the plane just depends on x and y. Express the map U from the triangle to the plane in complex/polar coordinates (x + iy). The constraint (on that triangle) for conformality is just dU/dx + i dU/dy = 0.

    Equations to convert this to a standard matrix solution.

• Packing: Arranging the texture pieces
  • Orient the pieces vertically.
  • Keep a curvy "horizon line".
  • Snug the next texture piece up to the horizon line.