Embedding assumption #4
Description
Currently it is assumed that the optimization is done on a mesh that has no initial flips. This assumption ensures that for open meshes the degrees of freedom of a symmetric mesh are easy to identify (i.e., the edges of a single copy) and the length dependencies are all trivial (i.e., identifications).
However, this complicates several other elements of the optimization, such as approximating the symmetric dirichlet energy.
With the current refactor to encapsulate the differentiable map from independent coordinates to redundant halfedge coordinates, it should now be fairly straightforward to incorporate a general initial symmetric mesh for optimization with dependent diagonal lengths handled correctly. The change would amount to an additional Jacobian matrix computation for the independent to dependent variable map.
The independent variables would be edges of type 1 and 3 (those crossing the symmetry line), with edges of type 2 identified with 1 and diagonal lengths computed using Ptolemy's formula.