noamaig / euclidean_orbifolds

Matlab implementation of the 2015 Siggraph Asia paper, "Orbifold Tutte Embeddings".

Orbifold Tutte Embeddings

Matlab code implementing the Siggraph Asia 2015 paper, "Orbifold Tutte Embeddings".

An extension of Tutte's embedding to domains with sphere topology, and also to disks, where the boundary is free to move.

Main high-level functions:

• flatten_sphere - embed a sphere mesh to a sphere orbifold.
• flatten_disk - embed a disk mesh to a disk orbifold.
• map_spheres - compute a map between two sphere meshes using the orbifold embeddings.
• map_disks - compute a map between two disk meshes using the orbifold embeddings.

There are 17 Euclidean orbifolds, of which only the 4 sphere orbifolds, and 2 disk orbifolds are implemented here. The 11 other cases can be easily implemented, if one wishes, by extending the cutting mechanism (needs to produce a cut-graph that opens the mesh into a disk) and the appropriate boundary conditions.

Example scripts:

• script_embed_sphere - map a spherical mesh to the Euclidean orbifold of type I.
• script_embed_sphere_4_points - map a spherical mesh to the Euclidean orbifold of type IV, first with the initial embedding, and then after modifying the configuration so as to minimize conformal distortion.
• script_embed_disk - map a mesh with disk topology to a disk orbifold.
• script_surface_map_spheres - use the sphere orbifold embedding to compute a seamless, conformal map between two human heads which interpolates 3 given correspondences, and then visualize the map with a cute animation of a morph between the two meshes.
• script_surface_map_disk - use the disk orbifold embedding to compute a conformal map between two disk-meshes which interpolates 3 designated points on the boundary.

The code is provided as-is for academic use only and without any guarantees. Please contact the author to report any bugs. Written by Noam Aigerman.