Grooming with Cartan: vector field processing on triangle meshes

While dealing with scalar fields on surface meshes has been a staple of geometry processing, the need for discrete tangent vector fields on triangulated surface has grown steadily over the last two decades: they are crucial to encode both directions and sizing on surfaces as commonly required in tasks such as texture synthesis, non-photorealistic rendering, digital grooming (i.e., creating hair and fur on characters), and meshing. In this talk, we explain how Cartan's moving frame method can be easily discretized on triangle meshes, giving rise to intuitive notions of parallel transport, connection, holonomy, and torsion. We show how to combine these definitions to design tangent vector fields (or frame fields as well) on discrete surfaces with full control of the singularities through only linear algebra. We also show how the same ideas can be exploited in higher dimensions to generalize the well-known Isomap approach for nonlinear dimensionality reduction to non geodesically convex sampled domains, removing a long-standing limitation of manifold learning.