  While neural representations for static 3D shapes are widely studied,
representations for deformable surfaces are limited to be template-dependent or
lack efficiency. We introduce Canonical Deformation Coordinate Space (CaDeX), a
unified representation of both shape and nonrigid motion. Our key insight is
the factorization of the deformation between frames by continuous bijective
canonical maps (homeomorphisms) and their inverses that go through a learned
canonical shape. Our novel deformation representation and its implementation
are simple, efficient, and guarantee cycle consistency, topology preservation,
and, if needed, volume conservation. Our modelling of the learned canonical
shapes provides a flexible and stable space for shape prior learning. We
demonstrate state-of-the-art performance in modelling a wide range of
deformable geometries: human bodies, animal bodies, and articulated objects.
