Integrable PolyVector Fields

Olga Diamanti, Amir Vaxman, Daniele Panozzo, Olga Sorkine-Hornung

Integrable Polyvector Fields

Parameterization using integrable fields. Given a set of user-provided constraints (left), a smooth tangent field can be created by interpolation, but the subsequent parameterization might not be well aligned with this field (middle). We optimize for an integrable field that respects the constraints, while exactly corresponding to a parameterization (right).


We present a framework for designing curl-free tangent vector fields on discrete surfaces. Such vector fields are gradients of locally-defined scalar functions, and this property is beneficial for creating surface parameterizations, since the gradients of the parameterization coordinate functions are then exactly aligned with the designed fields.

We introduce a novel definition for discrete curl between unordered sets of vectors (PolyVectors), and devise a curl-eliminating continuous optimization that is independent of the matchings between them. Our algorithm naturally places the singularities required to satisfy the user-provided alignment constraints, and our fields are the gradients of an inversion-free parameterization by design.


accompanying video (with narration)


We thank Marco Tarini for providing the software for the visualization of the vector fields, and Christian Schüller for discussions on optimization with barriers. This work was supported in part by the ERC Starting Grant iModel (StG-2012-306877), the Lise-Meitner grant M1618-N25 and grant P23735-N13 of the Austrian Science Fund (FWF).