## Identifying Vector Field Singularities using a Discrete Hodge Decomposition

in: Visualization and Mathematics III, H.C. Hege and K. Polthier (Eds.),
Springer Verlag (2003), pp. 113-134.

Konrad Polthier and Eike Preuß

### Abstract

We derive a Hodge decomposition of discrete vector fields on polyhedral
surfaces, and apply it to the identification of vector field singularities. This
novel approach allows us to easily detect and analyze singularities as critical
points of corresponding potentials. Our method uses a global variational
approach to independently compute two potentials whose gradient respectively
co-gradient are rotation-free respectively divergence-free components of the
vector field. The sinks and sources respectively vortices are then automatically
identified as the critical points of the corresponding scalar-valued potentials.
The global nature of the decomposition avoids the approximation problem of the
Jacobian and higher order tensors used in local methods, while the two
potentials plus a harmonic flow component are an exact decomposition of the
vector field containing all information.

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