Power-Linear Polar Directional Fields

Fig. 1. We compute rotationally-smooth directional fields by prescribing their topology explicitly. Our fields (left) are piecewise power-linear, and therefore allow for singularities of any index defined at any point on the mesh, including on edges and inside faces. This generalizes common piecewise-constant approaches (right). Our approach results in smoother fields in comparison, even for uneven and coarse meshes.

We introduce a novel method for directional-field design on meshes, enabling users to specify singularities at any location on a mesh. Our method uses a piecewise power-linear representation for phase and scale, offering precise control over field topology. The resulting fields are smooth and accommodate any singularity index and field symmetry. With this representation, we mitigate the artifacts caused by coarse or uneven meshes. We showcase our approach on meshes with diverse topologies and triangle qualities.