Robust Field-aligned Global Parametrization : Supplement 1 , Proofs and Algorithmic Details

The immediate consequence of this definition is that the input cross-field and the one computed from the parametrization as above should have identical singularity indices as well as equal holonomies on topologically equivalent loops. Torus with a single 3-5 pair. The nonexistence of a quadrangulation with a single 3-5 pair for a torus has been proved in [Barnette et al. 1971]. Remarkably, [Jucovič and Trenkler 1973] shows that for any choice of non-regular vertex valences, constrained by the formula for Euler’s characteristics, and any genus other than 1, a quadrangulation with vertices with these valences exists (if vertices of valence 4 can be added); and for tori, the 3-5 pair is the only exception. We note that this does not guarantee that for any vector field there is a quadrangulation consistent with it, as the topology of the field also includes holonomies along non-contractible loops. A torus with field rotation π/2 along a loop. If the field on a torus has no singularities, but undergoes π/2 rotation around a noncontractible loop, there is no quadrangulation with the same field topology. [Kurth 1986] shows that any regular quadrangulation of a torus is obtained from a fundamental parallelogram with vertices at integer points in the plane, with opposite sides glued together. We observe that u and v gradient fields as a result are globally defined on the torus, and turning number of either field is zero along any loop. We conclude that regular quadrangulations are not compatible with the field. As the field has no singularities, no other quadrangulation can be compatible.