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Input: Mesh of triangles . For some triangles, we have a confident estimate of the local min / max curvature directions (e.g. computed using the Shape Operator) given by an angle relative to some reference edge .

Smooth Cross Field
We propagate a smooth cross field over the entire surface. For each edge we can define a smoothness energy

where  and are the cross field directions given as an angle with the reference edges and , respectively. denotes the angle between the two reference edges.
is an integer variable that allows rotations of the cross fields.
For the entire mesh, we construct the system

We solve this system for the real variables and the integer variables . For those triangles where an input cross field direction is known, we fix the value: .

Parametrization
For each triangle corner in the mesh, compute coordinates such that for each triangle,

becomes small. Globally:


In order to obtain a disk topology for parametrization, the surface needs to be cut open. As an additional constraint, we thus introduce the requirement, that the parametrization should be consistent across boundary edges. That means that the transformation of parameters across an edge should be a grid automorphism.

Specifically, given a boundary edge between two triangles and we constrain the coordinates at the opposite sides of the vertices to be equal up to grid automorphisms. That means:


The integer variable is known from the smooth cross field. For , , we need to find integer solutions.

In order to ensure that the resulting mesh will be a quad mesh, we snap singularities to integer positions.

We can introduce additional hard alignment constraints in order to snap feature edges to integer iso-lines. If an edge should be snapped to a -integer-iso-line, we constrain the values of for vertices of to be integer.

Solving Mixed-Integer Problems
In general: NP-Hard. We approximate a solution by relaxing the problem: Find a real-valued solution. Snap integer variables to nearest possible value, repeat.