Selectable validity criteria

[sebmestrallet]

Instead of a single set of validity criteria (ours), it would be great to allow the user to choose Eppstein and Mumford 2010 ("simple orthogonal polyhedra") or He et al. 2024.

Ours

• all charts must have a valence of at least 4
• all boundaries must be between charts assigned to different axes. Optional: or between charts of the same axis if the solid angle is greater than 180°
• for each corner, the incident boundaries must not be associated to the same axes, they must be associable in pairs, or make a 𝑋𝑌𝑍 trio (to be revised, see Better validity criterion on corners #17)

Eppstein and Mumford 2010

(the usual validity criteria until 2023, see Mestrallet et al. 2023 section 2.2)

three-dimensional polyhedra with the topology of a sphere in which three mutually-perpendicular edges meet at each vertex

Which can be derived as:

• all charts must have a valence of at least 4
• all boundaries must be between charts assigned to different axes
• all corners must have a valence of 3

He et al. 2024

Validity criteria based on the Gauss-Bonnet theorem, that is the link between geometry (through Gauss curvature) and topology (through the Euler characteristic).

They count the number of $i$-connected corners (valence of $i$), the number of $V_i$ corners (quad-mesh valence of $i$) and the number of chart corners $T_i$ (having a polycube angle of $\frac{\pi}{2} i$).