Boolean Operations using Generalized Winding Numbers

The generalized winding number function measures insideness for arbitrary oriented triangle meshes. Exploiting this, I similarly generalize binary boolean operations to act on such meshes. The resulting operations for union, intersection, difference, etc. avoid volumetric discretization or pre-processing. 1 Booleans & Classic Winding Numbers If A ⊂ R and B ⊂ R are open subregions of space, then their union contains all points in A or B, their intersection all points in A and B, and the difference of B fromA all points inA but not B. Written in set notation, we have respectively: A ∪ B = { p | p ∈ A or p ∈ B } , (1) A ∩ B = { p | p ∈ A and p ∈ B } , (2) A \ B = { p | p ∈ A and p / ∈ B } . (3) Meanwhile, the winding number function wA : R \ ∂A → Z determines for every point whether it is inside the set A purely by examining the set’s oriented boundary ∂A. The winding number integrates the signed surface area of ∂A projected onto a unit ball around a given point p, or in polar coordinates and w.l.o.g. p = 0: