Computing IGA-suitable planar parameterizations by PolySquare-enhanced domain partition

We propose an algorithm for computing injective IGA-suitable planar parameterizations with as uniform and orthogonal as possible iso-parametric structure. Central to this approach is a PolySquare-enhanced domain partition procedure that decomposes the input complex planar domain into coarse and square-like quad patches. First, we triangulate the input planar domain that may be multiply connected, and deform it to be a PolySquare-like structure by optimizing a boundary alignment energy. Then, the PolySquare-like structure is pixelated to make the input domain be a quadrilateral mesh that is decimated to generate a coarse patch layout, where each patch is an approximately squared quadrilateral. Finally, the sparse patches are subdivided to represent the input domain and produce the resulting partition. We parameterize each patch with continuous constraints to find the parameterization of the input domain. Compared with existing IGA-suitable planar parameterization methods, our method produces better parameterizations and fewer patches than the ones that also use domain partition strategy. We demonstrate the superiority of our method over various complex domains, including an example containing 30 holes.