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Any closed manifold of genus g can be cut open to form a topological disk andthen mapped to a regular polygon with 4g sides. This construction is called thecanonical polygonal schema of the manifold, and is a key ingredient for manyapplications in graphics and engineering, where a parameterization between twoshapes with same topology is often needed. The sides of the 4g-gon define onthe manifold a system of loops, which all intersect at a single point and aredisjoint elsewhere. Computing a shortest system of loops of this kind isNP-hard. A computationally tractable alternative consists in computing a set ofshortest loops that are not fully disjoint in polynomial time, using the greedyhomotopy basis algorithm proposed by Erickson and Whittlesey, and then detachthem in post processing via mesh refinement. Despite this operation isconceptually simple, known refinement strategies do not scale well for highgenus shapes, triggering a mesh growth that may exceed the amount of memoryavailable in modern computers, leading to failures. In this paper we studyvarious local refinement operators to detach cycles in a system of loops, andshow that there are important differences between them, both in terms of meshcomplexity and preservation of the original surface. We ultimately propose twonovel refinement approaches: the former minimizes the number of new elements inthe mesh, possibly at the cost of a deviation from the input geometry. Thelatter allows to trade mesh complexity for geometric accuracy, boundingdeviation from the input surface. Both strategies are trivial to implement, andexperiments confirm that they allow to realize canonical polygonal schemas evenfor extremely high genus shapes where previous methods fail.

Obtaining a Canonical Polygonal Schema from a Greedy Homotopy Basis with Minimal Mesh Refinement