  Any closed manifold of genus g can be cut open to form a topological disk and
then mapped to a regular polygon with 4g sides. This construction is called the
canonical polygonal schema of the manifold, and is a key ingredient for many
applications in graphics and engineering, where a parameterization between two
shapes with same topology is often needed. The sides of the 4g-gon define on
the manifold a system of loops, which all intersect at a single point and are
disjoint elsewhere. Computing a shortest system of loops of this kind is
NP-hard. A computationally tractable alternative consists in computing a set of
shortest loops that are not fully disjoint in polynomial time, using the greedy
homotopy basis algorithm proposed by Erickson and Whittlesey, and then detach
them in post processing via mesh refinement. Despite this operation is
conceptually simple, known refinement strategies do not scale well for high
genus shapes, triggering a mesh growth that may exceed the amount of memory
available in modern computers, leading to failures. In this paper we study
various local refinement operators to detach cycles in a system of loops, and
show that there are important differences between them, both in terms of mesh
complexity and preservation of the original surface. We ultimately propose two
novel refinement approaches: the former minimizes the number of new elements in
the mesh, possibly at the cost of a deviation from the input geometry. The
latter allows to trade mesh complexity for geometric accuracy, bounding
deviation from the input surface. Both strategies are trivial to implement, and
experiments confirm that they allow to realize canonical polygonal schemas even
for extremely high genus shapes where previous methods fail.
