A generative classification of mesh refinement rules with lattice transformations

We give a classification of the subdivision refinement rules using sequences of similar lattices. Our work expands and unifies recent results in the classification of primal triangular subdivision [1], and results on the refinement of quadrilateral lattices [14] [13]. In the examples we concentrate on the cases with low ratio of similarity and find new univariate and bivariate refinement rules with the lowest possible such ratio, showing that this very low ratio usually comes at the expense of symmetry.