Supplement to Planar Interpolation with Extreme Deformation, opology Change and Dynamics

Given compact non-self-intersecting planar shapes, there are a pair of ambiguities characterized by topological (in-)equivalence. When shapes are topologically inequivalent, one shape can not be continuously morphed to the other as there is no bi-continuous map (homeomorphism) between them. Thus we cannot build an isomorphism between the shapes’ standard topology. Nevertheless, one can develop a quotient set, P/ ∼, by introducing an equivalence relation, defined as homeomorphism, to the set of compact non-selfintersecting planar shapes, P. P/ ∼ can then be generated by two elements, equivalence class of disk, s (simply connected), and annulus, m (multiply connected) under the operations, disjoint union ( ⊔ ) and connected sum (#) as shown in Figure 1. Thus we require two topological operations, cutting and opening, to convert input shapes into an equivalence class in P/ ∼. Locations and shapes of these cuts and openings are defined on the domain, which then determine both how and where shapes will open and split.