Block meshes: Topologically robust shape modeling with graphs embedded on 3-manifolds

•Shape algebra describes structures in higher-dimensional manifolds with minimum operations.•Our unified framework represents all topologically distinct shapes in 3D, from solids to surfaces and curves.•The algebra models shapes via simpler, more powerful and topologically robust algorithms.

We present a unifying framework to represent all topologically distinct shapes in 3D, from solids to surfaces and curves. This framework can be used to build a universal and modular system for the visualization, design, and construction of shapes, amenable to a broad range of science, engineering, architecture, and design applications. Our unifying framework uses 3-space immersions of higher-dimensional-manifolds, which facilitate our goal of topological robustness.We demonstrate that a specific type of orientable 2-manifold mesh, which we call a CMM-pattern coverable mesh, can be used to represent structures in higher-dimensional manifolds, which we call block meshes. Moreover, the framework includes a set of operations that can preserve CMM-pattern coverability. In this sense, CMM-pattern-coverable meshes provide an algebraization of shape processing that (1) supports a generalized mesh representation for blocks that may not necessarily be solids, and (2) requires a minimal set of operations that transform CMM-pattern-coverable meshes to CMM-pattern-coverable meshes.

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