Tree Structure for Contractible Regions in ℝ3

摘要

For contractible regions ωin ℝ3 with generic smooth boundary, we determine the global structure of the Blum medial axis M. We give an algorithm for decomposing M into “irreducible components” which are attached to each other along “fin curves”. The attaching cannot be described by a tree structure as in the 2D case. However, a simplified but topologically equivalent medial structure ̂ M with the same irreducible components can be described by a two level tree structure. The top level describes the simplified form of the attaching, and the second level tree structure for each irreducible component specifies how to construct the component by attaching smooth medial sheets to the network of Y-branch curves. The conditions for these structures are complete in the sense that any region whose Blum medial axis satisfies the conditions is contractible.