| Summary: | 碩士 === 國立清華大學 === 資訊工程學系 === 102 === An unfolding of a polyhedron is a cutting of the polyhedron's surface so that its surface can be flattened into a single connected flat patch on the plane without any self-overlapping. There exist many unfolding methods with different cuttings. The two main method of this thesis are an edge-unfolding which is a cutting of the polyhedron's surface along its edges and a grid-unfolding which is a cutting of the polyhedron's refined surface along its edges.
A one-layer lattice polyhedron is a polyhedron of height one, whose surface is composed of unit squares as its faces. We consider the unfolding problem on the one-layer lattice polyhedron of genus greater than zero. We propose two edge-unfolding algorithms for one-layer lattice polyhedra with three holes and one-layer lattice polyhedra with rectangular boundary and four holes. Besides, we propose four grid-unfolding algorithms for one-layer lattice polyhedra with monotone boundary and rectangular holes, one-layer lattice polyhedra with rectangular holes, one-layer lattice polyhedra with monotone boundary and holes, and one-layer lattice polyhedra with monotone holes. The basic idea of these algorithms is that we use different paths to connect the flattened patches of each hole of the given polyhedron so that no self-overlapping can occur in the final flattened patch.
We leave the question open whether any of the general one-layer lattice polyhedra with holes can be edge-unfolded and the question whether any of the general lattice polyhedra without holes can be edge-unfolded.
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