Continuous Flattening and Reversing of Convex Polyhedral Linkages
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916534200303616 |
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| author | Demaine, Erik D. Demaine, Martin L. Hecher, Markus Lin, Rebecca Luo, Victor H. Nara, Chie |
| author_facet | Demaine, Erik D. Demaine, Martin L. Hecher, Markus Lin, Rebecca Luo, Victor H. Nara, Chie |
| contents | We prove two results about transforming any convex polyhedron, modeled as a linkage L of its edges. First, if we subdivide each edge of L in half, then L can be continuously flattened into a plane. Second, if L is equilateral and we again subdivide each edge in half, then L can be reversed, i.e., turned inside-out. A linear number of subdivisions is optimal up to constant factors, as we show (nonequilateral) examples that require a linear number of subdivisions. For nonequilateral linkages, we show that more subdivisions can be required: even a tetrahedron can require an arbitrary number of subdivisions to reverse. For nonequilateral tetrahedra, we provide an algorithm that matches this lower bound up to constant factors: logarithmic in the aspect ratio. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_15130 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Continuous Flattening and Reversing of Convex Polyhedral Linkages Demaine, Erik D. Demaine, Martin L. Hecher, Markus Lin, Rebecca Luo, Victor H. Nara, Chie Computational Geometry Computational Complexity Discrete Mathematics Data Structures and Algorithms 68R10, 68Q17, 68U05 G.2.2; F.2.2; I.3.5 We prove two results about transforming any convex polyhedron, modeled as a linkage L of its edges. First, if we subdivide each edge of L in half, then L can be continuously flattened into a plane. Second, if L is equilateral and we again subdivide each edge in half, then L can be reversed, i.e., turned inside-out. A linear number of subdivisions is optimal up to constant factors, as we show (nonequilateral) examples that require a linear number of subdivisions. For nonequilateral linkages, we show that more subdivisions can be required: even a tetrahedron can require an arbitrary number of subdivisions to reverse. For nonequilateral tetrahedra, we provide an algorithm that matches this lower bound up to constant factors: logarithmic in the aspect ratio. |
| title | Continuous Flattening and Reversing of Convex Polyhedral Linkages |
| topic | Computational Geometry Computational Complexity Discrete Mathematics Data Structures and Algorithms 68R10, 68Q17, 68U05 G.2.2; F.2.2; I.3.5 |
| url | https://arxiv.org/abs/2412.15130 |