The Analytic Arc Cover Problem and its Applications to Contiguous Art Gallery, Polygon Separation, and Shape Carving
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866908415421317120 |
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| author | Robson, Eliot W. Spalding-Jamieson, Jack Zheng, Da Wei |
| author_facet | Robson, Eliot W. Spalding-Jamieson, Jack Zheng, Da Wei |
| contents | We show the following problems are in $\textsf{P}$:
1. The contiguous art gallery problem -- a variation of the art gallery problem where each guard can protect a contiguous interval along the boundary of a simple polygon. This was posed at the open problem session at CCCG '24 by Thomas C. Shermer.
2. The polygon separation problem for line segments -- For two sets of line segments $S_1$ and $S_2$, find a minimum-vertex convex polygon $P$ that completely contains $S_1$ and does not contain or cross any segment of $S_2$.
3. Minimizing the number of half-plane cuts to carve a 3D polytope.
To accomplish this, we study the analytic arc cover problem -- an interval set cover problem over the unit circle with infinitely many implicitly-defined arcs, given by a function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_15567 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Analytic Arc Cover Problem and its Applications to Contiguous Art Gallery, Polygon Separation, and Shape Carving Robson, Eliot W. Spalding-Jamieson, Jack Zheng, Da Wei Computational Geometry We show the following problems are in $\textsf{P}$: 1. The contiguous art gallery problem -- a variation of the art gallery problem where each guard can protect a contiguous interval along the boundary of a simple polygon. This was posed at the open problem session at CCCG '24 by Thomas C. Shermer. 2. The polygon separation problem for line segments -- For two sets of line segments $S_1$ and $S_2$, find a minimum-vertex convex polygon $P$ that completely contains $S_1$ and does not contain or cross any segment of $S_2$. 3. Minimizing the number of half-plane cuts to carve a 3D polytope. To accomplish this, we study the analytic arc cover problem -- an interval set cover problem over the unit circle with infinitely many implicitly-defined arcs, given by a function. |
| title | The Analytic Arc Cover Problem and its Applications to Contiguous Art Gallery, Polygon Separation, and Shape Carving |
| topic | Computational Geometry |
| url | https://arxiv.org/abs/2412.15567 |