On The Topology of Polygonal Meshes
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908748127141888 |
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| author | Bærentzen, Andreas |
| author_facet | Bærentzen, Andreas |
| contents | This paper is an introductory and informal exposition on the topology of polygonal meshes. We begin with a broad overview of topological notions and discuss how homeomorphisms, homotopy, and homology can be used to characterise topology. We move on to define polygonal meshes and make a distinction between intrinsic topology and extrinsic topology which depends on the space in which the mesh is immersed. A distinction is also made between quantitative topological properties and qualitative properties. Next, we outline proofs of the Euler and the Euler-Poincaré formulas. The Betti numbers are then defined in terms of the Euler-Poincaré formula and other mesh statistics rather than as cardinalities of the homology groups which allows us to avoid abstract algebra. Finally, we discuss how it is possible to cut a polygonal mesh such that it becomes a topological disc. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_11618 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On The Topology of Polygonal Meshes Bærentzen, Andreas History and Overview Graphics Geometric Topology 57 I.3.5 This paper is an introductory and informal exposition on the topology of polygonal meshes. We begin with a broad overview of topological notions and discuss how homeomorphisms, homotopy, and homology can be used to characterise topology. We move on to define polygonal meshes and make a distinction between intrinsic topology and extrinsic topology which depends on the space in which the mesh is immersed. A distinction is also made between quantitative topological properties and qualitative properties. Next, we outline proofs of the Euler and the Euler-Poincaré formulas. The Betti numbers are then defined in terms of the Euler-Poincaré formula and other mesh statistics rather than as cardinalities of the homology groups which allows us to avoid abstract algebra. Finally, we discuss how it is possible to cut a polygonal mesh such that it becomes a topological disc. |
| title | On The Topology of Polygonal Meshes |
| topic | History and Overview Graphics Geometric Topology 57 I.3.5 |
| url | https://arxiv.org/abs/2511.11618 |