Curvature of an Arbitrary Surface for Discrete Gravity and for $d=2$ Pure Simplicial Complexes
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911356599402496 |
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| author | Chamseddine, Ali H. Malaeb, Ola Najem, Sara |
| author_facet | Chamseddine, Ali H. Malaeb, Ola Najem, Sara |
| contents | We propose a computation of curvature of arbitrary two-dimensional surfaces of three-dimensional objects, which is a contribution to discrete gravity with potential applications in network geometry. We begin by linking each point of the surface in question to its four closest neighbors, forming quads. We then focus on the simplices of $d=2$, or triangles embedded in these quads, which make up a pure simplicial complex with $d=2$. This allows us to numerically compute the local metric along with zweibeins, which subsequently leads to a derivation of discrete curvature defined at every triangle or face. We provide an efficient algorithm with $\mathcal{O}(N \log{N})$ complexity that first orients two-dimensional surfaces, solves the nonlinear system of equations of the spin-connections resulting from the torsion condition, and returns the value of curvature at each face. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_04375 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Curvature of an Arbitrary Surface for Discrete Gravity and for $d=2$ Pure Simplicial Complexes Chamseddine, Ali H. Malaeb, Ola Najem, Sara General Relativity and Quantum Cosmology High Energy Physics - Lattice High Energy Physics - Theory Computational Physics We propose a computation of curvature of arbitrary two-dimensional surfaces of three-dimensional objects, which is a contribution to discrete gravity with potential applications in network geometry. We begin by linking each point of the surface in question to its four closest neighbors, forming quads. We then focus on the simplices of $d=2$, or triangles embedded in these quads, which make up a pure simplicial complex with $d=2$. This allows us to numerically compute the local metric along with zweibeins, which subsequently leads to a derivation of discrete curvature defined at every triangle or face. We provide an efficient algorithm with $\mathcal{O}(N \log{N})$ complexity that first orients two-dimensional surfaces, solves the nonlinear system of equations of the spin-connections resulting from the torsion condition, and returns the value of curvature at each face. |
| title | Curvature of an Arbitrary Surface for Discrete Gravity and for $d=2$ Pure Simplicial Complexes |
| topic | General Relativity and Quantum Cosmology High Energy Physics - Lattice High Energy Physics - Theory Computational Physics |
| url | https://arxiv.org/abs/2409.04375 |