Analyzing Time-Varying Scalar Fields using Piecewise-Linear Morse-Cerf Theory
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912459559796736 |
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| author | Dhar, Amritendu Chakraborty, Apratim Natarajan, Vijay |
| author_facet | Dhar, Amritendu Chakraborty, Apratim Natarajan, Vijay |
| contents | Morse-Cerf theory considers a one-parameter family of smooth functions defined on a manifold and studies the evolution of their critical points with the parameter. This paper presents an adaptation of Morse-Cerf theory to a family of piecewise-linear (PL) functions. The vertex diagram and Cerf diagram are introduced as representations of the evolution of critical points of the PL function. The characterization of a crossing in the vertex diagram based on the homology of the lower links of vertices leads to the definition of a topological descriptor for time-varying scalar fields. An algorithm for computing the Cerf diagram and a measure for comparing two Cerf diagrams are also described together with experimental results on time-varying scalar fields. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_00725 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Analyzing Time-Varying Scalar Fields using Piecewise-Linear Morse-Cerf Theory Dhar, Amritendu Chakraborty, Apratim Natarajan, Vijay Graphics Computational Geometry I.3.5 Morse-Cerf theory considers a one-parameter family of smooth functions defined on a manifold and studies the evolution of their critical points with the parameter. This paper presents an adaptation of Morse-Cerf theory to a family of piecewise-linear (PL) functions. The vertex diagram and Cerf diagram are introduced as representations of the evolution of critical points of the PL function. The characterization of a crossing in the vertex diagram based on the homology of the lower links of vertices leads to the definition of a topological descriptor for time-varying scalar fields. An algorithm for computing the Cerf diagram and a measure for comparing two Cerf diagrams are also described together with experimental results on time-varying scalar fields. |
| title | Analyzing Time-Varying Scalar Fields using Piecewise-Linear Morse-Cerf Theory |
| topic | Graphics Computational Geometry I.3.5 |
| url | https://arxiv.org/abs/2507.00725 |