Toroidal density-equalizing map for genus-one surfaces
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909904224124928 |
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| author | Yao, Shunyu Choi, Gary P. T. |
| author_facet | Yao, Shunyu Choi, Gary P. T. |
| contents | Density-equalizing map is a shape deformation technique originally developed for cartogram creation and sociological data visualization on planar geographical maps. In recent years, there has been an increasing interest in developing density-equalizing mapping methods for surface and volumetric domains and applying them to various problems in geometry processing and imaging science. However, the existing surface density-equalizing mapping methods are only applicable to surfaces with relatively simple topologies but not surfaces with topological holes. In this work, we develop a novel algorithm for computing density-equalizing maps for toroidal surfaces. In particular, different shape deformation effects can be easily achieved by prescribing different population functions on the torus and performing diffusion-based deformations on a planar domain with periodic boundary conditions. Furthermore, the proposed toroidal density-equalizing mapping method naturally leads to an effective method for computing toroidal parameterizations of genus-one surfaces with controllable shape changes, with the toroidal area-preserving parameterization being a prime example. Experimental results are presented to demonstrate the effectiveness of our proposed methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_16833 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Toroidal density-equalizing map for genus-one surfaces Yao, Shunyu Choi, Gary P. T. Graphics Computational Geometry Numerical Analysis Density-equalizing map is a shape deformation technique originally developed for cartogram creation and sociological data visualization on planar geographical maps. In recent years, there has been an increasing interest in developing density-equalizing mapping methods for surface and volumetric domains and applying them to various problems in geometry processing and imaging science. However, the existing surface density-equalizing mapping methods are only applicable to surfaces with relatively simple topologies but not surfaces with topological holes. In this work, we develop a novel algorithm for computing density-equalizing maps for toroidal surfaces. In particular, different shape deformation effects can be easily achieved by prescribing different population functions on the torus and performing diffusion-based deformations on a planar domain with periodic boundary conditions. Furthermore, the proposed toroidal density-equalizing mapping method naturally leads to an effective method for computing toroidal parameterizations of genus-one surfaces with controllable shape changes, with the toroidal area-preserving parameterization being a prime example. Experimental results are presented to demonstrate the effectiveness of our proposed methods. |
| title | Toroidal density-equalizing map for genus-one surfaces |
| topic | Graphics Computational Geometry Numerical Analysis |
| url | https://arxiv.org/abs/2410.16833 |