# Toroidal density-equalizing map for genus-one surfaces

arXiv: https://arxiv.org/abs/2410.16833 · HTML: https://mlsanity.com/paper/2410.16833/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Oct 2024.

## Built on by (most cited first, 2 of 2)

- [Learning-based density-equalizing map](https://mlsanity.com/paper/2506.10027/) (2025)
- [Learning Task-Aware Sampling with Shared Saliency through Density-Equalizing Mappings](https://mlsanity.com/paper/2606.12869/) (2026)

## Abstract (arXiv)

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.