Toroidal density-equalizing map for genus-one surfaces

TL;DR

This work extends diffusion-based density-equalizing maps to genus-one surfaces by formulating density diffusion on a planar domain with periodic torus boundaries and a toroidal projection, enabling controllable deformations via a prescribed population $P$. It then develops a toroidal density-equalizing parameterization framework, producing a toroidal map $f=g\circ h$ from any genus-one surface to a target torus with adjustable area changes, including an area-preserving variant. Through numerical experiments on toroidal and genus-one models, the method demonstrates effective density equalization, consistent toroidal maps across different cut paths, and substantial area-distortion reductions for parameterizations, with practical texture-mapping demonstrations. Overall, the approach broadens diffusion-based mapping techniques to topologies with holes, offering flexible toroidal deformations and robust genus-one parameterizations for visualization and geometry processing.

Abstract

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.

Figures (17)

• Figure 1: An illustration of the proposed toroidal density-equalizing map and toroidal parameterization methods. (a) Using the proposed toroidal density-equalizing map (TDEM) algorithm, we can achieve shape deformations on a toroidal surface with different regions enlarged or shrunk based on some given density information. (b) Using the proposed toroidal density-equalizing parameterization method, we can effectively map a genus-one surface onto a prescribed torus with controllable area changes.
• Figure 2: An illustration of density-equalizing maps. Given a planar domain $\mathcal{D}$ with some high-density regions (yellow) and some low-density regions (purple), the density-equalizing map will produce a shape deformation based on the density diffusion process. In particular, the high-density regions will expand and the low-density regions will shrink.
• Figure 3: An illustration of the periodic boundary constraints in the planar density diffusion and mapping process. Throughout the planar density diffusion and mapping process, periodic boundary constraints need to be enforced for both the top and bottom boundaries (red) and both the left and right boundaries (blue). Specifically, for every pair of corresponding boundary vertices (the black dot and the white dot), a virtual copy of the neighboring nodes and triangles will be considered in the density diffusion process so that the density information can be exchanged along the boundaries. The boundaries are then updated consistently so that they only differ by a translation, which ensures that they will be mapped back to a consistent location on the torus under the toroidal projection.
• Figure 4: Outline of the proposed toroidal density-equalizing parameterization method. Given a genus-one surface with some prescribed population, we first compute an initial parameterization of it onto a torus. Then, we apply the inverse toroidal projection to map the torus onto the plane. We then apply our TDEM algorithm to obtain the deformed planar map with the periodic condition satisfied. Finally, we apply the toroidal projection to map the planar mapping result onto the torus, yielding the desired toroidal density-equalizing parameterization.
• Figure 5: Mapping a torus with the prescribed population $P(T) = 2 - \cos(u)$, where $(u,v)$ is the face centroid of every triangle $T$ under the inverse toroidal projection $\phi^{-1}$. The top-left figure panel shows the original toroidal surface color-coded with the population. The bottom-left panel shows the planar representation of the original toroidal surface under $\phi^{-1}$. The bottom-right panel shows the planar mapping result. The top-right panel shows the final TDEM result. The blue dotted lines represent the cut paths.