Topology-Preserving Line Densification for Creating Contiguous Cartograms

TL;DR

This work tackles topology violations in contiguous cartograms caused by discretizing region boundaries. It introduces 5FCarto, a topology-preserving line-densification pipeline that combines a graded quadtree, constrained Delaunay triangulation, and line densification to ensure polygons remain connected and non-overlapping under density-equalizing map projections. Across 32 real-world maps and a 2024 US election cartogram, 5FCarto achieves zero intersections and superior area accuracy with competitive running times compared to state-of-the-art baselines. The approach provides an open-source, robust framework for producing topology-preserving cartograms suitable for geovisualization and quantitative analysis.

Abstract

Cartograms depict geographic regions with areas proportional to quantitative data. However, when created using density-equalizing map projections, cartograms may exhibit invalid topologies if boundary polygons are drawn using only a finite set of vertices connected by straight lines. Here we introduce a method for topology-preserving line densification that guarantees that cartogram regions remain connected and non-overlapping when using density-equalizing map projections. By combining our densification technique with a flow-based cartogram generator, we present a robust framework for strictly topology-preserving cartogram construction. Quantitative evaluations demonstrate that the proposed algorithm produces cartograms with greater accuracy and speed than alternative methods while maintaining comparable shape fidelity.

Figures (10)

• Figure 1: Maps of Belgium. (a) Belgium’s regions on an equal-area map, abbreviated as BRU (Brussels), VLG (Flanders), and WAL (Wallonia). (b) Contiguous cartogram representing population. As Brussels is the most densely populated region, its area is enlarged in the cartogram.
• Figure 2: Illustration of line densification when approximating a curve with a finite number of points. Panel (a) depicts the ground truth, showing the precise geometry of two curves, $C_1$ and $C_2$. Panel (b) illustrates finite approximations, $A_1$ and $A_2$, that use a sparse distribution of points along the respective curves. Topology is violated as $A_1$ intersects itself and also $A_2$. Panel (c) demonstrates line densification resulting in $A_1'$ and $A_2'$ preserving the true topology.
• Figure 3: Illustration of a topology violation caused by projecting polylines. The figure depicts two polylines, one colored blue (ABC) and the other red (DE). Panel (a) illustrates the original geometry, while panel (b) shows the situation after transformation via a density-equalizing map projection. The transformed black grid lines indicate the projection. The projected positions of the polyline vertices A to E are interpolated on the basis of the quadrilaterals formed by the projected grid lines in panel (b). Although the blue and red polylines do not intersect before the transformation, they cross each other afterward, resulting in an invalid topology.
• Figure 4: Illustration of the quadtree splitting criteria. The squares that are delineated by thin gray lines indicate a rasterized density grid, while those outlined in black denote quadtree leaf nodes. The quadtree is constructed by recursively splitting an $n \times n$ cell containing the largest difference between maximum and minimum density into four $(n/2) \times (n/2)$ cells until a predetermined number of leaf nodes (7 in this example) is reached. The squares in the raster grid that contain the maximum and minimum density values are annotated with those values. Panel (a) depicts the initial $4 \times 4$ cell, which is split due to having the largest internal density difference (22) and because the current number of leaf nodes (1) has not reached the target (7). Panel (b) illustrates the resulting subdivision with 4 leaf nodes, where again the node with the largest density difference (12, bottom-right) is selected for further splitting, as the target number of leaf nodes has still not been met. Panel (c) shows the final subdivision with 7 leaf nodes, at which point the desired number of leaf nodes has been reached, marking the completion of the quadtree.
• Figure 5: Illustration of the triangulation process. (a) Original graded quadtree, as obtained in Figure \ref{['fig:quadtree_explanation-c']}. (b) Quadtree corners projected according to the density grid from Figure \ref{['fig:quadtree_explanation']}. The projected quadtree cell edges (black solid lines) serve as constraints for computing a Delaunay triangulation, $T_{\text{temp}}$, of the projected corners. The additional triangulation edges are represented by dashed lines. (c) After unprojecting the edges, the final constrained Delaunay triangulation, $T$, comprises both the solid black lines, as well as the red, green and blue dashed lines. Although the yellow dashed line represents the Delaunay triangulation of the transformed quadtree, we do not insert it into the original quadtree because it overlaps with a boundary of a quadtree leaf node. Thus, we initially only add the blue dashed line as a constraint in the upper right leaf node. Subsequently, the green dashed line is added to complete the Delaunay triangulation under the specified constraints.