Topology-Preserving Line Densification

• The paper demonstrates that piecewise-affine projection methods preserve topology by maintaining non-crossing boundaries and correct vertex incidences during deformation.
• It employs adaptive graded quadtree construction and constrained Delaunay triangulation to achieve high numerical robustness and near-perfect area accuracy (error ≲10⁻⁷).
• Flow-based cartogram frameworks and divergence-free vector field models integrate with densification algorithms to ensure robust topology preservation across complex mappings.

Topology-preserving line densification refers to a family of algorithms and mathematical methodologies designed to refine the discretization of curves or line-like structures under continuous deformations, guaranteeing that essential topological properties—such as connectivity, non-self-intersection, and adjacency—are exactly preserved. This problem arises in various contexts, notably in the generation of contiguous cartograms using density-equalizing map projections and in the densification of field lines in divergence-free vector fields with preservation of their knot/link topology (Miaji et al., 11 Nov 2025, Brenier, 2013).

1. Mathematical Principles and Topological Constraints

The critical challenge in topology-preserving line densification is to guarantee that after a deformation map $\Pi:\mathbb{R}^2 \to \mathbb{R}^2$, all polygonal boundaries remain non-overlapping, simple (Jordan), and maintain the correct incidences at shared endpoints. In the context of cartogram construction, $\Pi$ is a density-equalizing projection, whose Jacobian determinant at each point $(x,y)$ satisfies

$$J_\Pi(x,y) = \det(D\Pi(x,y)) = \frac{\rho(x,y)}{\bar\rho}\,,$$

where $\rho$ is the prescribed density and $\bar\rho$ its average over the domain. Continuity and global invertibility (diffeomorphism) theoretically preclude region overlaps or inversions. However, representing region boundaries by a finite set of vertices and straight line segments introduces the risk that topological invariants are violated after projection.

To formalize the admissible class of densifications, topology preservation is enforced by three conditions (Miaji et al., 11 Nov 2025):

• P1. No Self-Intersections: Each polygonal boundary $A_i$ is simple.
• P2. No Unintended Overlaps: Distinct regions intersect only at prescribed shared endpoints.
• P3. Winding Order: Cyclic ordering at junctions where multiple regions meet remains consistent after deformation.

In divergence-free vector fields, preservation of the knot or link type of field lines underlines similar mathematical requirements: the diffeomorphic flow must transport each integral line without introducing reconnections or singularities (Brenier, 2013).

2. Densification Algorithms for Piecewise-Linear Boundaries

A robust solution, as in 5FCarto (Miaji et al., 11 Nov 2025), utilizes geometric and combinatorial structures to densify boundaries:

• Adaptive Graded Quadtree Construction: The density grid is recursively partitioned, with cell splitting guided by the maximum local density disparity. Grading ensures that neighboring cells differ in quadtree depth by at most one, which bounds the minimum triangle angles in subsequent triangulations and avoids slivers.
• Constrained Delaunay Triangulation: All cell corners of the quadtree are triangulated, with cell edges as hard constraints. The minimum projected triangle angle is maximized, which is essential for numerical robustness and well-conditioned affine approximations.
• Intersection-Based Sampling: Polygon edges are sampled at each intersection with the Delaunay triangulation's edges. This strategy guarantees that, within every triangle, the post-projection mapping is affine and orientation-preserving.
• Piecewise-Affine Projection: The deformation mapping $\Pi$ is approximated by an affine map $f_\tau(x) = A_\tau x + b_\tau$ on each triangle $\tau$, ensuring consistency on shared triangle edges and thus precluding cracks or overlaps at triangle boundaries.
• Vertex Simplification (Optional): After projection, the number of vertices can be reduced to a user-defined budget (e.g., via CGAL's simplification), balancing complexity and fidelity.

The overall complexity is $O(N\log N + p)$ where $N$ is the number of raster cells and $p$ is the number of input polygon vertices, due to efficient grading and local sampling properties.

3. Integration with Flow-Based Deformation and Cartograms

Flow-based cartogram frameworks, such as density-equalizing map projections (Gastner–Newman diffusion), provide the smooth invertible $\Pi$ upon which topology-preserving densification operates (Miaji et al., 11 Nov 2025). In the 5FCarto pipeline, densification occurs just before the projection step, ensuring that each boundary refinement is robust to the continuous but potentially complex deformation imposed by $\Pi$.

Mathematically, the piecewise-affine approximation ensures that lines that were non-crossing and adjacent before deformation remain so after projection (see Lemma 1 in (Miaji et al., 11 Nov 2025)), since affine maps preserve intersection properties within each triangle. For divergence-free vector fields, the PDE

$$\partial_t B + \nabla \cdot (B \otimes v - v \otimes B) = 0, \quad v = P(\nabla \cdot (B \otimes B)), \quad \nabla \cdot B = 0$$

gives rise to a volume-preserving flow $\dot{X}(t,a) = v(t, X(t,a))$, and field lines are advected purely by this diffeomorphism; densification simply refines the sampling of these lines as they move without altering their topology (Brenier, 2013).

4. Theoretical Guarantees

The central correctness guarantee is that the densification algorithm, under smooth $\Pi$ and graded triangulation (minimum angle $\gtrsim 18.4^\circ$), strictly enforces all topological constraints:

• Non-crossing and Adjacency: Any two non-crossing polylines mapped by a piecewise-affine, orientation-preserving $f$ remain non-crossing and maintain shared endpoints.
• Winding Order: Orientation (winding) at multiple-junction points is preserved since each affine map's determinant remains positive across all triangles.
• Global Continuity: By sharing projected positions on triangle boundaries, the densified polygons avoid gaps or overlaps at triangle interfaces.

These claims are grounded in the proof that the piecewise-affine map is continuous, injective, and shares values along any shared triangle edge (implication of the Hadamard theorem applied locally and globally) (Miaji et al., 11 Nov 2025).

For the divergence-free vector field setting, the topology of dense sets of field lines is preserved by the underlying flow map, since it is globally volume-preserving and invertible for all time. No field-line reconnections or topological changes are possible in the continuous or the proposed discrete numerics, provided divergence is maintained to machine tolerance (Brenier, 2013).

5. Empirical Evaluation and Practical Performance

Empirical studies on 32 real-world maps of varying density disparity demonstrate that topology-preserving densification achieves near-perfect area accuracy and provably zero intersection defects under even extreme deformations (Miaji et al., 11 Nov 2025). Key quantitative findings are:

Method	Max Area Error (G4)	Self-Intersections	Overlaps	Time (G4)	Shape Deformation*
5FCarto	$\lesssim 10^{-7}$	0	0	0.62 s	Comparable to BFB
F4Carto	$\approx 10$	0	$\approx 600$	3.74 s	Lower, but inaccurate
BFB	$\approx 1$	$\approx 200$	$\approx 130$	30 s	Comparable

*Shape deformation averaged over Fréchet, Hausdorff, Symmetric Difference metrics.

For challenging cases, such as the 2024 US election cartogram, 5FCarto produced $0$ overlaps, with $E_{\max}$ on the order $6.2 \times 10^{-3}$ and sub-second runtimes. Competing algorithms could not match both topological robustness and area accuracy.

6. Implementation Practices and Limitations

Efficient implementation involves:

• Quadtree data structures graded to depth-difference at most $1$ for triangle quality.
• CGAL’s constrained Delaunay triangulation for geometric robustness.
• Adaptive vertex simplification tools for memory and rendering efficiency.

A noteworthy practical caveat is that, due to floating-point arithmetic, minuscule overlaps may still occur in degenerate cases; mitigation includes use of higher-precision types or polygon-repair utilities. Extremely convoluted boundaries or highly nonuniform density may require further refinement of the quadtree or denser sampling along curves.

In the vector field context, spatial discretization with divergence-free reconstruction (staggered grids or discrete exterior calculus) and careful time integration (volume-preserving Runge–Kutta, with arc-length monitoring for curve densification), guarantee that the sampled field lines remain topologically faithful (Brenier, 2013).

7. Relation to Broader Topology-Preserving Flows

The principle of topology preservation by diffeomorphic transport extends from cartograms to physical transport in divergence-free vector fields. The magnetic relaxation flow admits dissipative solutions, which are curves of maximal slope for the magnetic energy in the $L^2$ metric, yet their topological content is constant because the pressure-projected velocity field is purely advective. Densification of field lines in this context serves to increase geometric resolution without changing the homeomorphism class of the underlying bundle of lines (Brenier, 2013).

• "Topology-Preserving Line Densification for Creating Contiguous Cartograms" (Miaji et al., 11 Nov 2025)
• "Topology-preserving diffusion of divergence-free vector fields and magnetic relaxation" (Brenier, 2013)