Conformal mapping in linear time

Given any $ε>0$ and any planar region $Ω$ bounded by a simple n-gon $P$ we construct a ($1 + ε)$-quasiconformal map between $Ω$ and the unit disk in time $C(ε)n$. One can take $ C(ε) = C + C \log (1/ε) \log \log (1/ε)$.

💡 Research Summary

**
The paper presents a deterministic algorithm that computes a (1 + ε)‑quasiconformal map between any simply‑connected planar region Ω bounded by a simple n‑gon and the unit disk D in time proportional to n, with a modest dependence on the prescribed accuracy ε. The main result (Theorem 1) states that the map can be obtained in O(n · p log p) operations, where p = O(log 1/ε). Consequently the overall constant factor can be written as C(ε) = C + C·log(1/ε)·log log(1/ε), independent of n and the geometry of Ω.

The algorithm proceeds in several conceptual layers. First, an initial approximation of the pre‑image of the polygon vertices on the unit circle is produced by a new “ι‑map”. The ι‑map is built from a medial‑axis decomposition of the polygon: the domain is covered by a collection of interior disks whose boundaries touch the polygon in at least two points, and the boundary of this union is mapped to the unit circle by following orthogonal circular arcs. Computing the medial axis of an n‑gon is known to be linear‑time (Chin‑Snoeyink‑Wang), so the ι‑map yields an n‑tuple w that is within a fixed quasiconformal distance K ≤ 7.82 of the true pre‑images z.

Next the polygon is split into “thick” and “thin” parts, mirroring the thick‑thin decomposition of hyperbolic manifolds. Thin parts arise when the extremal length between two edges is very small; they fall into a finite catalog of shapes (hyperbolic quadrilaterals and parabolic sectors). Because each thin shape admits an explicit conformal map (e.g., via elementary Schwarz–Christoffel formulas), they can be handled in constant time. The remaining thick region is treated analytically.

The thick region is mapped to the upper half‑plane H and then partitioned into O(n) Whitney–Carleson pieces (disks, Carleson squares, and “arches”). On each piece the target map is represented by a p‑term power (or Laurent) series. Fast Fourier Transform techniques allow evaluation of a p‑term series in O(p log p) time, and the fast multipole method (FMM) enables rapid convolution of series across all pieces. The series are glued together by a smooth partition of unity, producing a global map F:H→Ω whose Beltrami coefficient μ = ∂̄F/∂F satisfies ‖μ‖∞ = O(ε).

To improve the quasiconformal distortion, the algorithm solves the Beltrami equation ∂̄H = μ approximately. Using the FMM to apply the Beurling transform to μ, each Newton‑type iteration reduces the error quadratically. Since p grows only logarithmically with 1/ε, the dominant cost is the final iteration, giving a total work of O(n log 1/ε · log log 1/ε). After O(log log 1/ε) iterations the map becomes (1 + ε)‑quasiconformal.

The paper also discusses how to bring the initial ι‑map within the ε₀‑radius required for the Newton iteration. This is achieved by constructing a short chain of intermediate domains (each a finite union of disks) linking Ω to the unit disk. Each step in the chain is a simple quasiconformal deformation with bounded dilatation, and the number of steps is O(1/ε₀).

Overall, the contribution is a complete pipeline: (1) linear‑time medial‑axis preprocessing, (2) thin‑part explicit handling, (3) thick‑part power‑series representation on a Whitney decomposition, (4) fast evaluation via FFT and FMM, (5) Newton‑type quasiconformal correction. The resulting algorithm runs in linear time up to a polylogarithmic factor in 1/ε, a dramatic improvement over classical methods that require solving large linear systems or performing iterative Schwarz–Christoffel parameter updates with super‑linear complexity. The work opens the door to practical, provably accurate conformal mapping for large‑scale polygonal meshes and suggests extensions to domains bounded by circular arcs or more general piecewise‑smooth curves.