Polygon Atlas: Geometry, Tiling, and Learning

• Polygon Atlas is a concept encompassing local coordinate charts and overlapping 2D patches to model complex surfaces and tiling patterns.
• It underpins neural surface reconstruction methods like AtlasNet, which map simple 2D domains to 3D shapes using learned parameterizations.
• Applications extend to combinatorial tiling, polygon aggregation in cartography, and blockchain protocols, highlighting its interdisciplinary impact.

“Polygon atlas” denotes several distinct but structurally related concepts in current research. In the most classical sense, an atlas is a family of local coordinate charts whose domains cover a manifold, so that globally non-Euclidean or hole-bearing objects can be represented by overlapping locally Euclidean patches. In graphics and geometric learning, the same idea becomes a collection of 2D parameter domains whose images cover a 3D surface, often with polygonal or tessellatable connectivity. In combinatorics, tilings, and algebra, an atlas can mean a finite catalog of legal polygonal neighborhoods or a hierarchy of polygon equations. In geospatial and vision pipelines, the term also appears in workflows that generate, preserve, pack, or aggregate polygonal representations directly rather than passing through raster masks or fixed global parameterizations (Cohn et al., 2021, Groueix et al., 2018, Fernique et al., 2022, Dimakis et al., 2014).

1. Atlas as a covering of geometric or polygonal structure

In differential-geometric terms, a $d$-dimensional smooth manifold $\mathcal M$ is a set such that every point $p\in\mathcal M$ lies in a neighborhood $U$ equipped with a diffeomorphism $\phi:U\to\mathbb R^d$. The pair $(U,\phi)$ is a coordinate chart, and a family of charts whose domains cover the manifold is an atlas. This formulation is explicit in topologically informed manifold learning, where the data are samples $X=\{x_1,\ldots,x_m\}\subset\mathbb R^n$ from a smooth manifold $\mathcal M$, and the goal is not one global map but an atlas $\mathcal A=\{(U_i,\phi_i)\}_{i\in I}$ that preserves local latent structure while respecting global topology (Cohn et al., 2021).

Surface-generation work adopts the same language in a more concrete parameterization setting. AtlasNet models a shape as the union of $N$ learned parameterizations $\mathcal M$0, each mapping a simple 2D domain, typically $\mathcal M$1, into $\mathcal M$2. Each patch is a chart, and the reconstructed surface is the union of chart images. Minimal Neural Atlas keeps the chart-map viewpoint but replaces fixed square domains by learned occupied subsets $\mathcal M$3, so that chart topology and boundary are themselves learnable rather than prescribed a priori (Groueix et al., 2018, Low et al., 2022).

In production graphics, the atlas is often the mesh’s 2D UV parameterization. A human-in-the-loop segmentation pipeline for 3D assets assumes a valid non-overlapping UV map and projects per-view labels back into this UV domain to form a unified segmented atlas in texture space. Here the atlas is neither an abstract manifold cover nor a neural latent representation, but a 2D parameterized control surface for downstream material assignment, semantic labeling, and texture-space editing (Kühn et al., 16 Jun 2026).

A different formal meaning appears in tiling theory. For geometrical Penrose tilings, a $\mathcal M$4-atlas is the set of all $\mathcal M$5-maps, where a $\mathcal M$6-map is the radius-$\mathcal M$7 polygonal neighborhood around a vertex in the edge graph. The central theorem is that geometrical Penrose tilings are characterized by their $\mathcal M$8-atlas: every tiling by thin and fat rhombi whose $\mathcal M$9-maps all belong to the finite atlas $p\in\mathcal M$0 is a geometrical Penrose tiling (Fernique et al., 2022).

2. Topology-aware and neural atlas construction

A central reason atlas methods are needed is that a single Euclidean chart cannot globally parameterize manifolds with holes. Topologically-Informed Atlas Learning makes this failure mode explicit for spaces such as cylinders, tori, spheres, cyclic trajectories, articulated-object configuration spaces, and gait cycles. Its construction is bottom-up: build a neighborhood graph $p\in\mathcal M$1; initialize many tiny overlapping chart domains as subgraphs; greedily merge two charts only if their overlap has precisely one connected component and contains no atomic cycle longer than a threshold $p\in\mathcal M$2; then embed each final chart independently with ISOMAP. The topological motivation is de Rham cohomology and a Mayer–Vietoris argument: if two charts and their intersection have trivial de Rham cohomology, then their union does as well (Cohn et al., 2021).

The implemented tests are only graph proxies for topology. Connectedness of the overlap approximates trivial $p\in\mathcal M$3, and absence of long atomic cycles approximates trivial $p\in\mathcal M$4. The method therefore targets primarily $p\in\mathcal M$5-dimensional hole structure visible as loops in the neighborhood graph. The paper is explicit that it does not compute exact de Rham cohomology on the sampled graph, cannot detect higher-dimensional holes reliably, offers no convergence or sample-complexity theorem, and does not construct explicit transition maps $p\in\mathcal M$6 on overlaps. The result is an atlas of overlapping local embeddings rather than a globally stitched parameterization (Cohn et al., 2021).

AtlasNet approaches atlas construction from generative shape modeling rather than topology testing. A latent code is concatenated with a 2D point $p\in\mathcal M$7, and each patch decoder MLP outputs a surface point $p\in\mathcal M$8. Training uses Chamfer loss between point samples from all patches and target surface samples. Because the decoder is continuous in the 2D domain, each patch can be sampled at arbitrary resolution and meshed by transferring connectivity from a regular tessellation of the square. The representation is therefore inherently tessellatable and polygon-friendly. At the same time, AtlasNet does not explicitly prevent overlap, does not enforce watertightness, and does not optimize chart compatibility across seams (Groueix et al., 2018).

Minimal Neural Atlas pushes atlas learning closer to classical low-distortion parameterization. By learning both chart maps $p\in\mathcal M$9 and chart domains $U$0, it separates topology from geometry: the occupancy field $U$1 trims chart support, while the map $U$2 is regularized by a scaled Symmetric Dirichlet Energy derived from the first fundamental form $U$3. The method argues that a general surface admits a minimal atlas of $U$4 charts and empirically shows strong reconstruction and distortion results in that regime. Yet it also remains prone to seam and intersection artifacts, and it does not impose explicit transition-map consistency or bijectivity (Low et al., 2022).

3. Polygon-native generation, detection, and annotation

A separate line of work treats polygons themselves as the primary output. Polygonizer is an image-to-sequence model for building-footprint delineation that avoids the standard semantic-segmentation-to-polygonization pipeline. A modified ResNet50 produces an encoded tensor of shape $U$5, and a 3-layer stacked LSTM with Bahdanau attention emits a sequence of discrete coordinate tokens between a start token <s> and a stop token </s>. The model is trained with teacher forcing and negative log-likelihood, assumes one object per crop with ground-truth bounding boxes, and is evaluated not only by overlap metrics but also by maximum tangent angle error. In the reported comparison, Polygonizer achieves a maximum tangent angle error of $U$6, versus $U$7 for FFL, $U$8 for PolyWorld, and $U$9 for PolyBuilding, while also showing substantially stronger robustness under downsampling and erased-dropout perturbations (Khomiakov et al., 2023).

DPPD addresses a related but distinct problem: polygon object detection as a compromise between rectangular boxes and dense instance masks. Each object is represented by a center $\phi:U\to\mathbb R^d$0 and $\phi:U\to\mathbb R^d$1 deformable polar vertices $\phi:U\to\mathbb R^d$2, where both radii and angles are predicted. Training does not compare sparse predicted vertices directly to arbitrarily annotated polygons; instead, both prediction and ground truth are densely resampled into equal-spaced raypoints, and the resampling operation is fully differentiable. The regression loss combines an origin loss, a polar IoU loss, and a smoothness term. On crosswalk detection, DPPD-36 reports $\phi:U\to\mathbb R^d$3, $\phi:U\to\mathbb R^d$4, and $\phi:U\to\mathbb R^d$5, outperforming PolarMask-64 at $\phi:U\to\mathbb R^d$6, $\phi:U\to\mathbb R^d$7, and $\phi:U\to\mathbb R^d$8 while also running faster (Zheng et al., 2023).

Topology preservation becomes especially delicate when polygons encode holes by a single cyclic chain. For ring-type annotations $\phi:U\to\mathbb R^d$9, where outer and inner boundaries are linked by a bridge edge and a closure edge, standard augmentation can remove vertices under clipping and silently corrupt cyclic adjacency. A topology-preserving augmentation pipeline addresses this by rasterizing to a mask, applying geometric augmentation in mask space, identifying surviving vertices, projecting them back into index space, and reconnecting them in original cyclic order. Its metric, Cyclic Adjacency Preservation,

$(U,\phi)$0

measures whether cyclic successor relations survive augmentation. Reported CAP values for the proposed method range from $(U,\phi)$1 to $(U,\phi)$2 across rotation, cropping, scaling, flip, and rotation-plus-cropping, and downstream mIoU improves from $(U,\phi)$3 to $(U,\phi)$4 for YOLOv11-Seg and from $(U,\phi)$5 to $(U,\phi)$6 for Mask R-CNN (Laudari et al., 16 Mar 2026).

4. UV atlases, packing, aggregation, and reconstruction

When an atlas already exists as a UV parameterization, it can become the target space for human-guided semantic structure. A human-in-the-loop workflow for 3D assets first selects a compact set of rendered views by greedy set cover over sampled surface points, then uses SAM 2 and Label Studio for interactive view segmentation, and finally back-projects the masks through visible faces into UV space. The output is a unified segmented atlas, represented as a color-coded segmentation texture aligned with the mesh UVs. In the reported evaluation on eight cultural heritage objects, the fixed setup used eight rendered views per model, and annotation for the seven logged cases took approximately $(U,\phi)$7 to $(U,\phi)$8 minutes, with a mean of about $(U,\phi)$9 minutes (Kühn et al., 16 Jun 2026).

Atlas packing concerns a different optimization problem: arranging already defined charts in fixed texture space. TABI targets interactive GPU packing of irregular polygonal charts while narrowing the quality gap to much slower offline packers. It tests $X=\{x_1,\ldots,x_m\}\subset\mathbb R^n$0 discrete scales, uses local AABB and approximate OBB proxies for chart shape, compacts empty space horizontally and vertically, and balances row directions and row widths by knee-aware folding. On $X=\{x_1,\ldots,x_m\}\subset\mathbb R^n$1 atlases, the average runtime is $X=\{x_1,\ldots,x_m\}\subset\mathbb R^n$2 ms and the average $X=\{x_1,\ldots,x_m\}\subset\mathbb R^n$3 stretch is $X=\{x_1,\ldots,x_m\}\subset\mathbb R^n$4, compared with $X=\{x_1,\ldots,x_m\}\subset\mathbb R^n$5 for FastAtlas and $X=\{x_1,\ldots,x_m\}\subset\mathbb R^n$6 for GPU Chameleon; the method is also reported as $X=\{x_1,\ldots,x_m\}\subset\mathbb R^n$7 faster than Xatlas random on average (Gu et al., 8 Feb 2026).

In cartographic aggregation, the atlas problem becomes one of replacing many small polygons by a smaller set of representative regions. Bicriteria Polygon Aggregation defines the objective

$X=\{x_1,\ldots,x_m\}\subset\mathbb R^n$8

with $X=\{x_1,\ldots,x_m\}\subset\mathbb R^n$9 total area and $\mathcal M$0 total perimeter. The unrestricted problem allows arbitrary region boundaries, yet the optimal free boundary pieces are proved to be circular arcs of radius $\mathcal M$1 together with input polygon edges. This yields a polynomial-time algorithm via reduction to a subdivision induced by all relevant candidate arcs, with runtime $\mathcal M$2. The same paper gives polygon-only approximations with factors at most $\mathcal M$3 when region vertices may lie on input boundaries and at most $\mathcal M$4 when they must be input vertices, and reports that on real building-footprint data the approximate solutions are visually similar to the unrestricted optimum and typically within $\mathcal M$5 of its objective value (Blank et al., 15 Jul 2025).

A more foundational reconstruction problem arises when only rasterized geometry is available. For planar pixelations $\mathcal M$6, raw pixel unions can create false holes and distorted geometry. A Morse-theory-inspired algorithm instead reconstructs a PL approximation $\mathcal M$7 from columns, stack counts, jump points of the slice-count function, and spread-controlled sampling in regular regions, while replacing noisy critical intervals by topology-safe quadrilaterals. For generic planar PL sets, the resulting approximations recover geometric and topological invariants such as Betti numbers, area, perimeter, and curvature measures as $\mathcal M$8 (Rowekamp, 2011).

5. Finite atlases in tilings and algebra

In the theory of aperiodic tilings, an atlas can be a finite list of legal polygonal neighborhoods. For geometrical Penrose tilings, the decisive result is that the undecorated tilings are characterized by their $\mathcal M$9-atlas: the finite set $\mathcal A=\{(U_i,\phi_i)\}_{i\in I}$0 of radius-$\mathcal A=\{(U_i,\phi_i)\}_{i\in I}$1 vertex neighborhoods. The paper proves that $\mathcal A=\{(U_i,\phi_i)\}_{i\in I}$2 consists of exactly $\mathcal A=\{(U_i,\phi_i)\}_{i\in I}$3 $\mathcal A=\{(U_i,\phi_i)\}_{i\in I}$4-maps up to isometry. By contrast, the undecorated $\mathcal A=\{(U_i,\phi_i)\}_{i\in I}$5-atlas has $\mathcal A=\{(U_i,\phi_i)\}_{i\in I}$6 patterns up to isometry and is not sufficient, since there exists a periodic tiling whose $\mathcal A=\{(U_i,\phi_i)\}_{i\in I}$7-maps are all legal Penrose $\mathcal A=\{(U_i,\phi_i)\}_{i\in I}$8-maps. The atlas can be computed either through substitution and linear recurrence or via cut-and-project regions in the acceptance window (Fernique et al., 2022).

Polygon equations use the word in a different but equally systematic sense. Higher Tamari orders $\mathcal A=\{(U_i,\phi_i)\}_{i\in I}$9 realize an infinite hierarchy of $N$0-gon equations that generalize the pentagon equation, just as higher Bruhat orders realize simplex equations such as the Yang–Baxter and tetrahedron equations. For $N$1, one has maps $N$2, and the $N$3-gon equation is obtained by equating the compositions along the two maximal chains of $N$4. The paper’s three-color decomposition of higher Bruhat orders shows that an $N$5-simplex equation decomposes into the $N$6-gon equation, its dual, and a mixed compatibility equation, thereby placing polygon equations in a broader algebraic atlas parallel to simplex equations (Dimakis et al., 2014).

These two literatures make a useful distinction clear. In tilings, a polygon atlas is a finite admissibility rule on local neighborhoods; in algebra, it is a hierarchy of consistency equations controlled by posets and visualized on polyhedra such as tetrahedra, cubes, associahedra, and the Edelman–Reiner polyhedron. Neither use is a surface parameterization, but both preserve the central atlas idea of representing global structure through a controlled family of local configurations (Fernique et al., 2022, Dimakis et al., 2014).

6. Unrelated protocol use: Polygon Atlas on the blockchain

An unrelated use of the same name appears in blockchain market design. In this setting, Polygon Atlas refers to a sealed-bid, per-opportunity MEV auction mechanism implemented through FastLane on Polygon. Instead of public Priority Gas Auctions, a searcher detects an opportunity transaction, constructs a bundle containing the OppTx hash and a signed SolverOperation, submits it to a FastLane node, and competes within an auction window of roughly $N$7 ms. Searchers must maintain an atlETH bond, competitor bids are unobservable before submission, and the resulting problem is modeled as a partially observable first-price auction (Seoev et al., 16 Oct 2025).

A PPO-based reinforcement-learning bidder is proposed for this environment. The state includes route features and recent market context, the action is a continuous bid fraction $N$8, and evaluation uses Win Ratio and Maximum-Profit Capture. Over a December 2024–September 2025 dataset containing $N$9 opportunity transactions and $\mathcal M$00 unique searchers total, the history-conditioned agent reports $\mathcal M$01 MPC in the historical-participation setting and $\mathcal M$02 MPC when replacing the market leader, corresponding to the paper’s headline summary of roughly $\mathcal M$03 and $\mathcal M$04 profit capture, respectively (Seoev et al., 16 Oct 2025).

This protocol use is terminologically distinct from geometric and combinatorial atlas notions. The shared word “Atlas” does not indicate a common mathematical framework; it is simply a name collision between polygonal geometry, topology, graphics, and a market protocol on the Polygon network (Seoev et al., 16 Oct 2025).