From 5b0a5b290aef58feb1571d223776c7f796c941bb Mon Sep 17 00:00:00 2001 From: didericis Date: Tue, 19 May 2026 23:26:36 -0400 Subject: [PATCH] Add labelled sequencing example figure Embed a worked example of the canonical quadrilateral sequencing into the paper. The new figure shows the deep embedding of a 9-vertex triangulation with each quadrilateral filled by type (shallow diamond, deep diamond, S quad) and annotated with its sequence index and move code. The generator script renders the figure from a fixed Sage RNG seed for reproducibility. Co-Authored-By: Claude Opus 4.7 --- .../plane_depth_sequencing/example_figure.pdf | Bin 0 -> 20135 bytes papers/plane_depth_sequencing/paper.pdf | Bin 208392 -> 240307 bytes papers/plane_depth_sequencing/paper.tex | 8 + plane_depth_sequencing_figure.py | 221 ++++++++++++++++++ 4 files changed, 229 insertions(+) create mode 100644 papers/plane_depth_sequencing/example_figure.pdf create mode 100644 plane_depth_sequencing_figure.py diff --git a/papers/plane_depth_sequencing/example_figure.pdf b/papers/plane_depth_sequencing/example_figure.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6d3c4dcf4529fe39ebf3525643d5b49049909476 GIT binary patch literal 20135 zcmeHv1yohd*SCa%7X(Rzx`2T6ZMc*)lF|~=-Hjk2jYvo%Atl`iiV}*XNDG3bN|%5% z3i6!``aC|$`+r>TS8ILiyR2d6%$eA;_dYXw_W8|WQWTfugmNK3Ohu#M!fFr{3<292 zT7m=wz}%|t_9kF%F#~4i|3e1_so20;7+}6S1|m1FoQ$qCYbObNdJP z$OdSOen)b4ywSOb|CQvNJZpknQB`XkuUka!Yxe(q1y!LK3v?3F3-7 zk@>}2bNJbL$fI%&FaaZh(Is|h-lZo|y!X=BqQ^GRwUwJjbiWG;XYB9{F48_ITAs1* 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zg+~w-#7v$sqeV%?OXJy6W%Y=C+Y)9;l+J{o75|bMKJPVkGx}Fx03F?mCjl>9go3V8 zAcK-DgUU?x3rxMpfpo$|(r3>y7oS>rfgMuOjb;f9GKF~Cg=U;tWZL)T<0Q?ImExRp zpb(`VhCfYOgpaf%thp*)5rdL-R-#=d`!3#KRtlr=2Xee}97 z_?|`$oDo8;eSg_M*6^(Y_4u$COI4CvzIFH^^jMX@wzw3ler+>VS5j>e+gDO;8yHp= zZwXC7SG$+^3es*)%XXE&`q9N?_qwdg67OkUdnErX9DGQ5tV%3htnNE4W0tQ-|y a9cF<}i!28bA+vDvaB?D2Qc5UEBK{X7NW1j_ diff --git a/papers/plane_depth_sequencing/paper.tex b/papers/plane_depth_sequencing/paper.tex index 779aa55..b35fc4c 100644 --- a/papers/plane_depth_sequencing/paper.tex +++ b/papers/plane_depth_sequencing/paper.tex @@ -37,6 +37,7 @@ \documentclass{amsart} \usepackage{amssymb} +\usepackage{graphicx} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} @@ -255,6 +256,13 @@ At each step $n \geq 1$, the next quadrilateral $Q_{n+1}$ is chosen by the first That is, each move is consulted only when no higher-precedence move applies. \end{definition} +\begin{figure} +\centering +\includegraphics[width=0.85\textwidth]{example_figure.pdf} +\caption{The deep embedding $G'$ of a small maximal planar graph (drawn with one outer-cap face as the outer face), with each quadrilateral $Q_n$ of the canonical sequence labelled by its index and the move code (AD = anchor drop, LA = level add, J = join, RC = ring completion) of the move that produced it. Solid edges are non-level; dashed edges are level. Background colour encodes quadrilateral type: amber for shallow diamonds, teal for deep diamonds, pink for S quads. Outer-cycle vertices are blue, the outer-cap vertex $x^{*}$ is red. The move-code string for this example is $01211333$.} +\label{fig:example-sequence} +\end{figure} + Let $N$ denote the total number of quadrilaterals in the decomposition of $G'$; equivalently, $N = |F(G')|/2$, where $F(G')$ is the set of triangular faces of $G'$. \begin{theorem}[Termination and coverage] diff --git a/plane_depth_sequencing_figure.py b/plane_depth_sequencing_figure.py new file mode 100644 index 0000000..9920c87 --- /dev/null +++ b/plane_depth_sequencing_figure.py @@ -0,0 +1,221 @@ +"""Generate the labelled quadrilateral sequencing example figure for the paper. + +Renders the deep embedding G' of a small maximal planar graph, fills each +quadrilateral with a color encoding its type, and overlays the index it +occupies in the canonical sequence Q_1, ..., Q_N. The output is a PDF placed +next to paper.tex. +""" +import argparse +from pathlib import Path +from typing import Any + +import matplotlib +matplotlib.use("Agg") +import matplotlib.patches as patches # noqa: E402 pylint: disable=wrong-import-position +import matplotlib.pyplot as plt # noqa: E402 pylint: disable=wrong-import-position + +from sage.all import Graph, graphs # type: ignore[attr-defined] # pylint: disable=no-name-in-module,wrong-import-position # noqa: E402 +from sage.misc.randstate import set_random_seed # type: ignore[import-not-found] # noqa: E402 + +from lib.tutte_embedding import tutte_embedding # noqa: E402 +from plane_depth_sequencing import ( # noqa: E402 + _level_edge_of_face, + _quad_type, + _quad_vertices, + quadrilateral_sequencing, +) + + +def _planar_pos( + g_prime: Graph, + outer_face: list[Any], + layout: str, +) -> dict[Any, tuple[float, float]]: + """Compute positions for g_prime under either 'tutte' or 'sage_planar' layout.""" + if layout == "tutte": + return tutte_embedding(g_prime, outer_face) + if layout == "sage_planar": + g_prime.is_planar(set_embedding=True, set_pos=True) + pos = g_prime.get_pos() + assert pos is not None, "Sage failed to set planar positions" + return {v: (float(p[0]), float(p[1])) for v, p in pos.items()} + raise ValueError(f"Unknown layout: {layout}") + + +MOVE_NAMES = {0: "AD", 1: "LA", 2: "J", 3: "RC"} +TYPE_COLORS = { + "shallow_diamond": "#FFE0B2", + "deep_diamond": "#B2DFDB", + "s_quad": "#F8BBD0", +} + + +def _ordered_quad_vertices(quad: frozenset, depth_labelling: dict[Any, int]) -> list[Any]: + """Return the 4 quad vertices in cyclic order: e1, a, e2, b.""" + f1, f2 = list(quad) + level_edge = _level_edge_of_face(f1, depth_labelling) + e1, e2 = list(level_edge) + a = next(v for v in f1 if v not in level_edge) + b = next(v for v in f2 if v not in level_edge) + return [e1, a, e2, b] + + +def _pick_outer_cap_face( + g_prime: Graph, + outer_cap_vertex: Any, + outer_cycle: list[Any], +) -> list[Any]: + """Return the vertex list of an outer-cap face (the face used as outer in the plane drawing).""" + g_prime.is_planar(set_embedding=True) + embedding = g_prime.get_embedding() + outer_set = set(outer_cycle) + for face in g_prime.faces(embedding): + verts = [u for u, _ in face] + if outer_cap_vertex in verts and len(set(verts) & outer_set) == 2: + return verts + raise RuntimeError("No outer-cap face found in G' embedding") + + +def _draw_figure( + g: Graph, + outer_cycle: list[Any], + out_path: Path, + layout: str = "sage_planar", +) -> dict[str, Any]: + """Render and save the labelled sequencing figure. Returns summary stats.""" + result = quadrilateral_sequencing(g, outer_cycle) + g_prime: Graph = result["deep_embedding"] + depth_labelling: dict[Any, int] = result["depth_labelling"] + sequence = result["sequence"] + move_codes = result["move_codes"] + outer_cap_vertex = result["outer_cap_vertex"] + + outer_face = _pick_outer_cap_face(g_prime, outer_cap_vertex, outer_cycle) + pos = _planar_pos(g_prime, outer_face, layout) + + fig, ax = plt.subplots(figsize=(9, 9)) + + for i, quad in enumerate(sequence): + ordered = _ordered_quad_vertices(quad, depth_labelling) + poly_pts = [pos[v] for v in ordered] + qt = _quad_type(quad, depth_labelling) + polygon = patches.Polygon( + poly_pts, + closed=True, + facecolor=TYPE_COLORS[qt], + edgecolor="none", + alpha=0.65, + zorder=1, + ) + ax.add_patch(polygon) + + for u, v in g_prime.edges(labels=False): + x1, y1 = pos[u] + x2, y2 = pos[v] + if depth_labelling[u] == depth_labelling[v]: + ax.plot([x1, x2], [y1, y2], linestyle=(0, (3, 2)), color="#666666", linewidth=1.0, zorder=2) + else: + ax.plot([x1, x2], [y1, y2], "-", color="black", linewidth=1.0, zorder=2) + + outer_set = set(outer_cycle) + for v, (x, y) in pos.items(): + if v == outer_cap_vertex: + color = "#D32F2F" + elif v in outer_set: + color = "#1976D2" + else: + color = "black" + ax.scatter([x], [y], s=28, color=color, zorder=5, edgecolors="white", linewidths=0.8) + name = "$x^{*}$" if v == outer_cap_vertex else str(v) + ax.annotate( + name, + (x, y), + xytext=(7, 6), + textcoords="offset points", + fontsize=9, + zorder=8, + color="#222222", + bbox={"boxstyle": "round,pad=0.05", "facecolor": "white", "edgecolor": "none", "alpha": 0.85}, + ) + + for i, quad in enumerate(sequence): + ordered = _ordered_quad_vertices(quad, depth_labelling) + poly_pts = [pos[v] for v in ordered] + cx = sum(p[0] for p in poly_pts) / 4 + cy = sum(p[1] for p in poly_pts) / 4 + if i == 0: + label = "$Q_{1}$" + else: + label = f"$Q_{{{i + 1}}}^{{\\mathrm{{{MOVE_NAMES[move_codes[i - 1]]}}}}}$" + ax.text( + cx, + cy, + label, + ha="center", + va="center", + fontsize=12, + zorder=7, + bbox={"boxstyle": "round,pad=0.22", "facecolor": "white", "edgecolor": "#444444", "alpha": 0.95}, + ) + + legend_handles = [ + patches.Patch(color=TYPE_COLORS["shallow_diamond"], label="shallow diamond"), + patches.Patch(color=TYPE_COLORS["deep_diamond"], label="deep diamond"), + patches.Patch(color=TYPE_COLORS["s_quad"], label="S quad"), + ] + ax.legend(handles=legend_handles, loc="lower right", fontsize=10, framealpha=0.95) + + ax.set_aspect("equal") + ax.axis("off") + plt.tight_layout() + out_path.parent.mkdir(parents=True, exist_ok=True) + plt.savefig(out_path, format="pdf", bbox_inches="tight") + plt.close(fig) + + return { + "n_vertices_g": g.order(), + "n_vertices_gprime": g_prime.order(), + "n_quads": len(sequence), + "move_codes": move_codes, + "outer_cycle": outer_cycle, + "outer_cap_vertex": outer_cap_vertex, + "depth_labelling": depth_labelling, + } + + +def _build_example(seed: int, n: int) -> tuple[Graph, list[Any]]: + """Build a random triangulation and pick a deterministic outer cycle.""" + set_random_seed(seed) + g = graphs.RandomTriangulation(n) + g.is_planar(set_embedding=True) + embedding = g.get_embedding() + faces = g.faces(embedding) + outer_cycle = [u for u, _ in faces[0]] + return g, outer_cycle + + +def main() -> None: + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--seed", type=int, default=141) + parser.add_argument("--n", type=int, default=9) + parser.add_argument( + "--out", + type=Path, + default=Path("papers/plane_depth_sequencing/example_figure.pdf"), + ) + parser.add_argument("--layout", choices=("tutte", "sage_planar"), default="sage_planar") + args = parser.parse_args() + + g, outer_cycle = _build_example(args.seed, args.n) + summary = _draw_figure(g, outer_cycle, args.out, layout=args.layout) + + print(f"Wrote {args.out}") + print(f" |V(G)|={summary['n_vertices_g']}, |V(G')|={summary['n_vertices_gprime']}, " + f"N={summary['n_quads']}") + print(f" outer cycle: {summary['outer_cycle']}, x* = {summary['outer_cap_vertex']}") + code_str = "".join(str(c) for c in summary["move_codes"]) + print(f" move-code string: {code_str}") + + +if __name__ == "__main__": + main()