/**
 * Where a placed tilemap blocks (#132). A cell blocks when the tile lying there is solid – on any
 * layer, a hidden one too: hiding is how a map is looked at, and an invisible blocking layer is
 * a common thing to want. Only cells inside the grid count, the same ones that are drawn.
 *
 * The blocking cells are merged into rectangles rather than handed on one box each, so a wall is
 * one shape and the collision tests stay few – and so an object sliding along a wall does not
 * catch on the seam between two of its cells.
 *
 * A tile's collision scale (#140) draws the box in or out, but only on the sides *without* a
 * blocking neighbour: between two blocking cells the side stays flush whatever their values, so a
 * wall of bushes is a narrower wall and not a string of beads. A neighbour across the corner only
 * does not count – that corner stays drawn in.
 *
 * A tile whose picture is a frame with collision outlines (#141) blocks with those instead, as
 * polygons, never merged: see `solidPolygons`. A plain tile lays its outlines over the cell. An
 * autotile's cell is pieced together from four quarters of its block, picked by the neighbours – so
 * its outlines are cut along the same quarters, and each cell takes the pieces of exactly the
 * quarters it is drawn from (see `tileShape`).
 *
 * One-way tiles (#155) block only from above. They are worked out on their own, by the same rules –
 * each function takes `oneWay` – so a platform never merges with a wall into one rectangle, nor with
 * a platform of another angle (#159). The same holds for a tile's surface (#193): with a `material`,
 * a function looks only at the tiles of that friction and bounciness.
 */
import type { TileAsset, TilemapAsset } from './Asset';
import { tileQuadrants } from './autotile';
import { angleOf, apply, invert, type Mat, scaleOf } from './matrix';
import { boundaryOf, type BoundaryLine, clipToRect, convexPieces, signedArea } from './polygon';
import type { PropertyBag, Vec2 } from './Properties';
import { findAssetOf, type Project } from './Project';
import { findFrame } from './texture';
import {
  isTileOneWay,
  isTileSolid,
  readTile,
  TILE_ONE_WAY_ANGLE_KEY,
  tileCollisionScale,
  tileKind,
  type TileKind,
} from './tile';
import {
  cellOrigin,
  inGrid,
  neighboursOf,
  parseCellKey,
  tilemapColumns,
  tilemapRows,
  tilemapSize,
  tilemapTileSize,
} from './tilemap';
import { cellsTileIds, cellsVersion, identityToken, KeyedMemo } from './tilemapMemo';
import { type ColliderShape, oneWayAngle, type ShapeBounds } from '../properties/collider';
import { colliderMaterial } from '../properties/rigidbody';

/** A rectangle in cells: top-left corner and size. Whole cells at collision scale 1, parts of one otherwise. */
export interface CellRect {
  x: number;
  y: number;
  w: number;
  h: number;
}

/** A blocking cell and the collision scale of its tile; left out, the scale is 1. */
export interface SolidCell {
  x: number;
  y: number;
  scale?: number;
}

/** A cell that blocks with an outline (#141): the outline and its convex pieces, in cells of the map. */
export interface CellPolygon {
  x: number;
  y: number;
  outline: Vec2[];
  pieces: Vec2[][];
}

/** An outline and its convex pieces: 0 to 1 across a cell for a plain tile, across a quarter for an autotile. */
export interface TileOutline {
  outline: Vec2[];
  pieces: Vec2[][];
}

/** What one quarter of an autotile's block blocks with: all of it, or the parts of the outlines in it – maybe none. */
export type QuarterBlock = 'full' | TileOutline[];

/**
 * How a solid tile with outlines blocks: a plain tile with its outlines across the cell, an autotile
 * with a block of every quarter of its block, by `x,y` in quarters.
 */
export type TileShape =
  | { kind: 'plain'; outlines: TileOutline[] }
  | { kind: 'auto'; tileKind: Exclude<TileKind, 'none'>; quarters: Map<string, QuarterBlock> };

const SQUARE: TileOutline = {
  outline: [
    { x: 0, y: 0 },
    { x: 1, y: 0 },
    { x: 1, y: 1 },
    { x: 0, y: 1 },
  ],
  pieces: [
    [
      { x: 0, y: 0 },
      { x: 1, y: 0 },
      { x: 1, y: 1 },
      { x: 0, y: 1 },
    ],
  ],
};

/** A share of a quarter's area this close to all of it is all of it. */
const FULL = 1 - 1e-9;

/** The outlines that are polygons (see `polygonProblem`), with their pieces; the others collide as none. */
function validOutlines(outlines: Vec2[][]): TileOutline[] {
  return outlines
    .map((outline) => ({ outline, pieces: convexPieces(outline) }))
    .filter((outline) => outline.pieces.length > 0);
}

/**
 * How the tile blocks, or null when it blocks as a box – when its first picture has no outline that
 * is a polygon. Only the first picture counts: a blocking shape that twitched with the animation is
 * rarely wanted.
 *
 * An autotile's outlines are drawn over its whole block and cut into its quarters, since a cell is
 * drawn from four of them. Where a tile of the block has nothing drawn in it at all, its quarters
 * block whole – so the notches of a floor nobody outlined leave no hole at the inner corners of a
 * lake – but once anything is drawn in it, only what is drawn blocks, and a quarter can be left open.
 * The collision scale does not apply here: the outline is the shape.
 */
export function tileShape(project: Project, tile: TileAsset): TileShape | null {
  const id = tile.pictures[0];
  const frame = id ? findFrame(project, id)?.frame : undefined;
  if (!frame?.collision || frame.w <= 0 || frame.h <= 0) return null;
  const kind = tileKind(tile);
  const quad = kind === 'none' ? 0 : readTile(project, tile).quad;
  // Outlines are replaced, never edited in place, so the array stands for what is drawn.
  let known = shapes.get(frame.collision);
  if (!known) shapes.set(frame.collision, (known = new Map<string, TileShape | null>()));
  const key = `${kind}|${quad}|${frame.w}|${frame.h}`;
  if (!known.has(key)) known.set(key, workOutShape(frame.collision, frame.w, frame.h, kind, quad));
  return known.get(key)!;
}

const shapes = new WeakMap<Vec2[][], Map<string, TileShape | null>>();

function workOutShape(
  collision: Vec2[][],
  w: number,
  h: number,
  kind: TileKind,
  quad: number,
): TileShape | null {
  if (kind === 'none') {
    const across = collision.map((o) => o.map((p) => ({ x: p.x / w, y: p.y / h })));
    const outlines = validOutlines(across);
    return outlines.length > 0 ? { kind: 'plain', outlines } : null;
  }
  if (quad <= 0) return null;
  const outlines = validOutlines(collision.map((o) => o.map((p) => ({ x: p.x / quad, y: p.y / quad }))));
  if (outlines.length === 0) return null;
  const columns = Math.round(w / quad);
  const rows = Math.round(h / quad);
  const quarters = new Map<string, QuarterBlock>();
  for (let qy = 0; qy < rows; qy++)
    for (let qx = 0; qx < columns; qx++) quarters.set(`${qx},${qy}`, quarterBlock(outlines, qx, qy));
  for (let ty = 0; ty < rows; ty += 2)
    for (let tx = 0; tx < columns; tx += 2) {
      const keys = [`${tx},${ty}`, `${tx + 1},${ty}`, `${tx},${ty + 1}`, `${tx + 1},${ty + 1}`];
      if (keys.every((key) => (quarters.get(key) as QuarterBlock).length === 0))
        for (const key of keys) quarters.set(key, 'full');
    }
  return { kind: 'auto', tileKind: kind, quarters };
}

/** The parts of the outlines inside the quarter at (qx, qy), moved to 0 … 1 across it. */
function quarterBlock(outlines: readonly TileOutline[], qx: number, qy: number): QuarterBlock {
  const local = (p: Vec2) => ({ x: p.x - qx, y: p.y - qy });
  const parts: TileOutline[] = [];
  for (const { outline, pieces } of outlines) {
    const cut = pieces.map((piece) => clipToRect(piece, qx, qy, qx + 1, qy + 1)).filter((p) => p.length > 0);
    if (cut.length === 0) continue;
    // The pieces of one outline never overlap, so their areas add up to what it covers.
    if (cut.reduce((sum, piece) => sum + Math.abs(signedArea(piece)), 0) >= FULL) return 'full';
    const drawn = clipToRect(outline, qx, qy, qx + 1, qy + 1);
    parts.push({
      outline: (drawn.length > 0 ? drawn : cut[0]!).map(local),
      pieces: cut.map((p) => p.map(local)),
    });
  }
  return parts;
}

/** What blocks in each cell: the largest box scale of the solid tiles blocking as boxes, and the outlines of the others, in cells. */
interface CellBlock {
  x: number;
  y: number;
  box: number | null;
  polygons: TileOutline[];
}

/** An outline of 0 to 1 across its square laid over the square of side `size` at (x, y). */
function placed(outline: TileOutline, x: number, y: number, size: number): TileOutline {
  const place = (p: Vec2) => ({ x: x + p.x * size, y: y + p.y * size });
  return { outline: outline.outline.map(place), pieces: outline.pieces.map((piece) => piece.map(place)) };
}

/**
 * Which solid tiles a function looks at: false the ones blocking from everywhere, true every one-way
 * tile, a number the one-way tiles of that angle (#159) – so tiles of two angles never merge.
 */
export type OneWayFilter = boolean | number;

/** The angle a one-way tile lands things within, held to 0 … 90°. */
function tileOneWayAngle(tile: TileAsset): number {
  return oneWayAngle(tile.props[TILE_ONE_WAY_ANGLE_KEY]);
}

/** The surface of a solid tile (#193), read like a collider's. A map never moves, so no density. */
export interface TileMaterial {
  friction: number;
  bounciness: number;
}

export function tileMaterial(tile: TileAsset): TileMaterial {
  const { friction, bounciness } = colliderMaterial(tile.props);
  return { friction, bounciness };
}

function sameMaterial(a: TileMaterial, b: TileMaterial): boolean {
  return a.friction === b.friction && a.bounciness === b.bounciness;
}

/**
 * Which solid tiles a function looks at: those passing `oneWay`, and with a `material` only the
 * tiles of that surface – so ice and rubber never merge into one rectangle (#193).
 */
function passesFilter(tile: TileAsset, oneWay: OneWayFilter, material?: TileMaterial): boolean {
  if (material && !sameMaterial(tileMaterial(tile), material)) return false;
  if (!isTileOneWay(tile)) return oneWay === false;
  return oneWay === true || tileOneWayAngle(tile) === oneWay;
}

/** The solid tiles painted on the map, each once, in the order the layers first name them. */
function solidTiles(project: Project, asset: TilemapAsset): TileAsset[] {
  const ids = new Set(asset.layers.flatMap((layer) => cellsTileIds(layer.cells)));
  return [...ids].flatMap((id) => {
    const tile = findAssetOf(project, id, 'tile');
    return tile && isTileSolid(tile) ? [tile] : [];
  });
}

/** The surfaces of the solid tiles painted on the map that pass `oneWay`, each once. */
export function tileMaterials(
  project: Project,
  asset: TilemapAsset,
  oneWay: OneWayFilter = false,
): TileMaterial[] {
  const materials: TileMaterial[] = [];
  for (const tile of solidTiles(project, asset)) {
    if (!passesFilter(tile, oneWay)) continue;
    const material = tileMaterial(tile);
    if (!materials.some((known) => sameMaterial(known, material))) materials.push(material);
  }
  return materials.sort((a, b) => a.friction - b.friction || a.bounciness - b.bounciness);
}

/** The angles of the solid one-way tiles painted on the map, each once, smallest first. */
export function oneWayAngles(project: Project, asset: TilemapAsset): number[] {
  const angles = new Set(solidTiles(project, asset).filter(isTileOneWay).map(tileOneWayAngle));
  return [...angles].sort((a, b) => a - b);
}

type BlockingTile = { scale: number; shape: TileShape | null } | null;

const blocking = new KeyedMemo<readonly CellBlock[]>();

/**
 * Worked out once per content: the key holds the grid, the filter, the version of every layer's
 * cells and how each tile painted there blocks, which is all the answer depends on.
 */
function blockingCells(
  project: Project,
  props: PropertyBag,
  asset: TilemapAsset,
  oneWay: OneWayFilter,
  material?: TileMaterial,
): readonly CellBlock[] {
  const tiles = new Map<string, BlockingTile>();
  for (const layer of asset.layers)
    for (const id of cellsTileIds(layer.cells)) {
      if (tiles.has(id)) continue;
      const tile = findAssetOf(project, id, 'tile');
      tiles.set(
        id,
        tile && isTileSolid(tile) && passesFilter(tile, oneWay, material)
          ? { scale: tileCollisionScale(tile), shape: tileShape(project, tile) }
          : null,
      );
    }
  const key = [
    tilemapColumns(props),
    tilemapRows(props),
    oneWay,
    material ? `${material.friction}/${material.bounciness}` : '',
    asset.layers.map((layer) => cellsVersion(layer.cells)).join(','),
    [...tiles]
      .map(([id, t]) => (t ? `${id}:${t.scale}:${t.shape ? identityToken(t.shape) : 'box'}` : id))
      .join(','),
  ].join('|');
  return blocking.get(key, () => workOutBlockingCells(props, asset, tiles));
}

function workOutBlockingCells(
  props: PropertyBag,
  asset: TilemapAsset,
  tiles: Map<string, BlockingTile>,
): CellBlock[] {
  const tileOf = (id: string) => tiles.get(id) ?? null;
  const found = new Map<string, CellBlock>();
  for (const layer of asset.layers) {
    for (const [key, tileId] of Object.entries(layer.cells)) {
      const tile = tileId ? tileOf(tileId) : null;
      if (!tile) continue;
      let cell = found.get(key);
      if (!cell) {
        const at = parseCellKey(key);
        if (!at || !inGrid(props, at.x, at.y)) continue;
        found.set(key, (cell = { ...at, box: null, polygons: [] }));
      }
      const { shape, scale } = tile;
      if (!shape) {
        cell.box = Math.max(cell.box ?? 0, scale);
      } else if (shape.kind === 'plain') {
        // The collision scale draws the outline in or out about the cell's centre (#140).
        const at = 0.5 - scale / 2;
        for (const outline of shape.outlines)
          cell.polygons.push(placed(outline, cell.x + at, cell.y + at, scale));
      } else {
        const around = neighboursOf(props, layer, cell.x, cell.y, tileId);
        const quarters = tileQuadrants(shape.tileKind, around).map(
          (q) => shape.quarters.get(`${q.x},${q.y}`) ?? [],
        );
        if (quarters.every((q) => q === 'full')) {
          cell.box = Math.max(cell.box ?? 0, 1);
          continue;
        }
        for (const [corner, quarter] of quarters.entries()) {
          const x = cell.x + (corner % 2) / 2;
          const y = cell.y + Math.floor(corner / 2) / 2;
          for (const outline of quarter === 'full' ? [SQUARE] : quarter)
            cell.polygons.push(placed(outline, x, y, 0.5));
        }
      }
    }
  }
  return [...found.values()];
}

/**
 * Every cell of the map that blocks as a box, once each however many layers hold a solid tile
 * there. Where several do, the largest scale wins: each of them blocks, so the cell blocks as far
 * as any does. An autotile cell built only from quarters that block whole is a box too, at scale 1.
 * A cell whose only solid tiles block with outlines is left to `solidPolygons`.
 */
export function solidCells(
  project: Project,
  props: PropertyBag,
  asset: TilemapAsset,
  oneWay: OneWayFilter = false,
  material?: TileMaterial,
): Required<SolidCell>[] {
  return boxCells(blockingCells(project, props, asset, oneWay, material));
}

function boxCells(cells: readonly CellBlock[]): Required<SolidCell>[] {
  return cells.filter((cell) => cell.box !== null).map(({ x, y, box }) => ({ x, y, scale: box! }));
}

/**
 * The polygons of the cells that block with outlines: for a plain tile one per outline, for an
 * autotile one per part of an outline in each of its quarters, and a square for a quarter that
 * blocks whole. A tile blocking as a box on another layer of the same cell covers the whole cell
 * already, so there the outlines are left out.
 */
export function solidPolygons(
  project: Project,
  props: PropertyBag,
  asset: TilemapAsset,
  oneWay: OneWayFilter = false,
  material?: TileMaterial,
): readonly CellPolygon[] {
  return derived(polygonsOf, blockingCells(project, props, asset, oneWay, material), (cells) => {
    const polygons: CellPolygon[] = [];
    for (const cell of cells) {
      if (cell.box !== null) continue;
      for (const { outline, pieces } of cell.polygons)
        polygons.push({ x: cell.x, y: cell.y, outline, pieces });
    }
    return polygons.sort((a, b) => a.y - b.y || a.x - b.x);
  });
}

/**
 * The same cells hand out the same rectangles and polygons, the very same arrays – so whoever builds
 * on them can tell by identity that nothing changed (the physics does, every tick).
 */
function derived<T>(
  held: WeakMap<readonly CellBlock[], T>,
  cells: readonly CellBlock[],
  make: (cells: readonly CellBlock[]) => T,
): T {
  if (!held.has(cells)) held.set(cells, make(cells));
  return held.get(cells)!;
}

const polygonsOf = new WeakMap<readonly CellBlock[], readonly CellPolygon[]>();
const rectsOf = new WeakMap<readonly CellBlock[], readonly CellRect[]>();

/** A rectangle by its four sides, which is what the merge compares. */
interface Piece {
  x0: number;
  x1: number;
  y0: number;
  y1: number;
}

/**
 * The part of the map one cell blocks, as pieces. At scale 1 that is the cell. Below 1 the cell is
 * cut at the inset `e` into a 3 × 3 grid: the middle always blocks, a side strip only towards a
 * blocking neighbour, a corner only when both its sides *and* the cell across the corner block –
 * which is what closes the inner corner of an L cleanly instead of leaving a notch filled. Above 1
 * the cell grows by the same amount on every side without a blocking neighbour.
 */
function cellPieces(cell: SolidCell, blocks: (x: number, y: number) => boolean): Piece[] {
  const { x, y } = cell;
  const e = (1 - (cell.scale ?? 1)) / 2;
  const left = blocks(x - 1, y);
  const right = blocks(x + 1, y);
  const up = blocks(x, y - 1);
  const down = blocks(x, y + 1);
  if (e <= 0) {
    const g = -e;
    return [
      {
        x0: x - (left ? 0 : g),
        x1: x + 1 + (right ? 0 : g),
        y0: y - (up ? 0 : g),
        y1: y + 1 + (down ? 0 : g),
      },
    ];
  }
  // Bands along each axis: [start, end, neighbour side (-1, 0, 1)]; the same expressions on both
  // sides of a cut keep touching pieces exactly equal, so they merge.
  const bands = (at: number): [number, number, -1 | 0 | 1][] => [
    [at, at + e, -1],
    [at + e, at + (1 - e), 0],
    [at + (1 - e), at + 1, 1],
  ];
  const pieces: Piece[] = [];
  for (const [y0, y1, dy] of bands(y)) {
    for (const [x0, x1, dx] of bands(x)) {
      const side = (dx === 0 || (dx < 0 ? left : right)) && (dy === 0 || (dy < 0 ? up : down));
      if (side && (dx === 0 || dy === 0 || blocks(x + dx, y + dy))) pieces.push({ x0, x1, y0, y1 });
    }
  }
  return pieces;
}

/**
 * The cells merged into rectangles. Each cell is cut into its pieces (see `cellPieces`); pieces on
 * the same strip – same top and bottom – that touch or overlap join into one, and then strips of
 * exactly the same left and right that meet top to bottom join into one taller rectangle. At scale 1
 * every piece is a cell, so a run joins across a row and runs of the same columns stack. The result
 * is ordered top-down, then left to right.
 */
export function mergeCells(cells: readonly SolidCell[]): CellRect[] {
  const unique = new Map<string, SolidCell>();
  for (const cell of cells) unique.set(`${cell.x},${cell.y}`, cell);
  const blocks = (x: number, y: number) => unique.has(`${x},${y}`);
  const strips = new Map<string, Piece[]>();
  for (const cell of unique.values()) {
    for (const piece of cellPieces(cell, blocks)) {
      const key = `${piece.y0},${piece.y1}`;
      const strip = strips.get(key);
      if (strip) strip.push(piece);
      else strips.set(key, [piece]);
    }
  }
  const runs: Piece[] = [];
  for (const strip of strips.values()) {
    strip.sort((a, b) => a.x0 - b.x0);
    let run: Piece | null = null;
    for (const piece of strip) {
      if (run && piece.x0 <= run.x1) run.x1 = Math.max(run.x1, piece.x1);
      else runs.push((run = { ...piece }));
    }
  }
  runs.sort((a, b) => a.y0 - b.y0 || a.x0 - b.x0);
  const done: Piece[] = [];
  // Rectangles still open at the bottom, by the left and right they span.
  const open = new Map<string, Piece>();
  for (const run of runs) {
    const key = `${run.x0},${run.x1}`;
    const above = open.get(key);
    if (above && above.y1 === run.y0) {
      above.y1 = run.y1;
      continue;
    }
    if (above) done.push(above);
    open.set(key, run);
  }
  done.push(...open.values());
  return done
    .sort((a, b) => a.y0 - b.y0 || a.x0 - b.x0)
    .map((p) => ({ x: p.x0, y: p.y0, w: p.x1 - p.x0, h: p.y1 - p.y0 }));
}

/** The blocking rectangles of a map, in cells. */
export function solidRects(
  project: Project,
  props: PropertyBag,
  asset: TilemapAsset,
  oneWay: OneWayFilter = false,
  material?: TileMaterial,
): readonly CellRect[] {
  return derived(rectsOf, blockingCells(project, props, asset, oneWay, material), (cells) =>
    mergeCells(boxCells(cells)),
  );
}

/**
 * The outline of where the map blocks, in cells, as the scene view draws it (#173): only the outer
 * edges of the rectangles and outlined cells together, so the quarters an autotile is pieced from
 * and the rectangles of a merged wall do not show as strokes across what blocks. Drawing only –
 * collision keeps the pieces.
 */
export function solidBoundary(rects: readonly CellRect[], polygons: readonly CellPolygon[]): BoundaryLine[] {
  const rings = rects.map(({ x, y, w, h }) => [
    { x, y },
    { x: x + w, y },
    { x: x + w, y: y + h },
    { x, y: y + h },
  ]);
  return boundaryOf([...rings, ...polygons.map((polygon) => polygon.outline)]);
}

/**
 * One rectangle as it sits in the world: a box turning and growing with the placed map, the same
 * shape a collider on it would have (see `properties/collider.ts`).
 */
export function cellRectShape(props: PropertyBag, rect: CellRect, m: Mat): ColliderShape {
  const tile = tilemapTileSize(props);
  const corner = cellOrigin(props, rect.x, rect.y);
  const centre = apply(m, { x: corner.x + (rect.w * tile) / 2, y: corner.y + (rect.h * tile) / 2 });
  const scale = scaleOf(m);
  return {
    kind: 'box',
    x: centre.x,
    y: centre.y,
    w: rect.w * tile * scale,
    h: rect.h * tile * scale,
    angle: angleOf(m),
  };
}

/**
 * Which cells of a placed map can reach into a world rectangle (#184): the rectangle taken back
 * into the map's own units, as cell coordinates – a turned map gives the rectangle around its
 * turned self, which is wider than needed but never too narrow.
 */
export function cellsWithin(props: PropertyBag, m: Mat, bounds: ShapeBounds): CellBounds {
  const back = invert(m);
  const tile = tilemapTileSize(props);
  const size = tilemapSize(props);
  const corners = [
    { x: bounds.left, y: bounds.top },
    { x: bounds.right, y: bounds.top },
    { x: bounds.right, y: bounds.bottom },
    { x: bounds.left, y: bounds.bottom },
  ].map((p) => apply(back, p));
  const cell = (values: number[], half: number) => values.map((v) => (v + half) / tile);
  const xs = cell(
    corners.map((p) => p.x),
    size.w / 2,
  );
  const ys = cell(
    corners.map((p) => p.y),
    size.h / 2,
  );
  return {
    left: Math.min(...xs),
    top: Math.min(...ys),
    right: Math.max(...xs),
    bottom: Math.max(...ys),
  };
}

/** A rectangle in cell coordinates, as `cellsWithin` reports it. */
export interface CellBounds {
  left: number;
  top: number;
  right: number;
  bottom: number;
}

/** Whether a rectangle of cells touches the range at all – touching counts, so nothing near is lost. */
export function cellsMeet(range: CellBounds, x: number, y: number, w = 1, h = 1): boolean {
  return x <= range.right && x + w >= range.left && y <= range.bottom && y + h >= range.top;
}

/** An outlined cell as it sits in the world: a polygon turning and growing with the placed map. */
export function cellPolygonShape(props: PropertyBag, polygon: CellPolygon, m: Mat): ColliderShape {
  const world = (p: Vec2) => apply(m, cellOrigin(props, p.x, p.y));
  const centre = world({ x: polygon.x + 0.5, y: polygon.y + 0.5 });
  return {
    kind: 'polygon',
    x: centre.x,
    y: centre.y,
    angle: angleOf(m),
    outline: polygon.outline.map(world),
    pieces: polygon.pieces.map((piece) => piece.map(world)),
  };
}