/**
 * The face artwork: everything that used to be a hand-drawn PNG in GIMP is now
 * drawn from parameters onto a canvas, which is then used as the texture of the
 * two overlay planes that already exist on the model.
 *
 * The plates have a linear UV mapping fitted to the original artwork, so drawing
 * in "texture pixels" lands exactly where the original artwork did. Every constant
 * below was measured from the original textures (see README for the numbers), so
 * the defaults reproduce the 2022 artwork while every part of it stays editable.
 *
 * Texture space: the artwork WINDOW, 1024 x 380, origin at the top-left, y grows
 * downwards. `canvasLayout` then says where that window sits on the bigger canvas
 * a face plate actually needs (see below), and `Face` hands the bounds back as
 * `limits`, so the drawing can clamp against the real edge instead of assuming
 * the window is all there is.
 *
 * glTF stores v downwards too, and the textures are uploaded with flipY = false,
 * so no flipping is needed anywhere.
 */

const TAU = Math.PI * 2;
const clamp = (v, lo, hi) => Math.min(hi, Math.max(lo, v));
const HUGE = 4000;

/**
 * The artwork window: the rectangle the 2022 hand-drawn textures occupied. Every
 * constant below is written in this space.
 */
export const ART_WINDOW = { width: 1024, height: 380 };

/** Keep this much clear at the canvas border, so filtering has room to breathe. */
const EDGE_MARGIN = 8;

/** What a drawing falls back to when no plate bounds are supplied. */
const WINDOW_LIMITS = {
  top: 0, bottom: ART_WINDOW.height, left: 0, right: ART_WINDOW.width,
};

/**
 * The canvas a face plate's drawing needs, and where the artwork window sits on it.
 *
 * WHY: the plates are the *whole* front half of the body, so their UVs run far
 * outside the 0..1 the artwork was drawn in (measured on the shipped model: the
 * eye plate covers v -1.02..1.93, and 9 of 10 of its vertices sit within a
 * twentieth of the border). Everything outside 0..1 used to clamp to the canvas
 * edge row, so the moment a drawing reached the border - a tear dropped low, a
 * brow lifted, a thick mouth pushed down - that edge row was copied across the
 * rest of the plate and the mark stretched into a long smear.
 *
 * `Face` rewrites each plate's UVs so this range maps onto 0..1 instead, which
 * removes the clamp altogether. The map is affine, so the artwork keeps its exact
 * pixel size and position and merely shifts by `offset`; the canvas grows by the
 * same factor, and those extra rows are the room a moved tear, brow or mouth
 * needs (`limits` says where the real edge now is, in window coordinates).
 *
 * @param {{u0:number,u1:number,v0:number,v1:number}} range the plate's UV range
 * @param {{width:number,height:number}} [window] the artwork window
 */
export function canvasLayout(range, window = ART_WINDOW) {
  const u0 = Number.isFinite(range?.u0) ? range.u0 : 0;
  const u1 = Number.isFinite(range?.u1) ? range.u1 : 1;
  const v0 = Number.isFinite(range?.v0) ? range.v0 : 0;
  const v1 = Number.isFinite(range?.v1) ? range.v1 : 1;
  const du = Math.max(1e-6, u1 - u0);
  const dv = Math.max(1e-6, v1 - v0);
  const width = Math.max(1, Math.round(window.width * du));
  const height = Math.max(1, Math.round(window.height * dv));
  // `+ 0` turns a `-0` into a plain `0`, so the offsets compare cleanly.
  const offsetX = Math.round(-u0 * window.width) + 0;
  const offsetY = Math.round(-v0 * window.height) + 0;
  return {
    width,
    height,
    offsetX,
    offsetY,
    window,
    // The canvas edges in the artwork's own coordinates. `top` is negative when
    // the plate reaches above the window: that is the extra room.
    limits: {
      top: -offsetY + 0,
      bottom: height - offsetY,
      left: -offsetX + 0,
      right: width - offsetX,
    },
  };
}

/** Fixed layout of the eye plane, measured from the original `eyes-open.png`. */
export const EYE_LAYOUT = {
  radius: 84,
  irisRadius: 62.5,
  highlightRadius: 28,
  highlightOffset: { x: 29.5, y: -18 },
  // The model's left eye (its own left, +x) sits in the u > 0.5 half.
  eyes: [
    { key: 'right', cx: 267.5, cy: 240, towardNose: 1 },
    { key: 'left', cx: 755.5, cy: 240, towardNose: -1 },
  ],
  lidRadiusFactor: 1.35,
  lidStroke: 14,
  // Eyebrows are painted into the same texture as the eyes. The eye plane is a
  // flat-ish patch on a round head, so only its lower part is really outside
  // the head: a brow high on the forehead would be swallowed by it. The brow
  // therefore sits just above the eyeball (row 156) and the eyeball is drawn
  // over its lower edge afterwards.
  // The brow sits just above the eyeball, and the eyeball is drawn over its
  // lower edge. `lift` is that resting height, so `height: 0` is the brow's
  // natural place and a POSITIVE height raises it (a negative one sinks it
  // towards the eye, where the eyeball will cover it).
  brow: { lift: 114, length: 134, thickness: 15, curve: 0.14, offsetX: 6 },
  // A teardrop hangs well below the eye. The plate keeps the whole canvas, so
  // the drop can sit low without being clipped (measured with a ruler grid).
  tear: { size: 34, offsetX: 66, offsetY: 88 },
  closedLine: {
    // The single shut line is the *long* form of the eyelid, the same reach as
    // the arms of the "ぎゅっ" below, so switching between 1 and 3 lines does not
    // change how wide the eye reads.
    length: 205,
    offsetX: 6,
    offsetY: -10,
    slantDeg: 3.9,
    bow: 3,
    // 2- and 3-line shut eyes, copied from the original `eyes-close-tight`
    // artwork: three strokes sharing one vertex that points at the nose, opening
    // to a wide bird's foot / arrow shape. `spread` is half the height the arms
    // open to at the far end.
    armSpan: 205,
    spread: 62,
    vertex: 87.5,
    armX: 104,
  },
  arch: { edgesUp: 8, apexUp: 44 },
};

/**
 * ほっぺ (a manga blush) and 頭の模様 (a mark on the head), both drawn into the eye
 * plate next to the eyes and brows.
 *
 * They are deliberately *generic* manga devices, not a copy of any one character:
 * a pink patch crossed by short diagonal strokes is a stock convention, and the
 * head mark is one of a few simple shapes (a hook, a spiral, three strokes, a dot
 * row) that starts off. Both stay editable through the panel. The cheek positions
 * are tied to `EYE_LAYOUT` so they sit beside the eyes; the head mark sits up in
 * the plate's upper area, i.e. on the forehead above the brows.
 */
export const CHEEK_LAYOUT = {
  drop: 64,        // how far below the eye centre the patch sits (artwork px)
  outward: 66,     // how far outwards, away from the midline, it sits
  radius: 50,      // half-width of the patch
  squash: 0.55,    // half-height, as a fraction of the half-width (a flat ellipse)
  // The blush is *always* four strokes - the reference artwork has four - so
  // there is no line-count control any more. The strokes are horizontal and
  // hand-drawn: each wobbles a little and tapers to a point at both ends.
  lines: 4,
  lineWidth: 10,   // the stroke thickness at its widest (artwork px)
  spread: 0.62,    // how far the outermost stroke sits from the centre (x half-height)
  margin: 0.18,    // clear gap left between a stroke's end and the patch outline
  wobble: 0.32,    // sideways wander, as a fraction of the stroke thickness
};

/**
 * 頭の模様: a *hair-like covering* over the whole head, not a small forehead symbol.
 *
 * The face plate is a shell of the *whole* front half of the body, so the plate's
 * own edge is the head's front silhouette. Filling the plate from its top edge
 * down to an adjustable boundary therefore makes the covering's outline follow
 * the silhouette for free - there is no separate ellipse to keep in step.
 *
 * It is a generic device: a colour cap, a fringe, a side part, or a left/right
 * two-tone split. It is deliberately *not* a copy of any one character's hair.
 * It starts off.
 */
export const HEAD_MARK_LAYOUT = {
  // Where the boundary sits by default: on the forehead, just above the brows
  // (EYE_LAYOUT.eyes[0].cy is the eye row and the brows reach ~114px above it).
  cy: 120,
  reach: 190,      // artwork px the boundary moves per unit of `size`
};

/**
 * 頭の模様 on the *hair plate*: the head's cover when the model carries a plate
 * that wraps all the way round (material `hair-plate`, cylindrical UV).
 *
 * The front plate only reaches the head's silhouette, so hair drawn on it stops
 * at the front half. The hair plate's UV is `u` = angle round the head (front at
 * 0.5, back at the seam 0/1) and `v` = height (0 at the model's feet, 1 at the
 * top of the head), so the covering is everything *above* a boundary line: a cap
 * that is visible from behind as well.
 */
export const HAIR_WRAP_LAYOUT = {
  // The wrap's v runs top-of-head (0) to the model's feet (1) after the GLB's
  // V-flip, so `base` is how far down from the top the hairline sits at size 1.
  // `reach` is how much a unit of `size` moves it; `offsetScale` converts the
  // `offsetY` slider (artwork px) to the same units, so one slider drives both.
  base: 0.30,
  reach: 0.30,
  offsetScale: 0.30 / 190,
};

/**
 * ひげ: the mustaches and beards that hang under the nose. Drawn on the eye plate
 * at the nose, the same spot the snot bubble uses. `shape` picks a well-known
 * kind; `size` scales it about the nose, and the colours are adjustable.
 */
export const BEARD_LAYOUT = {
  cx: 511.5,
  cy: 352,        // just under the nose
  width: 240,     // base width of the widest beard at size 1
};

/**
 * 眼鏡 / サングラス, drawn into the same texture as the eyes and brows.
 *
 * Not measured off any artwork - the character has no glasses - but the numbers
 * are tied to the eyeball so a lens is always a little wider than the eye it
 * covers. The lens centre sits *below* the eye centre because the brow is drawn
 * first, just above the eyeball (see `EYE_LAYOUT.brow`): a lens wide enough to
 * read as glasses would otherwise cut straight through it.
 *
 * The two kinds get separate shapes on purpose. They used to share one ellipse
 * and differ only in how dark the lens was filled, which made them hard to tell
 * apart; the round 眼鏡 below and the wide, angular サングラス in
 * `SUNGLASSES_LAYOUT` now read as different objects at a glance.
 */
const GLASSES_LAYOUT = {
  widthFactor: 1.08,   // lens half-width, as a multiple of the eyeball radius
  heightFactor: 1.0,   // a circle, so the round 眼鏡 stays round (a flatter
                       // ellipse used to drift towards the shades below)
  drop: 20,            // the lens centre hangs this far below the eye centre (artwork px)
  bridgeRise: 8,       // how much the bridge arcs up over the nose
  bridgeWidth: 0.75,   // bridge thickness, as a fraction of the frame stroke
  templeLength: 1.5,   // temple stub length, as a multiple of the eyeball radius
  templeRise: 30,      // how far the temple climbs towards the side of the head
};

/**
 * サングラス: a long, pointed cat-eye rather than the round 眼鏡 lens. Each side is a
 * slim wedge whose *outer end starts low*, rises to a *high, pointed outer corner*
 * set further out (`tipX`), and whose top edge then sweeps back down towards the
 * nose - the "外側が長くとがって上に上がる" shape the reference shows. The inner end is
 * short, so the lens tapers inwards. The eye is allowed to poke out above and
 * below it, so the lens no longer has to cover the whole eyeball. A level bar
 * joins the two inner ends, so the pair still reads as one dark visor. The ratios
 * are of the eyeball radius, exactly like `GLASSES_LAYOUT`, so `scale` means the
 * same thing for both kinds.
 */
const SUNGLASSES_LAYOUT = {
  widthFactor: 1.6,    // a long lens: the outer end reaches well past the eyeball
  heightFactor: 0.58,  // slim, so the eye is free to show above and below it
  drop: 8,             // sits a little lower than the round pair
  corner: 0.05,        // corner rounding, as a fraction of the half-height; kept
                       // tiny so the outer corner stays pointed
  topSkew: 1.0,        // the pointed corner rises this far above the lens top (x half-height)
  innerTop: 0.42,      // the inner top is low, which is what makes the top edge climb
  innerBottom: 0.34,   // the inner bottom is pinched up towards the nose (x half-height)
  outerEndX: 0.75,     // the low outer end sits this far out (x half-width)
  outerBottom: 0.82,   // ...and this deep (x half-height)
  tipX: 1.22,          // the pointed corner juts this far out (x half-width)
  bridgeWidth: 1.15,   // a short, thick bar - thicker than the frame stroke
  bridgeLift: 0.25,    // the bar sits this far above the lens centre (x half-height)
  templeLength: 1.15,  // a short stub, like the round pair's but a little shorter
  templeRise: 24,
  // A heavy rim. It used to be slimmer than the round pair's stroke, which read
  // as a thin pair of shades; the reference look is a chunky frame.
  frameFactor: 1.6,
};

/** The mouth artwork is drawn on the `mouth-plate` shell, which covers the whole
 *  1024 x 380 canvas (see tools/build-face-plates.py), so the drawing no longer
 *  has to be squeezed into a band. The only limit left is the canvas itself. */
export const MOUTH_LAYOUT = {
  centreX: 511.5,
  chordY: 88,
  halfChord: 420.5,
  sag: 177,
  // With the ends and the depth both fixed - which is what the slider promises -
  // the only thing left to choose about the curve is *where* it bends. 1/3 is a
  // quadratic Bézier (a parabola): it is flattest at the apex and falls away
  // fastest near the ends. Pulling the cubic's control points in towards the ends
  // makes the middle of the smile straighter still and lets the fall happen near
  // the corners, which is what reads as a *gentler* curve at the same depth.
  sagBend: 0.2,
  thickness: 21,
  // Kept for reference: the 2022 plane only showed these rows.
  bandTop: 46,
  bandBottom: 332,
  // Only used to cap the *size* of the round "O" mouth, so the surprise face
  // stays a mouth and not a hole. The mouth's travel no longer stops here: the
  // drawing is given the plate's real edges through `limits`.
  safeBottom: 372,
  // The nose is a separate mesh that pokes out in front of the plate. The plate
  // is depth tested, so the nose hides whatever is drawn behind it - but a frown
  // arcs *up* into that hiding place, so slide the mouth down until the middle
  // of the arc clears the nose. Only the middle is checked: a smile curves away
  // from the nose, and checking its ends instead is what used to pin the whole
  // mouth in place and make the height slider do nothing.
  noseClear: 132,
  // The original hand-drawn mouth is a *shallow* arc with tall corner strokes
  // flicking up at the ends (measured from the artwork: the arc's own sag is
  // ~0.13 of its chord, while the corners reach ~90px above it). Keeping that
  // split is what makes `smile: 1` read as the original; making the arc itself
  // deep instead looked too steep.
  //
  // The tongue rises from the lip line to a rounded top just above the corner
  // strokes. Measured off the original: its crown is a *super-ellipse* -
  // `rise = height * (1 - |dx/half|^2.5)` - so it is steep-sided with a smooth,
  // almost flat top. It is convex upwards, but it is **not** a point.
  tongue: { pos: 0.84, width: 126, height: 115, crown: 2.5 },
  corner: { fromX: 26, fromY: 3, toX: 32, toY: 42, width: 11, curve: 26 },
  openRise: 64,
  // Was 0.25: opening the mouth used to flatten the smile to stay inside the
  // texture band. There is no band to stay inside now.
  openFlatten: 0,
};

/* ------------------------------------------------------------------ helpers */

function circlePath(ctx, cx, cy, r) {
  ctx.beginPath();
  ctx.arc(cx, cy, r, 0, TAU);
  ctx.closePath();
}

function fillCircle(ctx, cx, cy, r, color) {
  circlePath(ctx, cx, cy, r);
  ctx.fillStyle = color;
  ctx.fill();
}

/**
 * A heart, used for the "love" eyes. The path is wider than it is tall, which
 * is what makes it read as a heart rather than a blob at small sizes.
 */
function heartPath(ctx, cx, cy, r) {
  ctx.beginPath();
  ctx.moveTo(cx, cy + r * 0.80);
  ctx.bezierCurveTo(cx - r * 1.24, cy - r * 0.34, cx - r * 0.50, cy - r * 1.18, cx, cy - r * 0.40);
  ctx.bezierCurveTo(cx + r * 0.50, cy - r * 1.18, cx + r * 1.24, cy - r * 0.34, cx, cy + r * 0.80);
  ctx.closePath();
}

function fillHeart(ctx, cx, cy, r, color) {
  heartPath(ctx, cx, cy, r);
  ctx.fillStyle = color;
  ctx.fill();
}

/** A teardrop, used by the crying expression. */
function dropPath(ctx, cx, cy, size) {
  ctx.beginPath();
  ctx.moveTo(cx, cy - size * 1.32);
  ctx.bezierCurveTo(cx + size * 0.95, cy - size * 0.34, cx + size * 0.95, cy + size * 0.78, cx, cy + size * 0.78);
  ctx.bezierCurveTo(cx - size * 0.95, cy + size * 0.78, cx - size * 0.95, cy - size * 0.34, cx, cy - size * 1.32);
  ctx.closePath();
}

function strokeCircle(ctx, cx, cy, r, color, width) {
  circlePath(ctx, cx, cy, r);
  ctx.strokeStyle = color;
  ctx.lineWidth = width;
  ctx.stroke();
}

function tracePolyline(ctx, points, closed) {
  if (!points.length) return;
  ctx.beginPath();
  ctx.moveTo(points[0].x, points[0].y);
  for (let i = 1; i < points.length; i++) ctx.lineTo(points[i].x, points[i].y);
  if (closed) ctx.closePath();
}

function strokePolyline(ctx, points, { color, width, closed = false }) {
  if (points.length < 2) return;
  ctx.save();
  ctx.strokeStyle = color;
  ctx.lineWidth = width;
  ctx.lineJoin = 'round';
  ctx.lineCap = 'round';
  tracePolyline(ctx, points, closed);
  ctx.stroke();
  ctx.restore();
}

function fillPolygon(ctx, points, color) {
  if (points.length < 3) return;
  ctx.save();
  ctx.fillStyle = color;
  tracePolyline(ctx, points, true);
  ctx.fill();
  ctx.restore();
}

/**
 * Trace a closed polygon with rounded corners. Used for the squarish
 * サングラス lens: an ellipse cannot be angular, and a plain polygon has cusps.
 * Each corner is cut back by `radius` (clamped so short edges cannot overlap)
 * and joined with a quadratic through the original vertex.
 */
function traceRoundedPolygon(ctx, points, radius) {
  const n = points.length;
  ctx.beginPath();
  for (let i = 0; i < n; i++) {
    const prev = points[(i + n - 1) % n];
    const cur = points[i];
    const next = points[(i + 1) % n];
    const inLen = Math.hypot(cur.x - prev.x, cur.y - prev.y) || 1;
    const outLen = Math.hypot(next.x - cur.x, next.y - cur.y) || 1;
    const cut = Math.min(radius, inLen / 2, outLen / 2);
    const from = { x: cur.x + ((prev.x - cur.x) / inLen) * cut, y: cur.y + ((prev.y - cur.y) / inLen) * cut };
    const to = { x: cur.x + ((next.x - cur.x) / outLen) * cut, y: cur.y + ((next.y - cur.y) / outLen) * cut };
    if (i === 0) ctx.moveTo(from.x, from.y);
    else ctx.lineTo(from.x, from.y);
    ctx.quadraticCurveTo(cur.x, cur.y, to.x, to.y);
  }
  ctx.closePath();
}

/** Points along a quadratic Bézier, `steps` segments (steps + 1 points). */
function quadraticPoints(p0, p1, p2, steps) {
  const out = [];
  for (let i = 0; i <= steps; i++) {
    const t = i / steps;
    const u = 1 - t;
    out.push({
      x: u * u * p0.x + 2 * u * t * p1.x + t * t * p2.x,
      y: u * u * p0.y + 2 * u * t * p1.y + t * t * p2.y,
    });
  }
  return out;
}

/** A point on a cubic Bézier. */
function cubicPoint(p0, p1, p2, p3, t) {
  const u = 1 - t;
  const a = u * u * u;
  const b = 3 * u * u * t;
  const c = 3 * u * t * t;
  const d = t * t * t;
  return {
    x: a * p0.x + b * p1.x + c * p2.x + d * p3.x,
    y: a * p0.y + b * p1.y + c * p2.y + d * p3.y,
  };
}

/** Points along a cubic Bézier, `steps` segments (steps + 1 points). */
function cubicPoints(p0, p1, p2, p3, steps) {
  const out = [];
  for (let i = 0; i <= steps; i++) out.push(cubicPoint(p0, p1, p2, p3, i / steps));
  return out;
}

/**
 * The four control points of the mouth's centreline: the same two ends, the same
 * depth, but `bend` decides where the curve actually bends (see `sagBend`).
 */
function mouthControls(left, right, sag, bend) {
  const width = right.x - left.x;
  const y = left.y + (4 / 3) * sag;
  return {
    p0: left,
    p1: { x: left.x + width * bend, y },
    p2: { x: right.x - width * bend, y },
    p3: right,
  };
}

function rotate(points, pivot, degrees) {
  if (!degrees) return points;
  const a = (degrees * Math.PI) / 180;
  const cos = Math.cos(a);
  const sin = Math.sin(a);
  return points.map((p) => {
    const dx = p.x - pivot.x;
    const dy = p.y - pivot.y;
    return { x: pivot.x + dx * cos - dy * sin, y: pivot.y + dx * sin + dy * cos };
  });
}

/* --------------------------------------------------------------------- eyes */

/**
 * Intersect the canvas clip with the eyeball disc and the (possibly closed)
 * lids. Because the overlay plane is transparent and sits in front of the face,
 * clipping is all that is needed: whatever is clipped away shows the real
 * shaded face behind it. That is what makes the eyelid blend perfectly without
 * baking a face-coloured background into the texture.
 */
function clipEye(ctx, eye, open, lower, flat = false) {
  circlePath(ctx, eye.cx, eye.cy, eye.r);
  ctx.clip();

  const upper = 1 - clamp(open, 0, 1);
  if (upper > 0.0005) {
    const rl = eye.r * EYE_LAYOUT.lidRadiusFactor;
    const lowest = eye.cy - eye.r + 2 * eye.r * upper;
    ctx.beginPath();
    if (flat) {
      // A flat lid: a straight edge straight across the eye.
      ctx.rect(eye.cx - HUGE, lowest, HUGE * 2, HUGE);
    } else {
      const cy = lowest - rl;
      ctx.moveTo(eye.cx - HUGE, cy);
      ctx.lineTo(eye.cx - rl, cy);
      ctx.arc(eye.cx, cy, rl, Math.PI, 0, true); // lower semicircle: bulges down
      ctx.lineTo(eye.cx + HUGE, cy);
      ctx.lineTo(eye.cx + HUGE, cy + HUGE);
      ctx.lineTo(eye.cx - HUGE, cy + HUGE);
    }
    ctx.closePath();
    ctx.clip();
  }

  const bottom = clamp(lower, 0, 1);
  if (bottom > 0.0005) {
    const rl = eye.r * EYE_LAYOUT.lidRadiusFactor;
    const highest = eye.cy + eye.r - 2 * eye.r * bottom;
    ctx.beginPath();
    if (flat) {
      ctx.rect(eye.cx - HUGE, highest - HUGE, HUGE * 2, HUGE);
    } else {
      const cy = highest + rl;
      ctx.moveTo(eye.cx - HUGE, cy);
      ctx.lineTo(eye.cx - rl, cy);
      ctx.arc(eye.cx, cy, rl, Math.PI, 0, false); // upper semicircle: bulges up
      ctx.lineTo(eye.cx + HUGE, cy);
      ctx.lineTo(eye.cx + HUGE, cy - HUGE);
      ctx.lineTo(eye.cx - HUGE, cy - HUGE);
    }
    ctx.closePath();
    ctx.clip();
  }
}

function lidPath(ctx, eye, amount, lower, flat = false) {
  ctx.beginPath();
  if (flat) {
    // A straight lid edge; the caller clips it to the eyeball.
    const y = lower
      ? eye.cy + eye.r - 2 * eye.r * amount
      : eye.cy - eye.r + 2 * eye.r * (1 - amount);
    ctx.moveTo(eye.cx - eye.r * 2, y);
    ctx.lineTo(eye.cx + eye.r * 2, y);
    return;
  }
  const rl = eye.r * EYE_LAYOUT.lidRadiusFactor;
  if (!lower) {
    const lowest = eye.cy - eye.r + 2 * eye.r * (1 - amount);
    ctx.arc(eye.cx, lowest - rl, rl, Math.PI, 0, true);
  } else {
    const highest = eye.cy + eye.r - 2 * eye.r * amount;
    ctx.arc(eye.cx, highest + rl, rl, Math.PI, 0, false);
  }
}

/** The "eyes shut" artwork: a line, a chevron or a happy arch. */
function drawShutEye(ctx, eye, spec, style) {
  const layout = EYE_LAYOUT.closedLine;
  const { line, width } = style;

  if (spec.closed === 'chevron') {
    // The chevron points *at* the nose, matching the original artwork.
    const vertex = { x: eye.cx + eye.towardNose * layout.vertex, y: eye.cy + layout.offsetY - 8 };
    const armX = eye.cx - eye.towardNose * layout.armX;
    const spread = layout.spread * (spec.spread ?? 1);
    strokePolyline(ctx, [vertex, { x: armX, y: vertex.y - spread }], { color: line, width });
    strokePolyline(ctx, [vertex, { x: armX, y: vertex.y + spread }], { color: line, width });
    return;
  }

  if (spec.closed === 'three') {
    // A real "3": an upper bowl and a lower bowl that meet at a single pinch on
    // the nose side. Stacking two C's instead made the two lobes sit on top of
    // each other; joining them at a point is what makes it read as the digit.
    // `length` scales the whole shape.
    const r = 34 * (spec.length ?? 1);
    // Which way the digit faces. It used to follow the nose side, which made the
    // pair a mirror image; each eye can now be set either way, because a pair of
    // mirrored 3s does not always read the way you want.
    const dir = (spec.threeFlip ? -1 : 1) * eye.towardNose;
    const sx = eye.cx;
    const cy = eye.cy + layout.offsetY;
    const at = (x, y) => ({ x: sx + dir * r * x, y: cy + r * y });
    const pinch = at(0.72, 0);
    const upper = quadraticPoints(at(0.02, -1.62), at(2.05, -1.40), pinch, 26);
    const lower = quadraticPoints(pinch, at(2.05, 1.46), at(0.02, 1.78), 26);
    strokePolyline(ctx, upper, { color: line, width });
    strokePolyline(ctx, lower, { color: line, width });
    return;
  }

  if (spec.closed === 'arch') {
    const edges = { x: eye.r, y: EYE_LAYOUT.arch.edgesUp };
    const points = quadraticPoints(
      { x: eye.cx - edges.x, y: eye.cy + edges.y },
      { x: eye.cx, y: eye.cy + edges.y - EYE_LAYOUT.arch.apexUp * 2 },
      { x: eye.cx + edges.x, y: eye.cy + edges.y },
      24,
    );
    strokePolyline(ctx, points, { color: line, width });
    return;
  }

  // Default: the gently slanted line of the original artwork, mirrored so it
  // always slopes down towards the nose. With `closedLines` set to 2 or 3 the
  // strokes share one endpoint instead - a ">" or bird's-foot shape, which is
  // how manga draws a happily squeezed-shut eye.
  const count = clamp(Math.round(spec.closedLines ?? 1), 1, 3);
  if (count >= 2) {
    const vertex = { x: eye.cx + eye.towardNose * layout.vertex, y: eye.cy + layout.offsetY - 8 };
    // Reach as far as the single line does, not just a short stub: the arms are
    // the *long* form of the same eyelid the one-line version covers.
    const outX = -eye.towardNose * layout.armSpan;
    const arms = count === 2
      ? [{ x: outX, y: -layout.spread }, { x: outX, y: layout.spread }]
      : [
        { x: outX, y: -layout.spread },
        { x: -eye.towardNose * Math.hypot(layout.armSpan, layout.spread), y: 0 },
        { x: outX, y: layout.spread },
      ];
    for (const arm of arms) {
      strokePolyline(ctx, [vertex, { x: vertex.x + arm.x, y: vertex.y + arm.y }], { color: line, width });
    }
    return;
  }

  const half = (layout.length / 2) * (spec.length ?? 1);
  const midX = eye.cx - eye.towardNose * layout.offsetX;
  const midY = eye.cy + layout.offsetY;
  const angle = ((layout.slantDeg * (spec.slant ?? 1) * eye.towardNose) * Math.PI) / 180;
  const bow = (layout.bow ?? 0) * (spec.slant ?? 1);
  const from = { x: midX - Math.cos(angle) * half, y: midY - Math.sin(angle) * half };
  const to = { x: midX + Math.cos(angle) * half, y: midY + Math.sin(angle) * half };
  const points = quadraticPoints(from, { x: midX, y: midY + bow * 2 }, to, 20);
  strokePolyline(ctx, points, { color: line, width });
}

/**
 * A stroke whose width runs from `w0` at the first point to `w1` at the last.
 *
 * Used for a ゴルゴ13-style brow: a filled wedge that comes to a point at one end
 * reads as a heavy eyebrow, where a constant-width stroke reads as a soft arc.
 */
function taperedStroke(ctx, points, w0, w1, color) {
  const count = points.length;
  if (count < 2) return;
  const side = [[], []];
  for (let i = 0; i < count; i++) {
    const t = i / (count - 1);
    const w = (w0 + (w1 - w0) * t) / 2;
    const before = points[Math.max(0, i - 1)];
    const after = points[Math.min(count - 1, i + 1)];
    const dx = after.x - before.x;
    const dy = after.y - before.y;
    const len = Math.hypot(dx, dy) || 1;
    const nx = -dy / len;
    const ny = dx / len;
    side[0].push({ x: points[i].x + nx * w, y: points[i].y + ny * w });
    side[1].push({ x: points[i].x - nx * w, y: points[i].y - ny * w });
  }
  ctx.save();
  tracePolyline(ctx, [...side[0], ...side[1].reverse()], true);
  ctx.fillStyle = color;
  ctx.fill();
  ctx.restore();
}

/** An eyebrow: a short arc above the eye, mirrored between the two eyes so a
 *  positive `angle` always means "inner end down" (an angry brow). */
function drawBrow(ctx, eye, spec, style, limits = WINDOW_LIMITS) {
  const brow = style.brow;
  if (!brow?.enabled) return;
  const layout = EYE_LAYOUT.brow;
  // 0 is allowed: a zero-length stroke with round caps is a dot, which is what a
  // length of 0 is asking for.
  const length = Math.max(0, (brow.length ?? 1) * layout.length);
  const thickness = Math.max(1, (brow.thickness ?? 1) * layout.thickness);
  const cy = eye.cy - layout.lift - (brow.height ?? 0) - (spec.browHeight ?? 0);
  // `spacing` widens the gap between the two brows: a positive value moves each
  // one away from the nose. `layout.offsetX` is the resting inset from the
  // artwork (which the old code read off the wrong object, so it never applied).
  const cx = eye.cx + eye.towardNose * (layout.offsetX - (brow.spacing ?? 0));
  const angle = (((brow.angle ?? 0) + (spec.browAngle ?? 0)) * eye.towardNose * Math.PI) / 180;
  const half = length / 2;
  const from = { x: cx - Math.cos(angle) * half, y: cy - Math.sin(angle) * half };
  const to = { x: cx + Math.cos(angle) * half, y: cy + Math.sin(angle) * half };
  const bow = (brow.curve ?? layout.curve) * length;
  const points = quadraticPoints(from, { x: cx, y: cy - bow }, to, 16);
  // A brow raised with the height slider used to run off the top of the canvas,
  // where the clamped edge row was copied across the whole plate and the brow
  // smeared upwards. Keep the whole stroke inside the artwork instead.
  const top = limits.top + EDGE_MARGIN;
  const highest = Math.min(...points.map((q) => q.y)) - thickness / 2;
  if (highest < top) for (const q of points) q.y += top - highest;
  // `taper` is how wide the *inner* (nose-side) end is: 1 = a plain stroke, and
  // lower values turn the brow into a wedge that comes to a point at the nose.
  const taper = clamp(brow.taper ?? 1, 0, 1);
  const color = brow.color ?? style.line ?? '#55386e';
  // A tapered wedge has no shape at zero length, so a dot always takes the plain
  // round-capped stroke.
  if (taper >= 0.999 || length < 2) {
    strokePolyline(ctx, points, { color, width: thickness });
    return;
  }
  const innerIsTo = eye.towardNose > 0;
  taperedStroke(
    ctx,
    points,
    thickness * (innerIsTo ? 1 : taper),
    thickness * (innerIsTo ? taper : 1),
    color,
  );
}

/* ------------------------------------------------------- ほっぺ / 頭の模様 */

/**
 * ほっぺ: a soft pink patch crossed by short diagonal hatch strokes, the classic
 * manga blush. Drawn once per cheek; the caller draws these before the eyes.
 */
function drawCheek(ctx, cx, cy, spec) {
  const size = clamp(spec.size ?? 1, 0.2, 3);
  const r = CHEEK_LAYOUT.radius * size;
  const ry = r * CHEEK_LAYOUT.squash;

  ctx.save();
  ctx.beginPath();
  ctx.ellipse(cx, cy, r, ry, 0, 0, TAU);
  ctx.fillStyle = spec.color ?? '#f6a6b8';
  ctx.fill();
  // Clip the hatching to the patch, so a wobbly stroke keeps a clean edge.
  ctx.clip();

  // Four horizontal strokes, hand-drawn rather than ruled: each tapers to a
  // point at both ends and wanders a little. Each one stops short of the patch
  // outline - the margin is measured against the patch's own width at that
  // height, so a stroke on the narrow top or bottom cannot reach the edge.
  const thickness = Math.max(1.5, CHEEK_LAYOUT.lineWidth * size);
  ctx.fillStyle = spec.hatchColor ?? '#e0708f';
  for (let i = 0; i < CHEEK_LAYOUT.lines; i += 1) {
    const t = CHEEK_LAYOUT.lines === 1 ? 0 : (i / (CHEEK_LAYOUT.lines - 1)) * 2 - 1; // -1..1
    const dy = t * ry * CHEEK_LAYOUT.spread;
    const chord = r * Math.sqrt(Math.max(0, 1 - (dy / ry) ** 2));
    const half = chord - r * CHEEK_LAYOUT.margin;
    if (half < 4) continue;
    traceHandHatch(ctx, cx, cy + dy, half, thickness, i * 1.9);
  }
  ctx.restore();
}

/**
 * One hand-drawn hatch stroke: a lens-shaped mark that tapers to a point at both
 * ends, with a gentle wobble so it reads as drawn rather than printed.
 */
function traceHandHatch(ctx, cx, cy, half, thickness, seed) {
  const steps = 12;
  const top = [];
  const bottom = [];
  for (let i = 0; i <= steps; i += 1) {
    const u = i / steps;
    const x = cx + (u - 0.5) * 2 * half;
    const wobble = Math.sin(u * Math.PI * 1.6 + seed) * thickness * CHEEK_LAYOUT.wobble;
    const w = (thickness / 2) * Math.sin(u * Math.PI); // 0 at both ends
    top.push({ x, y: cy + wobble - w });
    bottom.push({ x, y: cy + wobble + w });
  }
  ctx.beginPath();
  ctx.moveTo(top[0].x, top[0].y);
  for (const point of top.slice(1)) ctx.lineTo(point.x, point.y);
  for (let i = bottom.length - 1; i >= 0; i -= 1) ctx.lineTo(bottom[i].x, bottom[i].y);
  ctx.closePath();
  ctx.fill();
}

/** The pair of cheeks: symmetric about the face's midline (see `EYE_LAYOUT`). */
function drawCheeks(ctx, spec) {
  if (!spec?.enabled) return;
  const drop = CHEEK_LAYOUT.drop + clamp(spec.offsetY ?? 0, -400, 400);
  const outward = CHEEK_LAYOUT.outward + clamp(spec.spacing ?? 0, -400, 400);
  for (const layout of EYE_LAYOUT.eyes) {
    // Outwards is away from the midline, i.e. the opposite of `towardNose`.
    const cx = layout.cx - layout.towardNose * outward;
    drawCheek(ctx, cx, layout.cy + drop, spec);
  }
}

/**
 * 頭の模様: a hair-like covering over the whole head.
 *
 * The face plate is a shell of the front half of the body, so its own edge is
 * the head's silhouette. This fills the plate from its top edge down to an
 * adjustable boundary, which is what makes the covering follow the head outline
 * without knowing the head's shape ahead of time. The boundary style, colour and
 * size are all adjustable, and one style (`split`) divides the head into two
 * colours along a vertical curve - a generic "two-tone head" device. It is not a
 * copy of any particular character's hair.
 */
function drawHeadCover(ctx, spec, limits) {
  const shape = spec?.shape ?? 'off';
  if (shape === 'off' || shape === 'none' || shape === 'wave') return;

  const size = clamp(spec.size ?? 1, 0.02, 3);
  const left = limits.left;
  const right = limits.right;
  const top = limits.top;
  const bottom = limits.bottom;
  const width = right - left;
  const cx = (left + right) / 2 + clamp(spec.offsetX ?? 0, -600, 600);
  const color = spec.color ?? '#8a4fe0';
  const color2 = spec.color2 ?? '#ffffff';

  ctx.save();
  // Stay on the plate: nothing may spill past the artwork's real edges.
  ctx.beginPath();
  ctx.rect(left, top, width, bottom - top);
  ctx.clip();

  if (shape === 'split') {
    // Retired: the two-tone head was dropped, but an old saved state may still
    // carry the shape, so treat it as off rather than drawing a two-tone head.
    ctx.restore();
    return;
  }

  // The boundary curve. `t` runs -1..1 across the plate, so each style is a
  // small shape function; the covering is everything above it.
  const base = HEAD_MARK_LAYOUT.cy + (size - 1) * HEAD_MARK_LAYOUT.reach
    + clamp(spec.offsetY ?? 0, -600, 600);
  const halfSpan = width / 2;
  const boundary = (x) => {
    const t = clamp((x - cx) / halfSpan, -1, 1);
    // まえがみ: a narrow, deep V pointing down the middle (Atom / Mickey).
    if (shape === 'fringe') return base + 175 * Math.max(0, 1 - Math.abs(t) * 1.6);
    // はちわれ: the same V upside down - a widow's peak.
    if (shape === 'fringe-up') return base - 320 * Math.max(0, 1 - Math.abs(t) * 1.5);
    // カーブ: a plain shallow curve, higher (more hair) round the sides.
    if (shape === 'curve') return base + 160 * t * t;
    // わけめ: a 7:3 parting - a step at 30% from the left.
    if (shape === 'side') return base + 60 - 320 / (1 + Math.exp(-(t + 0.4) * 10));
    // `cap`: a rounded helmet, lowest in the middle.
    return base - 100 * (1 - t * t);
  };

  const steps = 48;
  ctx.beginPath();
  ctx.moveTo(left, top);
  ctx.lineTo(right, top);
  for (let i = steps; i >= 0; i -= 1) {
    const x = left + (width * i) / steps;
    ctx.lineTo(x, boundary(x));
  }
  ctx.closePath();
  ctx.fillStyle = color;
  ctx.fill();

  if (shape === 'side') {
    // わけめ: cut a real parting in the hair, so the two halves read as combed
    // apart rather than as one smooth sweep.
    const partX = cx - halfSpan * 0.4;
    const gw = Math.max(5, width * 0.02) * (0.6 + size);
    ctx.save();
    ctx.globalCompositeOperation = 'destination-out';
    ctx.beginPath();
    ctx.moveTo(partX - gw * 0.16, top);
    ctx.lineTo(partX + gw * 0.16, top);
    ctx.lineTo(partX + gw * 0.7, boundary(partX) + 4);
    ctx.lineTo(partX - gw * 0.7, boundary(partX) + 4);
    ctx.closePath();
    ctx.fill();
    ctx.restore();
  }
  ctx.restore();
}

/**
 * 頭の模様 drawn onto the hair plate, which wraps right round the head.
 *
 * The plate's canvas is the cylindrical UV unrolled: `x` is the angle (front at
 * the middle, the seam at the back at both ends) and `y` is the height. The GLB
 * flips V on export, so the top of the head lands at the top of the canvas and
 * the covering is everything above the boundary line - the same direction as
 * `drawHeadCover`, which fills down from the top edge of the front plate.
 *
 * The same `shape` names are reused, with their front-to-back modulation tied to
 * how far the column leans towards the front (`t`).
 */
export function drawHeadCoverWrap(ctx, spec, layout) {
  const shape = spec?.shape ?? 'off';
  // `split` and `wave` are retired front-plate shapes; they have no wrap equal.
  if (shape === 'off' || shape === 'none' || shape === 'split' || shape === 'wave') return;

  const size = clamp(spec.size ?? 1, 0.02, 3);
  const W = layout.width;
  const H = layout.height;
  // The horizontal axis is the angle. The *front* sits at u = 0.5, which is the
  // middle of the artwork window (1024), not the middle of the canvas: after the
  // seam heal the canvas is wider than the window, so using W/2 put the pattern
  // a good way off centre.
  const span = (layout.window ?? ART_WINDOW).width;
  // After the GLB's V-flip the wrap's v is 0 at the top of the head and 1 at the
  // model's feet, so a *larger* boundary means more hair (it reaches further
  // down) - the same direction as `size` and `offsetY` on the front plate.
  const base = HAIR_WRAP_LAYOUT.base
    + (size - 1) * HAIR_WRAP_LAYOUT.reach
    + clamp(spec.offsetY ?? 0, -600, 600) * HAIR_WRAP_LAYOUT.offsetScale;
  const color = spec.color ?? '#8a4fe0';

  ctx.save();
  ctx.beginPath();
  ctx.rect(0, 0, W, H);
  ctx.clip();

  // `t` is -1..1 across the front (+-90 deg around the front); everything behind
  // is pinned to +-1, so the hairline runs flat round the back of the head.
  const half = span * 0.25;
  const cx = layout.offsetX + span * 0.5 + clamp(spec.offsetX ?? 0, -600, 600);
  const vBound = (x) => {
    const t = clamp((x - cx) / half, -1, 1);
    // まえがみ: a narrow, deep V pointing down the middle (Atom / Mickey).
    if (shape === 'fringe') return base + 0.15 * Math.max(0, 1 - Math.abs(t) * 1.6);
    // はちわれ: the same V upside down - a widow's peak.
    if (shape === 'fringe-up') return base - 0.18 * Math.max(0, 1 - Math.abs(t) * 1.5);
    // カーブ: a deep round cover - low at the sides, high in the middle, like the
    // blue of a certain robot cat's head.
    if (shape === 'curve') return base + 0.42 * t * t;
    // わけめ: a 7:3 part. The hairline is asymmetric across the *front* only and
    // comes back to the base level at the sides, so it is continuous at the back
    // seam (an angle-varying hairline there would ride the seam and step). The
    // parting line is the gap cut out below.
    if (shape === 'side') {
      const w = Math.max(0, 1 - t * t);   // 0 at the sides, 1 at the front
      return base + 0.18 * w * (0.5 + 0.5 * t);
    }
    return base - 0.07 * (1 - t * t);  // `cap`
  };

  const steps = Math.max(96, Math.ceil(W / 2));
  ctx.beginPath();
  ctx.moveTo(0, 0);
  for (let i = 0; i <= steps; i += 1) {
    const x = (W * i) / steps;
    ctx.lineTo(x, clamp(vBound(x), 0, 1) * H);
  }
  ctx.lineTo(W, 0);
  ctx.closePath();
  ctx.fillStyle = color;
  ctx.fill();

  if (shape === 'side') {
    // The parting: a thin wedge cut out at 30% (a 7:3 split), running from the
    // top of the head down to the hairline.
    const partX = span * 0.4;
    const gw = Math.max(8, span * 0.016) * (0.6 + size);
    ctx.save();
    ctx.globalCompositeOperation = 'destination-out';
    ctx.beginPath();
    ctx.moveTo(partX - gw * 0.5, 0);
    ctx.lineTo(partX + gw * 0.5, 0);
    ctx.lineTo(partX + gw * 1.2, clamp(vBound(partX), 0, 1) * H + 2);
    ctx.lineTo(partX - gw * 1.2, clamp(vBound(partX), 0, 1) * H + 2);
    ctx.closePath();
    ctx.fill();
    ctx.restore();
  }
  ctx.restore();
}

/**
 * The shared extras (ほっぺ and 頭の模様) are drawn once per face, before the eyes,
 * so the eyes, brows and glasses win any overlap and the cheeks are not doubled.
 */
function drawFaceExtras(ctx, style, lineOnly) {
  drawCheeks(ctx, style.cheeks);
  // The head covering fills out to the plate's real edges, so it needs `limits`.
  drawHeadCover(ctx, style.headMark, style.limits ?? WINDOW_LIMITS);
  drawBeard(ctx, style.beard, lineOnly, style.beards);
}

const BEARD_TINT_IDS = new WeakMap();
const BEARD_TINT_CACHE = new Map();
let beardTintSeq = 0;

/**
 * A copy of a beard drawing in the chosen colour.
 *
 * The supplied files are drawn in near-black, so repainting the ink with the
 * beard colour (`source-in` keeps the anti-aliased alpha) is what lets the colour
 * picker work on a drawing. Cached per (sprite, colour).
 */
function tintedSprite(image, color) {
  let id = BEARD_TINT_IDS.get(image);
  if (!id) {
    beardTintSeq += 1;
    id = beardTintSeq;
    BEARD_TINT_IDS.set(image, id);
  }
  const key = `${id}|${color}`;
  const cached = BEARD_TINT_CACHE.get(key);
  if (cached) return cached;
  const canvas = document.createElement('canvas');
  canvas.width = image.width;
  canvas.height = image.height;
  const ctx = canvas.getContext('2d');
  ctx.drawImage(image, 0, 0);
  ctx.globalCompositeOperation = 'source-in';
  ctx.fillStyle = color;
  ctx.fillRect(0, 0, canvas.width, canvas.height);
  BEARD_TINT_CACHE.set(key, canvas);
  return canvas;
}

/**
 * ひげ: a mustache or beard under the nose. Each kind is a small shape built on
 * the spot; `scotch` and `cat` are drawn with round-capped strokes, and `kaiser`
 * is a thick waving band with a curl at each tip. A kind that ships a drawing in
 * `assets/beards/` is drawn from that instead (see `textures`), tinted with the
 * beard colour. In a line drawing nothing is filled and only the outlines are inked.
 */
function drawBeard(ctx, spec, lineOnly = false, textures = null) {
  const shape = spec?.shape ?? 'off';
  if (shape === 'off' || shape === 'none') return;
  const size = clamp(spec.size ?? 1, 0.2, 3);
  const cx = BEARD_LAYOUT.cx;
  const cy = BEARD_LAYOUT.cy + clamp(spec.offsetY ?? 0, -600, 600);
  const w = BEARD_LAYOUT.width * size;
  const color = spec.color ?? '#3a2a4a';

  // A supplied drawing for this kind (assets/beards/<shape>.png) replaces the
  // built-in strokes. The drawings are trimmed to their ink, so `w` is the
  // beard's real width; it is centred on the nose and scaled by `size`, and it is
  // tinted with the beard colour (the files are drawn in near-black).
  const entry = textures?.[shape];
  if (entry?.image) {
    const sprite = tintedSprite(entry.image, color);
    // Slightly wider than the built-in base: a moustache drawn to the same width
    // as a beard's stroke sat mostly behind the nose.
    const drawW = w * 1.4;
    if (entry.half) {
      // One side, mirrored to make the pair. The drawing is the model's own right
      // side, which sits in the artwork's left half; `spacing` opens the middle
      // (a negative value brings the two halves together).
      const halfW = drawW / 2;
      const dh = (halfW * (sprite.height || 1)) / (sprite.width || 1);
      const gap = clamp(spec.spacing ?? 0, -400, 800);
      const y = cy - dh / 2;
      ctx.drawImage(sprite, cx - gap - halfW, y, halfW, dh);
      ctx.save();
      ctx.translate(cx + gap, cy);
      ctx.scale(-1, 1);
      // Draw into [-halfW, 0] so that, mirrored, it lands in [cx+gap, cx+gap+halfW].
      ctx.drawImage(sprite, -halfW, -dh / 2, halfW, dh);
      ctx.restore();
      return;
    }
    const dh = (drawW * (sprite.height || 1)) / (sprite.width || 1);
    ctx.drawImage(sprite, cx - drawW / 2, cy - dh / 2, drawW, dh);
    return;
  }

  const line = spec.line ?? '#55386e';
  const lw = Math.max(2, 9 * size);
  const stroke = (points, width) => strokePolyline(ctx, points, {
    color: lineOnly ? line : color, width,
  });

  if (shape === 'scotch') {
    // ちょび髭: two short dashes under the nose, angled down and out so their tips
    // clear the nose mesh (which hides anything drawn straight behind it).
    stroke([{ x: cx - w * 0.07, y: cy + w * 0.02 }, { x: cx - w * 0.5, y: cy + w * 0.17 }], lw * 1.8);
    stroke([{ x: cx + w * 0.07, y: cy + w * 0.02 }, { x: cx + w * 0.5, y: cy + w * 0.17 }], lw * 1.8);
    return;
  }
  if (shape === 'kaiser') {
    // カイゼル: a full, thick mustache whose ends turn up.
    stroke([
      { x: cx - w * 0.5, y: cy - w * 0.02 },
      { x: cx - w * 0.28, y: cy + w * 0.07 },
      { x: cx, y: cy + w * 0.09 },
      { x: cx + w * 0.28, y: cy + w * 0.07 },
      { x: cx + w * 0.5, y: cy - w * 0.02 },
    ], lw * 2.6);
    const curlAt = (dir) => {
      // A little inward-turning spiral at the tip: the Dalí curl.
      const ex = cx + dir * w * 0.5;
      const ey = cy - w * 0.02;
      const r0 = lw * 0.5;
      const r1 = w * 0.17;
      const steps = 22;
      const points = [];
      for (let i = 0; i <= steps; i += 1) {
        const t = i / steps;
        const a = -Math.PI / 2 + dir * Math.PI * 2 * 1.4 * t;
        const r = r0 + (r1 - r0) * t;
        points.push({ x: ex + dir * Math.cos(a) * r, y: ey + Math.sin(a) * r });
      }
      stroke(points, lw * 1.25);
    };
    curlAt(-1);
    curlAt(1);
    return;
  }
  if (shape === 'apron') {
    // A filled bib was once a kind here; kept as a no-op so a saved state that
    // still names it simply shows nothing rather than throwing.
    return;
  }
  // ねこ: three whiskers a side, growing out of the cheeks with a gap left in the
  // middle (like a cat's). The thickness is fixed, so `size` makes the whiskers
  // longer and wider-spread without making them fatter, and `spacing` opens the
  // gap between the left and right.
  const whisker = 8;
  const spacing = clamp(spec.spacing ?? 0, 0, 800);
  const inner = w * 0.30 + spacing;
  // `length` scales how far each whisker reaches out; `lineGap` opens the spacing
  // between the three lines of one side. Neither changes the line's thickness.
  const length = clamp(spec.length ?? 1, 0.2, 3);
  const gap = w * 0.16 * clamp(spec.lineGap ?? 1, 0.2, 3);
  const outer = inner + w * 0.38 * length;
  for (const dir of [-1, 1]) {
    for (let i = 0; i < 3; i += 1) {
      stroke([
        { x: cx + dir * inner, y: cy + (i - 1) * gap },
        { x: cx + dir * outer, y: cy + (i - 1) * gap * 1.5 },
      ], whisker);
    }
  }
}

/**
 * One lens of a pair of 眼鏡 / サングラス, plus this eye's half of the bridge and its
 * temple.
 *
 * Called once per eye from `drawEyes`, after the eyeball (or the shut-eye artwork)
 * so the lens sits in front of the eye, and before that eye's tear so a teardrop
 * falls past the lens. Nothing about a pair of glasses is per-eye, so the two
 * halves are mirrored from `eye.towardNose` and meet at the face's midline. The
 * kind picks the shape: a round 眼鏡 lens, or a wide angular サングラス one.
 *
 * `style.glasses` is the shared spec. In a line drawing the lens is left empty, so
 * the paper behind shows through exactly as the eyeball's white does (see the note
 * in `drawEyes`); the frames are then stroked in the line colour.
 *
 * @param {CanvasRenderingContext2D} ctx
 * @param {{cx:number,cy:number,r:number,towardNose:number}} eye
 * @param {boolean} line                 true in 線画 mode
 * @param {object} style                 the bag `drawEyes` received
 * @param {{top:number,bottom:number,left:number,right:number}} limits
 */
function drawGlasses(ctx, eye, line, style, limits) {
  const spec = style.glasses;
  if (!spec?.enabled) return;

  const r = eye.r;
  const scale = clamp(spec.scale ?? 1, 0.2, 3);
  // The two kinds share everything about *where* they sit but not their shape:
  // 眼鏡 is a round lens, サングラス a wide angular one (see the layouts above).
  const shades = spec.kind === 'sunglasses';
  const L = shades ? SUNGLASSES_LAYOUT : GLASSES_LAYOUT;
  const rx = r * L.widthFactor * scale;
  const ry = rx * L.heightFactor;
  const frame = Math.max(
    1,
    (spec.frameWidth ?? 1) * EYE_LAYOUT.lidStroke * (shades ? L.frameFactor : 1),
  );
  const colour = line ? (style.line ?? '#55386e') : (spec.frameColor ?? '#2a1e33');
  // The outward direction (towards the temple) for this eye. The angular lens is
  // built in a local frame whose u runs outwards, so the two sides mirror exactly.
  const out = -eye.towardNose;
  // The pointed outer corner makes the shades taller on that side; clamp against
  // the full height so even that point cannot reach the canvas border.
  const halfV = shades ? ry * Math.max(L.innerTop + L.topSkew, L.outerBottom) : ry;

  // Keep the lens (and so everything that hangs off it) inside the artwork. The
  // rest of the file clamps against `limits` for the same reason: a mark that
  // reaches the canvas border gets the edge row copied across the whole plate.
  const lensCx = clamp(eye.cx, limits.left + EDGE_MARGIN + rx, limits.right - EDGE_MARGIN - rx);
  const lensCy = clamp(
    eye.cy + L.drop + (spec.offsetY ?? 0),
    limits.top + EDGE_MARGIN + halfV,
    limits.bottom - EDGE_MARGIN - halfV,
  );
  // The face's midline: the two eyes are symmetric about it, so it is where the
  // two halves of the bridge meet and what `tilt` turns the pair about.
  const midX = (EYE_LAYOUT.eyes[0].cx + EYE_LAYOUT.eyes[1].cx) / 2;
  // Local (u, v) -> canvas, with u outwards and v downwards.
  const at = (u, v) => ({ x: lensCx + out * u, y: lensCy + v });

  ctx.save();
  // `tilt` leans the whole pair at once. Rotating about the midline keeps the
  // bridge centred between the lenses instead of swinging it off to one side.
  if (spec.tilt) {
    ctx.translate(midX, lensCy);
    ctx.rotate((clamp(spec.tilt, -90, 90) * Math.PI) / 180);
    ctx.translate(-midX, -lensCy);
  }

  // --- lens ------------------------------------------------------------
  // The one path is filled and then stroked, so the fill and the frame can never
  // drift apart.
  if (shades) {
    // A long cat-eye, not an ellipse: the outer end starts low, rises to a
    // pointed, lifted outer corner set further out, and the top edge sweeps back
    // towards the nose. The inner end is short, so the lens tapers inwards.
    const lens = [
      at(-rx, -ry * L.innerTop),                          // inner top (near the nose)
      at(rx * L.tipX, -ry * (L.innerTop + L.topSkew)),    // pointed, lifted outer corner
      at(rx * L.outerEndX, ry * L.outerBottom),           // low outer end
      at(-rx, ry * L.innerBottom),                        // inner bottom (pinched towards the nose)
    ];
    traceRoundedPolygon(ctx, lens, ry * L.corner);
    if (!line) {
      ctx.globalAlpha = clamp(spec.lensOpacity ?? 0, 0, 1);
      ctx.fillStyle = spec.lensColor ?? '#2b2433';
      ctx.fill();
      ctx.globalAlpha = 1;
    }
    ctx.strokeStyle = colour;
    ctx.lineWidth = frame;
    ctx.stroke();
  } else {
    ctx.beginPath();
    ctx.ellipse(lensCx, lensCy, rx, ry, 0, 0, TAU);
    ctx.closePath();
    if (!line) {
      ctx.globalAlpha = clamp(spec.lensOpacity ?? 0, 0, 1);
      ctx.fillStyle = spec.lensColor ?? '#2b2433';
      ctx.fill();
      ctx.globalAlpha = 1;
    }
    ctx.strokeStyle = colour;
    ctx.lineWidth = frame;
    ctx.stroke();
  }

  // --- bridge ----------------------------------------------------------
  // Each eye draws the half of the bridge from its own lens to the midline.
  const innerX = eye.towardNose > 0
    ? Math.min(lensCx + rx, midX)
    : Math.max(lensCx - rx, midX);
  if (shades) {
    // A short, thick, level bar high on the lenses: with the two lenses it reads
    // as one continuous visor across the eyes.
    const barY = lensCy - ry * L.bridgeLift;
    strokePolyline(ctx, [{ x: innerX, y: barY }, { x: midX, y: barY }], {
      color: colour,
      width: frame * L.bridgeWidth,
    });
  } else {
    // The two halves meet at the midline with a horizontal tangent, so the join is smooth.
    const noseX = lensCy - L.bridgeRise;
    strokePolyline(ctx, quadraticPoints(
      { x: innerX, y: lensCy },
      { x: (innerX + midX) / 2, y: noseX },
      { x: midX, y: noseX },
      12,
    ), { color: colour, width: frame * L.bridgeWidth });
  }

  // --- temple ----------------------------------------------------------
  // A stub outwards from the outer edge of the lens. It stays short on purpose:
  // the plate only reaches so far, and a temple that ran off it would smear. The
  // round pair's outer edge is a vertical line at the lens centre height. The
  // shades' outer end drops low, so their arm leaves from the *top* edge instead
  // - from the pointed outer corner - which is where a cat-eye's arm attaches.
  const outerX = shades
    ? lensCx - eye.towardNose * rx * L.tipX
    : lensCx - eye.towardNose * rx;
  const outerY = shades ? lensCy - ry * (L.innerTop + L.topSkew) : lensCy;
  const reach = clamp(
    outerX - eye.towardNose * r * L.templeLength,
    limits.left + EDGE_MARGIN,
    limits.right - EDGE_MARGIN,
  );
  strokePolyline(ctx, [
    { x: outerX, y: outerY },
    {
      x: eye.towardNose > 0 ? Math.min(reach, outerX) : Math.max(reach, outerX),
      y: clamp(outerY - L.templeRise, limits.top + EDGE_MARGIN, limits.bottom - EDGE_MARGIN),
    },
  ], { color: colour, width: frame });

  ctx.restore();
}

/**
 * Draw the pair of eyes into a 1024 x 380 canvas.
 *
 * @param {CanvasRenderingContext2D} ctx
 * @param {object} p
 * @param {'paint'|'line'} [p.mode]   paint = filled cartoon eyes, line = outline only
 * @param {object} p.eyes             `{ left, right }`, each `{ open, lookX, lookY, closed }`
 * @param {object} [p.style]          colours and shared shaping parameters
 */
export function drawEyes(ctx, p) {
  const mode = p.mode ?? 'paint';
  const line = mode === 'line';
  const style = p.style ?? {};
  const limits = style.limits ?? WINDOW_LIMITS;
  const white = style.white ?? '#ffffff';
  const irisColor = style.iris ?? '#150e1b';
  const lineColor = style.line ?? '#55386e';
  const lidWidth = style.lidWidth ?? EYE_LAYOUT.lidStroke;
  const irisScale = style.irisScale ?? 1;
  const lookMax = style.lookMax ?? 1;
  const highlightOn = style.highlight !== false;
  const heartScale = style.heartScale ?? 0.85;
  const heartColor = style.heartColor ?? '#e0344f';
  const irisRadius = EYE_LAYOUT.irisRadius * irisScale;
  const travel = Math.max(0, EYE_LAYOUT.radius - irisRadius);
  const heartTravel = EYE_LAYOUT.radius * 0.9;
  const lowerLid = clamp(style.lowerLid ?? 0, 0, 1);
  const lidFlat = (style.lidShape ?? 'curve') === 'flat';

  // ほっぺ and 頭の模様 belong to the whole face, so they are drawn once here
  // rather than inside the per-eye loop below. 鼻ちょうちん comes along too.
  drawFaceExtras(ctx, style, line);

  for (const layout of EYE_LAYOUT.eyes) {
    const spec = (p.eyes ?? {})[layout.key] ?? {};
    const open = clamp(spec.open ?? 1, 0, 1);
    const eye = {
      // `eyeX` slides this eye - and its lid, brow and tear - sideways on its own.
      cx: layout.cx + clamp(spec.eyeX ?? 0, -400, 400),
      cy: layout.cy,
      r: EYE_LAYOUT.radius,
      towardNose: layout.towardNose,
    };
    const lineStyle = { line: lineColor, width: lidWidth };
    const isHeart = (spec.irisShape ?? 'circle') === 'heart';

    drawBrow(ctx, eye, spec, style, limits);

    // --- tear ----------------------------------------------------------
    // A teardrop hangs below the eye, so it is drawn on top of everything else
    // (the opaque eyeball would otherwise cover it) and for a shut eye too. The
    // amount, the height and the tilt are all *per eye*, because one eye crying
    // while the other does not is a real expression.
    const drawTear = () => {
      const amount = clamp(spec.tear ?? 0, 0, 1.6);
      if (amount <= 0.01) return;
      const size = EYE_LAYOUT.tear.size * amount;
      const tearX = eye.cx + clamp(spec.tearX ?? 0, -400, 400) - layout.towardNose * EYE_LAYOUT.tear.offsetX;
      let tearY = eye.cy + EYE_LAYOUT.tear.offsetY + (spec.tearY ?? 0);
      // The drop hangs from `cy - 1.32r` to `cy + 0.78r`. Keep its lower tip
      // inside the artwork: a drop that reached the canvas border used to be
      // smeared down the chin by the clamp.
      const tearBottom = limits.bottom - EDGE_MARGIN;
      if (tearY + size * 0.78 > tearBottom) tearY = tearBottom - size * 0.78;
      ctx.save();
      if (spec.tearTilt) {
        ctx.translate(tearX, tearY);
        ctx.rotate((spec.tearTilt * Math.PI) / 180);
        ctx.translate(-tearX, -tearY);
      }
      dropPath(ctx, tearX, tearY, size);
      if (!line) {
        ctx.fillStyle = style.tearColor ?? '#8fd8ff';
        ctx.fill();
      }
      ctx.strokeStyle = lineColor;
      ctx.lineWidth = lidWidth * 0.55;
      ctx.stroke();
      ctx.restore();
    };

    if (open <= 0.02) {
      drawShutEye(ctx, eye, spec, lineStyle);
      // The glasses sit over the shut-eye artwork too, but under the tear below.
      drawGlasses(ctx, eye, line, style, limits);
      drawTear();
      continue;
    }

    // --- eyeball -------------------------------------------------------
    ctx.save();
    // A heart is not an eyeball, so once the eye is fully open it is not clipped
    // to the eyeball circle - that is what lets a heart be much larger than the
    // iris. While the lid is partly closed the clip is still needed.
    if (!isHeart || open < 0.999) clipEye(ctx, eye, open, lowerLid * (1 - open), lidFlat);

    const eyeTravel = isHeart ? heartTravel * lookMax : travel * lookMax;
    let dx = clamp(spec.lookX ?? 0, -1, 1) * eyeTravel;
    let dy = -clamp(spec.lookY ?? 0, -1, 1) * eyeTravel;
    const dist = Math.hypot(dx, dy);
    if (dist > eyeTravel && dist > 0) {
      dx = (dx / dist) * eyeTravel;
      dy = (dy / dist) * eyeTravel;
    }

    // The eyeball's white is only painted in the shaded styles: in line art the
    // paper behind shows through, which is what a line drawing wants. (The SVG
    // export traces the alpha mask, so a white fill would close the eye up
    // completely.)
    if (!line && !(isHeart && style.heartHideWhite)) fillCircle(ctx, eye.cx, eye.cy, eye.r, white);
    const irisX = eye.cx + dx;
    const irisY = eye.cy + dy;
    if (isHeart) {
      // A heart is wider than a circle, so cap it at the plane rather than the
      // eyeball; a red heart has no pupil highlight (it would sit on top of it).
      // A heart is not an eyeball, so it may grow well past the eye: the plate
      // keeps the whole canvas, and the eyes plate is drawn over the mouth, so a
      // big heart has room. The cap only stops it from running off the canvas.
      const heartRadius = Math.min(irisRadius * heartScale, EYE_LAYOUT.radius * 2.1);
      fillHeart(ctx, irisX, irisY, heartRadius, line ? lineColor : heartColor);
    } else {
      fillCircle(ctx, irisX, irisY, irisRadius, line ? lineColor : irisColor);
    }
    if (highlightOn && !isHeart) {
      const hx = irisX + layout.towardNose * EYE_LAYOUT.highlightOffset.x * irisScale;
      const hy = irisY + EYE_LAYOUT.highlightOffset.y * irisScale;
      const hr = EYE_LAYOUT.highlightRadius * irisScale;
      if (line) {
        // Punch the sparkle out of the iris rather than painting it white. On
        // screen the paper behind shows through, and in the SVG export (which
        // only sees alpha) it stays a hole in the pupil - both of which read as
        // the white glint of the original artwork.
        ctx.save();
        ctx.globalCompositeOperation = 'destination-out';
        fillCircle(ctx, hx, hy, hr, '#000000');
        ctx.restore();
      } else {
        fillCircle(ctx, hx, hy, hr, white);
      }
    }
    ctx.restore();

    // Between the eyeball and the tear: the lens covers the eye, and a tear still
    // falls in front of it.
    drawGlasses(ctx, eye, line, style, limits);

    drawTear();

    // --- lid strokes ---------------------------------------------------
    // Faded in as the lid starts to cover the eye, so opening the eye all the
    // way leaves the clean original artwork with no extra line.
    const lidFade = clamp((0.95 - open) / 0.1, 0, 1);
    if (lidFade > 0.01 || lowerLid > 0.01) {
      ctx.save();
      circlePath(ctx, eye.cx, eye.cy, eye.r);
      ctx.clip();
      ctx.globalAlpha = line ? 1 : lidFade;
      ctx.strokeStyle = lineColor;
      ctx.lineWidth = lidWidth;
      if (lidFade > 0.01) {
        lidPath(ctx, eye, open, false, lidFlat);
        ctx.stroke();
      }
      if (lowerLid > 0.01) {
        ctx.globalAlpha = line ? 1 : lowerLid;
        lidPath(ctx, eye, lowerLid, true, lidFlat);
        ctx.stroke();
      }
      ctx.restore();
    }

    // --- outline (line-art mode only) ---------------------------------
    if (line) strokeCircle(ctx, eye.cx, eye.cy, eye.r, lineColor, lidWidth * 0.75);
  }
}

/* -------------------------------------------------------------------- mouth */

/**
 * Draw the mouth into a 1024 x 380 canvas.
 *
 * @param {CanvasRenderingContext2D} ctx
 * @param {object} p
 * @param {'paint'|'line'} [p.mode]
 */
export function drawMouth(ctx, p) {
  const mode = p.mode ?? 'paint';
  const line = mode === 'line';
  const L = MOUTH_LAYOUT;
  const limits = p.limits ?? WINDOW_LIMITS;
  // The artwork's real edges, in the artwork's own coordinates, with a margin
  // kept clear. `top` is negative when the plate reaches above the window, which
  // is the room a moved mouth has to work with.
  const edgeLeft = limits.left + EDGE_MARGIN;
  const edgeRight = limits.right - EDGE_MARGIN;
  const edgeTop = limits.top + EDGE_MARGIN;
  const edgeBottom = limits.bottom - EDGE_MARGIN;

  const openAmount = clamp(p.open ?? 0, 0, 1);
  const thickness = L.thickness * clamp(p.thickness ?? 1, 0.2, 3);
  const cornerAmount = clamp(p.corners ?? 1, 0, 1.6);
  // The corner strokes flick out past the ends of the chord, so the ends cannot
  // use the whole canvas: a very wide mouth used to smear sideways off the plate.
  const cornerReach = L.corner.toX * cornerAmount;
  const halfChord = Math.min(
    L.halfChord * clamp(p.width ?? 1, 0.2, 1.6),
    Math.min(L.centreX - (edgeLeft + cornerReach), (edgeRight - cornerReach) - L.centreX),
  );

  // A negative smile bulges the line upwards, which reads as a frown.
  const sagRaw = L.sag * clamp(p.smile ?? 1, -0.55, 1.4);
  // A frown's middle is the highest point of the mouth, and that is exactly
  // where the nose is in the way, so slide the whole mouth down until the curve
  // is in the open. The push is worked out from the *base* position and the
  // height slider is added afterwards: adding it first made the push cancel the
  // slider exactly, which is why dragging ‟口の高さ” did nothing at all on a
  // frowning mouth like むっと.
  const base = L.chordY + Math.max(0, L.noseClear - (L.chordY + sagRaw));
  let y0 = base + (p.offsetY ?? 0);
  // The smile flattens slightly as the mouth opens, so the whole mouth keeps
  // fitting inside the band the mouth plane actually shows.
  let sag = sagRaw * (1 - L.openFlatten * openAmount);
  // Keep the whole mark on the canvas. The corner strokes sit *above* the ends of
  // the chord, so the allowance is not symmetric - padding the bottom by the
  // corner's depth used to cost most of the height slider's travel.
  const cornerRise = L.corner.toY * cornerAmount + thickness;
  const bottomEdge = y0 + Math.max(0, sag) + thickness;
  if (bottomEdge > edgeBottom) y0 -= bottomEdge - edgeBottom;
  const topEdge = y0 + Math.min(0, sag) - cornerRise;
  if (topEdge < edgeTop) y0 += edgeTop - topEdge;
  const pivot = { x: L.centreX, y: y0 + sag * 0.5 };
  const tilt = p.tilt ?? 0;

  const strokeColor = line ? (p.line ?? '#3a2a4a') : (p.color ?? '#ff1a44');
  const innerColor = p.innerColor ?? '#4a0f1e';
  const tongueColor = line ? (p.line ?? '#3a2a4a') : (p.tongueColor ?? '#ff2d2d');

  const left = { x: L.centreX - halfChord, y: y0 };
  const right = { x: L.centreX + halfChord, y: y0 };
  const lip = mouthControls(left, right, sag, L.sagBend);
  const arc = cubicPoints(lip.p0, lip.p1, lip.p2, lip.p3, 96);

  const tongueAmount = clamp(p.tongue ?? 1, 0, 1.6);

  // --- a round "O" mouth (surprised, singing) ---------------------------
  const roundAmount = clamp(p.round ?? 0, 0, 1);
  if (roundAmount > 0.004) {
    let ry = L.sag * 0.42 * roundAmount * clamp(p.thickness ?? 1, 0.5, 2);
    // The nose hides the middle of the upper face, so keep the whole oval below
    // it and inside the plate. (`L.safeBottom` is only the round mouth's own size
    // limit; it no longer has to stay clear of the canvas edge.)
    ry = Math.min(ry, (L.safeBottom - thickness / 2 - L.noseClear) / 2.1);
    const rx = Math.min(ry * 1.15 * clamp(p.width ?? 1, 0.2, 1.6), L.halfChord * 0.8);
    const centreY = clamp(y0 + sag * 0.5, L.noseClear + ry * 1.02, edgeBottom - thickness / 2 - ry * 1.02);
    if (ry > 5) {
      ctx.save();
      ctx.translate(pivot.x, pivot.y);
      ctx.rotate((tilt * Math.PI) / 180);
      ctx.translate(-pivot.x, -pivot.y);
      ctx.beginPath();
      ctx.ellipse(L.centreX, centreY, rx, ry, 0, 0, TAU);
      if (!line) {
        ctx.fillStyle = innerColor;
        ctx.fill();
        if (tongueAmount > 0.01) {
          ctx.beginPath();
          ctx.ellipse(L.centreX, centreY + ry * 0.46, rx * 0.52, ry * 0.4, 0, 0, TAU);
          ctx.fillStyle = tongueColor;
          ctx.fill();
        }
      }
      ctx.strokeStyle = strokeColor;
      ctx.lineWidth = line ? thickness * 0.8 : thickness;
      ctx.stroke();
      ctx.restore();
    }
    return;
  }

  // The smile line is the upper lip. Opening the mouth drops the lower jaw
  // below it, and the room for that is limited by the texture band the mouth
  // plane shows - otherwise the jaw is silently clipped away.
  // Everything has to fit in the part of the plane that faces the camera.
  let rise = L.openRise * openAmount;
  const budget = edgeBottom - thickness / 2 - y0;
  if (sag + rise > budget) {
    rise = Math.max(0, budget - Math.min(sag, budget));
    if (sag + rise > budget) sag = Math.max(-80, budget - rise);
  }
  const jawCubic = mouthControls(left, right, sag + rise, L.sagBend);
  const jaw = cubicPoints(jawCubic.p0, jawCubic.p1, jawCubic.p2, jawCubic.p3, 96);
  const jawAt = (t) => cubicPoint(jawCubic.p0, jawCubic.p1, jawCubic.p2, jawCubic.p3, t);
  const lipAt = (t) => cubicPoint(lip.p0, lip.p1, lip.p2, lip.p3, t);
  const openShape = rotate([...arc, ...jaw.slice(1, -1).reverse()], pivot, tilt);

  const tonguePos = clamp(p.tonguePos ?? L.tongue.pos, 0.05, 0.95);
  const tongueWidth = L.tongue.width * tongueAmount;
  const tongueHeight = L.tongue.height * tongueAmount;

  /**
   * The tongue, rising from `curve` to `height` above its middle.
   *
   * `pointAt(t)` is that same curve as a function of t, so the tongue's foot can
   * sit along it without this needing to know what kind of curve it is.
   */
  function drawTongue(pointAt, curve, height) {
    const arcLength = Math.max(1, chordLength(curve));
    const dt = clamp(tongueWidth / 2 / arcLength, 0.01, 0.45);
    const t0 = clamp(tonguePos - dt, 0, 1);
    const t1 = clamp(tonguePos + dt, 0, 1);
    const base = [];
    for (let i = 0; i <= 16; i++) base.push(pointAt(t0 + ((t1 - t0) * i) / 16));
    const mid = pointAt(clamp(tonguePos, 0, 1));
    const half = tongueWidth / 2;
    // The foot runs left to right; the crown comes back right to left, so the two
    // together are already a closed ring.
    const footRight = base[base.length - 1];
    const footLeft = base[0];
    const crown = [];
    const steps = 26;
    for (let i = 0; i <= steps; i++) {
      const f = i / steps;
      const dx = 1 - 2 * f;
      const footY = footRight.y + (footLeft.y - footRight.y) * f;
      crown.push({
        x: mid.x + dx * half,
        y: footY - height * (1 - Math.pow(Math.abs(dx), L.tongue.crown)),
      });
    }
    const shaped = rotate([...base, ...crown], pivot, tilt);
    if (line) {
      strokePolyline(ctx, shaped, { color: strokeColor, width: thickness * 0.8, closed: true });
      return;
    }
    tracePolyline(ctx, shaped, true);
    ctx.fillStyle = tongueColor;
    ctx.fill();
  }

  if (openAmount > 0.004) {
    if (!line) fillPolygon(ctx, openShape, innerColor);
    // The tongue rises from the lower lip *into* the mouth, so it has to arrive
    // with the opening: at a hair's width the jaw curve is a sliver, and the
    // tongue used to burst straight out of it and sit on the chin.
    const tongueOpen = clamp((openAmount - 0.06) / 0.24, 0, 1);
    if (tongueAmount > 0.01 && tongueOpen > 0.01) {
      ctx.save();
      tracePolyline(ctx, openShape, true);
      ctx.clip();
      // Inside an open mouth the tongue sits on the lower jaw.
      drawTongue(jawAt, jaw, tongueHeight * tongueOpen * (1 + openAmount * 0.3));
      ctx.restore();
    }
    // One outline around the whole mouth reads as lips; the smile stroke alone
    // would leave the lower edge as a bare colour change.
    strokePolyline(ctx, openShape, { color: strokeColor, width: thickness, closed: true });
  } else {
    // Closed lips - but the tongue still pokes over the lip. That is what the
    // original artwork does (the mouth reads as a smile with the tongue showing,
    // like ペコちゃん), and it is why a plain line looked wrong there. The lip
    // line goes on afterwards, so the tongue comes out from under it.
    if (tongueAmount > 0.01) {
      drawTongue(lipAt, arc, tongueHeight);
    }
    strokePolyline(ctx, rotate(arc, pivot, tilt), { color: strokeColor, width: thickness });
  }

  // --- corner marks ---------------------------------------------------
  if (cornerAmount > 0.01) {
    const C = L.corner;
    const cornerAngle = p.cornerAngle ?? 0;
    for (const [end, side] of [[left, -1], [right, 1]]) {
      const from = { x: end.x + C.fromX * side * cornerAmount, y: end.y - C.fromY * cornerAmount };
      const to = { x: end.x - C.toX * side * cornerAmount, y: end.y - C.toY * cornerAmount };
      // A smooth curve, not a three-point kink. `mid` is the control point, so it
      // is pulled twice as far as the bow should reach; both corners bow the same
      // way (downwards on screen), which is what the original artwork does.
      const control = {
        x: (from.x + to.x) / 2,
        y: (from.y + to.y) / 2 + (C.curve ?? 26) * cornerAmount,
      };
      const curve = quadraticPoints(from, control, to, 16);
      // `cornerAngle` tilts the whole mark around the mouth corner, mirrored so
      // both sides move together.
      const shaped = cornerAngle ? rotate(curve, end, cornerAngle * side) : curve;
      strokePolyline(ctx, rotate(shaped, pivot, tilt), {
        color: line ? strokeColor : (p.cornerColor ?? '#725497'),
        // The *length* of the corner mark follows `corners`, but its thickness
        // does not: `corners: 0.71` is the length that matches the original, and
        // the original's stroke is the full thickness.
        width: C.width,
      });
    }
  }
}

/** Approximate length of a polyline. */
function chordLength(points) {
  let total = 0;
  for (let i = 1; i < points.length; i++) {
    total += Math.hypot(points[i].x - points[i - 1].x, points[i].y - points[i - 1].y);
  }
  return total;
}