PDF underlay placement and PDF import

/** Is this one cubic an arc of a circle (within tolerance)? → { cx, cy, r, a0, a1 (CCW radians) } | null */
export function cubicArc(p0, c1, c2, p3, tol = 0.002) {
  const at = (t) => { const u = 1 - t; return [u * u * u * p0[0] + 3 * u * u * t * c1[0] + 3 * u * t * t * c2[0] + t * t * t * p3[0], u * u * u * p0[1] + 3 * u * u * t * c1[1] + 3 * u * t * t * c2[1] + t * t * t * p3[1]]; };
  const m = at(0.5);
  const circ = circleThrough(p0, m, p3);
  if (!circ) return null;
  for (const t0 of [0.25, 0.75, 0.1, 0.9]) { const q = at(t0); if (Math.abs(Math.hypot(q[0] - circ.cx, q[1] - circ.cy) - circ.r) > tol * circ.r) return null; }
  // direction: the angle from p0 through m to p3 along the shorter way
  const a0 = Math.atan2(p0[1] - circ.cy, p0[0] - circ.cx), am = Math.atan2(m[1] - circ.cy, m[0] - circ.cx), a1 = Math.atan2(p3[1] - circ.cy, p3[0] - circ.cx);
  const norm = (a) => ((a % (2 * Math.PI)) + 2 * Math.PI) % (2 * Math.PI);
  const ccw = norm(am - a0) < norm(a1 - a0);
  return ccw ? { ...circ, a0: norm(a0), a1: norm(a1) } : { ...circ, a0: norm(a1), a1: norm(a0) };
}

// the outline as drawing points, curves flattened
const pts = [];
let cursor = null;
for (const o of ops) {
  if (o[0] === 'M' || o[0] === 'L') { const q = P(o[1], o[2]); pts.push({ x: q.x, y: q.y, b: 0 }); cursor = [o[1], o[2]]; }
  else {
    const p0 = cursor;
    for (let s = 1; s <= 10; s++) {
      const t0 = s / 10, u = 1 - t0;
      const x = u * u * u * p0[0] + 3 * u * u * t0 * o[1] + 3 * u * t0 * t0 * o[3] + t0 * t0 * t0 * o[5];
      const y = u * u * u * p0[1] + 3 * u * u * t0 * o[2] + 3 * u * t0 * t0 * o[4] + t0 * t0 * t0 * o[6];
      const q = P(x, y); pts.push({ x: q.x, y: q.y, b: 0 });
    }
    cursor = [o[5], o[6]];
  }
}
if (pts.length < 2) continue;

const cubics = ops.filter((o) => o[0] === 'C').length;
if (cubics === ops.length - 1 && cubics >= 1) {
  let cur = [ops[0][1], ops[0][2]], arcs = [], okAll = true;
  for (const o of ops.slice(1)) { const a = cubicArc(cur, [o[1], o[2]], [o[3], o[4]], [o[5], o[6]]); if (!a) { okAll = false; break; } arcs.push(a); cur = [o[5], o[6]]; }
  if (okAll) {
    const same = arcs.every((a) => Math.abs(a.r - arcs[0].r) < 1e-3 * arcs[0].r && Math.hypot(a.cx - arcs[0].cx, a.cy - arcs[0].cy) < 1e-3 * arcs[0].r);
    if (same) {
      const c0 = P(arcs[0].cx, arcs[0].cy), r = arcs[0].r * scale;
      const span = arcs.reduce((s, a) => s + (((a.a1 - a.a0) % (2 * Math.PI)) + 2 * Math.PI) % (2 * Math.PI), 0);
      if (arcs.length >= 3 && sub.closed || Math.abs(span - 2 * Math.PI) < 1e-3) { if (add({ type: 'circle', ...base, cx: c0.x, cy: c0.y, r })) counts.circles++; continue; }
      if (arcs.length === 1) {
        const a = arcs[0];
        const a0 = flip ? Math.PI - a.a1 + rot0 : a.a0 + rot0, a1 = flip ? Math.PI - a.a0 + rot0 : a.a1 + rot0;
        const n2 = (x) => ((x % (2 * Math.PI)) + 2 * Math.PI) % (2 * Math.PI);
        if (add({ type: 'arc', ...base, cx: c0.x, cy: c0.y, r, a0: n2(a0), a1: n2(a1) })) counts.arcs++;
        continue;
      }
    }
  }
}
if (pts.length === 2 && !sub.closed) { if (add({ type: 'line', ...base, x1: pts[0].x, y1: pts[0].y, x2: pts[1].x, y2: pts[1].y })) counts.lines++; continue; }