Zweispaltiger Aufsatz mit nummerierten Gleichungen

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// ─── 2 · Content ────────────────────────────────────────────────────────────
const markdown = String.raw`---
Markdown-Beispiel · 136 Zeilen · content.en.md
title: "The period of a pendulum at large amplitudes"
author: "Marta Oyelaran, Tomás Heikkinen and Priya Anand"
publishDate: "18 September 2026"
---

# The period of a pendulum \\ at large amplitudes {style="article" kicker="Research article · Classical mechanics" authors="Marta Oyelaran¹, Tomás Heikkinen² and Priya Anand¹" affiliations="¹ Department of Physics, Northgate College, Dunmore · ² Horology Workshop, Harrow Hill Institute" received="12 March 2026" accepted="30 June 2026"}

:::callout{type="abstract"}
Every textbook gives the period of a pendulum as $T_0 = 2\pi\sqrt{L/g}$ and adds that it holds for small swings. We ask how small. The exact period, written with a complete elliptic integral and computed in three lines with the arithmetic–geometric mean, is longer than the textbook value by 0.19% at 10°, 1.7% at 30° and 18% at 90°. A one-metre pendulum timed with a photogate at amplitudes from 5° to 90° follows the exact curve to within 0.08%, the limit set by reading the amplitude. We give the series a student can check by hand, and a rule of thumb for the teaching laboratory.

:::paragraphs{style="keywords"}
**Keywords** :chip[pendulum]{style="keyword"} :chip[elliptic integral]{style="keyword"} :chip[arithmetic–geometric mean]{style="keyword"} :chip[photogate]{style="keyword"}
:::
:::

## Introduction

Galileo is said to have noticed that a lamp swinging on a long chain seems to take the same time over every swing, wide or narrow, and in the *Two New Sciences* he stated the rule that textbooks still teach: the period of a pendulum depends on its length, not on how far it swings [1]. A generation later Huygens, who had invented the pendulum clock in 1656, showed that this is only nearly true [2]. A bob on a circular arc takes longer over a wide swing than over a narrow one, and he fitted his clocks with curved cheeks that made the bob follow a cycloid instead, the one curve on which the period does not depend on the amplitude at all. The cheeks brought errors of their own, however, and clockmakers soon settled for a simpler remedy of keeping the swing of the pendulum small and steady.

A pendulum timed in a teaching laboratory bears out both Galileo’s rule and Huygens’ correction. Every introductory course gives the period as

$$T_0 = 2\pi\sqrt{\frac{L}{g}} \, . \tag{1}$$

Here $L$ is the length from the pivot to the centre of the bob and $g$ the acceleration of free fall. For a one-metre pendulum the formula predicts 2.006 s, and a careful student will find a period close to that. Then someone pulls the bob far out to the side, well past the small angles of the textbook, and the measured period grows. By how much, and from what amplitude the growth matters, is the subject of this article.

In Section 2 we derive the exact period and show how to compute it on a pocket calculator. Section 3 describes a bench experiment with a photogate timer, Section 4 compares the two, and Section 5 turns to the teaching laboratory and to clocks. The geometry and the symbols are those of :ref{id="geometry" style="full"}.

## Theory

### The equation of motion

A bob of mass $m$ on a light rigid rod of length $L$, displaced by an angle $\theta$ from the vertical, feels a torque $-mgL\sin\theta$ about the pivot. We neglect the mass of the rod, the size of the bob and the drag of the air until Section 5. With the moment of inertia $mL^2$, the equation of motion is

$$\ddot\theta + \frac{g}{L}\,\sin\theta = 0 . \tag{2}$$

When the swing is small, $\sin\theta$ can be replaced by $\theta$. Equation (2) then becomes the equation of a harmonic oscillator, with an angular frequency $\omega_0 = \sqrt{g/L}$ and the period $T_0 = 2\pi/\omega_0$ of Eq. (1). Nothing in that solution depends on the amplitude $\theta_0$. This is Galileo’s isochronism, and it is exact only in the limit $\theta_0 \to 0$.

### The exact period

Equation (2) can be integrated once. Multiplying it by $\dot\theta$ and integrating from the turning point, where the bob is momentarily at rest at $\theta = \theta_0$, gives the conservation of energy:

$$\tfrac{1}{2}\,L\,\dot\theta^{2} = g\,(\cos\theta - \cos\theta_0) . \tag{3}$$

Separating the variables and integrating over a quarter of a swing, with the substitution $\sin(\theta/2) = k\sin\phi$ and the modulus $k = \sin(\theta_0/2)$, gives the exact period

$$T = 4\sqrt{\frac{L}{g}}\;K(k) . \tag{4}$$

Here $K$ is the complete elliptic integral of the first kind, as tabulated by Legendre [3] and treated in every course of analysis [4]:

$$K(k) = \int_0^{\pi/2}\frac{d\phi}{\sqrt{1-k^{2}\sin^{2}\phi}} . \tag{5}$$

Since $K(0) = \pi/2$, the small-amplitude limit gives back $T_0$. The ratio $T/T_0 = 2K(k)/\pi$ depends on the amplitude alone, so everything that follows holds for a pendulum of any length.

Elliptic integrals have a reputation for needing tables, but Gauss found that $K$ follows from the arithmetic–geometric mean. Start from $a_0 = 1$ and $b_0 = \cos(\theta_0/2)$, and repeat $a_{n+1} = (a_n + b_n)/2$ and $b_{n+1} = \sqrt{a_n b_n}$ until the two agree. Their common limit $M$ gives

$$T = \frac{T_0}{M\bigl(1,\,\cos(\theta_0/2)\bigr)} . \tag{6}$$

The iteration converges so fast that three steps, three rows of a spreadsheet, give ten correct digits even at 90° [5].

### The series in the amplitude

Expanding $K$ in powers of the amplitude gives the series of the classic texts [4,6], with $\theta_0$ in radians:

$$T = T_0\left(1 + \frac{\theta_0^{2}}{16} + \frac{11\,\theta_0^{4}}{3072} + \cdots\right) . \tag{7}$$

The next term is $173\,\theta_0^6/737\,280$, but the first correction alone is worth remembering. It says that the true period exceeds the textbook one by 1% when $\theta_0^2/16 = 0.01$, at 0.4 rad or 23°, and by 0.1% at 7°. For a one-metre pendulum, whose $T_0$ is 2.0059 s, a swing of 30° lasts 2.0408 s: the 35 ms difference adds up to a full second in just under a minute, well within reach of a stopwatch. Timing a pendulum was for two centuries the way to measure $g$ [7], and plotting $T_0^2$ against the length is still a classic exercise: a student who times the bob at an amplitude of 30° will find $g$ too small by 3.4%.

## Method

A brass bob 48 mm across, of mass 0.49 kg, hangs from a steel wire 0.3 mm in diameter, clamped between two hardened jaws so that the pivot is a sharp edge rather than a loop. The effective length, from the edge of the jaws to the centre of the bob, is $L = 1.000 \pm 0.001$ m, and a gravimetric survey of the building gives $g = 9.812$ m s^−2^. With these values the textbook formula predicts 2.0059 s.

A photogate at the bottom of the swing records each passage of the bob to 10 µs. For each amplitude we release the bob from a V-shaped holder set with a protractor, and time it over ten full periods. The amplitude is read again at the end of the run, and we report the mean of the two readings, which are known to ±0.5°. We stop at 90°, where the tension in the wire falls to zero at the turning points. Apart from the timer, the whole apparatus cost less than \$40, most of it for the bob.

## Results

:ref{id="runs" style="full"} lists the measured periods beside the exact prediction of Eq. (6), and :ref{id="period" style="full"} plots both, as the ratio to $T_0$, against the amplitude. The measurements follow the exact curve over the whole range. The residuals scatter on both sides of zero with no trend, and they grow with the amplitude as they should if their source is the reading of the angle: at 60°, an error of 0.5° in $\theta_0$ moves the prediction by 0.1%.

The textbook value, by contrast, falls further behind with every degree. It is only 0.05% short at 5°, but the true period exceeds it by 0.8% at 20°, 7% at 60° and 18% at 90°, where a pendulum that should swing to and fro in two seconds takes 2.37 s. The first two terms of Eq. (7) do far better: they stay within 0.1% up to 40°, and within 1% up to 70°. With its third term as well, Eq. (7) is within 0.01% of the exact period up to 45°, and within 0.4% even at 90°.

## Discussion

For the teaching laboratory the first term of Eq. (7) suggests a rule that needs no radians. Square the amplitude in degrees and divide by 50: the result is the excess of the true period over $T_0$ in parts per thousand. At 30° the rule gives 18 against an exact 17.4, and at 60° it gives 72 against 73.2. At 90° it gives 162 against 180, 10% low.

The period is not the only thing that changes. :ref{id="phase" style="full"} draws the motion in the phase plane, with one closed orbit for each amplitude: Eq. (3) solved for the angular velocity, in units of $\omega_0$. Small swings trace ellipses, round which the state of the pendulum turns at the steady rate $\omega_0$. Wide swings stretch into lemon shapes, flattened above and below, and most of the added period goes on the slow crawl round their ends, where the bob hangs near its turning points with little torque to bring it back. The motion is no longer a cosine either. At 90° a third harmonic of 1.5% of the fundamental flattens the swing at its extremes, against 0.02% at 10°. At 180° the orbit becomes the separatrix and the period grows without bound, since a pendulum balanced upside down never falls.

The same series explains why clockmakers kept their pendulums swinging through only a few degrees. At an amplitude of 2° the circular error is small, but it changes with the amplitude: a clock whose swing falls from 2.1° to 2° as its oil thickens gains 0.7 s a day. Huygens’ cheeks removed the error in principle [2], but it was the small, steady swing of the escapements of the next two centuries that removed it in practice.

Two effects we did not model deserve a word. The bob is not a point, and a sphere of radius $r$ on a wire of length $L$ swings like a simple pendulum of length $L\,(1 + \tfrac{2}{5}\,r^2/L^2)$, a correction of 0.023% in length and half that in period, below our scatter. Air drag lowers the amplitude by up to 4% over a run at large angles, as Newton measured and Stokes explained [8,9], which is why we report the mean amplitude of each run rather than the release angle.

## Conclusions

The period of a simple pendulum grows with the amplitude, slowly at first and then fast: by 0.19% at 10°, 1.7% at 30° and 18% at 90°. Three steps of the arithmetic–geometric mean give the exact value, and a one-metre pendulum on a bench confirms it to within the accuracy of a protractor. Below 23° the textbook formula is good to 1%; beyond that, a student can square the amplitude in degrees and divide by 50 to see how far off it is.

## Author contributions {style="back"}

M. O. and P. A. built the apparatus and timed the runs, T. H. wrote on clocks, and all three revised the article.

## Acknowledgements {style="back"}

We thank the Northgate College workshop, who made the pivot jaws, and two referees for their comments.

## Data availability {style="back"}

The photogate records of every run, and a spreadsheet that evaluates Eq. (6) in three rows, are published with this article as supplementary material.

## Competing interests {style="back"}

The authors declare no competing interests.

## References {style="back"}

:::paragraphs{style="references"}
[1] G. Galilei, *Discorsi e dimostrazioni matematiche intorno a due nuove scienze* (L. Elzevir, Leiden, 1638).

[2] C. Huygens, *Horologium oscillatorium* (F. Muguet, Paris, 1673).

[3] A.-M. Legendre, *Traité des fonctions elliptiques et des intégrales eulériennes*, vol. 2 (Huzard-Courcier, Paris, 1826).

[4] E. T. Whittaker and G. N. Watson, *A Course of Modern Analysis*, 4th ed. (Cambridge University Press, 1927).

[5] C. G. Carvalhaes and P. Suppes, Approximations for the period of the simple pendulum based on the arithmetic-geometric mean, *Am. J. Phys.* **76**, 1150–1154 (2008).

[6] R. A. Nelson and M. G. Olsson, The pendulum: rich physics from a simple system, *Am. J. Phys.* **54**, 112–121 (1986).

[7] H. Kater, An account of experiments for determining the length of the pendulum vibrating seconds in the latitude of London, *Phil. Trans. R. Soc. Lond.* **108**, 33–102 (1818).

[8] I. Newton, *Philosophiæ naturalis principia mathematica* (Royal Society, London, 1687), Book II.

[9] G. G. Stokes, On the effect of the internal friction of fluids on the motion of pendulums, *Trans. Camb. Phil. Soc.* **9**, 8–106 (1851).
:::

:::paragraphs{style="colophon"}
**Cite as** M. Oyelaran, T. Heikkinen and P. Anand, The period of a pendulum at large amplitudes, *Measure* **7**, 213–216 (2026), doi:10.5555/measure.7.3.213.

*Measure* is a fictional journal set for the Postext Cookbook: its authors, institutions and measurements are invented, while the physics and the references are real. Set in STIX Two Text, Schibsted Grotesk and Azeret Mono (SIL OFL). Text and figures: CC BY 4.0.
:::
`; // content.<lang>.md, inlined by the Cookbook

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const agm = (a, b) => [1, 2, 3].reduce(([x, y]) => [(x + y) / 2, Math.sqrt(x * y)], [a, b])[0];
const exact = (deg) => T0 / agm(1, Math.cos((deg * Math.PI) / 360)); // Eq. (6)
const runs = [[5, 2.0069], [10, 2.0096], [20, 2.0210], [30, 2.0412], [45, 2.0864],
  [60, 2.1517], [75, 2.2458], [90, 2.3658]]; // amplitude (°), measured period (s): simulated
const fixed = (x, n, sign) => (sign && x >= 0 ? '+' : '') + x.toFixed(n).replace('-', '−');
const cell = (content) => ({ content, align: 'right' }); // headerRowCount marks row 0
const table = { headerRowCount: 1, columnWidths: [1.25, 1.35, 1, 1.1, 1.2], rows: [
  ['Amplitude', 'Measured', 'Eq. (6)', 'Over *T*~0~', 'Residual'],
  ...runs.map(([deg, T]) => [`${deg}°`, fixed(T, 4), fixed(exact(deg), 4),
    `${fixed((exact(deg) / T0 - 1) * 100, 2)}%`, `${fixed((T / exact(deg) - 1) * 100, 3, 1)}%`]),
].map((row) => row.map(cell)) }; // *T*~0~ is plain markup: the header stays in its sans
// The figure labels are MathJax paths too: the same italic θ as the text, and vector in the PDF.
const R = (x) => Math.round(x * 100) / 100; // coordinates to 0.01 mm keep the SVG short
function tex(markup, x, y, size, anchor = 0, color = 'ink') { // anchor 0 left, .5 centre, 1 right
  const r = renderMath(markup, false, 100); // paths in MathJax units: 1000 to the em
  const k = size / 1000;
  return `<g transform="translate(${R(x - anchor * r.viewBox.width * k)} ${R(y)}) scale(${k})" `
    + `fill="${palette[color]}">${r.paths.map((p) => `<path d="${p.d}"/>`).join('')}</g>`;
}
const PX_PER_MM = 10; // the drawings are in mm, declared to the engine at 10 px to the mm
const figure = (id, w, h, caption, altText, placement) => ({ id, typeId: 'figure', kind: 'svg',
  svg: { fileId: `${id}.svg`, width: w * PX_PER_MM, height: h * PX_PER_MM }, caption, altText,
  placement, createdAt: 0, updatedAt: 0 });
const resources = [ // each is placed where the text first cites it with :ref; sizes in mm
  figure('strobe', 136, 70, '', 'A swinging pendulum, lit by flashes.'), // uncited: the band's
  figure('geometry', 84, 60, 'The pendulum and its symbols: the length *L* from the pivot to '
    + 'the centre of the bob, the mass *m* and the amplitude, the angle between the vertical '
    + 'and either turning point of the swing.', // the caption face has no Greek: no θ here
  'A bob on a rod hanging from a pivot, pulled aside by an angle.', { position: 'top' }),
  { id: 'runs', typeId: 'table', kind: 'table', table: { model: table }, createdAt: 0,
    updatedAt: 0, caption: 'Measured and predicted periods of the one-metre pendulum, in seconds, '
    + 'and their excess and residual in per cent.',
    note: 'Amplitudes to ±0.5°. The measurements are simulated for this example.' },
  figure('period', 84, 60, 'The period against the amplitude, as a ratio to *T*~0~: Eq. (6) '
    + '(solid), the first two terms of Eq. (7) (dashed) and the measurements (dots).',
  'The period grows with the amplitude, slowly, then fast; the measured points sit on the curve.'),
  figure('phase', 176, 76, 'The phase plane of the pendulum: one orbit for each amplitude from '
    + '30° to 150° in steps of 30°, the 90° orbit of our widest runs in red, and the separatrix '
    + 'of a swing to 180° dashed.', 'Nested closed orbits growing into lemon shapes.',
  { position: 'top', span: 'page' }), // cited on page 3, it opens page 4 across both columns
];
// #endregion

// #region art: the swinging pendulum, the geometry, the plot and the phase plane
const rad = (deg) => (deg * Math.PI) / 180;
const svg = (w, h, body) => `<svg xmlns="http://www.w3.org/2000/svg" width="${w * PX_PER_MM}" `
  + `height="${h * PX_PER_MM}" viewBox="0 0 ${w} ${h}">${body}</svg>`;
const line = (x1, y1, x2, y2, color, width, extra = '') => `<line x1="${R(x1)}" y1="${R(y1)}" `
  + `x2="${R(x2)}" y2="${R(y2)}" stroke="${palette[color]}" stroke-width="${width}" ${extra}/>`;
const dot = (x, y, r, color, extra = '') => `<circle cx="${R(x)}" cy="${R(y)}" r="${r}" `
  + `fill="${palette[color]}" ${extra}/>`;
const arc = (cx, cy, r, from, to, color, width, extra = '') => `<path d="M${R(cx + r * Math.sin(
  from))} ${R(cy + r * Math.cos(from))}A${r} ${r} 0 0 0 ${R(cx + r * Math.sin(to))} ${R(cy + r
  * Math.cos(to))}" fill="none" stroke="${palette[color]}" stroke-width="${width}" ${extra}/>`;
const path = (points, color, width, extra = '') => `<path d="${points.map(([x, y], i) =>
  `${i ? 'L' : 'M'}${R(x)} ${R(y)}`).join('')}" fill="none" stroke="${palette[color]}" `
  + `stroke-width="${width}" ${extra}/>`;
function strobe() { // 136 × 70 mm on the band: one half swing, lit at equal times
  const [px, len, amp] = [74, 60, rad(68)]; // pivot x, rod length (mm), amplitude
  let out = arc(px, 0, len, -amp, amp, 'mist', 0.35, 'stroke-dasharray="1 1.6" '
    + 'stroke-opacity="0.7"');
  for (let i = 8; i >= 0; i--) { // nine flashes at equal times: the bob bunches up where it
    const th = amp * Math.cos((Math.PI * (i + 0.5)) / 9); // slows down; the red one last
    const [x, y, last] = [px + len * Math.sin(th), len * Math.cos(th), i === 0];
    out += line(px, 0, x, y, last ? 'paper' : 'mist', last ? 0.5 : 0.3,
      `stroke-opacity="${last ? 0.9 : 0.4}"`)
      + dot(x, y, 4, last ? 'series' : 'mist', `fill-opacity="${last ? 1 : 0.34}"`);
  }
  return svg(136, 70, out + dot(px, 0, 1.1, 'paper'));
}
function geometry() { // 84 × 60 mm, one column
  const [px, py, len, a] = [42, 6, 44, rad(38)];
  const [bx, by] = [px + len * Math.sin(a), py + len * Math.cos(a)];
  return svg(84, 60, `<rect x="${px - 14}" y="${py - 3}" width="28" height="3" `
    + `fill="${palette.rule}"/>`
    + line(px, py, px, py + len + 6, 'muted', 0.3, 'stroke-dasharray="1.2 1"')
    + arc(px, py, len, -a, a, 'rule', 0.4, 'stroke-dasharray="1.2 1"')
    + arc(px, py, 12, 0, a, 'journal', 0.4)
    + line(px, py, 2 * px - bx, by, 'rule', 0.4) + dot(2 * px - bx, by, 3.4, 'rule')
    + line(px, py, bx, by, 'ink', 0.6) + dot(px, py, 0.9, 'ink') + dot(bx, by, 3.4, 'series')
    + tex('\\theta_0', px + 3.4, py + 18.4, 4.2)
    + tex('L', (px + bx) / 2 + 3.2 * Math.cos(a), (py + by) / 2 - 3.2 * Math.sin(a) + 1.4, 4.2)
    + tex('m', bx + 5, by + 1.5, 4.2));
}
function period() { // 84 × 60 mm: T/T0 against the amplitude
  const [x0, y0, w, h] = [13, 8, 66, 40];
  const X = (deg) => x0 + (deg / 90) * w;
  const Y = (ratio) => y0 + h - ((ratio - 1) / 0.2) * h;
  const curve = (f) => Array.from({ length: 91 }, (_, d) => [X(d), Y(f(d))]);
  let out = '';
  for (const r of [1, 1.05, 1.1, 1.15, 1.2]) {
    out += line(x0, Y(r), x0 + w, Y(r), 'rule', 0.2)
      + tex(r.toFixed(2), x0 - 1.8, Y(r) + 1.1, 3.1, 1, 'muted');
  }
  for (const d of [0, 30, 60, 90]) out += tex(`${d}^\\circ`, X(d), y0 + h + 4.6, 3.1, 0.5, 'muted');
  return svg(84, 60, out + path(curve((d) => exact(d) / T0), 'journal', 0.6)
    + path(curve((d) => 1 + rad(d) ** 2 / 16), 'muted', 0.4, 'stroke-dasharray="1.4 1"')
    + runs.map(([d, T]) => dot(X(d), Y(T / T0), 1, 'series')).join('')
    + tex('\\theta_0', x0 + w, y0 + h + 9.5, 3.6, 1) + tex('T/T_0', x0 - 1.8, y0 - 4.2, 3.6, 1));
}
function phase() { // 176 × 76 mm, across the page: the orbits of Eq. (3)
  const [cx, cy, sx, sy] = [88, 39, 24, 16]; // origin, and mm per radian and per unit of θ̇/ω0
  let out = line(cx - 84, cy, cx + 84, cy, 'rule', 0.3)
    + line(cx, cy - 37, cx, cy + 37, 'rule', 0.3);
  for (const [x, label] of [[-Math.PI, '-\\pi'], [Math.PI, '\\pi']]) {
    out += line(cx + sx * x, cy, cx + sx * x, cy + 1.2, 'muted', 0.3)
      + tex(label, cx + sx * x, cy + 5, 3.1, 0.5, 'muted');
  }
  for (const deg of [30, 60, 90, 120, 150, 180]) {
    const a = rad(deg);
    const v = (th) => Math.sqrt(Math.max(0, 2 * (Math.cos(th) - Math.cos(a)))); // Eq. (3)
    const orbit = Array.from({ length: 241 }, (_, i) => { // over the top, then back under
      const [u, s] = i < 120 ? [i / 120, 1] : [(i - 120) / 120, -1];
      const th = a * Math.sin(Math.PI * (u - 0.5));
      return [cx + sx * th * s, cy - sy * v(th) * s];
    });
    out += deg === 90 ? path(orbit, 'series', 0.7) : path(orbit, 'journal', deg === 180 ? 0.35
      : 0.45, deg === 180 ? 'stroke-dasharray="1.4 1"' : '');
  }
  return svg(176, 76, out + tex('\\theta', cx + 84, cy - 1.6, 3.8, 1)
    + tex('\\dot\\theta/\\omega_0', cx + 1.6, cy - 34.4, 3.8));
}
// #endregion

// ─── 3 · Fonts ──────────────────────────────────────────────────────────────
const FONTS = { // every face the pages paint, loaded before the first build
  'STIX Two Text': ['400', '400i', '700'],
  'Schibsted Grotesk': ['400', '400i', '500', '600', '700', '700i'],
  'Azeret Mono': ['400', '600'],
};

// ─── 4 · Build & show ───────────────────────────────────────────────────────
for (const [id, draw] of Object.entries({ strobe, geometry, period, phase })) {
  await loadSvg(`${id}.svg`, draw()); // registered for the canvas, kept as bytes for the PDF
}
const doc = await buildDocumentWithFonts({ markdown, resources }, config(), kitFonts(FONTS));
showPages(doc, { title: 'Two-column paper with numbered equations' });
offerPdf(() => renderToPdf(doc, { fontProvider: fontsourceProvider, resourceBytes: imageBytes }),
  `${RECIPE}.pdf`); // text in the Fontsource faces; formulas and figures as vector paths

Kit · core, fonts, viewer, pdf, images: in jedem Rezept gleich · 315 Zeilen
// ─── Kit ── helpers shared by every Cookbook recipe · postext.dev/cookbook ─────

// ─── Kit · core v1 ── the same in every recipe · postext.dev/cookbook
function mm(value) { return { value, unit: 'mm' }; }
function pt(value) { return { value, unit: 'pt' }; }
function em(value) { return { value, unit: 'em' }; }
/** The sample language's string: t({ en: 'Figure', es: 'Figura' }). */
function t(strings) { return strings[LANG] ?? Object.values(strings)[0]; }
/** A file in this recipe's assets folder, served from the Postext repo by jsDelivr. */
function asset(file) { return `https://cdn.jsdelivr.net/gh/drnachio/postext@main/cookbook/${RECIPE}/assets/${file}`; }

// ─── Kit · fonts v3 ── the same in every recipe · postext.dev/cookbook
// Postext loads the faces a config asks for before layout (prepareFonts,
// buildDocumentWithFonts) and asks the kit for the files: Fontsource's, the
// files the PDF embeds too.

/** The options of buildDocumentWithFonts / prepareFonts / withLoadedFonts:
 *  the Fontsource resolver, and the faces of FONTS loaded with the ones the
 *  config names. FONTS = { 'Family Name': ['400', '400i', '700'] }. */
function kitFonts(FONTS = {}) {
  const faces = Object.entries(FONTS).flatMap(([family, specs]) => specs.map((spec) =>
    ({ family, weight: parseInt(spec, 10), style: spec.endsWith('i') ? 'italic' : 'normal' })));
  return { resolve: fontsourceResolver, faces };
}

/** A face's Fontsource files, in a fixed order: latin, then latin-ext and greek
 *  when `text` needs them and the family ships them (kitSubsetsFor), each with
 *  its unicode-range. Null for a weight or slant Fontsource does not ship. */
async function fontsourceResolver(family, weight, style, { text = '' } = {}) {
  const id = fontsourceId(family);
  const meta = await fontsourceMeta(family);
  if (meta && !(meta.weights.includes(weight) && meta.styles.includes(style))) return null;
  kitStatus('Loading fonts…');
  const at = await fontsourceAt(id);
  return ['latin', ...kitSubsetsFor(text, meta)].map((subset) => ({
    source: `${at}files/${id}-${subset}-${weight}-${style}.woff2`,
    unicodeRange: kitRanges()[subset],
  }));
}

/** The unicode-range of each Fontsource file the kit loads (a function: the
 *  kit is inlined after the pen's code, which runs before a const would be set). */
function kitRanges() {
  return {
    latin: 'U+0000-00FF,U+0131,U+0152-0153,U+02BB-02BC,U+02C6,U+02DA,U+02DC,U+0304,U+0308,U+0329,'
      + 'U+2000-206F,U+20AC,U+2122,U+2191,U+2193,U+2212,U+2215,U+FEFF,U+FFFD',
    'latin-ext': 'U+0100-02BA,U+02BD-02C5,U+02C7-02CC,U+02CE-02D7,U+02DD-02FF,U+0304,U+0308,U+0329,'
      + 'U+1D00-1DBF,U+1E00-1E9F,U+1EF2-1EFF,U+2020,U+20A0-20AB,U+20AD-20C0,U+2113,U+2C60-2C7F,U+A720-A7FF',
    greek: 'U+0370-03FF',
  };
}

/** For the cjk, arabic and comics blocks, which add their own files: the
 *  faces of `faces` = { 'Family Name': ['400', '400i', '700'] }. `text` is the sample:
 *  č ł † α χ also load latin-ext and greek files (kitSubsetsFor). With
 *  `optional`, a face Fontsource does not ship is skipped instead of failing.
 *  The files are added once all have loaded, in the order asked (latin, then
 *  latin-ext, then greek), whatever order they arrive in: the last one added
 *  wins a character two files hold. Resolves to the number of faces added. */
async function loadFonts(faces, text = '', { optional = false } = {}) {
  kitStatus('Loading fonts…');
  const ranges = kitRanges();
  const jobs = [];
  for (const [family, specs] of Object.entries(faces)) {
    const id = fontsourceId(family);
    const todo = [...new Set(specs)].map((spec) => [parseInt(spec, 10), spec.endsWith('i') ? 'italic' : 'normal'])
      .filter(([weight, style]) => !hasFace(family, weight, style)); // before any await
    const at = fontsourceAt(id);
    const meta = optional || /[^\0-ÿ]/u.test(text) ? await fontsourceMeta(family) : null;
    const subsets = ['latin', ...kitSubsetsFor(text, meta)];
    for (const [weight, style] of todo) {
      if (optional && !(meta?.weights.includes(weight) && meta.styles.includes(style))) continue;
      for (const subset of subsets) {
        const url = `${await at}files/${id}-${subset}-${weight}-${style}.woff2`;
        const face = new FontFace(family, `url(${url}) format('woff2')`,
          { weight: String(weight), style, unicodeRange: ranges[subset] });
        jobs.push(face.load().catch(() => {
          if (subset === 'latin' && !optional) throw new Error(`Fontsource has no ${family} ${weight} ${style}`);
        }));
      }
    }
  }
  const ready = await Promise.all(jobs).catch((error) => { kitFail(error); throw error; });
  return ready.filter((face) => face && document.fonts.add(face)).length;
}

/** A loaded FontFace covers this family, weight and style (fonts.check() would
 *  also say yes for families nobody declared). */
function hasFace(family, weight, style) {
  for (const face of document.fonts) {
    if (face.status !== 'loaded' || face.style !== style) continue;
    if (face.family.replace(/^["']|["']$/g, '') !== family) continue;
    const [low, high = low] = face.weight.split(' ').map(Number);
    if (weight >= low && weight <= high) return true;
  }
  return false;
}

/** The files beyond latin `text` needs that `meta`'s family ships. */
function kitSubsetsFor(text, meta) {
  return [[/[Ā-˿ᴀ-ᶿḀ-ỿ†ℓⱠ-Ɀ꜠-ꟿ]/u, 'latin-ext'], [/[Ͱ-Ͽ]/u, 'greek']]
    .filter(([re, x]) => re.test(text) && meta?.subsets?.includes(x)).map(([, x]) => x);
}

/** Fontsource's id for a family: 'Source Serif 4' → 'source-serif-4'. */
function fontsourceId(family) { return family.toLowerCase().replace(/\s+/g, '-'); }

/** A package's folder at the one version @5 names now (#766). */
function fontsourceAt(id, u = `https://cdn.jsdelivr.net/npm/@fontsource/${id}@`) {
  return (fontsourceAt.c ??= {})[id] ??= fetch(u + '5/package.json').then((r) => r.json())
    .then((p) => p.version || 5, () => 5).then((v) => `${u}${v}/`);
}

/** The family's Fontsource metadata (weights, styles, subsets), or null. */
function fontsourceMeta(family) {
  fontsourceMeta.cache ??= new Map();
  const id = fontsourceId(family);
  if (!fontsourceMeta.cache.has(id)) {
    fontsourceMeta.cache.set(id, fetch(`https://api.fontsource.org/v1/fonts/${id}`)
      .then((res) => (res.ok ? res.json() : null), () => null));
  }
  return fontsourceMeta.cache.get(id);
}

// ─── Kit · viewer v2 ── the same in every recipe · postext.dev/cookbook
/** The pages as spreads on a dark desk, page 1 alone, then verso | recto,
 *  each painted when it scrolls near. Other options go to renderPageToCanvas. */
function showPages(docs, { title, width = 460, ...render } = {}) {
  const root = viewer(title);
  const pages = [docs].flat().flatMap((doc) =>
    doc.pages.map((page) => ({ doc, page, n: (doc.pageIndexOffset ?? 0) + page.index })));
  const spreads = [];
  let verso = null;
  for (const p of pages) {
    if (p.n % 2 === 1) { if (verso) spreads.push([verso, null]); verso = p; }
    else { spreads.push([verso, p]); verso = null; }
  }
  if (verso) spreads.push([verso, null]);
  const density = Math.min(window.devicePixelRatio || 1, 2);
  showPages.painter?.disconnect();
  const painter = new IntersectionObserver((entries) => {
    for (const { isIntersecting, target } of entries) {
      if (!isIntersecting) continue;
      painter.unobserve(target);
      const { doc, page } = target.postext;
      renderPageToCanvas(page, doc, target, { ...render, scale: (width * density) / page.width });
    }
  }, { rootMargin: '800px' });
  showPages.painter = painter;
  root.replaceChildren(...spreads.map((pair) => {
    const spread = document.createElement('div');
    spread.className = 'pt-spread';
    for (const p of pair) {
      const figure = document.createElement('figure');
      if (p) {
        const label = p.page.pageLabel || String(p.n + 1);
        const canvas = document.createElement('canvas');
        canvas.postext = p;
        canvas.style.aspectRatio = `${p.page.width} / ${p.page.height}`;
        canvas.setAttribute('role', 'img');
        canvas.setAttribute('aria-label', `Page ${label}`);
        const folio = document.createElement('figcaption');
        folio.textContent = label;
        figure.append(canvas, folio);
        painter.observe(canvas);
      } else figure.className = 'pt-blank';
      spread.append(figure);
    }
    return spread;
  }));
  kitStatus(`${pages.length} ${pages.length === 1 ? 'page' : 'pages'}`);
  document.documentElement.dataset.postext = 'ready';
  return pages.length;
}

/** The desk, the bar and the error reporting, created once. */
function viewer(title) {
  if (!document.getElementById('pt-kit')) {
    document.head.insertAdjacentHTML('beforeend', `<style id="pt-kit">
      :root { color-scheme: dark; }
      body { margin: 0; background: #0e1014; color: #b9bcc4; font: 13px/1.45 system-ui, sans-serif; }
      #pt-bar { position: sticky; top: 0; z-index: 1; display: flex; flex-wrap: wrap; align-items: center;
        gap: 6px 16px; padding: 10px 16px; background: rgb(14 16 20 / .92); backdrop-filter: blur(6px);
        border-bottom: 1px solid #23262d; }
      #pt-bar strong { color: #f4f1ea; font-weight: 600; }
      #pt-actions { display: flex; gap: 12px; margin-left: auto; }
      #pt-actions a, #pt-actions button { color: #d8a21a; font: inherit; background: none; border: 0; padding: 0; cursor: pointer; }
      #pages { display: grid; justify-items: center; gap: 48px; padding: 32px 16px 72px; }
      .pt-spread { display: flex; }
      .pt-spread figure { margin: 0; width: min(460px, 44vw); }
      .pt-spread canvas { display: block; width: 100%; background: #fff;
        box-shadow: 0 1px 2px rgb(0 0 0 / .5), 0 22px 44px -16px rgb(0 0 0 / .8); }
      .pt-spread figure:first-child canvas { box-shadow: inset -14px 0 14px -14px rgb(0 0 0 / .18), 0 1px 2px rgb(0 0 0 / .5), 0 22px 44px -16px rgb(0 0 0 / .8); }
      .pt-spread figcaption { margin-top: 10px; text-align: center; font: 600 10px/1 system-ui, sans-serif;
        letter-spacing: .18em; text-transform: uppercase; color: #6c7079; }
      .pt-blank { visibility: hidden; }
      @media (max-width: 760px) {
        .pt-spread { flex-direction: column; gap: 32px; }
        .pt-spread figure { width: min(460px, 92vw); }
        .pt-blank { display: none; }
      }
    </style>`);
    document.body.insertAdjacentHTML('afterbegin',
      '<header id="pt-bar"><strong id="pt-title"></strong><span id="pt-status" role="status"></span><span id="pt-actions"></span></header>');
    document.getElementById('pt-title').textContent = document.title || 'Postext';
    addEventListener('error', (event) => kitFail(event.error ?? event.message));
    addEventListener('unhandledrejection', (event) => kitFail(event.reason));
  }
  if (title) document.getElementById('pt-title').textContent = title;
  return document.getElementById('pages')
    ?? document.body.appendChild(Object.assign(document.createElement('main'), { id: 'pages' }));
}

function kitStatus(text) {
  viewer();
  document.getElementById('pt-status').textContent = text;
}

function kitFail(error) {
  document.documentElement.dataset.postext = 'error';
  kitStatus(`Error: ${error?.message ?? error}`);
}

// ─── Kit · pdf v3 ── the same in every recipe that exports a PDF
/** The Fontsource files the screen used, as TrueType, with their range and
 *  place: the nearest weight the family ships, upright with no italic;
 *  latin, then what its letters need (kitSubsetsFor). */
async function fontsourceProvider(family, weight, style, request) {
  const id = fontsourceId(family);
  const meta = await fontsourceMeta(family);
  const weights = meta?.weights?.length ? meta.weights : [400, 700];
  const w = weights.reduce((a, b) => (Math.abs(b - weight) < Math.abs(a - weight) ? b : a));
  const s = style === 'italic' && meta && !meta.styles.includes('italic') ? 'normal' : style;
  const text = String.fromCodePoint(...(request?.codePoints ?? []));
  const files = await Promise.all(['latin', ...kitSubsetsFor(text, meta)].map(async (subset, order) => {
    const res = await fetch(`${await fontsourceAt(id)}files/${id}-${subset}-${w}-${s}.woff2`);
    if (!res.ok) throw new Error(`Fontsource has no ${family} ${w} ${s} ${subset}`);
    return Object.assign(await decompressWoff2(new Uint8Array(await res.arrayBuffer())),
      { unicodeRange: kitRanges()[subset], order });
  }));
  return files.length === 1 ? files[0] : files;
}

/** A "Build the PDF" button; then "Open the PDF" (a new tab: CodePen's frame
 *  shows no PDFs) and a download link. */
function offerPdf(makePdf, filename) {
  viewer();
  const button = Object.assign(document.createElement('button'), { type: 'button', textContent: 'Build the PDF' });
  button.dataset.postextPdf = filename;
  button.addEventListener('click', async () => {
    button.disabled = true;
    button.textContent = 'Building the PDF…';
    try {
      const bytes = await makePdf();
      const url = URL.createObjectURL(new Blob([bytes], { type: 'application/pdf' }));
      const size = `${Math.max(1, Math.round(bytes.length / 1024))} KB`;
      button.replaceWith(
        Object.assign(document.createElement('a'), { href: url, target: '_blank', rel: 'noopener', textContent: 'Open the PDF ↗' }),
        Object.assign(document.createElement('a'), { href: url, download: filename, textContent: `Download ${filename} · ${size}` }));
    } catch (error) {
      button.disabled = false;
      button.textContent = 'Build the PDF';
      kitFail(error);
    }
  });
  document.getElementById('pt-actions').append(button);
}

// ─── Kit · images v3 ── recipes with pictures · postext.dev/cookbook
/** Registers a photo or PNG for the canvas and keeps its bytes for the PDF.
 *  fetch → ImageBitmap never taints the canvas (a plain cross-origin <img> would). */
async function loadImage(fileId, url) {
  const res = await fetch(url);
  if (!res.ok) throw new Error(`Image not found (${res.status}): ${url}`);
  const bytes = new Uint8Array(await res.arrayBuffer());
  registerResourceImage(fileId, await createImageBitmap(new Blob([bytes])));
  (loadImage.bytes ??= new Map()).set(fileId, bytes);
}

/** Registers SVG markup as a vector image, its fonts inside (inlineSvgFonts, ≥ 1.25).
 *  `options` go to registerResourceImage: { singleInk: true } tints it on the canvas. */
async function loadSvg(fileId, svg, options) {
  if (typeof inlineSvgFonts === 'function') svg = await inlineSvgFonts(svg, svgFaces);
  const img = new Image();
  img.src = `data:image/svg+xml;charset=utf-8,${encodeURIComponent(svg)}`;
  await img.decode();
  registerResourceImage(fileId, img, options);
  (loadImage.bytes ??= new Map()).set(fileId, new TextEncoder().encode(svg));
}

/** A face's Fontsource files: nearest weight; latin + what the SVG sets. */
async function svgFaces(family, weight, style, { codePoints = [] } = {}) {
  const [id, m] = [fontsourceId(family), await fontsourceMeta(family)];
  const w = m?.weights.reduce((a, b) => (Math.abs(b - weight) < Math.abs(a - weight) ? b : a)) ?? weight;
  const s = m?.styles.includes(style) === false ? 'normal' : style;
  return Promise.all(['latin', ...kitSubsetsFor(String.fromCodePoint(...codePoints), m)].map(async (x) => new Uint8Array(
    await (await fetch(`${await fontsourceAt(id)}files/${id}-${x}-${w}-${s}.woff2`)).arrayBuffer())));
}

/** renderToPdf({ resourceBytes: imageBytes }) */
function imageBytes(fileId) { return loadImage.bytes?.get(fileId); }

/** renderToHtml({ resourceImageUrl: imageUrl }) */
function imageUrl(fileId) {
  const bytes = imageBytes(fileId);
  if (!bytes) return undefined;
  imageUrl.urls ??= new Map();
  if (!imageUrl.urls.has(fileId)) {
    const type = /\.svg$/i.test(fileId) ? 'image/svg+xml' : /\.png$/i.test(fileId) ? 'image/png' : 'image/jpeg';
    imageUrl.urls.set(fileId, URL.createObjectURL(new Blob([bytes], { type })));
  }
  return imageUrl.urls.get(fileId);
}

// ─── /Kit ───────────────────────────────────────────────────────────────────────