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unifying_electro_magnetism

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unifying_electro_magnetism
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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Why does the field point this way? β€” Vector field analyzer</title>
<style>
  :root {
    --bg: #0a0a12;
    --panel: #12121e;
    --border: #2a2a3a;
    --text: #e8e8f0;
    --dim: #8888a0;
    --accent: #6cf;
    --good: #4caf50;
    --bad: #ff5566;
    --warn: #ffb454;
  }
  * { box-sizing: border-box; }
  body {
    margin: 0;
    background: var(--bg);
    color: var(--text);
    font-family: ui-sans-serif, system-ui, -apple-system, sans-serif;
    font-size: 14px;
    line-height: 1.55;
  }
  header { padding: 16px 24px; border-bottom: 1px solid var(--border); }
  header h1 { margin: 0 0 4px; font-size: 18px; font-weight: 600; }
  header p { margin: 0; color: var(--dim); font-size: 13px; max-width: 900px; }
  main {
    display: grid;
    grid-template-columns: minmax(0, 1fr) minmax(0, 1fr);
    gap: 16px;
    padding: 16px;
    max-width: 1440px;
    margin: 0 auto;
  }
  @media (max-width: 940px) { main { grid-template-columns: 1fr; } }
  .panel {
    background: var(--panel);
    border: 1px solid var(--border);
    border-radius: 8px;
    overflow: hidden;
  }
  .panel-header {
    padding: 9px 14px;
    border-bottom: 1px solid var(--border);
    display: flex;
    align-items: center;
    justify-content: space-between;
    gap: 10px;
    flex-wrap: wrap;
  }
  .panel-header h2 {
    margin: 0; font-size: 12px; font-weight: 600;
    text-transform: uppercase; letter-spacing: 0.06em; color: var(--dim);
  }
  .controls { display: flex; gap: 8px; align-items: center; flex-wrap: wrap; }
  select, input[type="text"] {
    background: #0e0e18;
    color: var(--text);
    border: 1px solid var(--border);
    border-radius: 4px;
    padding: 4px 8px;
    font-size: 12px;
    font-family: ui-monospace, monospace;
  }
  select:focus, input:focus { outline: 1px solid var(--accent); }
  .canvas-wrap { position: relative; background: #06060c; line-height: 0; }
  canvas { display: block; width: 100%; height: auto; touch-action: none; }
  #field { cursor: crosshair; }

  .overlay {
    position: absolute; inset: 0;
    display: flex; align-items: center; justify-content: center;
    background: rgba(0,0,0,0.55); color: var(--dim); font-size: 13px;
    pointer-events: none; opacity: 0; transition: opacity 0.2s;
  }
  .overlay.show { opacity: 1; }

  .readout {
    padding: 12px 14px;
    font-size: 12.5px;
    border-top: 1px solid var(--border);
    display: grid;
    grid-template-columns: repeat(2, minmax(0, 1fr));
    gap: 6px 14px;
    font-family: ui-monospace, monospace;
  }
  .readout .k { color: var(--dim); }
  .readout .v { color: var(--text); }

  #explanation { padding: 14px 16px; font-size: 13px; max-height: 520px; overflow-y: auto; }
  #explanation h3 { margin: 0 0 10px; font-size: 13px; color: var(--accent); font-weight: 600; }
  #explanation p { margin: 8px 0; }
  .verdict {
    padding: 8px 12px; border-radius: 4px; margin: 10px 0;
    font-weight: 600; font-size: 13px;
  }
  .verdict.bad { background: rgba(255,85,102,0.12); color: #ff8a99; border-left: 3px solid #ff5566; }
  .verdict.good { background: rgba(76,175,80,0.12); color: #8fdc92; border-left: 3px solid #4caf50; }
  .verdict.warn { background: rgba(255,180,84,0.12); color: #ffcf8a; border-left: 3px solid #ffb454; }
  #explanation table {
    width: 100%; border-collapse: collapse;
    font-family: ui-monospace, monospace; font-size: 11.5px; margin: 8px 0;
  }
  #explanation th, #explanation td {
    text-align: left; padding: 3px 6px; border-bottom: 1px solid var(--border);
  }
  #explanation th { color: var(--dim); font-weight: 500; }
  .jac {
    font-family: ui-monospace, monospace; font-size: 12.5px;
    background: #0e0e18; border: 1px solid var(--border); border-radius: 4px;
    padding: 8px 10px; margin: 8px 0; display: inline-block; line-height: 1.9;
  }
  .note {
    color: var(--dim); font-size: 12px; border-top: 1px solid var(--border);
    padding-top: 10px; margin-top: 14px; line-height: 1.6;
  }
  .hint {
    padding: 9px 14px; font-size: 11.5px; color: var(--dim);
    border-top: 1px solid var(--border); background: rgba(255,255,255,0.02);
    line-height: 1.5;
  }
  .hint b { color: var(--text); }
  .legend { display: flex; gap: 12px; flex-wrap: wrap; font-size: 11px; color: var(--dim); }
  .legend span { display: inline-flex; align-items: center; gap: 5px; }
  .swatch { width: 10px; height: 10px; border-radius: 2px; display: inline-block; }
  kbd {
    background: #1e1e2e; border: 1px solid var(--border); border-radius: 3px;
    padding: 1px 5px; font-size: 10.5px; font-family: ui-monospace, monospace;
  }
  .crosshair-legend {
    position: absolute; bottom: 8px; left: 8px; font-size: 10.5px;
    color: var(--dim); background: rgba(0,0,0,0.5); padding: 3px 7px;
    border-radius: 4px; pointer-events: none; font-family: ui-monospace, monospace;
  }
</style>
</head>
<body>
<header>
  <h1>Why does the field point this way?</h1>
  <p>Pick a vector field. Hover any point to see the local vector, the numerically estimated Jacobian, and a plain-language diagnosis: is this point a source, a sink, a rotation, a saddle? The color shows the direction of the field; the arrows show magnitude and heading.</p>
</header>
<main>
  <div class="panel">
    <div class="panel-header">
      <h2>Vector field</h2>
      <div class="controls">
        <select id="preset"></select>
        <input type="text" id="custom" size="26" placeholder="f(x,y)=..., g(x,y)=..." style="display:none">
      </div>
    </div>
    <div class="canvas-wrap">
      <canvas id="field" width="640" height="640"></canvas>
      <div class="overlay" id="renderingOverlay">Rendering…</div>
      <div class="crosshair-legend" id="coordReadout">hover the field</div>
    </div>
    <div class="hint">
      <b>Arrows</b> = direction + relative magnitude. <b>Color</b> = heading (hue) &amp; magnitude (brightness).
      Scroll to zoom, drag to pan, double-click to reset. <kbd>Shift</kbd>+scroll changes arrow density.
    </div>
    <div class="readout" id="readout">
      <div class="k">f(x,y)</div><div class="v" id="rv-f">β€”</div>
      <div class="k">g(x,y)</div><div class="v" id="rv-g">β€”</div>
      <div class="k">div F = βˆ‚f/βˆ‚x + βˆ‚g/βˆ‚y</div><div class="v" id="rv-div">β€”</div>
      <div class="k">curl F = βˆ‚g/βˆ‚x βˆ’ βˆ‚f/βˆ‚y</div><div class="v" id="rv-curl">β€”</div>
      <div class="k">speed |F|</div><div class="v" id="rv-mag">β€”</div>
      <div class="k">det J = λ₁λ₂</div><div class="v" id="rv-det">β€”</div>
      <div class="k">trace J = λ₁+Ξ»β‚‚</div><div class="v" id="rv-trace">β€”</div>
      <div class="k">eigenvalues</div><div class="v" id="rv-eig">β€”</div>
    </div>
  </div>
  <div class="panel">
    <div class="panel-header">
      <h2>Local analysis</h2>
      <div class="controls legend">
        <span><i class="swatch" style="background:#4caf50"></i>outward</span>
        <span><i class="swatch" style="background:#ff5566"></i>inward</span>
        <span><i class="swatch" style="background:#6cf"></i>field</span>
      </div>
    </div>
    <div class="canvas-wrap">
      <canvas id="zoom" width="640" height="480"></canvas>
    </div>
    <div id="explanation">
      <p style="color: var(--dim)">Hover over the field to analyze a point.</p>
    </div>
  </div>
</main>

<script>
"use strict";

// ---------------------------------------------------------------- config
const W = 640, H = 640;
const ZW = 640, ZH = 480;
const GLYPH_GRID_BASE = 26;     // arrows across when density = 1
const EPS = 1e-5;               // finite-difference step

const fieldCanvas = document.getElementById("field");
const fieldCtx = fieldCanvas.getContext("2d");
const zoomCanvas = document.getElementById("zoom");
const zoomCtx = zoomCanvas.getContext("2d");
const explanationEl = document.getElementById("explanation");
const overlayEl = document.getElementById("renderingOverlay");
const coordReadout = document.getElementById("coordReadout");
const presetSel = document.getElementById("preset");
const customInput = document.getElementById("custom");

// ---------------------------------------------------------------- view
let view = { xmin: -3, xmax: 3, ymin: -3, ymax: 3 };
const HOME_VIEW = { ...view };
let density = 1.0;
let hoverP = null;
let renderToken = 0;
let dragging = false;
let dragStart = null;

// ---------------------------------------------------------------- presets
// Each field is a pair of functions f(x,y), g(x,y).
const PRESETS = {
  "source β€” radial out":        { f: (x, y) => x,                  g: (x, y) => y },
  "sink β€” radial in":           { f: (x, y) => -x,                 g: (x, y) => -y },
  "rotation β€” CCW":             { f: (x, y) => -y,                 g: (x, y) => x },
  "rotation + spiral out":      { f: (x, y) => -y + 0.3 * x,       g: (x, y) => x + 0.3 * y },
  "saddle β€” hyperbolic":        { f: (x, y) => x,                  g: (x, y) => -y },
  "shear β€” parallel slide":     { f: (x, y) => y,                  g: (x, y) => 0 },
  "uniform β€” constant":         { f: (x, y) => 1,                  g: (x, y) => 0.5 },
  "dipole β€” two charges":       { f: (x, y) => cx(x, y),           g: (x, y) => cy(x, y) },
  "quadrupole β€” 4 poles":       { f: (x, y) => x * x - y * y,      g: (x, y) => 2 * x * y },
  "Duffing β€” nonlinear":        { f: (x, y) => y,                  g: (x, y) => x - x * x * x - 0.3 * y },
  "pendulum phase flow":        { f: (x, y) => y,                  g: (x, y) => -Math.sin(x) - 0.2 * y },
  "van der Pol":                { f: (x, y) => y,                  g: (x, y) => (1 - x * x) * y - x },
  "custom (type below)":        { f: (x, y) => x,                  g: (x, y) => y },
};

// dipole helper: sum of a source at (-1.5,0) and a sink at (1.5,0)
function cx(x, y) {
  const dx1 = x + 1.5, dy1 = y, r1 = dx1 * dx1 + dy1 * dy1 + 0.25;
  const dx2 = x - 1.5, dy2 = y, r2 = dx2 * dx2 + dy2 * dy2 + 0.25;
  return dx1 / r1 - dx2 / r2;
}
function cy(x, y) {
  const dx1 = x + 1.5, dy1 = y, r1 = dx1 * dx1 + dy1 * dy1 + 0.25;
  const dx2 = x - 1.5, dy2 = y, r2 = dx2 * dx2 + dy2 * dy2 + 0.25;
  return dy1 / r1 - dy2 / r2;
}

let F = PRESETS["source β€” radial out"].f;
let G = PRESETS["source β€” radial out"].g;
let currentName = "source β€” radial out";

// ---------------------------------------------------------------- palette
function lerp(a, b, t) { return a + (b - a) * t; }
function headingColor(angle, mag, magMax) {
  // angle in radians -> hue 0..360
  let a = angle / (2 * Math.PI) * 360;
  if (a < 0) a += 360;
  const t = Math.max(0, Math.min(1, mag / (magMax || 1)));
  const light = 25 + 45 * t;
  const sat = 70 + 20 * t;
  return `hsl(${a}, ${sat}%, ${light}%)`;
}

// ---------------------------------------------------------------- field eval
function fieldAt(x, y) {
  let fx = F(x, y), fy = G(x, y);
  if (!isFinite(fx)) fx = 0;
  if (!isFinite(fy)) fy = 0;
  return { f: fx, g: fy };
}

// numerical Jacobian [[df/dx, df/dy],[dg/dx, dg/dy]]
function jacobianAt(x, y) {
  const h = EPS;
  const fp = fieldAt(x + h, y), fm = fieldAt(x - h, y);
  const gp = fieldAt(x, y + h), gm = fieldAt(x, y - h);
  return {
    a: (fp.f - fm.f) / (2 * h),  // βˆ‚f/βˆ‚x
    b: (gp.f - gm.f) / (2 * h),  // βˆ‚f/βˆ‚y
    c: (fp.g - fm.g) / (2 * h),  // βˆ‚g/βˆ‚x
    d: (gp.g - gm.g) / (2 * h),  // βˆ‚g/βˆ‚y
  };
}

// ---------------------------------------------------------------- arrow glyphs
function drawArrow(ctx, x0, y0, x1, y1, color, width, headScale) {
  const dx = x1 - x0, dy = y1 - y0;
  const len = Math.hypot(dx, dy);
  if (len < 0.5) return;
  const ux = dx / len, uy = dy / len;
  const head = Math.min(Math.max(3, len * 0.35), 9 * headScale);
  ctx.strokeStyle = color;
  ctx.fillStyle = color;
  ctx.lineWidth = width;
  ctx.beginPath();
  ctx.moveTo(x0, y0);
  ctx.lineTo(x1 - ux * head * 0.6, y1 - uy * head * 0.6);
  ctx.stroke();
  ctx.beginPath();
  ctx.moveTo(x1, y1);
  ctx.lineTo(x1 - ux * head - uy * head * 0.5, y1 - uy * head + ux * head * 0.5);
  ctx.lineTo(x1 - ux * head + uy * head * 0.5, y1 - uy * head - ux * head * 0.5);
  ctx.closePath();
  ctx.fill();
}

// ---------------------------------------------------------------- field render
function toScreen(x, y) {
  return {
    sx: (x - view.xmin) / (view.xmax - view.xmin) * W,
    sy: (view.ymax - y) / (view.ymax - view.ymin) * H,
  };
}
function toWorld(sx, sy) {
  return {
    x: view.xmin + (sx / W) * (view.xmax - view.xmin),
    y: view.ymax - (sy / H) * (view.ymax - view.ymin),
  };
}

function renderField() {
  const token = ++renderToken;
  overlayEl.classList.add("show");

  const ctx = fieldCtx;
  // background gradient by speed sampled coarsely
  const bg = ctx.createImageData(W, H);
  const data = bg.data;
  const stepPx = 4; // coarse fill

  // determine a magnitude scale for the background by sampling
  const sampleN = 30;
  let magMax = 0;
  for (let i = 0; i <= sampleN; i++) {
    for (let j = 0; j <= sampleN; j++) {
      const wx = view.xmin + (i / sampleN) * (view.xmax - view.xmin);
      const wy = view.ymin + (j / sampleN) * (view.ymax - view.ymin);
      const v = fieldAt(wx, wy);
      const m = Math.hypot(v.f, v.g);
      if (isFinite(m) && m > magMax) magMax = m;
    }
  }
  if (magMax === 0) magMax = 1;

  let yPx = 0;
  function stepBg() {
    if (token !== renderToken) return;
    const t0 = performance.now();
    while (yPx < H && performance.now() - t0 < 12) {
      for (let xPx = 0; xPx < W; xPx += stepPx) {
        const w = toWorld(xPx, yPx);
        const v = fieldAt(w.x, w.y);
        const mag = Math.hypot(v.f, v.g);
        const t = Math.min(1, mag / magMax);
        // shade blue-ish dark -> bright by magnitude
        const r = Math.round(6 + 18 * t);
        const g = Math.round(8 + 26 * t);
        const b = Math.round(16 + 60 * t);
        for (let k = 0; k < stepPx && xPx + k < W; k++) {
          const idx = (yPx * W + xPx + k) * 4;
          data[idx] = r; data[idx + 1] = g; data[idx + 2] = b; data[idx + 3] = 255;
        }
      }
      // copy the same row for the block
      for (let dy = 1; dy < stepPx && yPx + dy < H; dy++) {
        const src = (yPx * W) * 4;
        data.copyWithin(((yPx + dy) * W) * 4, src, src + W * 4);
      }
      yPx += stepPx;
    }
    ctx.putImageData(bg, 0, 0);
    if (yPx < H) requestAnimationFrame(stepBg);
    else {
      drawGlyphs(magMax);
      overlayEl.classList.remove("show");
      drawHoverMarker();
    }
  }
  requestAnimationFrame(stepBg);
}

function drawGlyphs(magMax) {
  const ctx = fieldCtx;
  const n = Math.max(6, Math.round(GLYPH_GRID_BASE * density));
  const cell = W / n;
  const spacing = cell;
  const maxLen = spacing * 0.95;

  // sample field to find max in view for scaling
  let localMax = 0, minPos = Infinity;
  const samples = [];
  for (let i = 0; i < n; i++) {
    for (let j = 0; j < n; j++) {
      const sx = (i + 0.5) * cell;
      const sy = (j + 0.5) * cell;
      const w = toWorld(sx, sy);
      const v = fieldAt(w.x, w.y);
      const m = Math.hypot(v.f, v.g);
      samples.push({ sx, sy, v, m, w });
      if (m > localMax) localMax = m;
    }
  }
  if (localMax === 0) localMax = 1;

  // scale factor so the largest glyph is ~maxLen
  const scale = maxLen / localMax;

  // world-per-pixel to convert vector components into screen direction
  const wppX = (view.xmax - view.xmin) / W;
  const wppY = (view.ymax - view.ymin) / H;

  for (const s of samples) {
    if (s.m < 1e-9) continue;
    // vector in screen space: vx_screen = f / wppX, vy_screen = -g / wppY
    let vxs = s.v.f / wppX;
    let vys = -s.v.g / wppY;
    const slen = Math.hypot(vxs, vys);
    if (slen < 1e-9) continue;
    const drawLen = Math.min(slen * scale * wppX * (1 / wppX) * wppX, maxLen);
    // normalize direction, set length
    const ux = vxs / slen, uy = vys / slen;
    const L = Math.min(maxLen, (s.m / localMax) * maxLen);
    const x0 = s.sx - ux * L / 2;
    const y0 = s.sy - uy * L / 2;
    const x1 = s.sx + ux * L / 2;
    const y1 = s.sy + uy * L / 2;
    const angle = Math.atan2(s.v.g, s.v.f);
    const col = headingColor(angle, s.m, localMax);
    drawArrow(ctx, x0, y0, x1, y1, col, 1.4 + 0.8 * (s.m / localMax), 1);
  }
}

// ---------------------------------------------------------------- hover marker
function drawHoverMarker() {
  if (!hoverP) return;
  const ctx = fieldCtx;
  const { sx, sy } = toScreen(hoverP.x, hoverP.y);
  ctx.strokeStyle = "#ffffff";
  ctx.lineWidth = 1.5;
  ctx.beginPath();
  ctx.arc(sx, sy, 9, 0, 2 * Math.PI);
  ctx.stroke();
  ctx.beginPath();
  ctx.moveTo(sx - 14, sy); ctx.lineTo(sx - 5, sy);
  ctx.moveTo(sx + 5, sy);  ctx.lineTo(sx + 14, sy);
  ctx.moveTo(sx, sy - 14); ctx.lineTo(sx, sy - 5);
  ctx.moveTo(sx, sy + 5);  ctx.lineTo(sx, sy + 14);
  ctx.stroke();
  ctx.fillStyle = "#fff";
  ctx.font = "10px ui-monospace, monospace";
  ctx.fillText(`(${hoverP.x.toFixed(2)}, ${hoverP.y.toFixed(2)})`, sx + 16, sy - 10);
}

// ---------------------------------------------------------------- zoomed local view
function drawLocalView(x, y) {
  const ctx = zoomCtx;
  ctx.clearRect(0, 0, ZW, ZH);
  ctx.fillStyle = "#06060c";
  ctx.fillRect(0, 0, ZW, ZH);

  const span = 1.0;                        // world half-width of the zoom window
  const pxPerW = ZW / (2 * span);          // px per world unit (x)
  const pyPerW = ZH / (2 * span);          // px per world unit (y)

  const toZ = (wx, wy) => ({
    sx: ZW / 2 + (wx - x) * pxPerW,
    sy: ZH / 2 - (wy - y) * pyPerW,
  });

  // axes
  ctx.strokeStyle = "#1e1e2e";
  ctx.lineWidth = 1;
  ctx.beginPath();
  const ax = toZ(0, 0);
  ctx.moveTo(0, ax.sy); ctx.lineTo(ZW, ax.sy);
  ctx.moveTo(ax.sx, 0); ctx.lineTo(ax.sx, ZH);
  ctx.stroke();
  ctx.fillStyle = "#3a3a4a";
  ctx.font = "10px ui-monospace, monospace";
  ctx.fillText("x", ZW - 16, ax.sy - 5);
  ctx.fillText("y", ax.sx + 5, 12);

  // vector field glyphs in the local window
  const gn = 15;
  let mMax = 0;
  const cells = [];
  for (let i = 0; i < gn; i++) {
    for (let j = 0; j < gn; j++) {
      const wx = x - span + ((i + 0.5) / gn) * 2 * span;
      const wy = y - span + ((j + 0.5) / gn) * 2 * span;
      const v = fieldAt(wx, wy);
      const m = Math.hypot(v.f, v.g);
      cells.push({ wx, wy, v, m });
      if (m > mMax) mMax = m;
    }
  }
  if (mMax === 0) mMax = 1;
  const cellSize = Math.min(ZW / gn, ZH / gn);
  const maxLen = cellSize * 0.9;

  for (const c0 of cells) {
    const p = toZ(c0.wx, c0.wy);
    if (c0.m < 1e-9) {
      ctx.fillStyle = "#2a2a3a";
      ctx.beginPath(); ctx.arc(p.sx, p.sy, 1.2, 0, 2 * Math.PI); ctx.fill();
      continue;
    }
    const vxs = c0.v.f * (pxPerW / pyPerW) * 0; // unused
    // direction in screen space
    const ddx = c0.v.f / 1;   // world x-comp
    const ddy = c0.v.g;
    let sx = ddx, sy = -ddy;
    const sl = Math.hypot(sx, sy);
    const ux = sx / sl, uy = sy / sl;
    const L = Math.min(maxLen, (c0.m / mMax) * maxLen);
    const x0 = p.sx - ux * L / 2, y0 = p.sy - uy * L / 2;
    const x1 = p.sx + ux * L / 2, y1 = p.sy + uy * L / 2;
    const angle = Math.atan2(c0.v.g, c0.v.f);
    drawArrow(ctx, x0, y0, x1, y1, headingColor(angle, c0.m, mMax), 1.2, 0.9);
  }

  // the point itself
  const p = toZ(x, y);
  ctx.fillStyle = "#fff";
  ctx.beginPath(); ctx.arc(p.sx, p.sy, 3.5, 0, 2 * Math.PI); ctx.fill();

  // local vector at the point, drawn bold
  const v = fieldAt(x, y);
  const mag = Math.hypot(v.f, v.g);
  const J = jacobianAt(x, y);
  const div = J.a + J.d;
  const curl = J.c - J.b;

  // draw the local field vector from the point
  if (mag > 1e-12) {
    const ux = v.f / mag, uy = -v.g / mag;
    const L = Math.min(110, 40 + 30 * Math.log10(1 + mag));
    drawArrow(ctx, p.sx, p.sy, p.sx + ux * L, p.sy + uy * L, "#6cf", 2.5, 1.4);
    // trace streamline a little forward and backward
    traceStreamline(ctx, x, y, pxPerW, pyPerW, toZ, 1);
    traceStreamline(ctx, x, y, pxPerW, pyPerW, toZ, -1);
  }

  // divergence halo: green outward, red inward
  const haloC = div > 0.05 ? "rgba(76,175,80,0.85)" : div < -0.05 ? "rgba(255,85,102,0.85)" : "rgba(255,180,84,0.85)";
  ctx.strokeStyle = haloC;
  ctx.lineWidth = 2;
  ctx.beginPath();
  ctx.arc(p.sx, p.sy, 13, 0, 2 * Math.PI);
  ctx.stroke();

  // curl indicator: arc showing rotational tendency
  if (Math.abs(curl) > 0.05) {
    ctx.strokeStyle = "rgba(160,120,255,0.8)";
    ctx.lineWidth = 2;
    ctx.beginPath();
    const start = curl > 0 ? -0.5 : Math.PI + 0.5;
    const end = curl > 0 ? Math.PI + 0.5 : -0.5;
    ctx.arc(p.sx, p.sy, 20, start, end, false);
    ctx.stroke();
    // arrowhead
    const ang = curl > 0 ? Math.PI + 0.5 : -0.5;
    const hx = p.sx + 20 * Math.cos(ang), hy = p.sy + 20 * Math.sin(ang);
    drawArrow(ctx, hx, hy, hx + (curl > 0 ? -1 : 1) * 6, hy + 1, "rgba(160,120,255,0.9)", 2, 1);
  }

  // legend text
  ctx.fillStyle = "#8888a0";
  ctx.font = "10.5px ui-monospace, monospace";
  ctx.fillText("local window: Β±1.0 world units", 10, ZH - 10);
  ctx.fillText(`div=${div.toFixed(3)}  curl=${curl.toFixed(3)}`, 10, ZH - 26);
}

function traceStreamline(ctx, x0, y0, pxPerW, pyPerW, toZ, dir) {
  const dt = 0.02 * dir;
  let x = x0, y = y0;
  ctx.strokeStyle = dir > 0 ? "rgba(108,204,255,0.55)" : "rgba(108,204,255,0.25)";
  ctx.lineWidth = 1.6;
  ctx.beginPath();
  let first = true;
  for (let i = 0; i < 120; i++) {
    const v = fieldAt(x, y);
    const m = Math.hypot(v.f, v.g);
    if (m < 1e-9) break;
    // RK2-ish step, step size scaled so movement isn't huge
    const sp = 0.03 / Math.max(0.2, Math.min(3, m));
    const dx = (v.f / m) * sp * dir;
    const dy = (v.g / m) * sp * dir;
    const nx = x + dx, ny = y + dy;
    const p = toZ(nx, ny);
    if (first) { ctx.moveTo(p.sx, p.sy); first = false; }
    else ctx.lineTo(p.sx, p.sy);
    x = nx; y = ny;
    if (Math.abs(x - x0) > 1.6 || Math.abs(y - y0) > 1.6) break;
  }
  ctx.stroke();
}

// ---------------------------------------------------------------- explanation
function fmt(x, d = 4) {
  if (Math.abs(x) < 1e-12) return "0";
  if (Math.abs(x) >= 1e4 || Math.abs(x) < 1e-3) return x.toExponential(2);
  return x.toFixed(d);
}

function classify(x, y, f, g, J) {
  const div = J.a + J.d;
  const curl = J.c - J.b;
  const det = J.a * J.d - J.b * J.c;
  const tr = J.a + J.d;
  const disc = tr * tr - 4 * det;

  // eigenvalues
  let l1, l2, eigText;
  if (Math.abs(disc) < 1e-9) {
    l1 = l2 = tr / 2;
    eigText = `λ₁ = Ξ»β‚‚ = ${fmt(l1, 4)}`;
  } else if (disc > 0) {
    l1 = (tr + Math.sqrt(disc)) / 2;
    l2 = (tr - Math.sqrt(disc)) / 2;
    eigText = `λ₁ = ${fmt(l1, 4)}, Ξ»β‚‚ = ${fmt(l2, 4)}`;
  } else {
    const re = tr / 2, im = Math.sqrt(-disc) / 2;
    l1 = re; l2 = re;
    eigText = `Ξ» = ${fmt(re, 4)} Β± ${fmt(im, 4)}i`;
  }

  const mag = Math.hypot(f, g);
  let title, klass, verdict, why;

  const nearZero = mag < 1e-3;

  if (nearZero && Math.hypot(div, curl) > 1e-3) {
    klass = "fixed point";
    title = "Fixed point (stationary point)";
    if (det < 0) title = "Saddle point";
    else if (l1.imag !== undefined) {
      title = div > 0 ? "Spiral source (unstable focus)" : div < 0 ? "Spiral sink (stable focus)" : "Center (pure rotation)";
    } else {
      if (Math.abs(disc) < 1e-9) title = tr < 0 ? "Stable node (degenerate)" : "Unstable node (degenerate)";
      else if (div > 0) title = "Source node (unstable node)";
      else if (div < 0) title = "Sink node (stable node)";
      else title = "Saddle-like (det<0 or trace 0)";
    }
  } else if (nearZero) {
    klass = "flat";
    title = "Zero vector here";
  } else {
    klass = "regular point";
    title = "Regular point";
  }

  // build verdict + reasoning tuned to actual numbers
  const reasons = [];

  if (Math.abs(div) > 0.05) {
    if (div > 0) {
      verdict = { type: "bad", text: `Source-like at this point β€” div F = ${fmt(div, 3)} > 0 (field expands)` };
      reasons.push(`Positive divergence means the field pushes <i>outward</i> on average here: a tiny disk of test particles would grow in area. Since div = βˆ‚f/βˆ‚x + βˆ‚g/βˆ‚y = ${fmt(J.a, 3)} + ${fmt(J.d, 3)} = ${fmt(div, 3)}, both partials are mostly positive, so the flow stretches.`);
    } else {
      verdict = { type: "good", text: `Sink-like at this point β€” div F = ${fmt(div, 3)} < 0 (field converges)` };
      reasons.push(`Negative divergence means the field pulls <i>inward</i> on average here: a tiny disk of test particles would shrink in area. div = ${fmt(J.a, 3)} + ${fmt(J.d, 3)} = ${fmt(div, 3)}.`);
    }
  } else if (Math.abs(curl) > 0.05) {
    const dir = curl > 0 ? "counter-clockwise" : "clockwise";
    verdict = { type: "warn", text: `Irotational-like: curl F = ${fmt(curl, 3)} (${dir} rotation)` };
    reasons.push(`Curl measures local spin. Here curl = βˆ‚g/βˆ‚x βˆ’ βˆ‚f/βˆ‚y = ${fmt(J.c, 3)} βˆ’ ${fmt(J.b, 3)} = ${fmt(curl, 3)}, so a small paddle wheel placed here would turn ${dir}. The field is locally rotational even though it may not look like a circle.`);
  } else {
    verdict = { type: "warn", text: `Locally trivial β€” div β‰ˆ ${fmt(div, 3)}, curl β‰ˆ ${fmt(curl, 3)} (very little expansion or spin here)` };
    reasons.push(`Both the divergence and the curl are near zero, so at this scale the field looks like a simple translation: it mostly just carries points along without stretching or spinning them.`);
  }

  // field value explanation
  reasons.push(`The vector here is <b>F = (${fmt(f, 3)}, ${fmt(g, 3)})</b>, magnitude ${fmt(mag, 3)}. Its direction is atan2(g, f) = ${(Math.atan2(g, f) * 180 / Math.PI).toFixed(1)}Β°, which is why the arrow points that way.`);

  // fixed point extra
  if (klass === "fixed point") {
    reasons.push(`This is (or is extremely close to) an equilibrium point, because F β‰ˆ 0 there. The behaviour of nearby trajectories is decided entirely by the Jacobian J, whose eigenvalues are ${eigText}. Trace = ${fmt(tr, 3)}, determinant = ${fmt(det, 3)}.`);
    if (det < 0) {
      reasons.push(`Because det J < 0, the two eigenvalues have opposite signs: the point attracts along one direction and repels along another β€” a <b>saddle</b>. Trajectories near a saddle swing past and leave.`);
    } else if (Math.abs(disc) < 1e-9 && Math.abs(tr) < 1e-6) {
      reasons.push(`Trace β‰ˆ 0 and the discriminant is β‰ˆ 0: this is a degenerate borderline case. Numerically it is hard to pin down, so treat the classification as tentative.`);
    } else if (disc < 0) {
      reasons.push(`Because the discriminant is negative, the eigenvalues are a complex pair, so nearby trajectories <b>spiral</b>. The real part (${fmt(tr / 2, 3)}) decides stability: negative β†’ spiral in, positive β†’ spiral out.`);
    } else {
      reasons.push(`Both eigenvalues are real, so nearby trajectories approach or leave along two straight directions (the eigenvectors) rather than spiralling.`);
    }
  }

  return { title, klass, verdict, reasons, div, curl, det, tr, eigText, l1, l2, disc, mag };
}

function explain(x, y) {
  const v = fieldAt(x, y);
  const J = jacobianAt(x, y);
  const info = classify(x, y, v.f, v.g, J);
  const modeNames = { name: currentName };

  let html = `<h3>${currentName}</h3>`;
  html += `<p><b>point</b> (x, y) = (${fmt(x, 3)}, ${fmt(y, 3)})</p>`;

  html += `<div class="verdict ${info.verdict.type}">${info.verdict.text}</div>`;
  html += `<p><b>${info.title}</b>${info.klass === "regular point" ? " β€” the field is smooth and non-zero here." : "."}</p>`;

  // Jacobian box
  html += `<div class="jac">J = ⎑ ${fmt(J.a, 3)}  ${fmt(J.b, 3)} ⎀<br>&nbsp;&nbsp;&nbsp;&nbsp; ⎣ ${fmt(J.c, 3)}  ${fmt(J.d, 3)} ⎦</div>`;

  html += `<p>${info.reasons.join("</p><p>")}</p>`;

  html += `<table>`;
  html += `<tr><th>quantity</th><th>value</th><th>meaning</th></tr>`;
  html += `<tr><td>div F</td><td>${fmt(info.div, 3)}</td><td>${info.div > 0.05 ? "expanding" : info.div < -0.05 ? "converging" : "area-preserving"}</td></tr>`;
  html += `<tr><td>curl F</td><td>${fmt(info.curl, 3)}</td><td>${info.curl > 0.05 ? "CCW spin" : info.curl < -0.05 ? "CW spin" : "no spin"}</td></tr>`;
  html += `<tr><td>det J</td><td>${fmt(info.det, 3)}</td><td>${info.det > 0 ? "orientation kept" : info.det < 0 ? "orientation flipped" : "singular"}</td></tr>`;
  html += `<tr><td>trace J</td><td>${fmt(info.tr, 3)}</td><td>${info.tr > 0.05 ? "net growth" : info.tr < -0.05 ? "net decay" : "neutral"}</td></tr>`;
  html += `<tr><td>eigenvalues</td><td colspan="2">${info.eigText}</td></tr>`;
  html += `</table>`;

  html += `<div class="note">`;
  html += `<b>Why these numbers come from two derivatives.</b> The whole local story is carried by the Jacobian, the 2Γ—2 matrix of partial derivatives of (f, g). Its <i>trace</i> is the divergence (area growth), its <i>antisymmetric</i> part is the curl (rotation), and its <i>determinant</i> tells you whether the map preserves or flips orientation. The eigenvalues then split behaviour into just a few cases β€” node, focus, saddle, center β€” which is why physicists can classify an entire flow by looking at a single point.`;
  html += `</div>`;

  return html;
}

function updateReadout(x, y) {
  const v = fieldAt(x, y);
  const J = jacobianAt(x, y);
  const div = J.a + J.d;
  const curl = J.c - J.b;
  const det = J.a * J.d - J.b * J.c;
  const tr = J.a + J.d;
  const disc = tr * tr - 4 * det;
  let eig;
  if (Math.abs(disc) < 1e-9) eig = `${fmt(tr / 2, 3)} (double)`;
  else if (disc > 0) eig = `${fmt((tr + Math.sqrt(disc)) / 2, 2)}, ${fmt((tr - Math.sqrt(disc)) / 2, 2)}`;
  else eig = `${fmt(tr / 2, 2)} Β± ${fmt(Math.sqrt(-disc) / 2, 2)}i`;

  document.getElementById("rv-f").textContent = fmt(v.f, 4);
  document.getElementById("rv-g").textContent = fmt(v.g, 4);
  document.getElementById("rv-div").textContent = fmt(div, 4);
  document.getElementById("rv-curl").textContent = fmt(curl, 4);
  document.getElementById("rv-mag").textContent = fmt(Math.hypot(v.f, v.g), 4);
  document.getElementById("rv-det").textContent = fmt(det, 4);
  document.getElementById("rv-trace").textContent = fmt(tr, 4);
  document.getElementById("rv-eig").textContent = eig;
}

// ---------------------------------------------------------------- interaction
function updatePoint(x, y) {
  hoverP = { x, y };
  drawLocalView(x, y);
  explanationEl.innerHTML = explain(x, y);
  updateReadout(x, y);
  coordReadout.textContent = `(${x.toFixed(3)}, ${y.toFixed(3)})`;
}

fieldCanvas.addEventListener("mousemove", (e) => {
  if (dragging) return;
  const rect = fieldCanvas.getBoundingClientRect();
  const sx = (e.clientX - rect.left) / rect.width * W;
  const sy = (e.clientY - rect.top) / rect.height * H;
  const w = toWorld(sx, sy);
  updatePoint(w.x, w.y);
});

fieldCanvas.addEventListener("mouseleave", () => {
  hoverP = null;
  coordReadout.textContent = "hover the field";
});

fieldCanvas.addEventListener("wheel", (e) => {
  e.preventDefault();
  const rect = fieldCanvas.getBoundingClientRect();
  const pxFrac = (e.clientX - rect.left) / rect.width;
  const pyFrac = (e.clientY - rect.top) / rect.height;
  if (e.shiftKey) {
    density = Math.max(0.4, Math.min(3, density * (e.deltaY < 0 ? 1.15 : 0.87)));
    renderField();
    return;
  }
  const cx = view.xmin + pxFrac * (view.xmax - view.xmin);
  const cy = view.ymax - pyFrac * (view.ymax - view.ymin);
  const factor = e.deltaY < 0 ? 0.8 : 1.25;
  const nw = (view.xmax - view.xmin) * factor;
  const nh = (view.ymax - view.ymin) * factor;
  view.xmin = cx - pxFrac * nw;
  view.xmax = view.xmin + nw;
  view.ymax = cy + pyFrac * nh;
  view.ymin = view.ymax - nh;
  renderField();
}, { passive: false });

fieldCanvas.addEventListener("mousedown", (e) => {
  dragging = true;
  dragStart = { x: e.clientX, y: e.clientY, view: { ...view } };
});
window.addEventListener("mousemove", (e) => {
  if (!dragging) return;
  const rect = fieldCanvas.getBoundingClientRect();
  const dx = (e.clientX - dragStart.x) / rect.width * (dragStart.view.xmax - dragStart.view.xmin);
  const dy = (e.clientY - dragStart.y) / rect.height * (dragStart.view.ymax - dragStart.view.ymin);
  view.xmin = dragStart.view.xmin - dx;
  view.xmax = dragStart.view.xmax - dx;
  view.ymin = dragStart.view.ymin + dy;
  view.ymax = dragStart.view.ymax + dy;
  renderField();
});
window.addEventListener("mouseup", () => { dragging = false; });

fieldCanvas.addEventListener("dblclick", () => {
  view = { ...HOME_VIEW };
  renderField();
});

// ---------------------------------------------------------------- presets UI
function buildPresets() {
  presetSel.innerHTML = "";
  for (const name of Object.keys(PRESETS)) {
    const o = document.createElement("option");
    o.value = name;
    o.textContent = name;
    presetSel.appendChild(o);
  }
  presetSel.value = currentName;
}
presetSel.addEventListener("change", () => {
  currentName = presetSel.value;
  customInput.style.display = currentName.startsWith("custom") ? "inline-block" : "none";
  if (!currentName.startsWith("custom")) {
    F = PRESETS[currentName].f;
    G = PRESETS[currentName].g;
  }
  hoverP = null;
  explanationEl.innerHTML = '<p style="color: var(--dim)">Hover over the field to analyze a point.</p>';
  renderField();
});

// custom parser: "f(x,y)=..., g(x,y)=..." using Function() on a whitelist of names
function parseCustom(text) {
  // expected: f = ..., g = ...   OR   f(x,y)=..., g(x,y)=...
  const parts = text.split(",").map(s => s.trim());
  let fExpr = null, gExpr = null;
  for (const p of parts) {
    let m = p.match(/^f\s*(\(x,y\))?\s*=\s*(.+)$/i);
    if (m) { fExpr = m[2]; continue; }
    m = p.match(/^g\s*(\(x,y\))?\s*=\s*(.+)$/i);
    if (m) { gExpr = m[2]; continue; }
  }
  if (!fExpr || !gExpr) return null;
  const allowed = /^[0-9a-zA-Z_+\-*/^().,\s]*$/;
  if (!allowed.test(fExpr) || !allowed.test(gExpr)) return null;
  try {
    const fSrc = fExpr.replace(/\^/g, "**");
    const gSrc = gExpr.replace(/\^/g, "**");
    const f = new Function("x", "y", "with(Math){return (" + fSrc + ");}");
    const g = new Function("x", "y", "with(Math){return (" + gSrc + ");}");
    // smoke test
    const t1 = f(0.3, 0.4), t2 = g(0.3, 0.4);
    if (typeof t1 !== "number" || typeof t2 !== "number") return null;
    return { f, g };
  } catch (err) {
    return null;
  }
}

customInput.addEventListener("input", () => {
  const parsed = parseCustom(customInput.value);
  if (parsed) {
    F = parsed.f;
    G = parsed.g;
    currentName = "custom: " + customInput.value;
    renderField();
  } else {
    customInput.style.borderColor = "#ff5566";
    setTimeout(() => { customInput.style.borderColor = ""; }, 600);
  }
});

// ---------------------------------------------------------------- init
buildPresets();
renderField();
</script>
</body>
</html>