2e:null 2f:["$","title",null,{"children":"Electric field of a moving charge | Yao Liu | Observable"}] 34:T14f7,viewof simUI = { const div = document.createElement("div"); div.style.font = "12px sans-serif"; div.style.lineHeight = "1.5em"; const controls = document.createElement("div"); controls.style.marginBottom = "10px"; const mkSlider = (label, min, max, step, value) => { const row = document.createElement("div"); row.style.display = "flex"; row.style.alignItems = "center"; row.style.gap = "8px"; const lab = document.createElement("span"); lab.textContent = label; lab.style.width = "220px"; const input = document.createElement("input"); input.type = "range"; input.min = min; input.max = max; input.step = step; input.value = value; input.style.width = "260px"; const val = document.createElement("span"); val.textContent = value; input.oninput = () => { val.textContent = input.value; div.dispatchEvent(new CustomEvent("input")); }; row.append(lab, input, val); return {row, input, val}; }; const mkSelect = (label, options, value) => { const row = document.createElement("div"); row.style.display = "flex"; row.style.alignItems = "center"; row.style.gap = "8px"; const lab = document.createElement("span"); lab.textContent = label; lab.style.width = "220px"; const select = document.createElement("select"); select.style.width = "160px"; for (const opt of options) { const o = document.createElement("option"); o.value = opt; o.textContent = opt; if (opt === value) o.selected = true; select.appendChild(o); } select.oninput = () => div.dispatchEvent(new CustomEvent("input")); row.append(lab, select); return {row, select}; }; const motion = mkSelect( "default motion", ["mouse drag", "linear bounce", "oscillate", "circle", "square"], "mouse drag" ); const c = mkSlider("speed of light c (px/s)", 5, 100, 1, 20); const dotSpacing = mkSlider("spacing along emitted light (px)", 0.5, 20, 0.5, 2); const raysPerEmission = mkSlider("direction samples", 24, 360, 4, 120); const waveEvery = mkSlider("draw 1 wavefront every N emissions", 1, 40, 1, 10); const betaCap = mkSlider("max speed / c", 0.05, 0.98, 0.01, 0.75); const histSecs = mkSlider("history kept (s)", 0.5, 20.0, 0.1, 6.0); const motionSize = mkSlider("motion size (px)", 20, 260, 2, 120); const motionSpeedFrac = mkSlider("preset speed / max speed", 0.05, 1.00, 0.01, 0.55); const waveAlpha = mkSlider("wavefront alpha", 0.00, 1.00, 0.01, 0.22); const rayAlpha = mkSlider("fieldline alpha", 0.00, 1.00, 0.01, 0.50); const waveWidth = mkSlider("wavefront width", 0.5, 4.0, 0.1, 1.0); const rayWidth = mkSlider("fieldline width", 0.5, 4.0, 0.1, 1.2); const showWavefronts = document.createElement("input"); showWavefronts.type = "checkbox"; showWavefronts.checked = true; showWavefronts.oninput = () => div.dispatchEvent(new CustomEvent("input")); const rowWavefronts = document.createElement("div"); rowWavefronts.append(showWavefronts, document.createTextNode(" show wavefronts")); const showFieldlines = document.createElement("input"); showFieldlines.type = "checkbox"; showFieldlines.checked = true; showFieldlines.oninput = () => div.dispatchEvent(new CustomEvent("input")); const rowFieldlines = document.createElement("div"); rowFieldlines.append(showFieldlines, document.createTextNode(" show fieldlines")); const showRestCircle = document.createElement("input"); showRestCircle.type = "checkbox"; showRestCircle.checked = false; showRestCircle.oninput = () => div.dispatchEvent(new CustomEvent("input")); const rowRest = document.createElement("div"); rowRest.append(showRestCircle, document.createTextNode(" show local rest-frame emission circle")); controls.append( motion.row, c.row, dotSpacing.row, raysPerEmission.row, waveEvery.row, betaCap.row, histSecs.row, motionSize.row, motionSpeedFrac.row, waveAlpha.row, rayAlpha.row, waveWidth.row, rayWidth.row, rowWavefronts, rowFieldlines, rowRest ); const canvas = document.createElement("canvas"); canvas.width = width; canvas.height = Math.min(760, Math.round(width * 0.62)); canvas.style.border = "1px solid #ccc"; canvas.style.touchAction = "none"; canvas.style.display = "block"; div.append(controls, canvas); div.value = { get motion() { return motion.select.value; }, get c() { return +c.input.value; }, get dotSpacing() { return +dotSpacing.input.value; }, get raysPerEmission() { return +raysPerEmission.input.value; }, get waveEvery() { return +waveEvery.input.value; }, get betaCap() { return +betaCap.input.value; }, get histSecs() { return +histSecs.input.value; }, get motionSize() { return +motionSize.input.value; }, get motionSpeedFrac() { return +motionSpeedFrac.input.value; }, get waveAlpha() { return +waveAlpha.input.value; }, get rayAlpha() { return +rayAlpha.input.value; }, get waveWidth() { return +waveWidth.input.value; }, get rayWidth() { return +rayWidth.input.value; }, get showWavefronts() { return showWavefronts.checked; }, get showFieldlines() { return showFieldlines.checked; }, get showRestCircle() { return showRestCircle.checked; }, canvas }; return div; }35:T3d5a,{ const sim = viewof simUI.value; const canvas = sim.canvas; const ctx = canvas.getContext("2d"); const W = canvas.width; const H = canvas.height; const eps = 1e-9; const add = ([ax, ay], [bx, by]) => [ax + bx, ay + by]; const sub = ([ax, ay], [bx, by]) => [ax - bx, ay - by]; const mul = ([ax, ay], s) => [ax * s, ay * s]; const dot = ([ax, ay], [bx, by]) => ax * bx + ay * by; const norm2 = v => dot(v, v); const norm = v => Math.sqrt(norm2(v)); const unit = v => { const n = norm(v); return n < eps ? [1, 0] : [v[0] / n, v[1] / n]; }; const perp = ([x, y]) => [-y, x]; const cross2 = ([ax, ay], [bx, by]) => ax * by - ay * bx; const clamp = (x, a, b) => Math.max(a, Math.min(b, x)); function gammaFromV(v, c) { const b2 = clamp(norm2(v) / (c * c), 0, 0.999999999); return 1 / Math.sqrt(1 - b2); } function aberrate2D(v, c, phi, frameAngle = 0) { const speed = norm(v); if (speed < eps) { const a = phi + frameAngle; return [Math.cos(a), Math.sin(a)]; } const epar = unit(v); const eperp = perp(epar); const local = [ Math.cos(phi + frameAngle), Math.sin(phi + frameAngle) ]; const nparPrime = dot(local, epar); const nperpPrime = dot(local, eperp); const beta = speed / c; const gamma = 1 / Math.sqrt(1 - beta * beta); const denom = 1 + beta * nparPrime; const npar = (nparPrime + beta) / denom; const nperp = nperpPrime / (gamma * denom); return unit(add(mul(epar, npar), mul(eperp, nperp))); } function thomasOmega2D(v, a, c) { const g = gammaFromV(v, c); return (g * g / (g + 1)) * (cross2(v, a) / (c * c)); } function catmullRomClosed(ctx, pts) { const n = pts.length; if (n < 3) return; ctx.beginPath(); ctx.moveTo(pts[0][0], pts[0][1]); for (let i = 0; i < n; i++) { const p0 = pts[(i - 1 + n) % n]; const p1 = pts[i]; const p2 = pts[(i + 1) % n]; const p3 = pts[(i + 2) % n]; const c1 = [ p1[0] + (p2[0] - p0[0]) / 6, p1[1] + (p2[1] - p0[1]) / 6 ]; const c2 = [ p2[0] - (p3[0] - p1[0]) / 6, p2[1] - (p3[1] - p1[1]) / 6 ]; ctx.bezierCurveTo(c1[0], c1[1], c2[0], c2[1], p2[0], p2[1]); } ctx.closePath(); } function strokeOpenCurveSegmented(ctx, pts, breakDist = 80) { if (pts.length < 2) return; let started = false; for (let i = 0; i < pts.length; i++) { const p = pts[i]; if (!started) { ctx.beginPath(); ctx.moveTo(p[0], p[1]); started = true; continue; } const q = pts[i - 1]; const d = norm(sub(p, q)); if (d > breakDist) { ctx.stroke(); ctx.beginPath(); ctx.moveTo(p[0], p[1]); } else { ctx.lineTo(p[0], p[1]); } } if (started) ctx.stroke(); } function wavefrontMayBeVisible(e, tNow, margin = 40) { const age = tNow - e.tEmit; if (age < 0) return false; const R = sim.c * age; const cx = e.x[0], cy = e.x[1]; if (cx + R < -margin) return false; if (cx - R > W + margin) return false; if (cy + R < -margin) return false; if (cy - R > H + margin) return false; return true; } let mouse = [W * 0.5, H * 0.5]; let dragging = false; let chargePos = [W * 0.35, H * 0.55]; let chargeVel = [0, 0]; let samples = []; let emissions = []; let nextEmissionId = 0; let tLast = performance.now() / 1000; let lastEmittedTime = NaN; let lastMotion = sim.motion; let motionAnchor = chargePos.slice(); let motionStartTime = tLast; function resetMotionAnchor(tNow) { motionAnchor = chargePos.slice(); motionStartTime = tNow; } function prescribedPos(tNow) { const A = Math.min(sim.motionSize, 0.42 * Math.min(W, H)); const vmax = sim.motionSpeedFrac * sim.betaCap * sim.c; const mode = sim.motion; const tau = Math.max(0, tNow - motionStartTime); const [ax, ay] = motionAnchor; if (mode === "oscillate") { const omega = vmax / Math.max(A, 1e-6); return [ax, ay + A * Math.sin(omega * tau)]; } if (mode === "linear bounce") { const period = 4 * A / Math.max(vmax, 1e-6); const u = ((tau / period) % 1 + 1) % 1; const tri = 1 - 4 * Math.abs(u - 0.5); return [ax + A * tri, ay]; } if (mode === "circle") { const omega = vmax / Math.max(A, 1e-6); return [ax + A * (Math.cos(omega * tau) - 1), ay + A * Math.sin(omega * tau)]; } if (mode === "square") { // Start exactly at current position = lower-left corner of the square path. // Then move right, up, left, down with sharp corners but no position jump. const side = 2 * A; const perimeter = 8 * A; const T = perimeter / Math.max(vmax, 1e-6); const u = ((tau / T) % 1 + 1) % 1; const s = u * perimeter; const x0 = ax; const y0 = ay; if (s < side) return [x0 + s, y0]; if (s < 2 * side) return [x0 + side, y0 + (s - side)]; if (s < 3 * side) return [x0 + side - (s - 2 * side), y0 + side]; return [x0, y0 + side - (s - 3 * side)]; } return null; } function pointerPos(ev) { const r = canvas.getBoundingClientRect(); return [ (ev.clientX - r.left) * canvas.width / r.width, (ev.clientY - r.top) * canvas.height / r.height ]; } canvas.addEventListener("pointerdown", ev => { dragging = true; mouse = pointerPos(ev); canvas.setPointerCapture(ev.pointerId); }); canvas.addEventListener("pointermove", ev => { mouse = pointerPos(ev); }); canvas.addEventListener("pointerup", ev => { dragging = false; canvas.releasePointerCapture(ev.pointerId); }); canvas.addEventListener("pointerleave", () => { dragging = false; }); function boundedMoveToward(x, target, maxStep) { const d = sub(target, x); const L = norm(d); if (L <= maxStep) return target.slice(); return add(x, mul(d, maxStep / Math.max(L, 1e-9))); } function keepInBox(x) { return [ clamp(x[0], 20, W - 20), clamp(x[1], 20, H - 20) ]; } function updateCharge(dt, tNow) { if (sim.motion !== lastMotion) { resetMotionAnchor(tNow); lastMotion = sim.motion; } let newPos; if (sim.motion === "mouse drag") { const vmax = sim.betaCap * sim.c; const target = dragging ? mouse : chargePos; const moved = boundedMoveToward(chargePos, target, vmax * dt); newPos = keepInBox(moved); } else { newPos = keepInBox(prescribedPos(tNow)); } const newVel = mul(sub(newPos, chargePos), 1 / Math.max(dt, 1e-6)); chargePos = newPos; chargeVel = newVel; samples.push({t: tNow, x: chargePos.slice()}); const cutoff = tNow - sim.histSecs; while (samples.length && samples[0].t < cutoff) samples.shift(); } function buildKinematics() { const out = []; for (let i = 1; i + 1 < samples.length; i++) { const s0 = samples[i - 1]; const s1 = samples[i]; const s2 = samples[i + 1]; const dt1 = s1.t - s0.t; const dt2 = s2.t - s1.t; if (dt1 <= 1e-6 || dt2 <= 1e-6) continue; const v = mul(sub(s2.x, s0.x), 1 / (s2.t - s0.t)); const v01 = mul(sub(s1.x, s0.x), 1 / dt1); const v12 = mul(sub(s2.x, s1.x), 1 / dt2); const a = mul(sub(v12, v01), 2 / Math.max(s2.t - s0.t, 1e-6)); out.push({ t: s1.t, x: s1.x.slice(), v, a }); } return out; } function buildFrameAngles(kin) { if (!kin.length) return []; const angles = new Array(kin.length).fill(0); for (let j = 1; j < kin.length; j++) { const k0 = kin[j - 1]; const k1 = kin[j]; const dt = k1.t - k0.t; const om0 = thomasOmega2D(k0.v, k0.a, sim.c); const om1 = thomasOmega2D(k1.v, k1.a, sim.c); angles[j] = angles[j - 1] + 0.5 * (om0 + om1) * dt; } return angles; } function maybeEmitFromKinematics(kin, angles) { if (!kin.length) return; const emitInterval = sim.dotSpacing / sim.c; const newestT = kin[kin.length - 1].t; if (!isFinite(lastEmittedTime)) { lastEmittedTime = newestT - emitInterval; const diag = Math.hypot(W, H); const nBack = Math.ceil(diag / Math.max(sim.dotSpacing, 1e-6)) + 8; emissions.length = 0; nextEmissionId = 0; const k0 = kin[kin.length - 1]; const frameAngle0 = angles[angles.length - 1]; for (let b = nBack; b >= 1; b--) { const tEmit = newestT - b * emitInterval; const dirs = []; for (let m = 0; m < sim.raysPerEmission; m++) { const phi = 2 * Math.PI * m / sim.raysPerEmission; dirs.push(aberrate2D(k0.v, sim.c, phi, frameAngle0)); } emissions.push({ id: nextEmissionId++, tEmit, x: k0.x.slice(), v: k0.v.slice(), a: k0.a.slice(), frameAngle: frameAngle0, dirs }); } } while (lastEmittedTime + emitInterval <= newestT) { const targetT = lastEmittedTime + emitInterval; let best = 0; let bestDist = Infinity; for (let j = 0; j < kin.length; j++) { const d = Math.abs(kin[j].t - targetT); if (d < bestDist) { bestDist = d; best = j; } } const k = kin[best]; const frameAngle = angles[best]; const dirs = []; for (let m = 0; m < sim.raysPerEmission; m++) { const phi = 2 * Math.PI * m / sim.raysPerEmission; dirs.push(aberrate2D(k.v, sim.c, phi, frameAngle)); } emissions.push({ id: nextEmissionId++, tEmit: k.t, x: k.x.slice(), v: k.v.slice(), a: k.a.slice(), frameAngle, dirs }); lastEmittedTime += emitInterval; } } function pruneOldEmissions(tNow) { let cutIndex = 0; while (cutIndex < emissions.length && !wavefrontMayBeVisible(emissions[cutIndex], tNow, 60)) { cutIndex++; } if (cutIndex > 2) emissions.splice(0, cutIndex - 2); } function pointOnEmission(e, labelIndex, tNow) { const age = tNow - e.tEmit; if (age < 0) return null; return add(e.x, mul(e.dirs[labelIndex], sim.c * age)); } function drawBackground() { ctx.clearRect(0, 0, W, H); ctx.fillStyle = "white"; ctx.fillRect(0, 0, W, H); ctx.strokeStyle = "#f0f0f0"; ctx.lineWidth = 1; for (let x = 0; x <= W; x += 50) { ctx.beginPath(); ctx.moveTo(x, 0); ctx.lineTo(x, H); ctx.stroke(); } for (let y = 0; y <= H; y += 50) { ctx.beginPath(); ctx.moveTo(0, y); ctx.lineTo(W, y); ctx.stroke(); } } function drawHistory() { if (samples.length < 2) return; ctx.save(); ctx.beginPath(); ctx.moveTo(samples[0].x[0], samples[0].x[1]); for (const s of samples) ctx.lineTo(s.x[0], s.x[1]); ctx.strokeStyle = "rgba(220,40,40,0.85)"; ctx.lineWidth = 3.8; ctx.stroke(); ctx.restore(); } function drawWavefront(e, tNow) { const age = tNow - e.tEmit; if (age < 0) return; if (!wavefrontMayBeVisible(e, tNow, 20)) return; const pts = []; for (let k = 0; k < e.dirs.length; k++) { const p = pointOnEmission(e, k, tNow); if (p) pts.push(p); } if (pts.length < 3) return; ctx.save(); ctx.strokeStyle = `rgba(70,110,200,${sim.waveAlpha})`; ctx.lineWidth = sim.waveWidth; catmullRomClosed(ctx, pts); ctx.stroke(); ctx.restore(); } function drawWavefronts(tNow) { if (!sim.showWavefronts || !emissions.length) return; for (const e of emissions) { if (e.id % sim.waveEvery !== 0) continue; drawWavefront(e, tNow); } } function drawFieldlines(tNow) { if (!sim.showFieldlines || emissions.length < 2) return; const nLabels = emissions[0].dirs.length; const breakDist = Math.max(20, 6 * sim.dotSpacing); ctx.save(); ctx.beginPath(); ctx.rect(0, 0, W, H); ctx.clip(); ctx.strokeStyle = `rgba(40,120,135,${sim.rayAlpha})`; ctx.lineWidth = sim.rayWidth; for (let k = 0; k < nLabels; k++) { const pts = []; for (const e of emissions) { const p = pointOnEmission(e, k, tNow); if (p) pts.push(p); } if (pts.length >= 2) strokeOpenCurveSegmented(ctx, pts, breakDist); } ctx.restore(); } function drawCharge() { ctx.beginPath(); ctx.arc(chargePos[0], chargePos[1], 10, 0, 2 * Math.PI); ctx.fillStyle = "rgb(230,30,30)"; ctx.fill(); ctx.beginPath(); ctx.arc(chargePos[0], chargePos[1], 10, 0, 2 * Math.PI); ctx.strokeStyle = "rgba(120,0,0,0.6)"; ctx.lineWidth = 1.5; ctx.stroke(); const vmag = norm(chargeVel); if (vmag > 1e-3) { const u = unit(chargeVel); ctx.beginPath(); ctx.moveTo(chargePos[0], chargePos[1]); ctx.lineTo(chargePos[0] + 32 * u[0], chargePos[1] + 32 * u[1]); ctx.strokeStyle = "rgba(120,0,0,0.8)"; ctx.lineWidth = 2.2; ctx.stroke(); } } function drawRestCircleIfWanted() { if (!sim.showRestCircle) return; const kin = buildKinematics(); if (!kin.length) return; const angles = buildFrameAngles(kin); const k = kin[kin.length - 1]; const frameAngle = angles[angles.length - 1]; const R = 26; ctx.beginPath(); ctx.arc(k.x[0], k.x[1], R, 0, 2 * Math.PI); ctx.strokeStyle = "rgba(80,80,160,0.25)"; ctx.lineWidth = 1; ctx.stroke(); for (let m = 0; m < 16; m++) { const phi = 2 * Math.PI * m / 16 + frameAngle; ctx.beginPath(); ctx.moveTo(k.x[0], k.x[1]); ctx.lineTo(k.x[0] + R * Math.cos(phi), k.x[1] + R * Math.sin(phi)); ctx.strokeStyle = "rgba(80,80,160,0.18)"; ctx.stroke(); } } function drawHUD(tNow, kin) { const last = kin.length ? kin[kin.length - 1] : null; ctx.fillStyle = "rgba(255,255,255,0.85)"; ctx.fillRect(10, 10, 540, 126); ctx.strokeStyle = "rgba(0,0,0,0.15)"; ctx.strokeRect(10, 10, 540, 126); const speed = norm(chargeVel); const beta = speed / sim.c; const gamma = gammaFromV(chargeVel, sim.c); ctx.fillStyle = "#111"; ctx.font = "12px sans-serif"; ctx.fillText(`motion = ${sim.motion}`, 20, 30); ctx.fillText(`t = ${tNow.toFixed(3)} s`, 20, 48); ctx.fillText(`|v| = ${speed.toFixed(2)} px/s β = ${beta.toFixed(3)} γ = ${gamma.toFixed(3)}`, 20, 66); ctx.fillText(`c = ${sim.c.toFixed(1)} px/s dotSpacing = ${sim.dotSpacing.toFixed(2)} px waveEvery = ${sim.waveEvery}`, 20, 84); ctx.fillText(`motionSize = ${sim.motionSize.toFixed(1)} px preset speed fraction = ${sim.motionSpeedFrac.toFixed(2)}`, 20, 102); if (last) { const amag = norm(last.a); const omg = thomasOmega2D(last.v, last.a, sim.c); ctx.fillText(`|a| = ${amag.toFixed(2)} px/s² ω_T = ${omg.toFixed(4)} rad/s emissions stored = ${emissions.length}`, 20, 120); } } let frame; function tick() { const tNow = performance.now() / 1000; const dt = Math.min(0.03, Math.max(1 / 240, tNow - tLast)); tLast = tNow; updateCharge(dt, tNow); const kin = buildKinematics(); const angles = buildFrameAngles(kin); maybeEmitFromKinematics(kin, angles); pruneOldEmissions(tNow); drawBackground(); drawFieldlines(tNow); drawWavefronts(tNow); drawHistory(); drawRestCircleIfWanted(); drawCharge(); drawHUD(tNow, kin); frame = requestAnimationFrame(tick); } tick(); invalidation.then(() => cancelAnimationFrame(frame)); return canvas; }36:T9f6,helpers = { const eps = 1e-9; const add = ([ax, ay], [bx, by]) => [ax + bx, ay + by]; const sub = ([ax, ay], [bx, by]) => [ax - bx, ay - by]; const mul = ([ax, ay], s) => [ax * s, ay * s]; const dot = ([ax, ay], [bx, by]) => ax * bx + ay * by; const norm2 = v => dot(v, v); const norm = v => Math.sqrt(norm2(v)); const unit = v => { const n = norm(v); return n < eps ? [1, 0] : [v[0] / n, v[1] / n]; }; const perp = ([x, y]) => [-y, x]; const cross2 = ([ax, ay], [bx, by]) => ax * by - ay * bx; const clamp = (x, a, b) => Math.max(a, Math.min(b, x)); function gammaFromV(v, c) { const b2 = clamp(norm2(v) / (c * c), 0, 0.999999999); return 1 / Math.sqrt(1 - b2); } function aberrate2D(v, c, phi, frameAngle = 0) { const speed = norm(v); if (speed < eps) { const a = phi + frameAngle; return [Math.cos(a), Math.sin(a)]; } const epar = unit(v); const eperp = perp(epar); const local = [ Math.cos(phi + frameAngle), Math.sin(phi + frameAngle) ]; const nparPrime = dot(local, epar); const nperpPrime = dot(local, eperp); const beta = speed / c; const gamma = 1 / Math.sqrt(1 - beta * beta); const denom = 1 + beta * nparPrime; const npar = (nparPrime + beta) / denom; const nperp = nperpPrime / (gamma * denom); return unit(add(mul(epar, npar), mul(eperp, nperp))); } function thomasOmega2D(v, a, c) { const g = gammaFromV(v, c); return (g * g / (g + 1)) * (cross2(v, a) / (c * c)); } function catmullRomClosed(ctx, pts) { const n = pts.length; if (n < 2) return; ctx.beginPath(); ctx.moveTo(pts[0][0], pts[0][1]); for (let i = 0; i < n; i++) { const p0 = pts[(i - 1 + n) % n]; const p1 = pts[i]; const p2 = pts[(i + 1) % n]; const p3 = pts[(i + 2) % n]; const c1 = [ p1[0] + (p2[0] - p0[0]) / 6, p1[1] + (p2[1] - p0[1]) / 6 ]; const c2 = [ p2[0] - (p3[0] - p1[0]) / 6, p2[1] - (p3[1] - p1[1]) / 6 ]; ctx.bezierCurveTo(c1[0], c1[1], c2[0], c2[1], p2[0], p2[1]); } ctx.closePath(); } function strokeOpenCurve(ctx, pts) { if (pts.length < 2) return; ctx.beginPath(); ctx.moveTo(pts[0][0], pts[0][1]); for (let i = 1; i < pts.length; i++) ctx.lineTo(pts[i][0], pts[i][1]); ctx.stroke(); } return { eps, add, sub, mul, dot, norm2, norm, unit, perp, cross2, clamp, gammaFromV, aberrate2D, thomasOmega2D, catmullRomClosed, strokeOpenCurve }; }37:[] 32:{"id":"56ec37cbbb16ac2c","slug":"electric-field-of-a-moving-charge","title":"Electric field of a moving charge","thumbnail":"56077fb5f18597f46ea3567f85c21afe6c028b645f9562107fea4a0f12b0a966","access_level":"public","create_time":"$D2024-05-17T08:49:51.125Z","update_time":"$D2026-03-06T07:25:18.746Z","view_time":null,"first_public_time":"$D2024-05-17T09:03:16.437Z","has_unpublished_edits":null,"fork":false,"stars":0,"starred":false,"owner":{"type":"team","id":"490427fcc4ec2c1d","login":"liuyao12","name":"Yao Liu","avatar_url":"https://avatars-next.observableusercontent.com/895964f8b86922c183ea757cd86944e8e9d463ef5ba08171fab539d4dc087c54","tier":"starter_2024"},"version":95,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"ae908de8dd38723a","login":"liuyao12","name":"Yao Liu","avatar_url":"https://avatars-next.observableusercontent.com/895964f8b86922c183ea757cd86944e8e9d463ef5ba08171fab539d4dc087c54","tier":null}],"members":[{"type":"team","id":"490427fcc4ec2c1d","login":"liuyao12","name":"Yao Liu","avatar_url":"https://avatars-next.observableusercontent.com/895964f8b86922c183ea757cd86944e8e9d463ef5ba08171fab539d4dc087c54","tier":"starter_2024","role":"editor"},{"type":"individual","id":"ae908de8dd38723a","login":"liuyao12","name":"Yao Liu","avatar_url":"https://avatars-next.observableusercontent.com/895964f8b86922c183ea757cd86944e8e9d463ef5ba08171fab539d4dc087c54","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"title":"Electric field of a moving charge","cells":[{"id":0,"value":"# Electric field of a moving charge\n\nSimulation of the radiation from a single point charge. 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