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| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 | import type { NumberLineState } from '../../types' import { numberToScreenX } from '../../numberLineTicks' /** * √3 Demo: "The Tallest Triangle" * * Visualises √3 ≈ 1.732 as the height of an equilateral triangle. * * An equilateral triangle with base from −1 to 1 (side length 2) * is constructed on the number line using compass-arc swings. * The height — from apex (0, √3) to the midpoint (0, 0) — is * highlighted, then rotated down (compass-arm style) to lie on the * number line, landing at √3. The construction itself is the proof: * equal compass arcs guarantee equal sides, and the height falls * out naturally. */ // ── Constants ──────────────────────────────────────────────────────── const SQRT3 = Math.sqrt(3) // 1.7320508075688772 // ── Utilities ──────────────────────────────────────────────────────── function mapRange(v: number, s: number, e: number): number { if (v <= s) return 0 if (v >= e) return 1 return (v - s) / (e - s) } function smoothstep(t: number): number { const c = Math.max(0, Math.min(1, t)) return c * c * (3 - 2 * c) } function easeInOut(t: number): number { const c = Math.max(0, Math.min(1, t)) return c < 0.5 ? 2 * c * c : 1 - (-2 * c + 2) ** 2 / 2 } // ── Phase timing ───────────────────────────────────────────────────── // Must align with sqrt3DemoNarration.ts segment boundaries. const PHASE = { // Seg 0: base — highlight segment from −1 to 1 baseBegin: 0.0, baseEnd: 0.12, // Seg 1: build — equilateral triangle sides swing up buildBegin: 0.12, buildEnd: 0.3, // Seg 2: height — dashed altitude drops from apex heightBegin: 0.3, heightEnd: 0.47, // Seg 3: rotate — compass swings height to number line rotateBegin: 0.47, rotateEnd: 0.65, // Seg 4: mystery — question mark at landing spot mysteryBegin: 0.65, mysteryEnd: 0.8, // Seg 5: reveal — star, label revealBegin: 0.8, revealEnd: 1.0, } as const // ── Colors ─────────────────────────────────────────────────────────── function triCol(isDark: boolean) { return isDark ? '#60a5fa' : '#3b82f6' } // blue (triangle) function heightCol(isDark: boolean) { return isDark ? '#fb923c' : '#ea580c' } // orange (height) function heightColBright(isDark: boolean) { return isDark ? '#fdba74' : '#f97316' } function resultCol(isDark: boolean) { return isDark ? '#34d399' : '#059669' } // green (√3) function textCol(isDark: boolean) { return isDark ? 'rgba(255,255,255,0.9)' : 'rgba(0,0,0,0.8)' } function subtextCol(isDark: boolean) { return isDark ? 'rgba(255,255,255,0.6)' : 'rgba(0,0,0,0.5)' } // ── Viewport ───────────────────────────────────────────────────────── export function sqrt3DemoViewport(cssWidth: number, cssHeight: number) { // Center on [−1, √3] with some padding; triangle reaches √3 ≈ 1.732 above axis const center = (SQRT3 - 1) / 2 + 0.1 // ≈ 0.466 const ppu = Math.min((cssWidth * 0.65) / (SQRT3 + 1.5), cssHeight * 0.25) return { center, pixelsPerUnit: ppu } } // ── Drawing helpers ────────────────────────────────────────────────── type ToX = (v: number) => number function hPx(val: number, ppu: number): number { return val * ppu } /** Draw the base segment from −1 to 1 highlighted on the number line */ function drawBase( ctx: CanvasRenderingContext2D, toX: ToX, axisY: number, isDark: boolean, alpha: number, progress: number ) { if (progress <= 0 || alpha <= 0) return const x0 = toX(-1) const x1 = toX(1) const currentX = x0 + (x1 - x0) * progress ctx.beginPath() ctx.moveTo(x0, axisY) ctx.lineTo(currentX, axisY) ctx.strokeStyle = triCol(isDark) ctx.lineWidth = 4 ctx.lineCap = 'round' ctx.globalAlpha = alpha ctx.stroke() // Endpoint dots ctx.beginPath() ctx.arc(x0, axisY, 4, 0, Math.PI * 2) ctx.fillStyle = triCol(isDark) ctx.globalAlpha = alpha * 0.9 ctx.fill() if (progress > 0.05) { ctx.beginPath() ctx.arc(currentX, axisY, 4, 0, Math.PI * 2) ctx.fill() } } /** Draw the equilateral triangle above the axis using compass-swing construction. * * Both sides swing up simultaneously — pivoting 2-unit copies of the base * from each vertex toward the apex — making equal length and equal angles * visually obvious. * * Progress 0–1: both arcs sweep together. * Progress ≥ 1: light fill of the completed triangle; arc traces hidden. */ function drawTriangle( ctx: CanvasRenderingContext2D, toX: ToX, axisY: number, ppu: number, isDark: boolean, alpha: number, /** 0-1: how much of the two sides are drawn (0=none, 1=complete) */ sideProgress: number, fillAlpha = 0.08 ) { if (alpha <= 0) return const x0 = toX(-1) const x1 = toX(1) const apexX = toX(0) const apexY = axisY - hPx(SQRT3, ppu) const radius = hPx(2, ppu) // 2 units — same length as the base // Fill (only when sides complete) if (sideProgress >= 1) { ctx.beginPath() ctx.moveTo(x0, axisY) ctx.lineTo(x1, axisY) ctx.lineTo(apexX, apexY) ctx.closePath() ctx.fillStyle = triCol(isDark) ctx.globalAlpha = alpha * fillAlpha ctx.fill() } // Both sides share the same eased progress if (sideProgress > 0) { const t = smoothstep(Math.min(1, sideProgress)) // ── Left swing: pivot at (-1, 0), arc from angle 0 → -π/3 (CCW) ── // Canvas angles: 0 = right vertex, -π/3 = apex (60° above horizontal) const leftAngle = -(Math.PI / 3) * t // Dashed arc trace (only during construction) if (sideProgress < 1 && t > 0.01) { ctx.beginPath() ctx.arc(x0, axisY, radius, 0, leftAngle, true) ctx.strokeStyle = triCol(isDark) ctx.lineWidth = 1.5 ctx.setLineDash([4, 3]) ctx.globalAlpha = alpha * 0.35 ctx.stroke() ctx.setLineDash([]) } // Solid segment: pivot → current endpoint const lEndX = x0 + radius * Math.cos(leftAngle) const lEndY = axisY + radius * Math.sin(leftAngle) ctx.beginPath() ctx.moveTo(x0, axisY) ctx.lineTo(lEndX, lEndY) ctx.strokeStyle = triCol(isDark) ctx.lineWidth = 2.5 ctx.lineCap = 'round' ctx.globalAlpha = alpha * 0.8 ctx.stroke() // ── Right swing: pivot at (1, 0), arc from angle π → 4π/3 (CW) ── // Canvas angles: π = left vertex, 4π/3 = apex (upper-left) const rightAngle = Math.PI + (Math.PI / 3) * t // Dashed arc trace (only during construction) if (sideProgress < 1 && t > 0.01) { ctx.beginPath() ctx.arc(x1, axisY, radius, Math.PI, rightAngle) ctx.strokeStyle = triCol(isDark) ctx.lineWidth = 1.5 ctx.setLineDash([4, 3]) ctx.globalAlpha = alpha * 0.35 ctx.stroke() ctx.setLineDash([]) } // Solid segment: pivot → current endpoint const rEndX = x1 + radius * Math.cos(rightAngle) const rEndY = axisY + radius * Math.sin(rightAngle) ctx.beginPath() ctx.moveTo(x1, axisY) ctx.lineTo(rEndX, rEndY) ctx.strokeStyle = triCol(isDark) ctx.lineWidth = 2.5 ctx.lineCap = 'round' ctx.globalAlpha = alpha * 0.8 ctx.stroke() } } /** Draw the height (altitude) line from apex to midpoint */ function drawHeight( ctx: CanvasRenderingContext2D, toX: ToX, axisY: number, ppu: number, isDark: boolean, alpha: number, /** 0-1: how much of the height line is drawn (from apex down) */ progress: number, bright = false ) { if (progress <= 0 || alpha <= 0) return const midX = toX(0) const apexY = axisY - hPx(SQRT3, ppu) const currentY = apexY + (axisY - apexY) * progress ctx.beginPath() ctx.moveTo(midX, apexY) ctx.lineTo(midX, currentY) ctx.strokeStyle = bright ? heightColBright(isDark) : heightCol(isDark) ctx.lineWidth = 3 ctx.lineCap = 'round' ctx.setLineDash([6, 4]) ctx.globalAlpha = alpha ctx.stroke() ctx.setLineDash([]) // Endpoint dot at bottom if (progress > 0.9) { ctx.beginPath() ctx.arc(midX, currentY, 4, 0, Math.PI * 2) ctx.fillStyle = bright ? heightColBright(isDark) : heightCol(isDark) ctx.globalAlpha = alpha * 0.9 ctx.fill() } } /** Draw the height line at a rotation angle during compass swing */ function drawRotatingHeight( ctx: CanvasRenderingContext2D, toX: ToX, axisY: number, ppu: number, isDark: boolean, alpha: number, /** Angle from vertical: 0 = straight up (height in triangle), π/2 = flat on number line */ angle: number, bright = false ) { if (alpha <= 0) return const ox = toX(0) const length = hPx(SQRT3, ppu) // At angle=0: endpoint is at (0, √3) → screen (toX(0), axisY - hPx(√3, ppu)) // At angle=π/2: endpoint is at (√3, 0) → screen (toX(√3), axisY) // Rotating clockwise from π/2 (up) toward 0 (right) in canvas coords const currentAngle = Math.PI / 2 - angle const endX = ox + length * Math.cos(currentAngle) const endY = axisY - length * Math.sin(currentAngle) ctx.beginPath() ctx.moveTo(ox, axisY) ctx.lineTo(endX, endY) ctx.strokeStyle = bright ? heightColBright(isDark) : heightCol(isDark) ctx.lineWidth = 3.5 ctx.lineCap = 'round' ctx.globalAlpha = alpha ctx.stroke() // Endpoint dot ctx.beginPath() ctx.arc(endX, endY, 4, 0, Math.PI * 2) ctx.fillStyle = bright ? heightColBright(isDark) : heightCol(isDark) ctx.globalAlpha = alpha * 0.9 ctx.fill() } /** Draw the compass arc (trace of height tip during rotation) */ function drawArc( ctx: CanvasRenderingContext2D, toX: ToX, axisY: number, ppu: number, isDark: boolean, alpha: number, /** 0-1: how much of the arc is drawn */ progress: number ) { if (progress <= 0 || alpha <= 0) return const ox = toX(0) const radius = hPx(SQRT3, ppu) // Arc from π/2 (straight up) to 0 (horizontal right) // Canvas: negative angles are above x-axis const startAngle = -Math.PI / 2 // straight up in canvas coords const endAngle = 0 // horizontal right const arcEnd = startAngle + (endAngle - startAngle) * progress ctx.beginPath() ctx.arc(ox, axisY, radius, startAngle, arcEnd) ctx.strokeStyle = heightCol(isDark) ctx.lineWidth = 1.5 ctx.setLineDash([4, 3]) ctx.globalAlpha = alpha * 0.5 ctx.stroke() ctx.setLineDash([]) } /** Draw a 5-pointed star */ function drawStar( ctx: CanvasRenderingContext2D, cx: number, cy: number, r: number, color: string, alpha: number ) { ctx.beginPath() for (let i = 0; i < 10; i++) { const angle = -Math.PI / 2 + (i * Math.PI) / 5 const rad = i % 2 === 0 ? r : r * 0.4 const px = cx + rad * Math.cos(angle) const py = cy + rad * Math.sin(angle) if (i === 0) ctx.moveTo(px, py) else ctx.lineTo(px, py) } ctx.closePath() ctx.fillStyle = color ctx.globalAlpha = alpha ctx.fill() } // ── Main render ────────────────────────────────────────────────────── export function renderSqrt3Overlay( ctx: CanvasRenderingContext2D, state: NumberLineState, cssWidth: number, cssHeight: number, isDark: boolean, revealProgress: number, opacity: number ): void { if (opacity <= 0) return const ppu = state.pixelsPerUnit const axisY = cssHeight / 2 const toX = (v: number) => numberToScreenX(v, state.center, ppu, cssWidth) ctx.save() // ── Seg 0: Base — highlight segment from −1 to 1 ── if (revealProgress >= PHASE.baseBegin) { const baseP = easeInOut(mapRange(revealProgress, PHASE.baseBegin, PHASE.baseEnd - 0.02)) if (baseP > 0) { drawBase(ctx, toX, axisY, isDark, opacity, baseP) } // "-1" and "1" labels if (revealProgress < PHASE.buildBegin + 0.05) { const labelA = smoothstep(mapRange(revealProgress, 0.03, 0.06)) * (1 - smoothstep(mapRange(revealProgress, PHASE.baseEnd - 0.02, PHASE.baseEnd + 0.03))) if (labelA > 0.01) { const fs = Math.max(11, Math.min(14, ppu * 0.12)) ctx.font = `${fs}px system-ui, sans-serif` ctx.fillStyle = triCol(isDark) ctx.textAlign = 'center' ctx.textBaseline = 'top' ctx.globalAlpha = opacity * labelA ctx.fillText('−1', toX(-1), axisY + 10) ctx.fillText('1', toX(1), axisY + 10) // "2 units" label in center ctx.fillStyle = textCol(isDark) ctx.globalAlpha = opacity * labelA * 0.8 ctx.fillText('2 units', toX(0), axisY + 10) } } } // ── Seg 1: Build — equilateral triangle sides swing up ── const triVisible = revealProgress >= PHASE.buildBegin if (triVisible) { const sideP = easeInOut( mapRange(revealProgress, PHASE.buildBegin + 0.02, PHASE.buildEnd - 0.02) ) if (sideP > 0) { drawTriangle(ctx, toX, axisY, ppu, isDark, opacity, sideP) } // "Perfect triangle!" label if (revealProgress >= PHASE.buildBegin + 0.08 && revealProgress < PHASE.heightBegin) { const la = smoothstep(mapRange(revealProgress, PHASE.buildBegin + 0.08, PHASE.buildEnd - 0.04)) * (1 - smoothstep(mapRange(revealProgress, PHASE.buildEnd - 0.02, PHASE.buildEnd))) if (la > 0.01) { const fs = Math.max(11, Math.min(14, ppu * 0.12)) ctx.font = `${fs}px system-ui, sans-serif` ctx.fillStyle = triCol(isDark) ctx.textAlign = 'center' ctx.textBaseline = 'bottom' ctx.globalAlpha = opacity * la const apexY2 = axisY - hPx(SQRT3, ppu) ctx.fillText('Perfect triangle!', toX(0), apexY2 - 8) } } } // ── Seg 2: Height — dashed altitude drops from apex ── const heightVisible = revealProgress >= PHASE.heightBegin if (heightVisible) { const heightP = easeInOut( mapRange(revealProgress, PHASE.heightBegin + 0.02, PHASE.heightEnd - 0.04) ) // Before rotation, draw the vertical height if (revealProgress < PHASE.rotateBegin && heightP > 0) { drawHeight(ctx, toX, axisY, ppu, isDark, opacity, heightP) } // "How tall?" label if (revealProgress >= PHASE.heightBegin + 0.04 && revealProgress < PHASE.rotateBegin) { const la = smoothstep(mapRange(revealProgress, PHASE.heightBegin + 0.04, PHASE.heightBegin + 0.08)) * (1 - smoothstep(mapRange(revealProgress, PHASE.heightEnd - 0.03, PHASE.heightEnd))) if (la > 0.01) { const fs = Math.max(11, Math.min(14, ppu * 0.12)) ctx.font = `${fs}px system-ui, sans-serif` ctx.fillStyle = heightCol(isDark) ctx.textAlign = 'left' ctx.textBaseline = 'middle' ctx.globalAlpha = opacity * la const midHeight = axisY - hPx(SQRT3 / 2, ppu) ctx.fillText('How tall?', toX(0) + 10, midHeight) } } } // ── Seg 3: Rotate — compass swings height to number line ── const rotateVisible = revealProgress >= PHASE.rotateBegin if (rotateVisible) { const rotateP = easeInOut( mapRange(revealProgress, PHASE.rotateBegin + 0.02, PHASE.rotateEnd - 0.02) ) const angle = rotateP * (Math.PI / 2) // from 0 (vertical) to π/2 (horizontal) // Draw the arc during rotation and mystery if (revealProgress >= PHASE.rotateBegin && revealProgress < PHASE.mysteryEnd) { drawArc(ctx, toX, axisY, ppu, isDark, opacity, rotateP) } // Draw the rotating height line drawRotatingHeight(ctx, toX, axisY, ppu, isDark, opacity, angle, rotateP >= 1) } // ── Seg 4: Mystery — question mark at landing spot ── if (revealProgress >= PHASE.mysteryBegin && revealProgress < PHASE.revealBegin) { const qP = smoothstep( mapRange(revealProgress, PHASE.mysteryBegin + 0.02, PHASE.mysteryBegin + 0.06) ) const qFade = 1 - smoothstep(mapRange(revealProgress, PHASE.mysteryEnd - 0.02, PHASE.mysteryEnd)) const qAlpha = qP * qFade if (qAlpha > 0.01) { const sqrt3X = toX(SQRT3) // Question mark const fs = Math.max(16, Math.min(24, ppu * 0.2)) ctx.font = `bold ${fs}px system-ui, sans-serif` ctx.fillStyle = heightCol(isDark) ctx.textAlign = 'center' ctx.textBaseline = 'bottom' ctx.globalAlpha = opacity * qAlpha ctx.fillText('?', sqrt3X, axisY - 14) // "Past 1, not quite 2" label const lfs = Math.max(10, Math.min(13, ppu * 0.11)) ctx.font = `${lfs}px system-ui, sans-serif` ctx.fillStyle = textCol(isDark) ctx.textAlign = 'center' ctx.textBaseline = 'top' ctx.globalAlpha = opacity * qAlpha * 0.8 ctx.fillText('Past 1, not quite 2...', (toX(1) + toX(SQRT3)) / 2 + 10, axisY + 10) } } // ── Seg 5: Reveal — star, label ── if (revealProgress >= PHASE.revealBegin) { const revealP = smoothstep(mapRange(revealProgress, PHASE.revealBegin, PHASE.revealEnd)) // Height line on the number line (horizontal) if (revealP > 0) { drawRotatingHeight(ctx, toX, axisY, ppu, isDark, opacity * revealP, Math.PI / 2, true) } // Star at √3 const sqrt3X = toX(SQRT3) const starY = axisY - 20 const starR = Math.max(6, Math.min(10, ppu * 0.06)) const starPulse = 1 + 0.1 * Math.sin(revealProgress * Math.PI * 8) drawStar(ctx, sqrt3X, starY, starR * starPulse * revealP, resultCol(isDark), opacity * revealP) // "√3" label if (revealP > 0.3) { const labelP = smoothstep(mapRange(revealP, 0.3, 0.7)) const fs = Math.max(18, Math.min(26, ppu * 0.22)) ctx.font = `bold ${fs}px system-ui, sans-serif` ctx.fillStyle = resultCol(isDark) ctx.textAlign = 'center' ctx.textBaseline = 'bottom' ctx.globalAlpha = opacity * labelP ctx.fillText('√3', sqrt3X, starY - starR - 4) } // "≈ 1.732" value if (revealP > 0.5) { const valP = smoothstep(mapRange(revealP, 0.5, 0.85)) const fs = Math.max(14, Math.min(18, ppu * 0.16)) ctx.font = `bold ${fs}px system-ui, sans-serif` ctx.fillStyle = resultCol(isDark) ctx.textAlign = 'center' ctx.textBaseline = 'top' ctx.globalAlpha = opacity * valP ctx.fillText('≈ 1.732', sqrt3X, axisY + 6) } // Subtitle if (revealP > 0.7) { const subP = smoothstep(mapRange(revealP, 0.7, 1.0)) const fs = Math.max(10, Math.min(13, ppu * 0.11)) ctx.font = `${fs}px system-ui, sans-serif` ctx.fillStyle = subtextCol(isDark) ctx.textAlign = 'center' ctx.textBaseline = 'top' ctx.globalAlpha = opacity * subP * 0.7 ctx.fillText('The tallest triangle', sqrt3X, axisY + 26) } } ctx.globalAlpha = 1 ctx.setLineDash([]) ctx.restore() } |