|
| 1 | +<!DOCTYPE html> |
| 2 | +<html> |
| 3 | +<head> |
| 4 | + <script src="https://cdn.plot.ly/plotly-latest.min.js"></script> |
| 5 | + <style> |
| 6 | + body { margin: 0; padding: 0; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif; } |
| 7 | + #plot { width: 100%; height: 100%; } |
| 8 | + </style> |
| 9 | +</head> |
| 10 | +<body> |
| 11 | + <div id="plot"></div> |
| 12 | + <script> |
| 13 | + // Create mesh for the cone boundary (det = 0) |
| 14 | + // For 2x2 SPD matrix [[a,b],[b,c]], boundary is b² = ac |
| 15 | + const n = 40; |
| 16 | + |
| 17 | + // Parametric surface: use (a, c) as parameters, b = ±sqrt(ac) on boundary |
| 18 | + const aVals = [], cVals = []; |
| 19 | + for (let i = 0; i < n; i++) { |
| 20 | + aVals.push(0.01 + (3.0 * i / (n - 1))); |
| 21 | + cVals.push(0.01 + (3.0 * i / (n - 1))); |
| 22 | + } |
| 23 | + |
| 24 | + // Upper boundary surface (b = +sqrt(ac)) |
| 25 | + const xUpper = [], yUpper = [], zUpper = []; |
| 26 | + for (let i = 0; i < n; i++) { |
| 27 | + const xRow = [], yRow = [], zRow = []; |
| 28 | + for (let j = 0; j < n; j++) { |
| 29 | + const a = aVals[i]; |
| 30 | + const c = cVals[j]; |
| 31 | + xRow.push(a); |
| 32 | + yRow.push(Math.sqrt(a * c)); |
| 33 | + zRow.push(c); |
| 34 | + } |
| 35 | + xUpper.push(xRow); |
| 36 | + yUpper.push(yRow); |
| 37 | + zUpper.push(zRow); |
| 38 | + } |
| 39 | + |
| 40 | + // Lower boundary surface (b = -sqrt(ac)) |
| 41 | + const xLower = [], yLower = [], zLower = []; |
| 42 | + for (let i = 0; i < n; i++) { |
| 43 | + const xRow = [], yRow = [], zRow = []; |
| 44 | + for (let j = 0; j < n; j++) { |
| 45 | + const a = aVals[i]; |
| 46 | + const c = cVals[j]; |
| 47 | + xRow.push(a); |
| 48 | + yRow.push(-Math.sqrt(a * c)); |
| 49 | + zRow.push(c); |
| 50 | + } |
| 51 | + xLower.push(xRow); |
| 52 | + yLower.push(yRow); |
| 53 | + zLower.push(zRow); |
| 54 | + } |
| 55 | + |
| 56 | + // Generate random SPD points INSIDE the cone (det > 0, so |b| < sqrt(ac)) |
| 57 | + // Using mulberry32 PRNG for reproducibility |
| 58 | + function mulberry32(a) { |
| 59 | + return function() { |
| 60 | + var t = a += 0x6D2B79F5; |
| 61 | + t = Math.imul(t ^ t >>> 15, t | 1); |
| 62 | + t ^= t + Math.imul(t ^ t >>> 7, t | 61); |
| 63 | + return ((t ^ t >>> 14) >>> 0) / 4294967296; |
| 64 | + } |
| 65 | + } |
| 66 | + |
| 67 | + const rand = mulberry32(42); |
| 68 | + const spdX = [], spdY = [], spdZ = []; |
| 69 | + for (let i = 0; i < 80; i++) { |
| 70 | + const a = 0.3 + rand() * 2.4; |
| 71 | + const c = 0.3 + rand() * 2.4; |
| 72 | + const bMax = Math.sqrt(a * c) * 0.85; // Stay well inside cone |
| 73 | + const b = (rand() * 2 - 1) * bMax; |
| 74 | + spdX.push(a); |
| 75 | + spdY.push(b); |
| 76 | + spdZ.push(c); |
| 77 | + } |
| 78 | + |
| 79 | + const surfaceUpper = { |
| 80 | + type: 'surface', |
| 81 | + x: xUpper, |
| 82 | + y: yUpper, |
| 83 | + z: zUpper, |
| 84 | + opacity: 0.35, |
| 85 | + colorscale: [[0, 'rgb(79, 140, 255)'], [1, 'rgb(79, 140, 255)']], |
| 86 | + showscale: false, |
| 87 | + name: 'Cone boundary (det=0)', |
| 88 | + hoverinfo: 'skip' |
| 89 | + }; |
| 90 | + |
| 91 | + const surfaceLower = { |
| 92 | + type: 'surface', |
| 93 | + x: xLower, |
| 94 | + y: yLower, |
| 95 | + z: zLower, |
| 96 | + opacity: 0.35, |
| 97 | + colorscale: [[0, 'rgb(79, 140, 255)'], [1, 'rgb(79, 140, 255)']], |
| 98 | + showscale: false, |
| 99 | + name: 'Cone boundary', |
| 100 | + hoverinfo: 'skip', |
| 101 | + showlegend: false |
| 102 | + }; |
| 103 | + |
| 104 | + const spdPoints = { |
| 105 | + type: 'scatter3d', |
| 106 | + mode: 'markers', |
| 107 | + x: spdX, |
| 108 | + y: spdY, |
| 109 | + z: spdZ, |
| 110 | + marker: { |
| 111 | + size: 4, |
| 112 | + color: 'rgb(238, 76, 44)', |
| 113 | + opacity: 0.9 |
| 114 | + }, |
| 115 | + name: 'SPD matrices (det > 0)', |
| 116 | + hovertemplate: 'a=%{x:.2f}<br>b=%{y:.2f}<br>c=%{z:.2f}<br>det=%{customdata:.2f}<extra></extra>', |
| 117 | + customdata: spdX.map((a, i) => a * spdZ[i] - spdY[i] * spdY[i]) |
| 118 | + }; |
| 119 | + |
| 120 | + const layout = { |
| 121 | + title: { |
| 122 | + text: '<b>SPD Manifold as a Cone in 3D</b><br><span style="font-size:13px;color:#666">2×2 symmetric matrices: SPD region is inside the cone (det > 0)</span>', |
| 123 | + font: { size: 18 } |
| 124 | + }, |
| 125 | + scene: { |
| 126 | + xaxis: { title: 'a (diagonal)', range: [0, 3.2] }, |
| 127 | + yaxis: { title: 'b (off-diagonal)', range: [-3, 3] }, |
| 128 | + zaxis: { title: 'c (diagonal)', range: [0, 3.2] }, |
| 129 | + camera: { |
| 130 | + eye: { x: 1.8, y: 1.4, z: 1.0 } |
| 131 | + }, |
| 132 | + aspectmode: 'cube' |
| 133 | + }, |
| 134 | + margin: { l: 10, r: 10, t: 60, b: 10 }, |
| 135 | + legend: { |
| 136 | + x: 0.01, |
| 137 | + y: 0.99, |
| 138 | + bgcolor: 'rgba(255,255,255,0.85)', |
| 139 | + bordercolor: 'rgba(0,0,0,0.1)', |
| 140 | + borderwidth: 1 |
| 141 | + }, |
| 142 | + paper_bgcolor: 'rgba(0,0,0,0)', |
| 143 | + plot_bgcolor: 'rgba(0,0,0,0)' |
| 144 | + }; |
| 145 | + |
| 146 | + Plotly.newPlot('plot', [surfaceUpper, surfaceLower, spdPoints], layout, {responsive: true}); |
| 147 | + </script> |
| 148 | +</body> |
| 149 | +</html> |
0 commit comments