41:null 42:["$","title",null,{"children":"Marsaglia Polar Method for Structured Random Sequences | Workspace Zero | Observable"}] 48:T461,chart = { const svg = d3.create("svg") .attr("viewBox", [0, 0, width, height]); const gDot = svg.append("g") .attr("fill", "none") .attr("stroke-linecap", "round"); gDot.append("ellipse") .attr("cx", x(0)) .attr("cy", y(0)) .attr("rx", Math.abs(x(1) - x(0))) .attr("ry", Math.abs(y(1) - y(0))) .attr("fill", "none") .attr("stroke", "steelblue") .attr("stroke-width", 3.0) .attr("vector-effect", "non-scaling-stroke"); const gx = svg.append("g"); const gy = svg.append("g"); let transform = d3.zoomIdentity; function zoomed({transform: t}) { transform = t; const zx = t.rescaleX(x).interpolate(d3.interpolateRound); const zy = t.rescaleY(y).interpolate(d3.interpolateRound); gDot.attr("transform", t); gx.call(xAxis, zx); gy.call(yAxis, zy); } const zoom = d3.zoom() .scaleExtent([0.33, 10]) .on("zoom", zoomed); svg.call(zoom); const node = svg.node(); node.gDot = gDot; node.reset = () => svg.transition().duration(1000).call(zoom.transform, d3.zoomIdentity); node.getTransform = () => transform; return node; } 49:T8ab,viewof AnimationTime = { const duration = 4000; // ms for one 0 → 1 pass const button = html``; const slider = html``; const label = html`1.00`; const root = html`

`; root.append(button, slider, label); let playing = false, raf = null; let pos = 1.0, dir = 1, last = null, dirty = false, busy = false; root.value = pos; function show() { slider.value = pos; label.value = pos.toFixed(2); } function emit() { root.value = pos; root.dispatchEvent(new CustomEvent("input")); // no bubbles, no loop back into slider } function frame(now) { if (!playing) return; if (last === null) last = now; pos += dir * ((now - last) / duration); last = now; if (pos >= 1) { pos = 2 - pos; dir = -1; } else if (pos <= 0) { pos = -pos; dir = 1; } pos = Math.min(1, Math.max(0, pos)); show(); if (!busy) { busy = true; emit(); // yield to the browser so taps can be handled before we emit again setTimeout(() => { busy = false; }, 0); } raf = requestAnimationFrame(frame); } function play() { if (playing) return; playing = true; last = null; button.textContent = "Pause"; raf = requestAnimationFrame(frame); } function pause() { if (!playing) return; playing = false; cancelAnimationFrame(raf); button.textContent = "Play"; emit(); } // pointerdown fires immediately on iOS, click can be delayed under load button.addEventListener("pointerdown", e => { e.preventDefault(); playing ? pause() : play(); }); slider.addEventListener("input", e => { e.stopPropagation(); if (playing) { playing = false; cancelAnimationFrame(raf); button.textContent = "Play"; } pos = +slider.value; label.value = pos.toFixed(2); emit(); }); invalidation.then(() => { playing = false; cancelAnimationFrame(raf); }); return root; } 4a:T14e1, ## From the Uniform Disc to a 2D Gaussian The method begins with points uniformly distributed over the unit disc. Points are generated in the enclosing square and rejected if they fall outside the unit circle. For a point with polar coordinates ${tex`(r,\theta)`} the rejection step produces a uniform distribution with respect to area. Consequently, the squared radius [ s=r^2 ] is uniformly distributed on ([0,1)). The polar angle (\theta) is retained unchanged, while the radius is transformed to produce the required Gaussian radial distribution. For a 2D standard normal distribution, [ p(x,y)=\frac{1}{2\pi}e^{-(x^2+y^2)/2}, ] the squared radius divided by two follows an exponential distribution: [ \frac{r'^2}{2}\sim\operatorname{Exponential}(1). ] The exponential CDF is [ F(e)=1-e^{-e}. ] Thus the radial transformation can be derived directly from the relationship between the uniform variable (s=r^2) and the exponential distribution. ## The Two Monotonic Radial Mappings Under the constraints of a monotonic radial transformation that: starts with a uniform distribution over the unit disc; preserves the polar angle (\theta); exactly preserves probability measure; produces an unbounded 2D Gaussian distribution; and maps the entire unit disc onto the entire radial extent (0\le r'<\infty); there are exactly two possible mappings, corresponding to the two possible orientations of the probability measure. Orientation-preserving mapping Matching the uniform CDF directly to the exponential CDF gives [ s=1-e^{-r'^2/2}. ] Solving for (r'), [ \boxed{ r'=\sqrt{-2\ln(1-s)} }. ] This mapping preserves radial ordering: [ s:0\rightarrow1 \qquad\Longrightarrow\qquad r':0\rightarrow\infty. ] Points near the origin remain near the origin, while points approaching the boundary of the unit disc are progressively transported into the Gaussian tails. Orientation-reversing mapping The alternative orientation reverses the radial ordering: [ s=e^{-r'^2/2}. ] Solving for (r'), [ \boxed{ r'=\sqrt{-2\ln s} }. ] This is the mapping used by the original Marsaglia–Bray construction. Here, [ s:0\rightarrow1 \qquad\Longrightarrow\qquad r':\infty\rightarrow0. ] Thus, points near the origin of the unit disc are transported into the distant Gaussian tails, while points near the boundary are transported toward the Gaussian origin. Both mappings produce exactly the same Gaussian probability distribution. They differ only in how the spatial structure of the input point set is transported into that distribution. ## Local Neighbourhood Distortion For a radial transformation [ r\mapsto r', ] the local radial stretch is [ \frac{dr'}{dr}, ] while the local tangential stretch is [ \frac{r'}{r}. ] Their ratio gives a dimensionless measure of the local anisotropy of the transformation: \frac{dr'/dr}{r'/r} \frac{r}{r'}\frac{dr'}{dr} } ] using the magnitude for the orientation-reversing mapping. An isotropic transformation has [ \alpha=1. ] Values greater than one indicate greater radial than tangential stretching, while values below one indicate greater tangential than radial stretching. Because stretching and compression should contribute symmetrically to an anisotropy measure, a convenient measure of deviation from isotropy is [ \boxed{ A(r)=|\ln\alpha(r)| }. ] This gives equal error to reciprocal distortions: a radial stretch of (2\times) and a radial compression of (1/2\times) both have anisotropy [ |\ln2|. ] ## Anisotropy of the Marsaglia Mapping For [ r'=\sqrt{-2\ln(r^2)}, ] the stretch factor is [ \boxed{ \alpha_M(r)=\frac{1}{-2\ln r} }. ] As (r\rightarrow0), [ \alpha_M(r)\rightarrow0. ] Thus an arbitrarily small neighbourhood around the input origin becomes increasingly compressed in the radial direction relative to the tangential direction. The mapping then passes through isotropy at [ r=e^{-1/2}\approx0.60653, ] where [ \alpha_M=1. ] Beyond this point, the radial direction becomes increasingly stretched relative to the tangential direction, with [ \alpha_M(r)\rightarrow\infty ] as (r\rightarrow1). Consequently, the standard mapping strongly distorts the local geometry of the input sequence, particularly around the origin. Anisotropy of the Inside-Out Mapping For [ r'=\sqrt{-2\ln(1-r^2)}, ] the stretch factor is [ \boxed{ \alpha_I(r)= \frac{r^2} {(1-r^2)[-\ln(1-r^2)]} }. ] Its behaviour near the origin is particularly important. Using [ -\ln(1-r^2)=r^2+\frac{r^4}{2}+\cdots, ] we obtain [ \alpha_I(r)\rightarrow1 \qquad\text{as}\qquad r\rightarrow0. ] Indeed, the radial mapping itself has the local expansion [ r'\approx\sqrt2,r. ] Thus, infinitesimally close to the origin, the transformation is simply an approximately uniform scaling by (\sqrt2): [ (x,y)\mapsto(\sqrt2x,\sqrt2y). ] The local neighbourhood geometry is therefore preserved to first order. A small blue-noise or low-discrepancy neighbourhood remains a recognisable neighbourhood after transformation rather than being stretched into a distant region of the Gaussian plane. The anisotropy then increases progressively toward the boundary, with [ \alpha_I(r)\rightarrow\infty ] as [ r\rightarrow1. ] The strongest distortion is consequently transferred toward the low-probability Gaussian tails rather than concentrated at the high-density origin. 4b:T6c9, ## Consequence for Structured Point Sequences This distinction is largely irrelevant for independent random samples. Independent samples have no spatial structure that needs to be preserved. For low-discrepancy, blue-noise, or otherwise deliberately structured point sequences, however, the relationships between neighbouring samples are valuable. A transformation can preserve the target probability density exactly while nevertheless destroying those relationships through severe local deformation. The standard Marsaglia mapping uses an inside-out orientation: the unit circle is transformed to the origin, while the neighbourhood surrounding the origin of the input disc toward arbitrarily distant regions of the Gaussian plane. Infinitesimal neighbourhoods there become increasingly anisotropic, with [ \alpha_M\rightarrow0. ] Because neighbourhoods that originally near to the unit disc boundary are squashed towards the origin, points that were originally far apart transform to very near locations near the origin. As a result, blue noise and minimum-distance properties are lost near the origin in the output (codomain). The reversed mapping instead leaves that neighbourhood approximately isotropic: [ \alpha_I\rightarrow1. ] It therefore preserves the local spatial structure around the origin while progressively stretching the input sequence toward the Gaussian tails. This is particularly significant because the origin corresponds to the highest-density region of the 2D Gaussian. The alternative mapping therefore places its strongest neighbourhood preservation where the greatest concentration of samples occurs, while allowing increasing deformation in regions containing progressively less probability mass. 4c:T747,function normalInverseCDF(p) { if (p < 0 || p > 1) { throw new RangeError("p must be in [0, 1]"); } if (p === 0) return -Infinity; if (p === 1) return Infinity; // Coefficients for the rational approximation. const a = [ -3.969683028665376e+01, 2.209460984245205e+02, -2.759285104469687e+02, 1.383577518672690e+02, -3.066479806614716e+01, 2.506628277459239e+00 ]; const b = [ -5.447609879822406e+01, 1.615858368580409e+02, -1.556989798598866e+02, 6.680131188771972e+01, -1.328068155288572e+01 ]; const c = [ -7.784894002430293e-03, -3.223964580411365e-01, -2.400758277161838e+00, -2.549732539343734e+00, 4.374664141464968e+00, 2.938163982698783e+00 ]; const d = [ 7.784695709041462e-03, 3.224671290700398e-01, 2.445134137142996e+00, 3.754408661907416e+00 ]; const pLow = 0.02425; const pHigh = 1 - pLow; if (p < pLow) { const q = Math.sqrt(-2 * Math.log(p)); return (((((c[0] * q + c[1]) * q + c[2]) * q + c[3]) * q + c[4]) * q + c[5]) / ((((d[0] * q + d[1]) * q + d[2]) * q + d[3]) * q + 1); } if (p <= pHigh) { const q = p - 0.5; const r = q * q; return (((((a[0] * r + a[1]) * r + a[2]) * r + a[3]) * r + a[4]) * r + a[5]) * q / (((((b[0] * r + b[1]) * r + b[2]) * r + b[3]) * r + b[4]) * r + 1); } const q = Math.sqrt(-2 * Math.log(1 - p)); const num = (((((c[0] * q + c[1]) * q + c[2]) * q + c[3]) * q + c[4]) * q + c[5]); const den = ((((d[0] * q + d[1]) * q + d[2]) * q + d[3]) * q + 1); return -(num / den); }4d:Ta40,function generateBlueNoise(count, minDistance = 0.9 / Math.sqrt(count)) { if (!Number.isFinite(count) || count <= 0) return new Float32Array(0); const cellSize = minDistance / Math.SQRT2; const gridSize = Math.max(1, Math.ceil(1 / cellSize)); const grid = new Int32Array(gridSize * gridSize).fill(-1); const minDist2 = minDistance * minDistance; const points = []; const active = []; function gridIndex(x, y) { const gx = Math.min(gridSize - 1, Math.floor(x / cellSize)); const gy = Math.min(gridSize - 1, Math.floor(y / cellSize)); return { gx, gy }; } function isValid(x, y) { if (x < 0 || x >= 1 || y < 0 || y >= 1) return false; const { gx, gy } = gridIndex(x, y); for (let cellY = -2; cellY <= 2; ++cellY) { const yy = gy + cellY; if (yy < 0 || yy >= gridSize) continue; for (let cellX = -2; cellX <= 2; ++cellX) { const xx = gx + cellX; if (xx < 0 || xx >= gridSize) continue; const i = grid[yy * gridSize + xx]; if (i < 0) continue; const dx = x - points[i][0]; const dy = y - points[i][1]; if (dx * dx + dy * dy < minDist2) return false; } } return true; } function addPoint(x, y) { const idx = points.length; points.push([x, y]); active.push(idx); const { gx, gy } = gridIndex(x, y); grid[gy * gridSize + gx] = idx; } addPoint(Math.random(), Math.random()); const k = 30; // candidates per active point, standard Bridson value while (active.length > 0 && points.length < count) { const activeIdx = Math.floor(Math.random() * active.length); const [px, py] = points[active[activeIdx]]; let found = false; for (let i = 0; i < k; ++i) { const angle = Math.random() * Math.PI * 2; const radius = minDistance * (1 + Math.random()); // [r, 2r] const x = px + Math.cos(angle) * radius; const y = py + Math.sin(angle) * radius; if (isValid(x, y)) { addPoint(x, y); found = true; if (points.length >= count) break; } } if (!found) { active.splice(activeIdx, 1); // dead end, retire this point } } const out = new Float32Array(points.length * 2); for (let i = 0; i < points.length; ++i) { out[i * 2] = points[i][0]; out[i * 2 + 1] = points[i][1]; } return out; }4e:[] 46:{"id":"477023cce85ca3ee","slug":"marsaglia-modified","title":"Marsaglia Polar Method for Structured Random Sequences","thumbnail":"8d327269a3a470dcc1b91e60937f1c3c08458b231027ba0ff5ce806c252ba0a3","access_level":"public","create_time":"$D2026-08-31T14:49:50.476Z","update_time":"$D2026-09-24T14:32:06.659Z","view_time":null,"first_public_time":"$D2026-09-10T15:25:47.523Z","has_unpublished_edits":null,"fork":{"id":"ee1708b80be94523","slug":"webgl-shader","title":"WebGL Shader","owner":{"id":"f35c755083683fe5","login":"observablehq","name":"Observable"}},"stars":0,"starred":false,"owner":{"type":"team","id":"1d8947a5759b2aa1","login":"workspace-zero","name":"Workspace Zero","avatar_url":"https://avatars-next.observableusercontent.com/d212b8480b2ee2adfe0118e6ffee7da7d959b8ebd27b29b1aad8f7d20c95f349","tier":"starter_2024","bio":"","following":false},"version":1268,"publish_version":1268,"role":null,"authored":false,"authors":[{"type":"individual","id":"a94ed6f572f9bac7","login":"mrmatthew","name":"Matthew","avatar_url":"https://avatars-next.observableusercontent.com/309fbe9c089b3096d9916c1084c6b1db7f1b2cd353dc779d2c7ca696b1ef7ca6","tier":null,"bio":"","following":false}],"members":[{"type":"individual","id":"a94ed6f572f9bac7","login":"mrmatthew","name":"Matthew","avatar_url":"https://avatars-next.observableusercontent.com/309fbe9c089b3096d9916c1084c6b1db7f1b2cd353dc779d2c7ca696b1ef7ca6","tier":null,"role":"editor"},{"type":"team","id":"1d8947a5759b2aa1","login":"workspace-zero","name":"Workspace Zero","avatar_url":"https://avatars-next.observableusercontent.com/d212b8480b2ee2adfe0118e6ffee7da7d959b8ebd27b29b1aad8f7d20c95f349","tier":"starter_2024","role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"titling":"auto","title":"Marsaglia Polar Method for Structured Random Sequences","cells":[{"id":0,"value":"# Marsaglia Polar Method for Structured Random Sequences\n\nThis notebook demonstrates an alternative to the Marsaglia–Bray polar method for transforming a structured uniform point sequence into samples from an unbounded 2D Gaussian distribution.\n\nThe standard Marsaglia–Bray method is statistically correct for independent random samples, but its radial transformation can significantly distort the local spatial structure of a low-discrepancy or blue-noise input sequence. The alternative presented here uses the complementary radial mapping, which preserves the local structure around the origin much more effectively.\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":351,"value":"viewof PointCountSlider = {\n const input = Inputs.range([1, 100], {value: 50, step: 1, label: \"Points\"});\n input.querySelector(\"input[type=number]\").style.display = \"none\";\n input.style.maxWidth = \"300px\";\n return input;\n}\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":548,"value":"SelectedPoints.length / 5","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":212,"value":"reset2 = {\n const button = Inputs.button(\"Centre\");\n button.addEventListener(\"click\", () => chart.reset());\n return button;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":589,"value":"$48","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":438,"value":"viewof Distribution = Inputs.select(\n DistributionTypes,\n {label: \"Distribution\", format: x => x.name}\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":561,"value":"viewof Sequence = Inputs.select(\n SequenceTypes,\n {label: \"Sequence\", format: x => x.name}\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":584,"value":"viewof Animation = Inputs.select(\n AnimationTypes,\n {label: \"Animation\", format: x => x.name}\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":581,"value":"$49","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":509,"value":"$4a","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":516,"value":"$4b","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":518,"value":"\n## Hypothesis\n\nThe hypothesis investigated by this notebook is therefore not that the alternative mapping produces a different Gaussian distribution. Both mappings produce exactly the same 2D Gaussian distribution.\n\nRather, the hypothesis is that the orientation-preserving mapping provides a more structure-preserving transport from a uniform low-discrepancy or blue-noise point sequence to the Gaussian distribution.\n\nIn particular, it is expected to preserve the spatial organisation of high-density neighbourhoods more effectively because its local transformation approaches an isotropic scaling at the origin, whereas the conventional Marsaglia mapping becomes infinitely anisotropic there.\n\nThis notebook evaluates this hypothesis through visualisation and quantitative analysis of the two mappings, including radial distortion, local anisotropy, neighbourhood structure, and spectral characteristics.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":5,"value":"viewof zoom = Inputs.range([0.01, 10], {value: 0.5, transform: Math.log, label: \"Zoom\"})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":8,"value":"import {shader} from \"@mbostock/shader\"","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":357,"value":"DefaultSpan = 3.0","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":216,"value":"x = d3.scaleLinear()\n .domain([-DefaultSpan, DefaultSpan])\n .range([0, width])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":593,"value":"{\n const time = easeInOut(AnimationTime)\n const pts = getRandomPoints(SelectedPoints, 1, time);\n\n chart.gDot.selectAll(\"path\")\n .data(pts)\n .join(\"path\")\n .attr(\"d\", d => `M${x(d[0])},${y(d[1])}h0`)\n .attr(\"stroke\", d => z(d[2]));\n\n return \"Update point data\"\n}\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":598,"value":"function easeInOut(t)\n{\n return 3*t*t - 2*t*t*t;\n}","pinned":true,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":218,"value":"y = d3.scaleLinear()\n .domain([-DefaultSpan * k, DefaultSpan * k])\n .range([height, 0])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":442,"value":"SelectedPoints =\n Distribution.value === \"modified\" ? pointsMarsagliaModified :\n Distribution.value === \"standard\" ? pointsMarsagliaReference :\n Distribution.value === \"independent\" ? pointsNormalInverse :\n Distribution.value === \"uniform-disc\" ? pointsUniformDisc :\n pointsUniformSquare","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":594,"value":"{\n const pointSize = 250 / Math.sqrt(PointCount);\n chart.gDot.attr(\"stroke-width\", pointSize);\n return \"Update strike width\"\n}\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":393,"value":"DistributionTypes = [\n {\n name: \"Normal (Polar Jones)\",\n value: \"modified\"\n },\n {\n name: \"Normal (Polar Marsaglia–Bray)\",\n value: \"standard\"\n },\n {\n name: \"Normal (Dual CDF)\",\n value: \"independent\",\n square: true\n },\n {\n name: \"Uniform Disc\",\n value: \"uniform-disc\"\n },\n {\n name: \"Uniform Square\",\n value: \"uniform-square\",\n square: true\n }\n]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":562,"value":"SequenceTypes = [\n {\n name: \"Blue Noise\",\n value: \"blue-noise\"\n },\n {\n name: \"White Noise\",\n value: \"white-noise\",\n },\n {\n name: \"Jittered Grid\",\n value: \"jitter-grid\"\n },\n {\n name: \"Regular Grid\",\n value: \"regular-grid\"\n },\n]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":583,"value":"AnimationTypes = [\n {\n name: \"Blend\",\n value: \"blend\"\n },\n {\n name: \"Radial\",\n value: \"radial\",\n },\n {\n name: \"Clock\",\n value: \"clock\"\n 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DensityScaleFactor);","pinned":false,"hidden":true,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":542,"value":"PointCountAdjusted","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":582,"value":"import {Scrubber} from \"@mbostock/scrubber\"","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":565,"value":"mutable zoomTransform = d3.zoomIdentity","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":220,"value":"z = d3.scaleOrdinal()\n .domain(data.map(d => d[2]))\n .range(d3.schemeCategory10)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":222,"value":"xAxis = (g, x) => g\n .attr(\"transform\", `translate(0,${height})`)\n .call(d3.axisTop(x).ticks(6))\n .call(g => g.select(\".domain\").attr(\"display\", \"none\"))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":214,"value":"data = {\n const random = d3.randomNormal(0, 0.2);\n const sqrt3 = Math.sqrt(3);\n return [].concat(\n getRandomPoints(SelectedPoints, 1),\n );\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":224,"value":"yAxis = (g, y) => g\n .call(d3.axisRight(y).ticks(6 * k))\n .call(g => g.select(\".domain\").attr(\"display\", \"none\"))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":226,"value":"grid = (g, x, y) => g\n .attr(\"stroke\", \"currentColor\")\n .attr(\"stroke-opacity\", 0.1)\n .call(g => g\n .selectAll(\".x\")\n .data(x.ticks(12))\n .join(\n enter => enter.append(\"line\").attr(\"class\", \"x\").attr(\"y2\", height),\n update => update,\n exit => exit.remove()\n )\n .attr(\"x1\", d => 0.5 + x(d))\n .attr(\"x2\", d => 0.5 + x(d)))\n .call(g => g\n .selectAll(\".y\")\n .data(y.ticks(12 * k))\n .join(\n enter => enter.append(\"line\").attr(\"class\", \"y\").attr(\"x2\", width),\n update => update,\n exit => exit.remove()\n )\n .attr(\"y1\", d => 0.5 + y(d))\n .attr(\"y2\", d => 0.5 + y(d)));","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":575,"value":"function lerp(a, b, t)\n{\n return (b - a) * t + a;\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":228,"value":"k = height / width","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":547,"value":"DensityScaleFactor = Distribution.square ? 1.0 : 4.0 / Math.PI;","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":230,"value":"height = width","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":573,"value":"function Animate(x0, x1, t, time, a)\n{\n var type = Animation.value;\n\n if (type == \"blend\")\n return lerp(x0, x1, time);\n\n if (type == \"radial\")\n return t > 1-time ? x1 : x0;\n\n if (type == \"clock\")\n return a > 1-time ? x1 : x0;\n\n return x1;\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":238,"value":"function getRandomPoint(points, i, color, time)\n{\n i *= 5;\n const x1 = points[i + 0];\n const y1 = points[i + 1];\n\n const t = points[i + 2];\n const x0 = points[i + 3];\n const y0 = points[i + 4];\n\n var a = Math.atan2(x1, -y1);\n a = (a / Math.PI) * 0.5 + 0.5;\n\n return [\n Animate(x0, x1, t, time, a),\n Animate(y0, y1, t, time, a),\n color,\n ];\n}\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":261,"value":"function getRandomPoints(points, color, time)\n{\n const result = [];\n\n for (let i = 0; i < points.length / 5; i++)\n {\n result.push(getRandomPoint(points, i, color, time));\n }\n\n return result;\n}\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":533,"value":"$4c","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":558,"value":"function generateGrid(count) {\n if (!Number.isFinite(count) || count <= 0) {\n return new Float32Array(0);\n }\n\n const n = Math.max(1, Math.round(Math.sqrt(count)));\n const out = new Float32Array(n * n * 2);\n\n const spacing = 1 / n;\n\n let i = 0;\n for (let y = 0; y < n; ++y) {\n for (let x = 0; x < n; ++x) {\n out[i++] = (x + 0.5) * spacing;\n out[i++] = (y + 0.5) * spacing;\n }\n }\n\n return out;\n}\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":559,"value":"function generateJitteredGrid(count) {\n if (!Number.isFinite(count) || count <= 0) {\n return new Float32Array(0);\n }\n\n const n = Math.max(1, Math.round(Math.sqrt(count)));\n const out = new Float32Array(n * n * 2);\n\n const spacing = 1 / n;\n\n let i = 0;\n for (let y = 0; y < n; ++y) {\n for (let x = 0; x < n; ++x) {\n out[i++] = (x + Math.random()) * spacing;\n out[i++] = (y + Math.random()) * spacing;\n }\n }\n\n return out;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":560,"value":"function generateWhiteNoise(count) {\n if (!Number.isFinite(count) || count <= 0) {\n return new Float32Array(0);\n }\n\n const n = Math.floor(count);\n const out = new Float32Array(n * 2);\n\n for (let i = 0; i < n; ++i) {\n out[i * 2] = Math.random();\n out[i * 2 + 1] = Math.random();\n }\n\n return out;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":557,"value":"$4d","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":327,"value":"function marsagliaTransformReference(points) {\n const N = points.length / 2;\n const result = new Float32Array(N * 5);\n let n = 0;\n\n for (let i = 0; i < N; i++) {\n var x = points[2*i];\n var y = points[2*i + 1];\n\n x = x * 2 - 1;\n y = y * 2 - 1;\n\n const s = x * x + y * y;\n const t = 1.0 - s;\n\n // Reject points outside the unit disk\n if (s >= 1.0 || s === 0.0)\n continue;\n\n // MarsagliaReference()\n const f = Math.sqrt(-2.0 * Math.log(s) / s);\n\n result[n++] = x * f;\n result[n++] = y * f;\n result[n++] = t;\n result[n++] = x;\n result[n++] = y;\n }\n\n return result.slice(0, n);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":335,"value":"function marsagliaTransformModified(points) {\n const N = points.length / 2;\n const result = new Float32Array(N * 5);\n let n = 0;\n\n for (let i = 0; i < N; i++) {\n var x = points[2*i + 0];\n var y = points[2*i + 1];\n\n x = x * 2 - 1;\n y = y * 2 - 1;\n\n const s = x * x + y * y;\n const t = s;\n\n // Reject points outside the unit disk\n if (s >= 1.0)\n continue;\n\n const f = Math.sqrt(-2.0 * Math.log1p(-s) / s);\n result[n++] = x * f;\n result[n++] = y * f;\n result[n++] = t;\n result[n++] = x;\n result[n++] = y;\n }\n\n return result.slice(0, n);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":538,"value":"function normalDistributionViaInverse(points) {\n const N = points.length / 2;\n const result = new Float32Array(N * 5);\n let n = 0;\n\n for (let i = 0; i < N; i++) {\n var x = points[2*i];\n var y = points[2*i + 1];\n\n var u = x * 2 - 1;\n var v = y * 2 - 1;\n\n var t = Math.max(Math.abs(u), Math.abs(v))\n\n result[n++] = normalInverseCDF(x);\n result[n++] = normalInverseCDF(y);\n result[n++] = t;\n result[n++] = u;\n result[n++] = v;\n }\n\n return result.slice(0, n);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":459,"value":"function uniformDisc(points) {\n const N = points.length / 2;\n const result = new Float32Array(N * 5);\n let n = 0;\n\n for (let i = 0; i < N; i++) {\n var x = points[2*i];\n var y = points[2*i + 1];\n\n x = x * 2 - 1;\n y = y * 2 - 1;\n\n const s = x * x + y * y;\n\n // Reject points outside the unit disk\n if (s >= 1.0)\n continue;\n\n result[n++] = x;\n result[n++] = y;\n result[n++] = 0.0;\n result[n++] = x;\n result[n++] = y;\n }\n\n return result.slice(0, n);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":554,"value":"function uniformSquare(points) {\n const N = points.length / 2;\n const result = new Float32Array(N * 5);\n let n = 0;\n\n for (let i = 0; i < N; i++) {\n var x = points[2*i];\n var y = points[2*i + 1];\n\n x = x * 2 - 1;\n y = y * 2 - 1;\n\n result[n++] = x;\n result[n++] = y;\n result[n++] = 0.0;\n result[n++] = x;\n result[n++] = y;\n }\n\n return result.slice(0, n);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":329,"value":"pointsMarsagliaReference = marsagliaTransformReference(pointsDynamic)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":340,"value":"pointsMarsagliaModified = marsagliaTransformModified(pointsDynamic)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":539,"value":"pointsNormalInverse = normalDistributionViaInverse(pointsDynamic)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":461,"value":"pointsUniformDisc = uniformDisc(pointsDynamic)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":555,"value":"pointsUniformSquare = uniformSquare(pointsDynamic)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":313,"value":"pointsDynamic =\n Sequence.value === \"regular-grid\" ? generateGrid(PointCountAdjusted) :\n Sequence.value === \"jitter-grid\" ? generateJitteredGrid(PointCountAdjusted) :\n Sequence.value === \"white-noise\" ? generateWhiteNoise(PointCountAdjusted) :\n Sequence.value === \"blue-noise\" ? generateBlueNoise(PointCountAdjusted) :\n []","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W4e","stdlib":"1"}} 47:"$46:body"