28:null 29:["$","title",null,{"children":"Domain Coloring | nxrix | Observable"}] 2e:T1543,cplot = ({ f = "", caption, pixelRatio, base = 2, domain = [ [-2, 2], [-2, 2] ], legends = ["abs", "arg"], zoom = true } = {}) => { const div = d3.create("div"); const iv = Inputs.disposal(div.node()); const cm = cmark(f, { pixelRatio, base, invalidation: iv }); const sl = d3.scaleLinear(); const createPlot = (xd, yd) => { const chart = Plot.plot({ width: 512 * 0.75, aspectRatio: 1, caption, style: { fontFamily: "var(--sans-serif)" }, x: { label: "Real", labelArrow: false, domain: xd, ticks: 5 }, y: { label: "Imaginary", labelArrow: false, domain: yd, ticks: 5 }, marks: [cm, Plot.frame()] }); div.html("").append(() => chart); return [chart.scale("x"), chart.scale("y")]; }; const [xs, ys] = createPlot(domain[0], domain[1]); if (zoom) { const z = d3 .zoom() .filter((e) => { if (e.touches || e.changedTouches) { const svg = div.select("svg").node(); const pts = d3.pointers(e, svg); const [x0, x1] = xs.range, [y0, y1] = ys.range; return pts.some( ([x, y]) => x >= Math.min(x0, x1) && x <= Math.max(x0, x1) && y >= Math.min(y0, y1) && y <= Math.max(y0, y1) ); } const [x, y] = d3.pointer(e, div.select("svg").node()); const [x0, x1] = xs.range, [y0, y1] = ys.range; return ( x >= Math.min(x0, x1) && x <= Math.max(x0, x1) && y >= Math.min(y0, y1) && y <= Math.max(y0, y1) ); }) .on("zoom", (e) => { const t = e.transform; const svg = div.select("svg").node(); const rect = svg.getBoundingClientRect(); const kx = +svg.getAttribute("width") / rect.width; const ky = +svg.getAttribute("height") / rect.height; const tu = d3.zoomIdentity.translate(t.x * kx, t.y * ky).scale(t.k); const x = sl.domain(xs.domain).range(xs.range); const xd = tu.rescaleX(x).domain(); const y = sl.domain(ys.domain).range(ys.range); const yd = tu.rescaleY(y).domain(); createPlot(xd, yd); }); div.call(z); } /*if (zoom) { const z = d3.zoom().on("zoom", (e) => { const t = e.transform; const x = sl.domain(xs.domain).range(xs.range); const xd = t.rescaleX(x).domain(); const y = sl.domain(ys.domain).range(ys.range); const yd = t.rescaleY(y).domain(); createPlot(xd, yd); }); div.call(z); }*/ const p = div.node(); const h = p.querySelector("svg").getAttribute("height") - 10; const e = html`

${p}

`; if (legends) { if (legends === true || legends?.includes("abs")) { const l = base === 10 ? ["10⁻³", "10⁻²", "10⁻¹", "1", "10", "10²", "10³"] : Math.abs(base - Math.E) < 1e-12 ? ["e⁻³", "e⁻²", "e⁻¹", "1", "e", "e²", "e³"] : [ `1/${base ** 3}`, `1/${base ** 2}`, `1/${base}`, "1", `${base}`, `${base ** 2}`, `${base ** 3}` ]; const abs = Plot.plot({ width: 55, height: h, x: { axis: null }, y: { axis: "right", domain: [0, 1], ticks: [0, 0.11111, 0.2, 0.33333, 0.5, 0.66666, 0.8, 0.88888, 1], tickFormat: (_, i) => ["0", ...l, "∞"][i] }, marks: [ Plot.text([{ label: "abs" }], { frameAnchor: "bottom", dx: -8, dy: 10, text: "label", textAnchor: "start", lineAnchor: "middle" }), Plot.raster({ x1: 0, x2: 1, y1: 0, y2: 1, fill: (x, y) => d3.interpolateGreys(1 - y) }), Plot.frame() ] }); if (caption) abs.style.marginTop = "1rem"; e.appendChild(abs); } if (legends === true || legends?.includes("arg")) { const arg = Plot.plot({ width: 55, height: h, x: { axis: null }, y: { axis: "right", domain: [-Math.PI, Math.PI], ticks: [-Math.PI, -Math.PI / 2, 0, Math.PI / 2, Math.PI], tickFormat: (_, i) => ["−π", "−π⁄2", "0", "π⁄2", "π"][i] }, marks: [ Plot.text([{ label: "arg" }], { frameAnchor: "bottom", dx: -8, dy: 10, text: "label", textAnchor: "start", lineAnchor: "middle" }), Plot.raster({ x1: 0, x2: 1, y1: -Math.PI, y2: Math.PI, fill: (x, y) => { const r0 = 0.08499547839164734 * 1.28 * 1.5; const a = y + 0.8936868 * Math.PI; return ( "rgb(" + oklab2rgb(0.625, r0 * Math.cos(a), r0 * Math.sin(a)) .map((v) => (v * 255 + 0.5) | 0) .join(",") + ")" ); } }), Plot.frame() ] }); if (caption) arg.style.marginTop = "1rem"; e.appendChild(arg); } } return html`

${e}

`; }2f:T22ad,cmark = { const vs = `#version 300 es\nlayout(location=0) in vec2 p;\nvoid main() { gl_Position = vec4(p,0,1); }`; const fs = `#version 300 es precision highp float; uniform vec4 iD; uniform vec2 iR; out vec4 fragColor; #define PI 3.14159265358979 #define TAU 6.28318530717959 vec2 cconj(vec2 z) { return vec2(z.x,-z.y); } vec2 cmul(vec2 a, vec2 b) { return vec2( a.x*b.x-a.y*b.y, a.x*b.y+a.y*b.x ); } vec2 cdiv(vec2 a, vec2 b) { return vec2( a.x*b.x+a.y*b.y, a.y*b.x-a.x*b.y )/dot(b,b); } vec2 cinv(vec2 z) { return cconj(z)/dot(z,z); } vec2 csqr(vec2 z) { return vec2( z.x*z.x-z.y*z.y, 2.0*z.x*z.y ); } vec2 cexp(vec2 z) { float e = exp(z.x); return e*vec2(cos(z.y),sin(z.y)); } vec2 clog(vec2 z) { return vec2( 0.5*log(dot(z,z)), atan(z.y,z.x) ); } vec2 cpow(vec2 a, vec2 b) { return cexp(cmul(b,clog(a))); } vec2 cpow(vec2 a, float b) { return cexp(clog(a)*b); } vec2 cpow(vec2 z, int p) { vec2 nz = z; for (int i=1;i { const s = gl.createShader(type); gl.shaderSource(s, src); gl.compileShader(s); //if (!gl.getShaderParameter(s, gl.COMPILE_STATUS)) // throw new Error(gl.getShaderInfoLog(s)); return s; }; const vss = makeShader(gl.VERTEX_SHADER, vs); invalidation.then(() => { gl.deleteShader(vss); gl.deleteBuffer(b); gl.deleteVertexArray(vao); //const ext = gl.getExtension("WEBGL_lose_context"); //ext?.loseContext(); }); const getProgram = (f, { pixelRatio, base }) => { const p = gl.createProgram(); const s = 1 / Math.log(base); const bs = Number.isInteger(s) ? `${s}.0` : `${s}`; const b = base === 2 || base === 1 || base <= 0 ? "log2(mag)" : Math.abs(base - Math.E) < 1e-12 ? "log(mag)" : `log(mag)*${bs}`; const fss = makeShader( gl.FRAGMENT_SHADER, fs .replace("//--f--//", f) .replace("//--b--//", b) .replace("//--r--//", "*2.0") ); gl.attachShader(p, vss); gl.attachShader(p, fss); gl.linkProgram(p); if (!gl.getProgramParameter(p, gl.LINK_STATUS)) { const msg = gl.getShaderInfoLog(fss).replace(/\0+$/, ""); gl.deleteProgram(p); throw new Error(msg); } gl.deleteShader(fss); return p; }; class cmark extends Plot.Mark { constructor({ f, x1, x2, y1, y2, pixelRatio, invalidation, base, rest }) { super(undefined, rest); this._d = [x1, x2, y1, y2]; this._r = pixelRatio; this.program = getProgram(f, { pixelRatio, base }); this.liD = gl.getUniformLocation(this.program, "iD"); this.liR = gl.getUniformLocation(this.program, "iR"); this.canvas2 = document.createElement("canvas"); this.ctx = this.canvas2.getContext("2d"); this._invalidation = invalidation; if (invalidation) invalidation.then(() => { gl.deleteProgram(this.program); }); } render(_i, scales, _v, d, ctx) { const w = d.width - d.marginLeft - d.marginRight; const h = d.height - d.marginTop - d.marginBottom; const dx = scales.x && scales.x.domain ? scales.x.domain() : [this._d[0], this._d[1]]; const dy = scales.y && scales.y.domain ? scales.y.domain() : [this._d[2], this._d[3]]; const sw = Math.max(1, Math.round(w * this._r)); const sh = Math.max(1, Math.round(h * this._r)); if (canvas.width !== sw) canvas.width = sw; if (canvas.height !== sh) canvas.height = sh; gl.viewport(0, 0, canvas.width, canvas.height); gl.useProgram(this.program); gl.bindVertexArray(vao); gl.uniform4f(this.liD, dx[0], dy[0], dx[1], dy[1]); gl.uniform2f(this.liR, canvas.width, canvas.height); gl.drawArrays(gl.TRIANGLES, 0, 3); if (this.canvas2.width !== canvas.width) this.canvas2.width = canvas.width; if (this.canvas2.height !== canvas.height) this.canvas2.height = canvas.height; this.ctx.drawImage(canvas, 0, 0); const NS = "http://www.w3.org/2000/svg"; const fo = document.createElementNS(NS, "foreignObject"); fo.setAttribute("width", w); fo.setAttribute("height", h); this.canvas2.style.width = `${w}px`; this.canvas2.style.height = `${h}px`; fo.appendChild(this.canvas2); const g = document.createElementNS(NS, "g"); g.setAttribute("transform", `translate(${d.marginLeft},${d.marginTop})`); g.appendChild(fo); if (!this._invalidation) Inputs.disposal(g).then(() => { gl.deleteProgram(this.program); }); return g; } } return function complex(f, options = {}) { const { x1 = -1, x2 = 1, y1 = -1, y2 = 1, pixelRatio = Math.max(devicePixelRatio, 2), invalidation, base = 2, ...rest } = options; return new cmark({ f, x1, x2, y1, y2, pixelRatio, invalidation, base, rest }); }; }30:[] 2c:{"id":"d86dd4ac53ccce8b","slug":"domain-coloring","title":"Domain Coloring","thumbnail":"72e0ce7b43caf8ba1f4cf29aa9bfe9cd3683b8a3345aa0aba8ce4f2b47b95a3a","access_level":"public","create_time":"$D2026-06-30T12:15:36.832Z","update_time":"$D2026-09-17T16:15:11.065Z","view_time":null,"first_public_time":"$D2026-07-04T05:05:59.253Z","has_unpublished_edits":null,"fork":false,"stars":9,"starred":false,"owner":{"type":"team","id":"d6840fab65eebcab","login":"nxrix","name":"nxrix","avatar_url":"https://avatars-next.observableusercontent.com/e64bf55bfe8edb531a8042147a84d2906d224f9102c11c06b67fae3be08b1a80","tier":"starter_2024"},"version":1437,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"9d8e4bfddc4592ef","login":"nxrix","name":"nxrix","avatar_url":"https://avatars-next.observableusercontent.com/e64bf55bfe8edb531a8042147a84d2906d224f9102c11c06b67fae3be08b1a80","tier":null}],"members":[{"type":"individual","id":"9d8e4bfddc4592ef","login":"nxrix","name":"nxrix","avatar_url":"https://avatars-next.observableusercontent.com/e64bf55bfe8edb531a8042147a84d2906d224f9102c11c06b67fae3be08b1a80","tier":null,"role":"editor"},{"type":"team","id":"d6840fab65eebcab","login":"nxrix","name":"nxrix","avatar_url":"https://avatars-next.observableusercontent.com/e64bf55bfe8edb531a8042147a84d2906d224f9102c11c06b67fae3be08b1a80","tier":"starter_2024","role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":3,"collection":{"id":"14ce59584b762c88","slug":"math","title":"Math","owner":{"id":"d6840fab65eebcab","login":"nxrix","name":"nxrix"}},"type":"notebook","body":{"title":"Domain Coloring","cells":[{"id":0,"value":"# Domain Coloring","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1268,"value":"Domain coloring is a visualization technique for complex-valued functions. Unlike real-valued functions, which can be graphed in 2D, a complex-valued function has both a complex input and a complex output, making it four-dimensional. Since we cannot directly visualize four dimensions, domain coloring provides a way to represent and understand the function's output using colors.\n\nInstead of plotting the output value in extra dimensions, we assign a color to each point in the complex plane based on the value of ${tex`f(z)`}. Typically, the hue (color) represents the argument (phase) of the complex number, while the brightness or saturation represents its magnitude (absolute value).","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":949,"value":"cplot({\n f: `\n vec2 f(vec2 z) {\n vec2 z2 = csqr(z);\n return cdiv(cmul(z2-vec2(1,0),csqr(z-vec2(2,1))),z2+2.0);\n }`,\n domain: [\n [-2, 3],\n [-2, 3]\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1273,"value":"A domain coloring graph of the function\n\n${tex`f(z)=\\dfrac{(z^2-1)(z-(2+i))^2}{z^2+2+2i}`}\n\nDarker spots mark magnitudes close to zero, while brighter spots correspond to larger magnitudes. The function has zeros at ${tex`z=\\pm1`} and ${tex`z=2+i`}, and poles at ${tex`z=\\pm\\sqrt{-2-2i}`}.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1033,"value":"cplot({\n f: `\n vec2 f(vec2 z) {\n return cpow(z,3)-vec2(1,0);\n }`,\n caption: \"f(z) = z^3-1\",\n legends: false\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1274,"value":"---","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":33,"value":"viewof f = Inputs.textarea({\n rows: 5,\n value: \"vec2 f(vec2 z) {\\n\" + \" return z;\\n\" + \"}\",\n spellcheck: false\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1057,"value":"cplot({\n f,\n legends: false\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":407,"value":"$2e","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":114,"value":"$2f","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":716,"value":"oklab2rgb = (L, A, B) => {\n let l = L + 0.3963377774 * A + 0.2158037573 * B;\n let m = L - 0.1055613458 * A - 0.0638541728 * B;\n let s = L - 0.0894841775 * A - 1.291485548 * B;\n l = l * l * l;\n m = m * m * m;\n s = s * s * s;\n let r = 4.0767416621 * l - 3.3077115913 * m + 0.2309699292 * s;\n let g = -1.2684380046 * l + 2.6097574011 * m - 0.3413193965 * s;\n let b = -0.0041960863 * l - 0.7034186147 * m + 1.707614701 * s;\n return [toSRGB(r), toSRGB(g), toSRGB(b)];\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":713,"value":"toSRGB = (x) =>\n clamp(x <= 0.0031308 ? 12.92 * x : 1.055 * Math.pow(x, 1 / 2.4) - 0.055)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":728,"value":"clamp = (x, a = 0, b = 1) => Math.min(b, Math.max(a, x))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W30","stdlib":"2","titling":"auto"}} 2d:"$2c:body"