42:null 43:["$","title",null,{"children":"Double Doyle Spiral -- Fixed! | Michael Carleton | Observable"}] 49:T52a,canvas = { const height = Math.ceil((width * screen.height) / screen.width); const context = DOM.context2d(width, height); const min_d = 1e-4; const max_d = 1e4; const mod_b = Math.hypot(...root.b); const color = d3["interpolate" + c_scheme]; // for (let t = 0; true; t = (t + 0.001) % 1) { let t = 0; context.save(); context.clearRect(0, 0, width, height); context.fillStyle = background; context.fillRect(0, 0, width, height); context.translate(width / 2, height / 2); context.scale(width / 4, width / 4); for ( let i = 0, q = crot([min_d, 0], -t * 2 * Math.PI); i < p; ++i, q = cmul(q, root.a) ) { context.beginPath(); for ( let c = q, d = Math.hypot(...c); d < max_d; c = cmul(c, root.b), d *= mod_b ) { const [cx, cy, cr] = transform_circle(...c, d * root.r); if (cr > 3) continue; context.moveTo(cx + cr, cy); context.arc(cx, cy, cr, 0, 2 * Math.PI, false); } if (c_scheme == "Original") context.fillStyle = `hsl(${(i / p) * 360}, ${sat}%, ${lum}%)`; else context.fillStyle = color(i / p); context.fill(); // if (cbforce.indexOf('l') >= 0) simulation.force("link", d3.forceLink(links).id(d => d.id)); context.globalAlpha = 1; } yield context.canvas; context.restore(); // } }4a:T1025,doyle = function (p, q) { var epsilon = 1e-10; var pow = Math.pow, sin = Math.sin, cos = Math.cos, pi = Math.PI; function _d(z, t, p, q) { // The square of the distance between z*e^(it) and z*e^(it)^(p/q). var w = pow(z, p / q), s = (p * t + 2 * pi) / q; return pow(z * cos(t) - w * cos(s), 2) + pow(z * sin(t) - w * sin(s), 2); } function ddz_d(z, t, p, q) { // The partial derivative of _d with respect to z. var w = pow(z, p / q), s = (p * t + 2 * pi) / q, ddz_w = (p / q) * pow(z, (p - q) / q); return ( 2 * (w * cos(s) - z * cos(t)) * (ddz_w * cos(s) - cos(t)) + 2 * (w * sin(s) - z * sin(t)) * (ddz_w * sin(s) - sin(t)) ); } function ddt_d(z, t, p, q) { // The partial derivative of _d with respect to t. var w = pow(z, p / q), s = (p * t + 2 * pi) / q, dds_t = p / q; return ( 2 * (z * cos(t) - w * cos(s)) * (-z * sin(t) + w * sin(s) * dds_t) + 2 * (z * sin(t) - w * sin(s)) * (z * cos(t) - w * cos(s) * dds_t) ); } function _s(z, t, p, q) { // The square of the sum of the origin-distance of z*e^(it) and // the origin-distance of z*e^(it)^(p/q). return pow(z + pow(z, p / q), 2); } function ddz_s(z, t, p, q) { // The partial derivative of _s with respect to z. var w = pow(z, p / q), ddz_w = (p / q) * pow(z, (p - q) / q); return 2 * (w + z) * (ddz_w + 1); } /* function ddt_s(z,t, p,q) { // The partial derivative of _s with respect to t. return 0; } */ function _r(z, t, p, q) { // The square of the radius-ratio implied by having touching circles // centred at z*e^(it) and z*e^(it)^(p/q). return _d(z, t, p, q) / _s(z, t, p, q); } function ddz_r(z, t, p, q) { // The partial derivative of _r with respect to z. return ( (ddz_d(z, t, p, q) * _s(z, t, p, q) - _d(z, t, p, q) * ddz_s(z, t, p, q)) / pow(_s(z, t, p, q), 2) ); } function ddt_r(z, t, p, q) { // The partial derivative of _r with respect to t. return ( (ddt_d(z, t, p, q) * _s(z, t, p, q)) / /* - _d(z,t,p,q) * ddt_s(z,t,p,q) */ // omitted because ddt_s is constant at zero pow(_s(z, t, p, q), 2) ); } // We want to find (z, t) such that: // _r(z,t,0,1) = _r(z,t,p,q) = _r(pow(z, p/q), (p*t + 2*pi)/q, 0,1) // // so we define functions _f and _g to be zero when these equalities hold, // and use 2d Newton-Raphson to find a joint root of _f and _g. function _f(z, t) { return _r(z, t, 0, 1) - _r(z, t, p, q); } function ddz_f(z, t) { return ddz_r(z, t, 0, 1) - ddz_r(z, t, p, q); } function ddt_f(z, t) { return ddt_r(z, t, 0, 1) - ddt_r(z, t, p, q); } function _g(z, t) { return _r(z, t, 0, 1) - _r(pow(z, p / q), (p * t + 2 * pi) / q, 0, 1); } function ddz_g(z, t) { return ( ddz_r(z, t, 0, 1) - ddz_r(pow(z, p / q), (p * t + 2 * pi) / q, 0, 1) * (p / q) * pow(z, (p - q) / q) ); } function ddt_g(z, t) { return ( ddt_r(z, t, 0, 1) - ddt_r(pow(z, p / q), (p * t + 2 * pi) / q, 0, 1) * (p / q) ); } function find_root(z, t) { for (;;) { var v_f = _f(z, t), v_g = _g(z, t); if (-epsilon < v_f && v_f < epsilon && -epsilon < v_g && v_g < epsilon) return { ok: true, z: z, t: t, r: Math.sqrt(_r(z, t, 0, 1)) }; var a = ddz_f(z, t), b = ddt_f(z, t), c = ddz_g(z, t), d = ddt_g(z, t); var det = a * d - b * c; if (-epsilon < det && det < epsilon) return { ok: false }; z -= (d * v_f - b * v_g) / det; t -= (a * v_g - c * v_f) / det; if (z < epsilon) return { ok: false }; } } var root = find_root(2, 0); if (!root.ok) throw "Failed to find root for p=" + p + ", q=" + q; var a = [root.z * cos(root.t), root.z * sin(root.t)], coroot = { z: pow(root.z, p / q), t: (p * root.t + 2 * pi) / q }, b = [coroot.z * cos(coroot.t), coroot.z * sin(coroot.t)]; console.log(a, b); return { a: a, b: b, r: root.r, mod_a: root.z, arg_a: root.t }; }4b:[] 47:{"id":"af6a2f56126839e5","slug":null,"title":"Double Doyle Spiral -- Fixed!","thumbnail":"b97f66ee7686be633bb16f6dfb9e76f9523206665a5bc3e074837287bfff7ed8","access_level":"public","create_time":"$D2023-01-09T12:17:39.888Z","update_time":"$D2023-01-09T13:12:19.795Z","view_time":null,"first_public_time":"$D2023-01-09T12:17:39.888Z","has_unpublished_edits":null,"fork":{"id":"8af2ce7402e01317","slug":"double-doyle-spiral","title":"Double Doyle Spiral","owner":{"id":"727607e20e3c61f9","login":"bchoatejr","name":""}},"stars":1,"starred":false,"owner":{"type":"team","id":"9e44154f9d838517","login":"micycle1","name":"Michael Carleton","avatar_url":"https://avatars-next.observableusercontent.com/61fa427332ad83b494cef3bb81806c5fed245c75e35a02787bf9712b61733757","tier":"starter_2024","bio":"2014 - I create a GitHub account.\r\n2017 - I learn how to use GitHub.","following":false},"version":552,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"b3858af0c0de20aa","login":"micycle1","name":"Michael Carleton","avatar_url":"https://avatars-next.observableusercontent.com/61fa427332ad83b494cef3bb81806c5fed245c75e35a02787bf9712b61733757","tier":null,"bio":"2014 - I create a GitHub account.\r\n2017 - I learn how to use GitHub.","following":false}],"members":[{"type":"individual","id":"b3858af0c0de20aa","login":"micycle1","name":"Michael Carleton","avatar_url":"https://avatars-next.observableusercontent.com/61fa427332ad83b494cef3bb81806c5fed245c75e35a02787bf9712b61733757","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":1,"collection":null,"type":"notebook","body":{"titling":"auto","title":"Double Doyle Spiral -- Fixed!","cells":[{"id":0,"value":"md`# Double Doyle Spiral -- Fixed!\nBased on work by [Robin Houston](http://bl.ocks.org/robinhouston/6098310)\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":438,"value":"viewof c_scheme = select({\n title: \"Color Scheme\",\n description: \"Color Scheme\",\n options: color_schemes,\n value: \"Original\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":344,"value":"fullscreen = {\n const button = html`