2a:null 2b:["$","title",null,{"children":"Jacobi's complete integral of first kind and elliptic functions | Dan S. Reznik | Observable"}] 30:T495,md`# Jacobi's complete integral of first kind and elliptic functions

Port to JS of [Toshio Fukushima](https://www.iau.org/administration/membership/individual/6802/)'s original Fortran90 [implementation](https://www.researchgate.net/publication/233903220_xgscdtxt_Fortran_program_package_to_compute_the_Jacobian_elliptic_functions_snum_cnum_dnum) (see **Usage** section below). His fast, precise algorithms based on polynomial interpolation are described in: * T. Fukushima (2009), *Fast Computation of Complete Elliptic Integrals and Jacobian Elliptic Functions*, Celest. Mech. Dyn. Astron.,105, pp. 305-328. * T. Fukushima (2012), "Precise and Fast Computation of Jacobian Elliptic Functions by Conditional Duplication", Numer. Math. DOI 10.1007/s00211-012-0498-0
Also related: * T. Fukushima (2013), "Numerical Computation of Inverse Complete Elliptic Integrals of First and Second Kind", [link](https://www.researchgate.net/publication/237006396_Numerical_Computation_of_Inverse_Complete_Elliptic_Integrals_of_First_and_Second_Kind)`31:T1184,jacobi = class { // Input: m = complementary parameter 0 <= m <= 1 static ellipticK(m) { //let P, Q; const D1 = 1.0 / 16.0, D2 = 1.0 / 32.0, D3 = 21.0 / 1024.0; const D4 = 31.0 / 2048.0, D5 = 6257.0 / 524288.0, D6 = 10293.0 / 1048576.0; const PIHALF = 1.57079633, PIINV = 0.318309886; const mc = 1.0 - m; if (mc < 1.05e-8) { return 1.38629436 - 0.5 * Math.log(mc); } if (mc < 0.1) { const nome = mc * (D1 + mc * (D2 + mc * (D3 + mc * (D4 + mc * (D5 + mc * D6))))) const mx = mc - 0.05; const kkc = 1.59100345 + mx * (0.41600074 + mx * (0.24579151 + mx * (0.17948148 + mx * 0.14455606))); return -kkc * PIINV * Math.log(nome); } if (m <= 0.1) { const mx = m - 0.05; return 1.59100345 + mx * (0.41600074 + mx * (0.24579151 + mx * (0.17948148 + mx * 0.14455606))); } if (m <= 0.2) { const mx = m - 0.15; return 1.63525673 + mx * (0.47119063 + mx * (0.30972841 + mx * (0.25220831 + mx * 0.22672562))); } if (m <= 0.3) { const mx = m - 0.25; return 1.68575035 + mx * (0.54173185 + mx * (0.40152444 + mx * (0.36964247 + mx * 0.37606072))); } if (m <= 0.4) { const mx = m - 0.35; return 1.74435060 + mx * (0.63486428 + mx * (0.53984256 + mx * (0.57189271 + mx * (0.67029514 + mx * 0.83258659)))); } if (m <= 0.5) { const mx = m - 0.45; return 1.81388394 + mx * (0.76316325 + mx * (0.76192861 + mx * (0.95107465 + mx * (1.31518067 + mx * 1.92856069)))); } if (m <= 0.6) { const mx = m - 0.55; return 1.89892491 + mx * (0.95052179 + mx * (1.15107759 + mx * (1.75023911 + mx * (2.95267681 + mx * 5.28580040)))); } if (m <= 0.7) { const mx = m - 0.65; return 2.00759840 + mx * (1.24845723 + mx * (1.92623466 + mx * (3.75128964 + mx * (8.11994455 + mx * (18.6657213 + mx * 44.6039248))))); } if (m <= 0.8) { const mx = m - 0.75; return 2.15651565 + mx * (1.79180564 + mx * (3.82675129 + mx * (10.3867247 + mx * (31.4033141 + mx * (100.923704 + mx * (337.326828 + mx * 1158.70793)))))); } if (m <= 0.85) { const mx = m - 0.825; return 2.31812262 + mx * (2.61692015 + mx * (7.89793508 + mx * (30.5023972 + mx * (131.486937 + mx * (602.984764 + mx * 2877.02462))))); } // all failed const mx = m - 0.875; return 2.47359617 + mx * (3.72762424 + mx * (15.6073930 + mx * (84.1285084 + mx * (506.981820 + mx * (3252.27706 + mx * (21713.2424 + mx * 149037.045)))))); } // Single precision subroutine to compute three Jacobian elliptic functions simultaneously // Inputs: u = argument, m, 0 < m <= 1, // For limited argument: 0 <= u < K/2 // Output: s = sn(u|m), c=cn(u|m), d=dn(u|m) // note: origial was rscd2(u,mc), where mc=1-m static sn_cn_dn_low(u, m) { const mc = 1.0 - m, uA = 1.76269 + mc * 1.16357, uT = 9.207e-4 - m * 4.604e-4; let u0 = u; let broke = false; let n; for (n = 0; n <= 20; n++) { if (u0 < uT) { broke = true; break; } u0 = u0 * 0.5; } if (!broke) console.log("(rscd2) Too large input argument: u=", u); const v = u0 * u0; let a = 1.0, b = v * 0.5, y, z, my, goto2 = false, j; if (u < uA) { for (j = 1; j <= n; j++) { y = b * (a * 2.0 - b); z = a * a; my = m * y; b = (y * 2.0) * (z - my); a = z * z - my * y; } } else for (j = 1; j <= n; j++) { y = b * (a * 2.0 - b); z = a * a; my = m * y; if (z < my * 2.0) { goto2 = true; // de-spaghetti-cizing code ;^) break; } b = (y * 2.0) * (z - my); a = z * z - my * y } if (!goto2) { b = b / a; y = b * (2.0 - b); return { u: u, m: m, sn: Math.sqrt(y), cn: 1.0 - b, dn: Math.sqrt(1.0 - m * y) }; } let c = a - b, mc2 = mc * 2.0, m2 = m * 2.0; for (let i = j; i <= n; i++) { let x = c * c; z = a * a; let w = m * x * x - mc * z * z; let xz = x * z; c = mc2 * xz + w; a = m2 * xz - w; } c = c / a; let x = c * c; return { u: u, m: m, sn: Math.sqrt(1.0 - x), cn: c, dn: Math.sqrt(mc + m * x) }; } static modulo(v,d) { return v>0 ? v%d : d-(-v%d); } // work in progress: allow for it to be called with 0=2) scd.sn *= -1; return scd; } }32:[{"id":"777c9ef735a0eef5b69e72662811b611ae684d329033f2ac61f7c7ec4851503678fe57dc8531adea9051d4bdf524c1447399ba49f752bc4ebfb1c72cb2a4fdbd","href":"https://static.observableusercontent.com/files/777c9ef735a0eef5b69e72662811b611ae684d329033f2ac61f7c7ec4851503678fe57dc8531adea9051d4bdf524c1447399ba49f752bc4ebfb1c72cb2a4fdbd","name":"elliptic_functions.png","type":"image/png","lastModified":1620781911286.003,"size":92419},{"id":"10e7a7074d18108f0cc0484699b3406c9e5548ccb819ee8140df7404d2c47b6f5e1c07bd606d8ea09f28e6322694d265db1728117db97ccde58a5659b186b68e","href":"https://static.observableusercontent.com/files/10e7a7074d18108f0cc0484699b3406c9e5548ccb819ee8140df7404d2c47b6f5e1c07bd606d8ea09f28e6322694d265db1728117db97ccde58a5659b186b68e","name":"carl-gustav.jpg","type":"image/jpeg","lastModified":1620829984699.98,"size":24104}] 2e:{"id":"17a39060995e71e9","slug":"jacobi-elliptic-functions","title":"Jacobi's complete integral of first kind and elliptic functions","thumbnail":"04f33dffcc4412d96747d4ae6b67b9f8d4ecf9288d8b43ed254fb614b2a97086","access_level":"public","create_time":"$D2021-05-10T23:32:37.721Z","update_time":"$D2021-05-30T17:08:28.633Z","view_time":null,"first_public_time":"$D2021-05-10T23:37:12.310Z","fork":false,"stars":3,"starred":false,"owner":{"type":"team","id":"5f70011df0350296","login":"dan-reznik","name":"Dan S. Reznik","avatar_url":"https://avatars-next.observableusercontent.com/34300050560dea2b36d63efd30e3f07e6f455717f9e917936080d33429d8b426","tier":"starter_2024"},"version":1028,"publish_version":1028,"role":null,"authored":false,"authors":[{"type":"individual","id":"bda3bb7cbffeec7d","login":"dan-reznik","name":"Dan S. Reznik","avatar_url":"https://avatars-next.observableusercontent.com/34300050560dea2b36d63efd30e3f07e6f455717f9e917936080d33429d8b426","tier":null}],"members":[{"type":"individual","id":"bda3bb7cbffeec7d","login":"dan-reznik","name":"Dan S. Reznik","avatar_url":"https://avatars-next.observableusercontent.com/34300050560dea2b36d63efd30e3f07e6f455717f9e917936080d33429d8b426","tier":null,"role":"editor"}],"comments":[{"id":"85546a694e3f7c4f","reply_to_id":null,"content":"Change the anchor @esperanc/masson-elliptic-lib to @esperanc/masson-non-trivial-math","create_time":"$D2021-05-13T13:39:42.506Z","update_time":null,"resolved":true,"user":{"type":"individual","id":"591dcad3c6121253","login":"esperanc","name":"Claudio Esperança","avatar_url":"https://avatars-next.observableusercontent.com/91db65fe4bf8e490e5ff1a548612f36ccc04fa7781189e7e4dc7ba6633a66e52","tier":null}},{"id":"32520eba01b06875","reply_to_id":"85546a694e3f7c4f","content":"Fix the link to Ricky Reusser's notebook: https://observablehq.com/@rreusser/periodic-planar-three-body-orbits","create_time":"$D2021-05-13T13:53:22.664Z","update_time":null,"resolved":true,"user":{"type":"individual","id":"591dcad3c6121253","login":"esperanc","name":"Claudio Esperança","avatar_url":"https://avatars-next.observableusercontent.com/91db65fe4bf8e490e5ff1a548612f36ccc04fa7781189e7e4dc7ba6633a66e52","tier":null}},{"id":"08953d5ad741b495","reply_to_id":null,"content":"For a specific value of m, a rational approximation can be very fast. See https://observablehq.com/@jrus/fast-peirce-quincuncial (you also might enjoy https://observablehq.com/@jrus/scpie)","create_time":"$D2021-05-30T07:01:09.935Z","update_time":null,"resolved":false,"user":{"type":"individual","id":"ddd7235babc79ce0","login":"jrus","name":"Jacob Rus","avatar_url":"https://avatars-next.observableusercontent.com/c33dbeed76527c50e64208b0e005e0b6e54a3bef2a14ea6a823f1c7de4867662","tier":null}},{"id":"2f3256655a760ac9","reply_to_id":"08953d5ad741b495","content":"Jacob thanks for the suggestion. However take a look at Fukushima's approach: piecewise polynomials w pre-calculated coefficients.","create_time":"$D2021-05-30T10:20:04.211Z","update_time":null,"resolved":false,"user":{"type":"individual","id":"bda3bb7cbffeec7d","login":"dan-reznik","name":"Dan S. Reznik","avatar_url":"https://avatars-next.observableusercontent.com/34300050560dea2b36d63efd30e3f07e6f455717f9e917936080d33429d8b426","tier":null}},{"id":"3a2b87048f54f44c","reply_to_id":"08953d5ad741b495","content":"I am thinking about computing the (complex-valued) *incomplete* integral for a specific value of *m* and its inverse (as in the code below in this notebook), including with good precision near the singular corners, where many of the existing approximations have some difficulty. Computing the complete integral (as in the papers above) is a slightly different problem (though I think a rational approximation can potentially be helpful there too). My notebook above is just the disk-to-square case; I thought about but didn’t ever get around to implementing a disk-to-arbitrary-rectangle code (and computing the coefficients in Javascript would take porting a bunch of other building blocks).","create_time":"$D2021-05-30T17:00:00.670Z","update_time":null,"resolved":false,"user":{"type":"individual","id":"ddd7235babc79ce0","login":"jrus","name":"Jacob Rus","avatar_url":"https://avatars-next.observableusercontent.com/c33dbeed76527c50e64208b0e005e0b6e54a3bef2a14ea6a823f1c7de4867662","tier":null}},{"id":"b771599460220e26","reply_to_id":"08953d5ad741b495","content":"Awesome, please keep me posted. I left the incomplete integral as a to-do port task, Fukushima has F90 code for that as well.","create_time":"$D2021-05-30T17:05:53.507Z","update_time":null,"resolved":false,"user":{"type":"individual","id":"bda3bb7cbffeec7d","login":"dan-reznik","name":"Dan S. Reznik","avatar_url":"https://avatars-next.observableusercontent.com/34300050560dea2b36d63efd30e3f07e6f455717f9e917936080d33429d8b426","tier":null}}],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":1,"collection":{"id":"6543aeb6f72c303d","slug":"elliptic-billiards","title":"Poncelet Phenomena","owner":{"id":"5f70011df0350296","login":"dan-reznik","name":"Dan S. Reznik"}},"type":"notebook","body":{"title":"Jacobi's complete integral of first kind and elliptic functions","cells":[{"id":0,"value":"$30","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":476,"value":"md`Related pages:\n* Dr. Fukushima's (福島登志夫) [page on ResearchGate](https://www.researchgate.net/profile/Toshio-Fukushima/research) with hundreds of amazing publications in Physics, Math, and Numerical Algorithms\n* Prof. [Claudio Esperança](https://www.lcg.ufrj.br/~esperanc/)'s packaging of Paul Masson's [mathematical physics](https://github.com/paulmasson/math) library into Observable, available at [@esperanc/masson-non-trivial-math](https://observablehq.com/@esperanc/masson-non-trivial-math)\n* Ricky Reusser's [Periodic Planar 3-Body Orbits](https://observablehq.com/@rreusser/periodic-planar-three-body-orbits) and his numerical integration [library](https://observablehq.com/@rreusser/integration)\n* Jacob Rus's page on [Peirce Quincuncial](https://observablehq.com/@jrus/fast-peirce-quincuncial) interpolation\n* Our own page on [Poncelet 3-Periodics in the Elliptic Billiard](https://observablehq.com/@dan-reznik/elliptic-billiard-triangle)`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":109,"value":"md`## Definitions:\nsource: [Wikipedia](https://en.wikipedia.org/wiki/Jacobi_elliptic_functions)\n* Jacobi's *incomplete* elliptic integral of the first kind ${tex.block`K(\\varphi,m)=\\int_0^{\\varphi}\\frac{d\\theta}{\\sqrt{1-m \\sin^2\\theta}}.`}\n* Jacobi's *complete* elliptic integral of the first kind ${tex`K(m)`}, illustrated below, is simply ${tex`K(\\pi/2,m)`}.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":732,"value":"vl.markLine()\n.data(compare_narrow.objects())\n.encode(\n vl.x().fieldQ(\"v\").title(\"m\"),\n vl.y().fieldQ(\"value\").scale({domain: [1.5, 4]}).title(\"K(m)\"),\n vl.color().fieldN(\"key\")\n)\n.title(\"Jacobi Complete Integral of 1st Kind: Masson vs Fukushima\")\n.width(500)\n.render()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":975,"value":"md`* The elliptic sine (sn), cosine (cn), and delta amplitude (dn) can be defined as *inverses* of the incomplete integral. Namely, let ${tex`u=K(\\varphi|m)`}. Then: ${tex.block`\n\\begin{aligned}\n\\text{sn}(u|m)&=\\sin\\varphi\\\\\n\\text{cn}(u|m)&=\\cos\\varphi\\\\\n\\text{dn}(u|m)&=\\sqrt{1- m\\sin^2\\varphi}\\\\\n\\end{aligned}.`}\nAs seen below, their period is ${tex`4\\,K(m)`}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":967,"value":"viewof glob_m = Range([.01, .99], {label: md`**glob_m**`, value: .7, step: .01, format: x => x.toFixed(2)})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":845,"value":"vl.markLine()\n.data(jacobi_samples_narrow.objects())\n.encode(\n vl.x().fieldQ(\"k\").axis({title:\"u/k\"}),\n vl.y().fieldQ(\"value\"),\n vl.color().fieldN(\"key\")\n)\n.title(\"Jacobi Elliptic Functions sn,cn,dn, m=\"+glob_m)\n.width(500)\n.render()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":79,"value":"md`## Usage:\nAll functionality is available thru static methods in the \\`jacobi\\` class, appearing at the end of this notebook. Import it with:\\n\n\n\\`\\`\\`js\nimport { jacobi } from \\'@dan-reznik/jacobi-elliptic-functions\\'\n\\`\\`\\`\nThe following two static methods are available:\\n\n1. \\`jacobi.ellipticK(m)\\`: returns the elliptic integral of the first kind, ${tex`0{\\leq}\\,m\\,{\\leq}1`}; \n1. \\`jacobi.sn_cn_dn(u,m)\\`: returns js object with fields \\`sn\\`, \\`cn\\`, and \\`dn\\`, representing elliptic sine, cosine, and delta evaluated at (u|m), ${tex`0{\\leq}\\,m\\,{\\leq}1`}. Recall these are periodic on ${tex`u`} with period ${tex`4\\,K(m)`}.\n\nExamples:` ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":298,"value":"jacobi.sn_cn_dn(1.713,glob_m)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":471,"value":"glob_k=jacobi.ellipticK(glob_m)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":166,"value":"md`## Test: Prof. Hellmuth Stachel's Jacobi parametrization of N-periodics in the elliptic billiard\nThis family is best parametrized via the so-called *universal measure* which requires Jacobi's elliptic functions. Details by Prof. [Hellmuth Stachel](https://www.geometrie.tuwien.ac.at/stachel/) (TU-Wien, Geometry) appear on page 34 of [a recent presentation](https://www.geometrie.tuwien.ac.at/fg3/pdf/slides_stachel.pdf) and [a recent preprint](https://arxiv.org/abs/2105.03624). Simulation and animation logic is implemented in a [companion page](https://observablehq.com/@dan-reznik/elliptic-billiard-triangle).`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":371,"value":"viewof a = Range([1.001, 3], {label: md`**a/b**`, value: 1.5, step: .001, format: x => x.toFixed(3)})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":368,"value":"{\n const kw = 1.8, height = 400, width=kw*height;\n const dr = new draw(height,width);\n const [ac,bc]=caustic_axes(a);\n while (true) {\n const tDeg_raw = (Date.now() / 20)/(360);\n const tDeg = (tDeg_raw - Math.trunc(tDeg_raw))*360;\n const ons = jacobi_three_periodic(a, tDeg, ac,bc);\n draw_orbit(dr,ons,ac,bc,[]);\n yield dr.canvas;\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":374,"value":"import {draw,jacobi_three_periodic,draw_orbit,caustic_axes} from '@dan-reznik/elliptic-billiard-triangle'","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":554,"value":"import {Range} from \"@observablehq/inputs\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":637,"value":"md`## Main code\n(kudos to Toshio Fukushima)`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":45,"value":"$31","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":646,"value":"md`## K(m) errors wrt. Paul Masson's library\nBelow we compute the relative deviation of Fukushima's implementation of Jacobi's complete elliptic integral of the 1st kind with respect to results with Paul Masson's library, ported by Claudio Esperança to Observable [here](https://observablehq.com/@esperanc/masson-non-trivial-math). Fortunately, the relative errors are in the ${tex`10^{-10}`} range.\n\nNote: K(m) is still implemented in single-precision mode. It will be upgraded at some point.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":648,"value":"import {masson} from '@esperanc/masson-non-trivial-math'","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":818,"value":"md`Compare them at ${tex`u=0.3`}:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":651,"value":"({masson: masson.ellipticK(.3),fukushima:jacobi.ellipticK(.3)})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":758,"value":"gen_01 = function(n) { return Array.from(Array(n).keys()).map(v=>v/n); }","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":823,"value":"md`Create sample table from 0 to 1.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":662,"value":"compare=gen_01(100).map(v=>({v:v,masson:masson.ellipticK(v),fukushima:jacobi.ellipticK(v)}))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":826,"value":"md`Fold it into long-narrow format so can be plotted with VL.`\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":673,"value":"compare_narrow=aq.from(compare).fold([\"masson\",\"fukushima\"])","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":792,"value":"md`Create derived relative error column.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":736,"value":"viewof compare_errors=aq\n.from(compare)\n.derive({ diff: d => (d.masson - d.fukushima)/d.fukushima })\n.view({height:240});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":657,"value":"vl.markLine()\n .data(compare_errors.objects())\n .encode(\n vl.x().fieldQ(\"v\"),\n vl.y().fieldQ(\"diff\").scale({type:'sqrt'})\n)\n .title(\"Relative errors Masson vs Fukushima\")\n .render()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1000,"value":"md`Create table of sn,cn,dn`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":837,"value":"jacobi_samples = {\n const k = jacobi.ellipticK(glob_m);\n const us = gen_01(360).map(v=>v*4*k);\n return us.map(u=>jacobi.sn_cn_dn(u,glob_m));\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":932,"value":"md`Narrow table for VL plot sn,cn,dn vs u/k`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":852,"value":"viewof jacobi_samples_narrow = \n aq.from(jacobi_samples)\n.fold([\"sn\",\"cn\",\"dn\"])\n.params({glob_k}) // don't like this\n.derive({k:(d,$)=>d.u/$.glob_k}).view({height:240});","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":658,"value":"import { vl } from \"@vega/vega-lite-api\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":666,"value":"import {aq, op} from '@uwdata/arquero'","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"}],"files":"$W32","stdlib":"1","titling":"auto"}} 2f:"$2e:body"