2b:null 2c:["$","title",null,{"children":"2D Lattices and Finite Subsets of a Circle | Christopher Henn | Observable"}] 31:T73e,{ const ctx = DOM.context2d(w, h); if (showLattice) { ctx.fillStyle = 'black'; for (let [x, y] of ps) { ctx.beginPath(); ctx.arc(xScale(x), yScale(y), 2.5, 0, 2 * Math.PI); ctx.fill(); } } const drawPolygon = cell => { const [x0, y0] = cell[0]; ctx.beginPath(); ctx.moveTo(xScale(x0), yScale(y0)); for (const [x, y] of cell) { ctx.lineTo(xScale(x), yScale(y)); } ctx.lineTo(xScale(x0), yScale(y0)); }; if (showVoronoi) { ctx.strokeStyle = 'rgba(0, 0, 0, 0.6)'; for (let cell of polygons) { drawPolygon(cell); ctx.stroke(); } } if (showVoronoiCell) { drawPolygon(voronoiOriginCell); ctx.fillStyle = 'rgba(85, 239, 196, 0.4)'; ctx.fill(); ctx.strokeStyle = 'rgba(0, 0, 0, 0.8)'; ctx.stroke(); } if (showClosestLines) { ctx.fillStyle = '#e84393'; ctx.beginPath(); ctx.arc(xScale(vertex[0]), yScale(vertex[1]), 5, 0, 2 * Math.PI); ctx.fill(); ctx.strokeStyle = '#e84393'; ctx.beginPath(); const [vx, vy] = vertex; for (let [x, y] of closestPs) { ctx.moveTo(xScale(vx - 100 * (x - vx)), yScale(vy - 100 * (y - vy))); ctx.lineTo(xScale(vx + 100 * (x - vx)), yScale(vy + 100 * (y - vy))); ctx.stroke(); } } if (showCircle) { ctx.strokeStyle = '#e84393'; ctx.beginPath(); ctx.arc(xScale(vertex[0]), yScale(vertex[1]), 100, 0, 2 * Math.PI) ctx.stroke(); // const [vx, vy] = [xScale(vertex[0]), yScale(vertex[1])]; // ctx.fillStyle = '#0984e3'; // for (let p of closestPs) { // const [x, y] = [xScale(p[0]), yScale(p[1])]; // ctx.beginPath(); // ctx.arc(vx + (x - vx), vy + (y - vy), 3, 0, 2 * Math.PI); // ctx.fill(); // } } return ctx.canvas; }32:T486,{ const lattice = html``; lattice.checked = showLattice; lattice.onchange = () => mutable showLattice = lattice.checked; const voronoi = html``; voronoi.checked = showVoronoi; voronoi.onchange = () => mutable showVoronoi = voronoi.checked; const voronoiCell = html``; voronoiCell.checked = showVoronoiCell; voronoiCell.onchange = () => mutable showVoronoiCell = voronoiCell.checked; const closestLines = html``; closestLines.checked = showClosestLines; closestLines.onchange = () => mutable showClosestLines = closestLines.checked; const circle = html``; circle.checked = showCircle; circle.onchange = () => mutable showCircle = circle.checked; return html` ${lattice} Show Lattice ${voronoi} Show Voronoi Tessellation ${voronoiCell} Show Origin Voronoi Cell ${closestLines} Show Closest Lines ${circle} Show Circle

`; }33:[] 2f:{"id":"13d7f7030a469b9f","slug":"2d-lattices-and-finite-subsets-of-a-circle","title":"2D Lattices and Finite Subsets of a Circle","thumbnail":"51404f54c3431bd6add7461dc4ba17d4bc76a467c0e6d441ea1b612ccc61fb88","access_level":"public","create_time":"$D2018-05-01T23:24:18.061Z","update_time":"$D2020-11-30T00:50:21.382Z","view_time":null,"first_public_time":"$D2018-05-02T19:19:03.561Z","fork":false,"stars":4,"starred":false,"owner":{"type":"team","id":"f3428374116cf9e8","login":"chnn","name":"Christopher Henn","avatar_url":"https://avatars-next.observableusercontent.com/78bb870dc1a41660b957b0e3e7c72e7a1393fc8fc056e370a3d26079dff4b2c9","tier":"starter_2024"},"version":268,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"c1d31c67ff33b8b2","login":"chnn","name":"Christopher Henn","avatar_url":"https://avatars-next.observableusercontent.com/78bb870dc1a41660b957b0e3e7c72e7a1393fc8fc056e370a3d26079dff4b2c9","tier":null}],"members":[{"type":"individual","id":"c1d31c67ff33b8b2","login":"chnn","name":"Christopher Henn","avatar_url":"https://avatars-next.observableusercontent.com/78bb870dc1a41660b957b0e3e7c72e7a1393fc8fc056e370a3d26079dff4b2c9","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":1,"collection":null,"type":"notebook","body":{"title":"2D Lattices and Finite Subsets of a Circle","cells":[{"id":0,"value":"md`# 2D Lattices and Finite Subsets of a Circle`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":6,"value":"md`A visualization of a homeomorphism from the space of unimodular lattices to the space of all subsets of the circle with at most 3 points. The homeomorphism is originally given by [Mostovoy](https://arxiv.org/abs/math/9911224v2).`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":51,"value":"$31","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":32,"value":"viewof i = html``","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":191,"value":"$32","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":38,"value":"orbitForA = orbit(a)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":109,"value":"ps = latticePoints(orbitForA[i], 10)\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":107,"value":"polygons = {\n const [x0, x1] = d3.extent(ps, p => p[0]);\n const [y0, y1] = d3.extent(ps, p => p[1]);\n \n return d3.voronoi().extent([[x0, y0], [x1, y1]])(ps).polygons();\n}\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":117,"value":"voronoiOriginCell = polygons.find(({data: [x, y]}) => x === 0 && y === 0)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":136,"value":"vertex = _.max(voronoiOriginCell, ([x, y]) => y)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":132,"value":"closestPs = {\n const isRectangular = voronoiOriginCell.length === 4;\n \n return _.sortBy(ps, p => distance(p, vertex)).slice(0, isRectangular ? 2 : 3);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":40,"value":"a = [[2, 1], [1, 1]]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":16,"value":"// viewof showLattice = html``\nmutable showLattice = true","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":23,"value":"mutable showVoronoi = true","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":27,"value":"mutable showVoronoiCell = true","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":29,"value":"mutable showClosestLines = true","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":171,"value":"mutable showCircle = false","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":53,"value":"w = 800","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":55,"value":"h = 600","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":57,"value":"xDomain = [-2, 2]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":59,"value":"yDomain = [-2, 2]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":61,"value":"xScale = d3.scaleLinear().domain(xDomain).range([0, w])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":65,"value":"yScale = d3.scaleLinear().domain(yDomain).range([h, 0])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":14,"value":"// {\n// const cell = voronoiCellForLattice(orbitForA[i]);\n\n// const ctx = DOM.context2d(400, 400);\n// const xScale = d3.scaleLinear().domain([-1, 1]).range([0, 400]);\n// const yScale = d3.scaleLinear().domain([-1, 1]).range([400, 0]);\n\n// const drawPolygon = cell => {\n// const [x0, y0] = cell[0];\n\n// ctx.beginPath();\n// ctx.moveTo(xScale(x0), yScale(y0));\n \n// for (const [x, y] of cell) {\n// ctx.lineTo(xScale(x), yScale(y));\n// }\n \n// ctx.lineTo(xScale(x0), yScale(y0));\n// ctx.stroke();\n// };\n \n// drawPolygon(cell);\n\n// return ctx.canvas;\n// }","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":42,"value":"import {\n orbit,\n latticePoints,\n distance\n}\nfrom \"@chnn/the-modular-flow-in-tex-mathcal-l-and-tex-mathbb-h\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":63,"value":"d3 = require(\"d3@5\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":144,"value":"_ = require(\"underscore\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"}],"files":"$W33","stdlib":"1","titling":"auto"}} 30:"$2f:body"