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<!DOCTYPE html> |
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<canvas width="960" height="500"></canvas> |
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<script src="https://d3js.org/d3.v4.js"></script> |
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<script src="https://d3js.org/d3-geo-projection.v2.min.js"></script> |
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<script src="https://d3js.org/topojson.v2.min.js"></script> |
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<script src="https://unpkg.com/complex.js"></script> |
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<script> |
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// Approximate \int _0 ^sm(z) dt / (1 - t^3)^(2/3) |
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// sm maps a triangle to a disc, sm^-1 does the opposite |
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function sm_1(s){ |
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var z = Complex(s), |
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w = Complex([-1/2, Math.sqrt(3)/2]), |
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k = Complex(0); |
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// rotate to have s ~= 1 |
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var rot = w.clone().pow(d3.scan([0,1,2].map( |
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i => -(z.clone().mul(w.clone().pow(i))).re |
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))); |
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var y = rot.clone().mul(z).mul(-1).add(1); |
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// var n = 3 |
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// w1 = gamma(1/n) * gamma(1 - 2/n) / n / gamma(1 - 1/n) |
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// https://bl.ocks.org/Fil/852557838117687bbd985e4b38ff77d4 |
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var w1 = 1.7666387502854533; |
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// McIlroy formula 5 p6 and table for F3 page 16 |
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var F0 = [ |
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1.44224957030741, |
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0.240374928384568, |
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0.0686785509670194, |
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0.0178055502507087, |
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0.00228276285265497, |
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-1.48379585422573e-3, |
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-1.64287728109203e-3, |
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-1.02583417082273e-3, |
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-4.83607537673571e-4, |
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-1.67030822094781e-4, |
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-2.45024395166263e-5, |
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2.14092375450951e-5, |
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2.55897270486771e-5, |
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1.73086854400834e-5, |
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8.72756299984649e-6, |
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3.18304486798473e-6, |
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4.79323894565283e-7 |
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-4.58968389565456e-7, |
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-5.62970586787826e-7, |
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-3.92135372833465e-7 |
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]; |
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var F = Complex(0); |
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for (var i = F0.length; i--;) { |
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F = Complex(F0[i]).add(F.mul(y)); |
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} |
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// /!\ mutates y |
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var k = Complex(w1).add(y.pow(1-2/3).mul(-1).mul(F)).mul(rot.pow(2)) |
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// when we are close to [0,0] we switch to another approximation: |
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// https://www.wolframalpha.com/input/?i=(-2%2F3+choose+k)++*+(-1)%5Ek++%2F+(k%2B1)+with+k%3D0,1,2,3,4 |
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// the difference is _very_ tiny but necessary |
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// if we want projection(0,0) === [0,0] |
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var n = z.abs(), m = 0.3; |
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if (n < m) { |
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var H0 = [ |
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1, 1/3, 5/27, 10/81, 22/243 //… |
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]; |
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var z3 = z.clone().pow(3); |
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var h = Complex(0); |
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for (i = H0.length; i--;) { |
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h = Complex(H0[i]).add(h.mul(z3)); |
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} |
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h = h.mul(z); |
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k.mul(n / m).add(h.mul(1 - n / m)); |
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} |
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return k.toVector(); |
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} |
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var canvas = d3.select("canvas"), |
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width = canvas.property("width"), |
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height = canvas.property("height"), |
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context = canvas.node().getContext("2d"); |
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// retina display |
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var devicePixelRatio = window.devicePixelRatio || 1; |
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canvas.style('width', canvas.attr('width')+'px'); |
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canvas.style('height', canvas.attr('height')+'px'); |
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canvas.attr('width', canvas.attr('width') * devicePixelRatio); |
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canvas.attr('height', canvas.attr('height') * devicePixelRatio); |
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context.scale(devicePixelRatio,devicePixelRatio); |
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function coxRawN(lambda, phi){ |
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var s = d3.geoLagrangeRaw(0.5)(lambda, phi); |
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var t = sm_1([-s[1] / 2, -s[0] / 2]); |
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return [-t[1], -t[0]] |
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} |
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projection = d3.geoProjection(coxRawN) |
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.rotate([-10.8,0]) |
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.scale(188.68216) |
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.translate([480, 500 * 1/3]) |
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.precision(0.01); |
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// the Sphere should go *exactly* to the vertices of the triangles |
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// because they're singular points |
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function sphere() { |
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var degrees = 180 / Math.PI, |
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c = 2 * Math.asin(1/Math.sqrt(5))*degrees; |
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return { |
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type: "Polygon", |
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coordinates: [ |
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[ [ 0,-90 ], [ -180, c ], [ 0, 90 ], [ 180, c ], [ 0,-90 ] ] |
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] |
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}; |
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} |
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var stream_ = projection.stream; |
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projection.stream = function(stream) { |
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var rotate = projection.rotate(), |
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rotateStream = stream_(stream), |
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sphereStream = (projection.rotate([0, 0]), stream_(stream)); |
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projection.rotate(rotate); |
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rotateStream.sphere = function() { d3.geoStream(sphere(), sphereStream); }; |
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return rotateStream; |
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}; |
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var path = d3.geoPath().projection(projection).context(context); |
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d3.json("https://unpkg.com/world-atlas@1/world/110m.json", function( |
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error, |
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world |
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) { |
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if (error) throw error; |
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var land = topojson.merge(world, world.objects.countries.geometries); |
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render = function() { |
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context.fillStyle = "#fff"; |
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context.fillRect(0, 0, width, height); |
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context.beginPath(); |
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path({type:"Sphere"}); |
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context.strokeStyle = "black"; |
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context.lineWidth = 1.5; |
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context.stroke(), context.closePath(); |
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context.beginPath(); |
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path(land); |
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context.lineWidth = 1; |
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context.strokeStyle = "#000"; |
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context.stroke(); |
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context.fillStyle = "#000"; |
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context.fill(); |
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context.closePath(); |
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context.beginPath(); |
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path(d3.geoGraticule()()); |
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context.strokeStyle = "#777"; |
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context.lineWidth = 0.5; |
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context.stroke(), context.closePath(); |
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}; |
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render(); |
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}); |
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</script> |
