Fil/.block

## .block
license: mit

## README.md

      
    Raw
  

              README.md
            
          
    Conformal projection of the Sphere within an equilateral triangle.
Wallpaper version, ignoring Antarctica
Original research by Philippe Rivière for d3-geo-projection, issue #118.
(See also the North polar aspect.)
Reference papers:


J. F. Cox, Représentation de la surface entière de la terre dans un triangle équilatéral. Bulletin de la Classe des Sciences, Académie Royale de Belgique, 5, 21: 66-71. Bruxelles, 1935.


J. Magis, Calcul du canevas de la représentation conforme de la sphère entière dans un triangle équilatéral. Bulletin Geodésique, 247-256. Paris, 1938.


L.P. Lee, Conformal Projections Based On Dixon Elliptic Functions, Cartographica: The International Journal for Geographic Information and Geovisualization > Vol. 13, # 1, June 1976.


M. Douglas McIlroy, Wallpaper maps, 2011.


 [
](https://github.com/Fil/) Questions and comments welcome on [gitter.im/d3](https://gitter.im/d3/d3), [twitter](https://twitter.com/@recifs) or [slack](https://d3js.slack.com).
<script>
  (function(i,s,o,g,r,a,m){i['GoogleAnalyticsObject']=r;i[r]=i[r]||function(){
  (i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o),
  m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m)
  })(window,document,'script','https://www.google-analytics.com/analytics.js','ga');
  ga('create', 'UA-58621-8', 'auto');
  ga('send', 'pageview');
</script>
forked from Fil's block: Cox Projection
forked from Fil's block: Cox Projection - no ATA

  
## index.html
<!DOCTYPE html>
<canvas width="960" height="500"></canvas>
<script src="https://d3js.org/d3.v4.js"></script>
<script src="https://d3js.org/d3-geo-projection.v2.min.js"></script>
<script src="https://d3js.org/topojson.v2.min.js"></script>
<script src="https://unpkg.com/complex.js"></script>

<script>
  // Approximate \int _0 ^sm(z)  dt / (1 - t^3)^(2/3)
  // sm maps a triangle to a disc, sm^-1 does the opposite
  function sm_1(s){
    var z = Complex(s),
        w = Complex([-1/2, Math.sqrt(3)/2]),
        k = Complex(0);

    // rotate to have s ~= 1
    var rot = w.clone().pow(d3.scan([0,1,2].map(
      i => -(z.clone().mul(w.clone().pow(i))).re
    )));

    var y = rot.clone().mul(z).mul(-1).add(1);

    // var n = 3
    // w1 = gamma(1/n) * gamma(1 - 2/n) / n / gamma(1 - 1/n)
    // https://bl.ocks.org/Fil/852557838117687bbd985e4b38ff77d4
    var w1 = 1.7666387502854533;

    // McIlroy formula 5 p6 and table for F3 page 16
    var F0 = [
      1.44224957030741,
      0.240374928384568,
      0.0686785509670194,
      0.0178055502507087,
      0.00228276285265497,
      -1.48379585422573e-3,
      -1.64287728109203e-3,
      -1.02583417082273e-3,
      -4.83607537673571e-4,
      -1.67030822094781e-4,
      -2.45024395166263e-5,
      2.14092375450951e-5,
      2.55897270486771e-5,
      1.73086854400834e-5,
      8.72756299984649e-6,
      3.18304486798473e-6,
      4.79323894565283e-7
      -4.58968389565456e-7,
      -5.62970586787826e-7,
      -3.92135372833465e-7
    ];

    var F = Complex(0);
    for (var i = F0.length; i--;) {
      F = Complex(F0[i]).add(F.mul(y));
    }

    // /!\ mutates y
    var k = Complex(w1).add(y.pow(1-2/3).mul(-1).mul(F)).mul(rot.pow(2))


    // when we are close to [0,0] we switch to another approximation:
    // https://www.wolframalpha.com/input/?i=(-2%2F3+choose+k)++*+(-1)%5Ek++%2F+(k%2B1)+with+k%3D0,1,2,3,4
    // the difference is _very_ tiny but necessary
    // if we want projection(0,0) === [0,0]
    var n = z.abs(), m = 0.3;
    if (n < m) {
      var H0 = [
        1, 1/3, 5/27, 10/81, 22/243 //…
       ];
      var z3 = z.clone().pow(3);
      var h = Complex(0);
      for (i = H0.length; i--;) {
        h = Complex(H0[i]).add(h.mul(z3));
      }
      h = h.mul(z);
      k.mul(n / m).add(h.mul(1 - n / m));
    }

    return k.toVector();
  }


  var canvas = d3.select("canvas"),
  width = canvas.property("width"),
  height = canvas.property("height"),
  context = canvas.node().getContext("2d");

  // retina display
  var devicePixelRatio = window.devicePixelRatio || 1;
  canvas.style('width', canvas.attr('width')+'px');
  canvas.style('height', canvas.attr('height')+'px');
  canvas.attr('width', canvas.attr('width') * devicePixelRatio);
  canvas.attr('height', canvas.attr('height') * devicePixelRatio);
  context.scale(devicePixelRatio,devicePixelRatio);

  function coxRaw(lambda, phi){
    var s = d3.geoLagrangeRaw(0.5)(lambda, phi);
    var t = sm_1([s[1] / 2, s[0] / 2]);
    return [t[1], t[0]]
  }

projection = d3.geoProjection(coxRaw)
  .rotate([-10.8,0])
  .fitExtent([[-5000, 0], [5000, 270]], {type:"Sphere"})
.precision(0.01);


// the Sphere should go *exactly* to the vertices of the triangles
// because they're singular points
function sphere() {
  var degrees = 180 / Math.PI,
      c = 2 * Math.asin(1/Math.sqrt(5))*degrees;
  return {
    type: "Polygon",
    coordinates: [
      [ [ 0,90 ],  [ -180, -c ], [ 0, -90 ], [ 180, -c ], [ 0,90 ] ]
    ]
  };
}


var path = d3.geoPath().projection(projection).context(context);

d3.json("https://unpkg.com/world-atlas@1/world/110m.json", function(
  error,
  world
) {
  if (error) throw error;

  var land1 = topojson.merge(world, world.objects.countries.geometries
                            .filter(d => d.id !== "010") // Antarctica
                           );
  var land = topojson.merge(world, world.objects.countries.geometries
                            //.filter(d => d.id !== "010") // Antarctica
                           );

  render = function() {

    for(var i = 0; i < 6; i++) {
      context.translate(480, 240);
      context.rotate((i+0.5) * Math.PI/3);

          context.beginPath();
    path({type:"Sphere"});
    context.strokeStyle = "#999";
      context.setLineDash([1,4]);
    context.lineWidth = 0.5;
    context.stroke(), context.closePath();

    context.lineWidth = 0.5;
      context.setLineDash([]);
    context.strokeStyle = "#030a67";
    context.fillStyle = "#c0c1d7";


      context.beginPath();
      path(land1);
      context.fill();
      context.stroke();
      context.closePath();
      context.resetTransform();
      context.scale(devicePixelRatio,devicePixelRatio);
    }


  };

  render();
});

</script>

## thumbnail.png

      
    Raw
  

              thumbnail.png
	<!DOCTYPE html>
	<canvas width="960" height="500"></canvas>
	<script src="https://d3js.org/d3.v4.js"></script>
	<script src="https://d3js.org/d3-geo-projection.v2.min.js"></script>
	<script src="https://d3js.org/topojson.v2.min.js"></script>
	<script src="https://unpkg.com/complex.js"></script>

	<script>
	// Approximate \int _0 ^sm(z) dt / (1 - t^3)^(2/3)
	// sm maps a triangle to a disc, sm^-1 does the opposite
	function sm_1(s){
	var z = Complex(s),
	w = Complex([-1/2, Math.sqrt(3)/2]),
	k = Complex(0);

	// rotate to have s ~= 1
	var rot = w.clone().pow(d3.scan([0,1,2].map(
	i => -(z.clone().mul(w.clone().pow(i))).re
	)));

	var y = rot.clone().mul(z).mul(-1).add(1);

	// var n = 3
	// w1 = gamma(1/n) * gamma(1 - 2/n) / n / gamma(1 - 1/n)
	// https://bl.ocks.org/Fil/852557838117687bbd985e4b38ff77d4
	var w1 = 1.7666387502854533;

	// McIlroy formula 5 p6 and table for F3 page 16
	var F0 = [
	1.44224957030741,
	0.240374928384568,
	0.0686785509670194,
	0.0178055502507087,
	0.00228276285265497,
	-1.48379585422573e-3,
	-1.64287728109203e-3,
	-1.02583417082273e-3,
	-4.83607537673571e-4,
	-1.67030822094781e-4,
	-2.45024395166263e-5,
	2.14092375450951e-5,
	2.55897270486771e-5,
	1.73086854400834e-5,
	8.72756299984649e-6,
	3.18304486798473e-6,
	4.79323894565283e-7
	-4.58968389565456e-7,
	-5.62970586787826e-7,
	-3.92135372833465e-7
	];

	var F = Complex(0);
	for (var i = F0.length; i--;) {
	F = Complex(F0[i]).add(F.mul(y));
	}

	// /!\ mutates y
	var k = Complex(w1).add(y.pow(1-2/3).mul(-1).mul(F)).mul(rot.pow(2))


	// when we are close to [0,0] we switch to another approximation:
	// https://www.wolframalpha.com/input/?i=(-2%2F3+choose+k)++*+(-1)%5Ek++%2F+(k%2B1)+with+k%3D0,1,2,3,4
	// the difference is _very_ tiny but necessary
	// if we want projection(0,0) === [0,0]
	var n = z.abs(), m = 0.3;
	if (n < m) {
	var H0 = [
	1, 1/3, 5/27, 10/81, 22/243 //…
	];
	var z3 = z.clone().pow(3);
	var h = Complex(0);
	for (i = H0.length; i--;) {
	h = Complex(H0[i]).add(h.mul(z3));
	}
	h = h.mul(z);
	k.mul(n / m).add(h.mul(1 - n / m));
	}

	return k.toVector();
	}


	var canvas = d3.select("canvas"),
	width = canvas.property("width"),
	height = canvas.property("height"),
	context = canvas.node().getContext("2d");

	// retina display
	var devicePixelRatio = window.devicePixelRatio \|\| 1;
	canvas.style('width', canvas.attr('width')+'px');
	canvas.style('height', canvas.attr('height')+'px');
	canvas.attr('width', canvas.attr('width') * devicePixelRatio);
	canvas.attr('height', canvas.attr('height') * devicePixelRatio);
	context.scale(devicePixelRatio,devicePixelRatio);

	function coxRaw(lambda, phi){
	var s = d3.geoLagrangeRaw(0.5)(lambda, phi);
	var t = sm_1([s[1] / 2, s[0] / 2]);
	return [t[1], t[0]]
	}

	projection = d3.geoProjection(coxRaw)
	.rotate([-10.8,0])
	.fitExtent([[-5000, 0], [5000, 270]], {type:"Sphere"})
	.precision(0.01);


	// the Sphere should go exactly to the vertices of the triangles
	// because they're singular points
	function sphere() {
	var degrees = 180 / Math.PI,
	c = 2 * Math.asin(1/Math.sqrt(5))*degrees;
	return {
	type: "Polygon",
	coordinates: [
	[ [ 0,90 ], [ -180, -c ], [ 0, -90 ], [ 180, -c ], [ 0,90 ] ]
	]
	};
	}


	var path = d3.geoPath().projection(projection).context(context);

	d3.json("https://unpkg.com/world-atlas@1/world/110m.json", function(
	error,
	world
	) {
	if (error) throw error;

	var land1 = topojson.merge(world, world.objects.countries.geometries
	.filter(d => d.id !== "010") // Antarctica
	);
	var land = topojson.merge(world, world.objects.countries.geometries
	//.filter(d => d.id !== "010") // Antarctica
	);

	render = function() {

	for(var i = 0; i < 6; i++) {
	context.translate(480, 240);
	context.rotate((i+0.5) * Math.PI/3);

	context.beginPath();
	path({type:"Sphere"});
	context.strokeStyle = "#999";
	context.setLineDash([1,4]);
	context.lineWidth = 0.5;
	context.stroke(), context.closePath();

	context.lineWidth = 0.5;
	context.setLineDash([]);
	context.strokeStyle = "#030a67";
	context.fillStyle = "#c0c1d7";


	context.beginPath();
	path(land1);
	context.fill();
	context.stroke();
	context.closePath();
	context.resetTransform();
	context.scale(devicePixelRatio,devicePixelRatio);
	}



	};

	render();
	});

	</script>