Equateur MMSH [UNLISTED]

.block

license: gpl-3.0
height: 600

README.md

L'équateur de Marseille s'appuie sur la Canebière et la prolonge dans les deux directions, jusqu'à faire le tour du monde.

Cette approche a été conçue et développée par Jean-Luc Arnaud dans « Marseille et la Canebière - Echelles du monde », 2017. <halshs-01484193>

Visualisation interactive par Philippe Rivière -- visionscarto.net

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Forked from mbostock's block: Versor Dragging II

forked from Fil's block: L’Équateur de Marseille

forked from 007Carto's block: Equateur MMSH

d3-geo-projection.js

// https://d3js.org/d3-geo-projection/ Version 2.1.0. Copyright 2017 Mike Bostock.
(function (global, factory) {
	typeof exports === 'object' && typeof module !== 'undefined' ? factory(exports, require('d3-geo'), require('d3-array')) :
	typeof define === 'function' && define.amd ? define(['exports', 'd3-geo', 'd3-array'], factory) :
	(factory((global.d3 = global.d3 || {}),global.d3,global.d3));
}(this, (function (exports,d3Geo,d3Array) { 'use strict';

var abs = Math.abs;
var atan = Math.atan;
var atan2 = Math.atan2;

var cos = Math.cos;
var exp = Math.exp;
var floor = Math.floor;
var log = Math.log;
var max = Math.max;
var min = Math.min;
var pow = Math.pow;
var round = Math.round;
var sign = Math.sign || function(x) { return x > 0 ? 1 : x < 0 ? -1 : 0; };
var sin = Math.sin;
var tan = Math.tan;

var epsilon = 1e-6;
var epsilon2 = 1e-12;
var pi = Math.PI;
var halfPi = pi / 2;
var quarterPi = pi / 4;
var sqrt1_2 = Math.SQRT1_2;
var sqrt2 = sqrt(2);
var sqrtPi = sqrt(pi);
var tau = pi * 2;
var degrees = 180 / pi;
var radians = pi / 180;

function sinci(x) {
  return x ? x / Math.sin(x) : 1;
}

function asin(x) {
  return x > 1 ? halfPi : x < -1 ? -halfPi : Math.asin(x);
}

function acos(x) {
  return x > 1 ? 0 : x < -1 ? pi : Math.acos(x);
}

function sqrt(x) {
  return x > 0 ? Math.sqrt(x) : 0;
}

function tanh(x) {
  x = exp(2 * x);
  return (x - 1) / (x + 1);
}

function sinh(x) {
  return (exp(x) - exp(-x)) / 2;
}

function cosh(x) {
  return (exp(x) + exp(-x)) / 2;
}

function arsinh(x) {
  return log(x + sqrt(x * x + 1));
}

function arcosh(x) {
  return log(x + sqrt(x * x - 1));
}

function airyRaw(beta) {
  var tanBeta_2 = tan(beta / 2),
      b = 2 * log(cos(beta / 2)) / (tanBeta_2 * tanBeta_2);

  function forward(x, y) {
    var cosx = cos(x),
        cosy = cos(y),
        siny = sin(y),
        cosz = cosy * cosx,
        k = -((1 - cosz ? log((1 + cosz) / 2) / (1 - cosz) : -0.5) + b / (1 + cosz));
    return [k * cosy * sin(x), k * siny];
  }

  forward.invert = function(x, y) {
    var r = sqrt(x * x + y * y),
        z = -beta / 2,
        i = 50, delta;
    if (!r) return [0, 0];
    do {
      var z_2 = z / 2,
          cosz_2 = cos(z_2),
          sinz_2 = sin(z_2),
          tanz_2 = tan(z_2),
          lnsecz_2 = log(1 / cosz_2);
      z -= delta = (2 / tanz_2 * lnsecz_2 - b * tanz_2 - r) / (-lnsecz_2 / (sinz_2 * sinz_2) + 1 - b / (2 * cosz_2 * cosz_2));
    } while (abs(delta) > epsilon && --i > 0);
    var sinz = sin(z);
    return [atan2(x * sinz, r * cos(z)), asin(y * sinz / r)];
  };

  return forward;
}

var airy = function() {
  var beta = halfPi,
      m = d3Geo.geoProjectionMutator(airyRaw),
      p = m(beta);

  p.radius = function(_) {
    return arguments.length ? m(beta = _ * radians) : beta * degrees;
  };

  return p
      .scale(179.976)
      .clipAngle(147);
};

function aitoffRaw(x, y) {
  var cosy = cos(y), sincia = sinci(acos(cosy * cos(x /= 2)));
  return [2 * cosy * sin(x) * sincia, sin(y) * sincia];
}

// Abort if [x, y] is not within an ellipse centered at [0, 0] with
// semi-major axis pi and semi-minor axis pi/2.
aitoffRaw.invert = function(x, y) {
  if (x * x + 4 * y * y > pi * pi + epsilon) return;
  var x1 = x, y1 = y, i = 25;
  do {
    var sinx = sin(x1),
        sinx_2 = sin(x1 / 2),
        cosx_2 = cos(x1 / 2),
        siny = sin(y1),
        cosy = cos(y1),
        sin_2y = sin(2 * y1),
        sin2y = siny * siny,
        cos2y = cosy * cosy,
        sin2x_2 = sinx_2 * sinx_2,
        c = 1 - cos2y * cosx_2 * cosx_2,
        e = c ? acos(cosy * cosx_2) * sqrt(f = 1 / c) : f = 0,
        f,
        fx = 2 * e * cosy * sinx_2 - x,
        fy = e * siny - y,
        dxdx = f * (cos2y * sin2x_2 + e * cosy * cosx_2 * sin2y),
        dxdy = f * (0.5 * sinx * sin_2y - e * 2 * siny * sinx_2),
        dydx = f * 0.25 * (sin_2y * sinx_2 - e * siny * cos2y * sinx),
        dydy = f * (sin2y * cosx_2 + e * sin2x_2 * cosy),
        z = dxdy * dydx - dydy * dxdx;
    if (!z) break;
    var dx = (fy * dxdy - fx * dydy) / z,
        dy = (fx * dydx - fy * dxdx) / z;
    x1 -= dx, y1 -= dy;
  } while ((abs(dx) > epsilon || abs(dy) > epsilon) && --i > 0);
  return [x1, y1];
};

var aitoff = function() {
  return d3Geo.geoProjection(aitoffRaw)
      .scale(152.63);
};

function armadilloRaw(phi0) {
  var sinPhi0 = sin(phi0),
      cosPhi0 = cos(phi0),
      sPhi0 = phi0 >= 0 ? 1 : -1,
      tanPhi0 = tan(sPhi0 * phi0),
      k = (1 + sinPhi0 - cosPhi0) / 2;

  function forward(lambda, phi) {
    var cosPhi = cos(phi),
        cosLambda = cos(lambda /= 2);
    return [
      (1 + cosPhi) * sin(lambda),
      (sPhi0 * phi > -atan2(cosLambda, tanPhi0) - 1e-3 ? 0 : -sPhi0 * 10) + k + sin(phi) * cosPhi0 - (1 + cosPhi) * sinPhi0 * cosLambda // TODO D3 core should allow null or [NaN, NaN] to be returned.
    ];
  }

  forward.invert = function(x, y) {
    var lambda = 0,
        phi = 0,
        i = 50;
    do {
      var cosLambda = cos(lambda),
          sinLambda = sin(lambda),
          cosPhi = cos(phi),
          sinPhi = sin(phi),
          A = 1 + cosPhi,
          fx = A * sinLambda - x,
          fy = k + sinPhi * cosPhi0 - A * sinPhi0 * cosLambda - y,
          dxdLambda = A * cosLambda / 2,
          dxdPhi = -sinLambda * sinPhi,
          dydLambda = sinPhi0 * A * sinLambda / 2,
          dydPhi = cosPhi0 * cosPhi + sinPhi0 * cosLambda * sinPhi,
          denominator = dxdPhi * dydLambda - dydPhi * dxdLambda,
          dLambda = (fy * dxdPhi - fx * dydPhi) / denominator / 2,
          dPhi = (fx * dydLambda - fy * dxdLambda) / denominator;
      lambda -= dLambda, phi -= dPhi;
    } while ((abs(dLambda) > epsilon || abs(dPhi) > epsilon) && --i > 0);
    return sPhi0 * phi > -atan2(cos(lambda), tanPhi0) - 1e-3 ? [lambda * 2, phi] : null;
  };

  return forward;
}

var armadillo = function() {
  var phi0 = 20 * radians,
      sPhi0 = phi0 >= 0 ? 1 : -1,
      tanPhi0 = tan(sPhi0 * phi0),
      m = d3Geo.geoProjectionMutator(armadilloRaw),
      p = m(phi0),
      stream_ = p.stream;

  p.parallel = function(_) {
    if (!arguments.length) return phi0 * degrees;
    tanPhi0 = tan((sPhi0 = (phi0 = _ * radians) >= 0 ? 1 : -1) * phi0);
    return m(phi0);
  };

  p.stream = function(stream) {
    var rotate = p.rotate(),
        rotateStream = stream_(stream),
        sphereStream = (p.rotate([0, 0]), stream_(stream));
    p.rotate(rotate);
    rotateStream.sphere = function() {
      sphereStream.polygonStart(), sphereStream.lineStart();
      for (var lambda = sPhi0 * -180; sPhi0 * lambda < 180; lambda += sPhi0 * 90) sphereStream.point(lambda, sPhi0 * 90);
      while (sPhi0 * (lambda -= phi0) >= -180) { // TODO precision?
        sphereStream.point(lambda, sPhi0 * -atan2(cos(lambda * radians / 2), tanPhi0) * degrees);
      }
      sphereStream.lineEnd(), sphereStream.polygonEnd();
    };
    return rotateStream;
  };

  return p
      .scale(218.695)
      .center([0, 28.0974]);
};

function augustRaw(lambda, phi) {
  var tanPhi = tan(phi / 2),
      k = sqrt(1 - tanPhi * tanPhi),
      c = 1 + k * cos(lambda /= 2),
      x = sin(lambda) * k / c,
      y = tanPhi / c,
      x2 = x * x,
      y2 = y * y;
  return [
    4 / 3 * x * (3 + x2 - 3 * y2),
    4 / 3 * y * (3 + 3 * x2 - y2)
  ];
}

augustRaw.invert = function(x, y) {
  x *= 3 / 8, y *= 3 / 8;
  if (!x && abs(y) > 1) return null;
  var x2 = x * x,
      y2 = y * y,
      s = 1 + x2 + y2,
      sin3Eta = sqrt((s - sqrt(s * s - 4 * y * y)) / 2),
      eta = asin(sin3Eta) / 3,
      xi = sin3Eta ? arcosh(abs(y / sin3Eta)) / 3 : arsinh(abs(x)) / 3,
      cosEta = cos(eta),
      coshXi = cosh(xi),
      d = coshXi * coshXi - cosEta * cosEta;
  return [
    sign(x) * 2 * atan2(sinh(xi) * cosEta, 0.25 - d),
    sign(y) * 2 * atan2(coshXi * sin(eta), 0.25 + d)
  ];
};

var august = function() {
  return d3Geo.geoProjection(augustRaw)
      .scale(66.1603);
};

var sqrt8 = sqrt(8);
var phi0 = log(1 + sqrt2);

function bakerRaw(lambda, phi) {
  var phi0 = abs(phi);
  return phi0 < quarterPi
      ? [lambda, log(tan(quarterPi + phi / 2))]
      : [lambda * cos(phi0) * (2 * sqrt2 - 1 / sin(phi0)), sign(phi) * (2 * sqrt2 * (phi0 - quarterPi) - log(tan(phi0 / 2)))];
}

bakerRaw.invert = function(x, y) {
  if ((y0 = abs(y)) < phi0) return [x, 2 * atan(exp(y)) - halfPi];
  var phi = quarterPi, i = 25, delta, y0;
  do {
    var cosPhi_2 = cos(phi / 2), tanPhi_2 = tan(phi / 2);
    phi -= delta = (sqrt8 * (phi - quarterPi) - log(tanPhi_2) - y0) / (sqrt8 - cosPhi_2 * cosPhi_2 / (2 * tanPhi_2));
  } while (abs(delta) > epsilon2 && --i > 0);
  return [x / (cos(phi) * (sqrt8 - 1 / sin(phi))), sign(y) * phi];
};

var baker = function() {
  return d3Geo.geoProjection(bakerRaw)
      .scale(112.314);
};

function berghausRaw(lobes) {
  var k = 2 * pi / lobes;

  function forward(lambda, phi) {
    var p = d3Geo.geoAzimuthalEquidistantRaw(lambda, phi);
    if (abs(lambda) > halfPi) { // back hemisphere
      var theta = atan2(p[1], p[0]),
          r = sqrt(p[0] * p[0] + p[1] * p[1]),
          theta0 = k * round((theta - halfPi) / k) + halfPi,
          alpha = atan2(sin(theta -= theta0), 2 - cos(theta)); // angle relative to lobe end
      theta = theta0 + asin(pi / r * sin(alpha)) - alpha;
      p[0] = r * cos(theta);
      p[1] = r * sin(theta);
    }
    return p;
  }

  forward.invert = function(x, y) {
    var r = sqrt(x * x + y * y);
    if (r > halfPi) {
      var theta = atan2(y, x),
          theta0 = k * round((theta - halfPi) / k) + halfPi,
          s = theta > theta0 ? -1 : 1,
          A = r * cos(theta0 - theta),
          cotAlpha = 1 / tan(s * acos((A - pi) / sqrt(pi * (pi - 2 * A) + r * r)));
      theta = theta0 + 2 * atan((cotAlpha + s * sqrt(cotAlpha * cotAlpha - 3)) / 3);
      x = r * cos(theta), y = r * sin(theta);
    }
    return d3Geo.geoAzimuthalEquidistantRaw.invert(x, y);
  };

  return forward;
}

var berghaus = function() {
  var lobes = 5,
      m = d3Geo.geoProjectionMutator(berghausRaw),
      p = m(lobes),
      projectionStream = p.stream,
      epsilon$$1 = 1e-2,
      cr = -cos(epsilon$$1 * radians),
      sr = sin(epsilon$$1 * radians);

  p.lobes = function(_) {
    return arguments.length ? m(lobes = +_) : lobes;
  };

  p.stream = function(stream) {
    var rotate = p.rotate(),
        rotateStream = projectionStream(stream),
        sphereStream = (p.rotate([0, 0]), projectionStream(stream));
    p.rotate(rotate);
    rotateStream.sphere = function() {
      sphereStream.polygonStart(), sphereStream.lineStart();
      for (var i = 0, delta = 360 / lobes, delta0 = 2 * pi / lobes, phi = 90 - 180 / lobes, phi0 = halfPi; i < lobes; ++i, phi -= delta, phi0 -= delta0) {
        sphereStream.point(atan2(sr * cos(phi0), cr) * degrees, asin(sr * sin(phi0)) * degrees);
        if (phi < -90) {
          sphereStream.point(-90, -180 - phi - epsilon$$1);
          sphereStream.point(-90, -180 - phi + epsilon$$1);
        } else {
          sphereStream.point(90, phi + epsilon$$1);
          sphereStream.point(90, phi - epsilon$$1);
        }
      }
      sphereStream.lineEnd(), sphereStream.polygonEnd();
    };
    return rotateStream;
  };

  return p
      .scale(87.8076)
      .center([0, 17.1875])
      .clipAngle(180 - 1e-3);
};

function mollweideBromleyTheta(cp, phi) {
  var cpsinPhi = cp * sin(phi), i = 30, delta;
  do phi -= delta = (phi + sin(phi) - cpsinPhi) / (1 + cos(phi));
  while (abs(delta) > epsilon && --i > 0);
  return phi / 2;
}

function mollweideBromleyRaw(cx, cy, cp) {

  function forward(lambda, phi) {
    return [cx * lambda * cos(phi = mollweideBromleyTheta(cp, phi)), cy * sin(phi)];
  }

  forward.invert = function(x, y) {
    return y = asin(y / cy), [x / (cx * cos(y)), asin((2 * y + sin(2 * y)) / cp)];
  };

  return forward;
}

var mollweideRaw = mollweideBromleyRaw(sqrt2 / halfPi, sqrt2, pi);

var mollweide = function() {
  return d3Geo.geoProjection(mollweideRaw)
      .scale(169.529);
};

var k = 2.00276;
var w = 1.11072;

function boggsRaw(lambda, phi) {
  var theta = mollweideBromleyTheta(pi, phi);
  return [k * lambda / (1 / cos(phi) + w / cos(theta)), (phi + sqrt2 * sin(theta)) / k];
}

boggsRaw.invert = function(x, y) {
  var ky = k * y, theta = y < 0 ? -quarterPi : quarterPi, i = 25, delta, phi;
  do {
    phi = ky - sqrt2 * sin(theta);
    theta -= delta = (sin(2 * theta) + 2 * theta - pi * sin(phi)) / (2 * cos(2 * theta) + 2 + pi * cos(phi) * sqrt2 * cos(theta));
  } while (abs(delta) > epsilon && --i > 0);
  phi = ky - sqrt2 * sin(theta);
  return [x * (1 / cos(phi) + w / cos(theta)) / k, phi];
};

var boggs = function() {
  return d3Geo.geoProjection(boggsRaw)
      .scale(160.857);
};

var parallel1 = function(projectAt) {
  var phi0 = 0,
      m = d3Geo.geoProjectionMutator(projectAt),
      p = m(phi0);

  p.parallel = function(_) {
    return arguments.length ? m(phi0 = _ * radians) : phi0 * degrees;
  };

  return p;
};

function sinusoidalRaw(lambda, phi) {
  return [lambda * cos(phi), phi];
}

sinusoidalRaw.invert = function(x, y) {
  return [x / cos(y), y];
};

var sinusoidal = function() {
  return d3Geo.geoProjection(sinusoidalRaw)
      .scale(152.63);
};

function bonneRaw(phi0) {
  if (!phi0) return sinusoidalRaw;
  var cotPhi0 = 1 / tan(phi0);

  function forward(lambda, phi) {
    var rho = cotPhi0 + phi0 - phi,
        e = rho ? lambda * cos(phi) / rho : rho;
    return [rho * sin(e), cotPhi0 - rho * cos(e)];
  }

  forward.invert = function(x, y) {
    var rho = sqrt(x * x + (y = cotPhi0 - y) * y),
        phi = cotPhi0 + phi0 - rho;
    return [rho / cos(phi) * atan2(x, y), phi];
  };

  return forward;
}

var bonne = function() {
  return parallel1(bonneRaw)
      .scale(123.082)
      .center([0, 26.1441])
      .parallel(45);
};

function bottomleyRaw(sinPsi) {

  function forward(lambda, phi) {
    var rho = halfPi - phi,
        eta = rho ? lambda * sinPsi * sin(rho) / rho : rho;
    return [rho * sin(eta) / sinPsi, halfPi - rho * cos(eta)];
  }

  forward.invert = function(x, y) {
    var x1 = x * sinPsi,
        y1 = halfPi - y,
        rho = sqrt(x1 * x1 + y1 * y1),
        eta = atan2(x1, y1);
    return [(rho ? rho / sin(rho) : 1) * eta / sinPsi, halfPi - rho];
  };

  return forward;
}

var bottomley = function() {
  var sinPsi = 0.5,
      m = d3Geo.geoProjectionMutator(bottomleyRaw),
      p = m(sinPsi);

  p.fraction = function(_) {
    return arguments.length ? m(sinPsi = +_) : sinPsi;
  };

  return p
      .scale(158.837);
};

var bromleyRaw = mollweideBromleyRaw(1, 4 / pi, pi);

var bromley = function() {
  return d3Geo.geoProjection(bromleyRaw)
      .scale(152.63);
};

// Azimuthal distance.
function distance(dPhi, c1, s1, c2, s2, dLambda) {
  var cosdLambda = cos(dLambda), r;
  if (abs(dPhi) > 1 || abs(dLambda) > 1) {
    r = acos(s1 * s2 + c1 * c2 * cosdLambda);
  } else {
    var sindPhi = sin(dPhi / 2), sindLambda = sin(dLambda / 2);
    r = 2 * asin(sqrt(sindPhi * sindPhi + c1 * c2 * sindLambda * sindLambda));
  }
  return abs(r) > epsilon ? [r, atan2(c2 * sin(dLambda), c1 * s2 - s1 * c2 * cosdLambda)] : [0, 0];
}

// Angle opposite a, and contained between sides of lengths b and c.
function angle(b, c, a) {
  return acos((b * b + c * c - a * a) / (2 * b * c));
}

// Normalize longitude.
function longitude(lambda) {
  return lambda - 2 * pi * floor((lambda + pi) / (2 * pi));
}

function chamberlinRaw(p0, p1, p2) {
  var points = [
    [p0[0], p0[1], sin(p0[1]), cos(p0[1])],
    [p1[0], p1[1], sin(p1[1]), cos(p1[1])],
    [p2[0], p2[1], sin(p2[1]), cos(p2[1])]
  ];

  for (var a = points[2], b, i = 0; i < 3; ++i, a = b) {
    b = points[i];
    a.v = distance(b[1] - a[1], a[3], a[2], b[3], b[2], b[0] - a[0]);
    a.point = [0, 0];
  }

  var beta0 = angle(points[0].v[0], points[2].v[0], points[1].v[0]),
      beta1 = angle(points[0].v[0], points[1].v[0], points[2].v[0]),
      beta2 = pi - beta0;

  points[2].point[1] = 0;
  points[0].point[0] = -(points[1].point[0] = points[0].v[0] / 2);

  var mean = [
    points[2].point[0] = points[0].point[0] + points[2].v[0] * cos(beta0),
    2 * (points[0].point[1] = points[1].point[1] = points[2].v[0] * sin(beta0))
  ];

  function forward(lambda, phi) {
    var sinPhi = sin(phi),
        cosPhi = cos(phi),
        v = new Array(3), i;

    // Compute distance and azimuth from control points.
    for (i = 0; i < 3; ++i) {
      var p = points[i];
      v[i] = distance(phi - p[1], p[3], p[2], cosPhi, sinPhi, lambda - p[0]);
      if (!v[i][0]) return p.point;
      v[i][1] = longitude(v[i][1] - p.v[1]);
    }

    // Arithmetic mean of interception points.
    var point = mean.slice();
    for (i = 0; i < 3; ++i) {
      var j = i == 2 ? 0 : i + 1;
      var a = angle(points[i].v[0], v[i][0], v[j][0]);
      if (v[i][1] < 0) a = -a;

      if (!i) {
        point[0] += v[i][0] * cos(a);
        point[1] -= v[i][0] * sin(a);
      } else if (i == 1) {
        a = beta1 - a;
        point[0] -= v[i][0] * cos(a);
        point[1] -= v[i][0] * sin(a);
      } else {
        a = beta2 - a;
        point[0] += v[i][0] * cos(a);
        point[1] += v[i][0] * sin(a);
      }
    }

    point[0] /= 3, point[1] /= 3;
    return point;
  }

  return forward;
}

function pointRadians(p) {
  return p[0] *= radians, p[1] *= radians, p;
}

function chamberlinAfrica() {
  return chamberlin([0, 22], [45, 22], [22.5, -22])
      .scale(380)
      .center([22.5, 2]);
}

function chamberlin(p0, p1, p2) { // TODO order matters!
  var c = d3Geo.geoCentroid({type: "MultiPoint", coordinates: [p0, p1, p2]}),
      R = [-c[0], -c[1]],
      r = d3Geo.geoRotation(R),
      p = d3Geo.geoProjection(chamberlinRaw(pointRadians(r(p0)), pointRadians(r(p1)), pointRadians(r(p2)))).rotate(R),
      center = p.center;

  delete p.rotate;

  p.center = function(_) {
    return arguments.length ? center(r(_)) : r.invert(center());
  };

  return p
      .clipAngle(90);
}

function collignonRaw(lambda, phi) {
  var alpha = sqrt(1 - sin(phi));
  return [(2 / sqrtPi) * lambda * alpha, sqrtPi * (1 - alpha)];
}

collignonRaw.invert = function(x, y) {
  var lambda = (lambda = y / sqrtPi - 1) * lambda;
  return [lambda > 0 ? x * sqrt(pi / lambda) / 2 : 0, asin(1 - lambda)];
};

var collignon = function() {
  return d3Geo.geoProjection(collignonRaw)
      .scale(95.6464)
      .center([0, 30]);
};

function craigRaw(phi0) {
  var tanPhi0 = tan(phi0);

  function forward(lambda, phi) {
    return [lambda, (lambda ? lambda / sin(lambda) : 1) * (sin(phi) * cos(lambda) - tanPhi0 * cos(phi))];
  }

  forward.invert = tanPhi0 ? function(x, y) {
    if (x) y *= sin(x) / x;
    var cosLambda = cos(x);
    return [x, 2 * atan2(sqrt(cosLambda * cosLambda + tanPhi0 * tanPhi0 - y * y) - cosLambda, tanPhi0 - y)];
  } : function(x, y) {
    return [x, asin(x ? y * tan(x) / x : y)];
  };

  return forward;
}

var craig = function() {
  return parallel1(craigRaw)
      .scale(249.828)
      .clipAngle(90);
};

var sqrt3 = sqrt(3);

function crasterRaw(lambda, phi) {
  return [sqrt3 * lambda * (2 * cos(2 * phi / 3) - 1) / sqrtPi, sqrt3 * sqrtPi * sin(phi / 3)];
}

crasterRaw.invert = function(x, y) {
  var phi = 3 * asin(y / (sqrt3 * sqrtPi));
  return [sqrtPi * x / (sqrt3 * (2 * cos(2 * phi / 3) - 1)), phi];
};

var craster = function() {
  return d3Geo.geoProjection(crasterRaw)
      .scale(156.19);
};

function cylindricalEqualAreaRaw(phi0) {
  var cosPhi0 = cos(phi0);

  function forward(lambda, phi) {
    return [lambda * cosPhi0, sin(phi) / cosPhi0];
  }

  forward.invert = function(x, y) {
    return [x / cosPhi0, asin(y * cosPhi0)];
  };

  return forward;
}

var cylindricalEqualArea = function() {
  return parallel1(cylindricalEqualAreaRaw)
      .parallel(38.58) // acos(sqrt(width / height / pi)) * radians
      .scale(195.044); // width / (sqrt(width / height / pi) * 2 * pi)
};

function cylindricalStereographicRaw(phi0) {
  var cosPhi0 = cos(phi0);

  function forward(lambda, phi) {
    return [lambda * cosPhi0, (1 + cosPhi0) * tan(phi / 2)];
  }

  forward.invert = function(x, y) {
    return [x / cosPhi0, atan(y / (1 + cosPhi0)) * 2];
  };

  return forward;
}

var cylindricalStereographic = function() {
  return parallel1(cylindricalStereographicRaw)
      .scale(124.75);
};

function eckert1Raw(lambda, phi) {
  var alpha = sqrt(8 / (3 * pi));
  return [
    alpha * lambda * (1 - abs(phi) / pi),
    alpha * phi
  ];
}

eckert1Raw.invert = function(x, y) {
  var alpha = sqrt(8 / (3 * pi)),
      phi = y / alpha;
  return [
    x / (alpha * (1 - abs(phi) / pi)),
    phi
  ];
};

var eckert1 = function() {
  return d3Geo.geoProjection(eckert1Raw)
      .scale(165.664);
};

function eckert2Raw(lambda, phi) {
  var alpha = sqrt(4 - 3 * sin(abs(phi)));
  return [
    2 / sqrt(6 * pi) * lambda * alpha,
    sign(phi) * sqrt(2 * pi / 3) * (2 - alpha)
  ];
}

eckert2Raw.invert = function(x, y) {
  var alpha = 2 - abs(y) / sqrt(2 * pi / 3);
  return [
    x * sqrt(6 * pi) / (2 * alpha),
    sign(y) * asin((4 - alpha * alpha) / 3)
  ];
};

var eckert2 = function() {
  return d3Geo.geoProjection(eckert2Raw)
      .scale(165.664);
};

function eckert3Raw(lambda, phi) {
  var k = sqrt(pi * (4 + pi));
  return [
    2 / k * lambda * (1 + sqrt(1 - 4 * phi * phi / (pi * pi))),
    4 / k * phi
  ];
}

eckert3Raw.invert = function(x, y) {
  var k = sqrt(pi * (4 + pi)) / 2;
  return [
    x * k / (1 + sqrt(1 - y * y * (4 + pi) / (4 * pi))),
    y * k / 2
  ];
};

var eckert3 = function() {
  return d3Geo.geoProjection(eckert3Raw)
      .scale(180.739);
};

function eckert4Raw(lambda, phi) {
  var k = (2 + halfPi) * sin(phi);
  phi /= 2;
  for (var i = 0, delta = Infinity; i < 10 && abs(delta) > epsilon; i++) {
    var cosPhi = cos(phi);
    phi -= delta = (phi + sin(phi) * (cosPhi + 2) - k) / (2 * cosPhi * (1 + cosPhi));
  }
  return [
    2 / sqrt(pi * (4 + pi)) * lambda * (1 + cos(phi)),
    2 * sqrt(pi / (4 + pi)) * sin(phi)
  ];
}

eckert4Raw.invert = function(x, y) {
  var A = y * sqrt((4 + pi) / pi) / 2,
      k = asin(A),
      c = cos(k);
  return [
    x / (2 / sqrt(pi * (4 + pi)) * (1 + c)),
    asin((k + A * (c + 2)) / (2 + halfPi))
  ];
};

var eckert4 = function() {
  return d3Geo.geoProjection(eckert4Raw)
      .scale(180.739);
};

function eckert5Raw(lambda, phi) {
  return [
    lambda * (1 + cos(phi)) / sqrt(2 + pi),
    2 * phi / sqrt(2 + pi)
  ];
}

eckert5Raw.invert = function(x, y) {
  var k = sqrt(2 + pi),
      phi = y * k / 2;
  return [
    k * x / (1 + cos(phi)),
    phi
  ];
};

var eckert5 = function() {
  return d3Geo.geoProjection(eckert5Raw)
      .scale(173.044);
};

function eckert6Raw(lambda, phi) {
  var k = (1 + halfPi) * sin(phi);
  for (var i = 0, delta = Infinity; i < 10 && abs(delta) > epsilon; i++) {
    phi -= delta = (phi + sin(phi) - k) / (1 + cos(phi));
  }
  k = sqrt(2 + pi);
  return [
    lambda * (1 + cos(phi)) / k,
    2 * phi / k
  ];
}

eckert6Raw.invert = function(x, y) {
  var j = 1 + halfPi,
      k = sqrt(j / 2);
  return [
    x * 2 * k / (1 + cos(y *= k)),
    asin((y + sin(y)) / j)
  ];
};

var eckert6 = function() {
  return d3Geo.geoProjection(eckert6Raw)
      .scale(173.044);
};

var eisenlohrK = 3 + 2 * sqrt2;

function eisenlohrRaw(lambda, phi) {
  var s0 = sin(lambda /= 2),
      c0 = cos(lambda),
      k = sqrt(cos(phi)),
      c1 = cos(phi /= 2),
      t = sin(phi) / (c1 + sqrt2 * c0 * k),
      c = sqrt(2 / (1 + t * t)),
      v = sqrt((sqrt2 * c1 + (c0 + s0) * k) / (sqrt2 * c1 + (c0 - s0) * k));
  return [
    eisenlohrK * (c * (v - 1 / v) - 2 * log(v)),
    eisenlohrK * (c * t * (v + 1 / v) - 2 * atan(t))
  ];
}

eisenlohrRaw.invert = function(x, y) {
  if (!(p = augustRaw.invert(x / 1.2, y * 1.065))) return null;
  var lambda = p[0], phi = p[1], i = 20, p;
  x /= eisenlohrK, y /= eisenlohrK;
  do {
    var _0 = lambda / 2,
        _1 = phi / 2,
        s0 = sin(_0),
        c0 = cos(_0),
        s1 = sin(_1),
        c1 = cos(_1),
        cos1 = cos(phi),
        k = sqrt(cos1),
        t = s1 / (c1 + sqrt2 * c0 * k),
        t2 = t * t,
        c = sqrt(2 / (1 + t2)),
        v0 = (sqrt2 * c1 + (c0 + s0) * k),
        v1 = (sqrt2 * c1 + (c0 - s0) * k),
        v2 = v0 / v1,
        v = sqrt(v2),
        vm1v = v - 1 / v,
        vp1v = v + 1 / v,
        fx = c * vm1v - 2 * log(v) - x,
        fy = c * t * vp1v - 2 * atan(t) - y,
        deltatDeltaLambda = s1 && sqrt1_2 * k * s0 * t2 / s1,
        deltatDeltaPhi = (sqrt2 * c0 * c1 + k) / (2 * (c1 + sqrt2 * c0 * k) * (c1 + sqrt2 * c0 * k) * k),
        deltacDeltat = -0.5 * t * c * c * c,
        deltacDeltaLambda = deltacDeltat * deltatDeltaLambda,
        deltacDeltaPhi = deltacDeltat * deltatDeltaPhi,
        A = (A = 2 * c1 + sqrt2 * k * (c0 - s0)) * A * v,
        deltavDeltaLambda = (sqrt2 * c0 * c1 * k + cos1) / A,
        deltavDeltaPhi = -(sqrt2 * s0 * s1) / (k * A),
        deltaxDeltaLambda = vm1v * deltacDeltaLambda - 2 * deltavDeltaLambda / v + c * (deltavDeltaLambda + deltavDeltaLambda / v2),
        deltaxDeltaPhi = vm1v * deltacDeltaPhi - 2 * deltavDeltaPhi / v + c * (deltavDeltaPhi + deltavDeltaPhi / v2),
        deltayDeltaLambda = t * vp1v * deltacDeltaLambda - 2 * deltatDeltaLambda / (1 + t2) + c * vp1v * deltatDeltaLambda + c * t * (deltavDeltaLambda - deltavDeltaLambda / v2),
        deltayDeltaPhi = t * vp1v * deltacDeltaPhi - 2 * deltatDeltaPhi / (1 + t2) + c * vp1v * deltatDeltaPhi + c * t * (deltavDeltaPhi - deltavDeltaPhi / v2),
        denominator = deltaxDeltaPhi * deltayDeltaLambda - deltayDeltaPhi * deltaxDeltaLambda;
    if (!denominator) break;
    var deltaLambda = (fy * deltaxDeltaPhi - fx * deltayDeltaPhi) / denominator,
        deltaPhi = (fx * deltayDeltaLambda - fy * deltaxDeltaLambda) / denominator;
    lambda -= deltaLambda;
    phi = max(-halfPi, min(halfPi, phi - deltaPhi));
  } while ((abs(deltaLambda) > epsilon || abs(deltaPhi) > epsilon) && --i > 0);
  return abs(abs(phi) - halfPi) < epsilon ? [0, phi] : i && [lambda, phi];
};

var eisenlohr = function() {
  return d3Geo.geoProjection(eisenlohrRaw)
      .scale(62.5271);
};

var faheyK = cos(35 * radians);

function faheyRaw(lambda, phi) {
  var t = tan(phi / 2);
  return [lambda * faheyK * sqrt(1 - t * t), (1 + faheyK) * t];
}

faheyRaw.invert = function(x, y) {
  var t = y / (1 + faheyK);
  return [x && x / (faheyK * sqrt(1 - t * t)), 2 * atan(t)];
};

var fahey = function() {
  return d3Geo.geoProjection(faheyRaw)
      .scale(137.152);
};

function foucautRaw(lambda, phi) {
  var k = phi / 2, cosk = cos(k);
  return [ 2 * lambda / sqrtPi * cos(phi) * cosk * cosk, sqrtPi * tan(k)];
}

foucautRaw.invert = function(x, y) {
  var k = atan(y / sqrtPi), cosk = cos(k), phi = 2 * k;
  return [x * sqrtPi / 2 / (cos(phi) * cosk * cosk), phi];
};

var foucaut = function() {
  return d3Geo.geoProjection(foucautRaw)
      .scale(135.264);
};

function gilbertForward(point) {
  return [point[0] / 2, asin(tan(point[1] / 2 * radians)) * degrees];
}

function gilbertInvert(point) {
  return [point[0] * 2, 2 * atan(sin(point[1] * radians)) * degrees];
}

var gilbert = function(projectionType) {
  if (projectionType == null) projectionType = d3Geo.geoOrthographic;
  var projection = projectionType(),
      equirectangular = d3Geo.geoEquirectangular().scale(degrees).precision(0).clipAngle(null).translate([0, 0]); // antimeridian cutting

  function gilbert(point) {
    return projection(gilbertForward(point));
  }

  if (projection.invert) gilbert.invert = function(point) {
    return gilbertInvert(projection.invert(point));
  };

  gilbert.stream = function(stream) {
    var s1 = projection.stream(stream), s0 = equirectangular.stream({
      point: function(lambda, phi) { s1.point(lambda / 2, asin(tan(-phi / 2 * radians)) * degrees); },
      lineStart: function() { s1.lineStart(); },
      lineEnd: function() { s1.lineEnd(); },
      polygonStart: function() { s1.polygonStart(); },
      polygonEnd: function() { s1.polygonEnd(); }
    });
    s0.sphere = s1.sphere;
    return s0;
  };

  function property(name) {
    gilbert[name] = function(_) {
      return arguments.length ? (projection[name](_), gilbert) : projection[name]();
    };
  }

  gilbert.rotate = function(_) {
    return arguments.length ? (equirectangular.rotate(_), gilbert) : equirectangular.rotate();
  };

  gilbert.center = function(_) {
    return arguments.length ? (projection.center(gilbertForward(_)), gilbert) : gilbertInvert(projection.center());
  };

  property("clipAngle");
  property("clipExtent");
  property("scale");
  property("translate");
  property("precision");

  return gilbert
      .scale(249.5);
};

function gingeryRaw(rho, n) {
  var k = 2 * pi / n,
      rho2 = rho * rho;

  function forward(lambda, phi) {
    var p = d3Geo.geoAzimuthalEquidistantRaw(lambda, phi),
        x = p[0],
        y = p[1],
        r2 = x * x + y * y;

    if (r2 > rho2) {
      var r = sqrt(r2),
          theta = atan2(y, x),
          theta0 = k * round(theta / k),
          alpha = theta - theta0,
          rhoCosAlpha = rho * cos(alpha),
          k_ = (rho * sin(alpha) - alpha * sin(rhoCosAlpha)) / (halfPi - rhoCosAlpha),
          s_ = gingeryLength(alpha, k_),
          e = (pi - rho) / gingeryIntegrate(s_, rhoCosAlpha, pi);

      x = r;
      var i = 50, delta;
      do {
        x -= delta = (rho + gingeryIntegrate(s_, rhoCosAlpha, x) * e - r) / (s_(x) * e);
      } while (abs(delta) > epsilon && --i > 0);

      y = alpha * sin(x);
      if (x < halfPi) y -= k_ * (x - halfPi);

      var s = sin(theta0),
          c = cos(theta0);
      p[0] = x * c - y * s;
      p[1] = x * s + y * c;
    }
    return p;
  }

  forward.invert = function(x, y) {
    var r2 = x * x + y * y;
    if (r2 > rho2) {
      var r = sqrt(r2),
          theta = atan2(y, x),
          theta0 = k * round(theta / k),
          dTheta = theta - theta0;

      x = r * cos(dTheta);
      y = r * sin(dTheta);

      var x_halfPi = x - halfPi,
          sinx = sin(x),
          alpha = y / sinx,
          delta = x < halfPi ? Infinity : 0,
          i = 10;

      while (true) {
        var rhosinAlpha = rho * sin(alpha),
            rhoCosAlpha = rho * cos(alpha),
            sinRhoCosAlpha = sin(rhoCosAlpha),
            halfPi_RhoCosAlpha = halfPi - rhoCosAlpha,
            k_ = (rhosinAlpha - alpha * sinRhoCosAlpha) / halfPi_RhoCosAlpha,
            s_ = gingeryLength(alpha, k_);

        if (abs(delta) < epsilon2 || !--i) break;

        alpha -= delta = (alpha * sinx - k_ * x_halfPi - y) / (
          sinx - x_halfPi * 2 * (
            halfPi_RhoCosAlpha * (rhoCosAlpha + alpha * rhosinAlpha * cos(rhoCosAlpha) - sinRhoCosAlpha) -
            rhosinAlpha * (rhosinAlpha - alpha * sinRhoCosAlpha)
          ) / (halfPi_RhoCosAlpha * halfPi_RhoCosAlpha));
      }
      r = rho + gingeryIntegrate(s_, rhoCosAlpha, x) * (pi - rho) / gingeryIntegrate(s_, rhoCosAlpha, pi);
      theta = theta0 + alpha;
      x = r * cos(theta);
      y = r * sin(theta);
    }
    return d3Geo.geoAzimuthalEquidistantRaw.invert(x, y);
  };

  return forward;
}

function gingeryLength(alpha, k) {
  return function(x) {
    var y_ = alpha * cos(x);
    if (x < halfPi) y_ -= k;
    return sqrt(1 + y_ * y_);
  };
}

// Numerical integration: trapezoidal rule.
function gingeryIntegrate(f, a, b) {
  var n = 50,
      h = (b - a) / n,
      s = f(a) + f(b);
  for (var i = 1, x = a; i < n; ++i) s += 2 * f(x += h);
  return s * 0.5 * h;
}

var gingery = function() {
  var n = 6,
      rho = 30 * radians,
      cRho = cos(rho),
      sRho = sin(rho),
      m = d3Geo.geoProjectionMutator(gingeryRaw),
      p = m(rho, n),
      stream_ = p.stream,
      epsilon$$1 = 1e-2,
      cr = -cos(epsilon$$1 * radians),
      sr = sin(epsilon$$1 * radians);

  p.radius = function(_) {
    if (!arguments.length) return rho * degrees;
    cRho = cos(rho = _ * radians);
    sRho = sin(rho);
    return m(rho, n);
  };

  p.lobes = function(_) {
    if (!arguments.length) return n;
    return m(rho, n = +_);
  };

  p.stream = function(stream) {
    var rotate = p.rotate(),
        rotateStream = stream_(stream),
        sphereStream = (p.rotate([0, 0]), stream_(stream));
    p.rotate(rotate);
    rotateStream.sphere = function() {
      sphereStream.polygonStart(), sphereStream.lineStart();
      for (var i = 0, delta = 2 * pi / n, phi = 0; i < n; ++i, phi -= delta) {
        sphereStream.point(atan2(sr * cos(phi), cr) * degrees, asin(sr * sin(phi)) * degrees);
        sphereStream.point(atan2(sRho * cos(phi - delta / 2), cRho) * degrees, asin(sRho * sin(phi - delta / 2)) * degrees);
      }
      sphereStream.lineEnd(), sphereStream.polygonEnd();
    };
    return rotateStream;
  };

  return p
      .rotate([90, -40])
      .scale(91.7095)
      .clipAngle(180 - 1e-3);
};

var ginzburgPolyconicRaw = function(a, b, c, d, e, f, g, h) {
  if (arguments.length < 8) h = 0;

  function forward(lambda, phi) {
    if (!phi) return [a * lambda / pi, 0];
    var phi2 = phi * phi,
        xB = a + phi2 * (b + phi2 * (c + phi2 * d)),
        yB = phi * (e - 1 + phi2 * (f - h + phi2 * g)),
        m = (xB * xB + yB * yB) / (2 * yB),
        alpha = lambda * asin(xB / m) / pi;
    return [m * sin(alpha), phi * (1 + phi2 * h) + m * (1 - cos(alpha))];
  }

  forward.invert = function(x, y) {
    var lambda = pi * x / a,
        phi = y,
        deltaLambda, deltaPhi, i = 50;
    do {
      var phi2 = phi * phi,
          xB = a + phi2 * (b + phi2 * (c + phi2 * d)),
          yB = phi * (e - 1 + phi2 * (f - h + phi2 * g)),
          p = xB * xB + yB * yB,
          q = 2 * yB,
          m = p / q,
          m2 = m * m,
          dAlphadLambda = asin(xB / m) / pi,
          alpha = lambda * dAlphadLambda,
          xB2 = xB * xB,
          dxBdPhi = (2 * b + phi2 * (4 * c + phi2 * 6 * d)) * phi,
          dyBdPhi = e + phi2 * (3 * f + phi2 * 5 * g),
          dpdPhi = 2 * (xB * dxBdPhi + yB * (dyBdPhi - 1)),
          dqdPhi = 2 * (dyBdPhi - 1),
          dmdPhi = (dpdPhi * q - p * dqdPhi) / (q * q),
          cosAlpha = cos(alpha),
          sinAlpha = sin(alpha),
          mcosAlpha = m * cosAlpha,
          msinAlpha = m * sinAlpha,
          dAlphadPhi = ((lambda / pi) * (1 / sqrt(1 - xB2 / m2)) * (dxBdPhi * m - xB * dmdPhi)) / m2,
          fx = msinAlpha - x,
          fy = phi * (1 + phi2 * h) + m - mcosAlpha - y,
          deltaxDeltaPhi = dmdPhi * sinAlpha + mcosAlpha * dAlphadPhi,
          deltaxDeltaLambda = mcosAlpha * dAlphadLambda,
          deltayDeltaPhi = 1 + dmdPhi - (dmdPhi * cosAlpha - msinAlpha * dAlphadPhi),
          deltayDeltaLambda = msinAlpha * dAlphadLambda,
          denominator = deltaxDeltaPhi * deltayDeltaLambda - deltayDeltaPhi * deltaxDeltaLambda;
      if (!denominator) break;
      lambda -= deltaLambda = (fy * deltaxDeltaPhi - fx * deltayDeltaPhi) / denominator;
      phi -= deltaPhi = (fx * deltayDeltaLambda - fy * deltaxDeltaLambda) / denominator;
    } while ((abs(deltaLambda) > epsilon || abs(deltaPhi) > epsilon) && --i > 0);
    return [lambda, phi];
  };

  return forward;
};

var ginzburg4Raw = ginzburgPolyconicRaw(2.8284, -1.6988, 0.75432, -0.18071, 1.76003, -0.38914, 0.042555);

var ginzburg4 = function() {
  return d3Geo.geoProjection(ginzburg4Raw)
      .scale(149.995);
};

var ginzburg5Raw = ginzburgPolyconicRaw(2.583819, -0.835827, 0.170354, -0.038094, 1.543313, -0.411435,0.082742);

var ginzburg5 = function() {
  return d3Geo.geoProjection(ginzburg5Raw)
      .scale(153.93);
};

var ginzburg6Raw = ginzburgPolyconicRaw(5 / 6 * pi, -0.62636, -0.0344, 0, 1.3493, -0.05524, 0, 0.045);

var ginzburg6 = function() {
  return d3Geo.geoProjection(ginzburg6Raw)
      .scale(130.945);
};

function ginzburg8Raw(lambda, phi) {
  var lambda2 = lambda * lambda,
      phi2 = phi * phi;
  return [
    lambda * (1 - 0.162388 * phi2) * (0.87 - 0.000952426 * lambda2 * lambda2),
    phi * (1 + phi2 / 12)
  ];
}

ginzburg8Raw.invert = function(x, y) {
  var lambda = x,
      phi = y,
      i = 50, delta;
  do {
    var phi2 = phi * phi;
    phi -= delta = (phi * (1 + phi2 / 12) - y) / (1 + phi2 / 4);
  } while (abs(delta) > epsilon && --i > 0);
  i = 50;
  x /= 1 -0.162388 * phi2;
  do {
    var lambda4 = (lambda4 = lambda * lambda) * lambda4;
    lambda -= delta = (lambda * (0.87 - 0.000952426 * lambda4) - x) / (0.87 - 0.00476213 * lambda4);
  } while (abs(delta) > epsilon && --i > 0);
  return [lambda, phi];
};

var ginzburg8 = function() {
  return d3Geo.geoProjection(ginzburg8Raw)
      .scale(131.747);
};

var ginzburg9Raw = ginzburgPolyconicRaw(2.6516, -0.76534, 0.19123, -0.047094, 1.36289, -0.13965,0.031762);

var ginzburg9 = function() {
  return d3Geo.geoProjection(ginzburg9Raw)
      .scale(131.087);
};

var squareRaw = function(project) {
  var dx = project(halfPi, 0)[0] - project(-halfPi, 0)[0];

  function projectSquare(lambda, phi) {
    var s = lambda > 0 ? -0.5 : 0.5,
        point = project(lambda + s * pi, phi);
    point[0] -= s * dx;
    return point;
  }

  if (project.invert) projectSquare.invert = function(x, y) {
    var s = x > 0 ? -0.5 : 0.5,
        location = project.invert(x + s * dx, y),
        lambda = location[0] - s * pi;
    if (lambda < -pi) lambda += 2 * pi;
    else if (lambda > pi) lambda -= 2 * pi;
    location[0] = lambda;
    return location;
  };

  return projectSquare;
};

function gringortenRaw(lambda, phi) {
  var sLambda = sign(lambda),
      sPhi = sign(phi),
      cosPhi = cos(phi),
      x = cos(lambda) * cosPhi,
      y = sin(lambda) * cosPhi,
      z = sin(sPhi * phi);
  lambda = abs(atan2(y, z));
  phi = asin(x);
  if (abs(lambda - halfPi) > epsilon) lambda %= halfPi;
  var point = gringortenHexadecant(lambda > pi / 4 ? halfPi - lambda : lambda, phi);
  if (lambda > pi / 4) z = point[0], point[0] = -point[1], point[1] = -z;
  return (point[0] *= sLambda, point[1] *= -sPhi, point);
}

gringortenRaw.invert = function(x, y) {
  if (abs(x) > 1) x = sign(x) * 2 - x;
  if (abs(y) > 1) y = sign(y) * 2 - y;
  var sx = sign(x),
      sy = sign(y),
      x0 = -sx * x,
      y0 = -sy * y,
      t = y0 / x0 < 1,
      p = gringortenHexadecantInvert(t ? y0 : x0, t ? x0 : y0),
      lambda = p[0],
      phi = p[1],
      cosPhi = cos(phi);
  if (t) lambda = -halfPi - lambda;
  return [sx * (atan2(sin(lambda) * cosPhi, -sin(phi)) + pi), sy * asin(cos(lambda) * cosPhi)];
};

function gringortenHexadecant(lambda, phi) {
  if (phi === halfPi) return [0, 0];

  var sinPhi = sin(phi),
      r = sinPhi * sinPhi,
      r2 = r * r,
      j = 1 + r2,
      k = 1 + 3 * r2,
      q = 1 - r2,
      z = asin(1 / sqrt(j)),
      v = q + r * j * z,
      p2 = (1 - sinPhi) / v,
      p = sqrt(p2),
      a2 = p2 * j,
      a = sqrt(a2),
      h = p * q,
      x,
      i;

  if (lambda === 0) return [0, -(h + r * a)];

  var cosPhi = cos(phi),
      secPhi = 1 / cosPhi,
      drdPhi = 2 * sinPhi * cosPhi,
      dvdPhi = (-3 * r + z * k) * drdPhi,
      dp2dPhi = (-v * cosPhi - (1 - sinPhi) * dvdPhi) / (v * v),
      dpdPhi = (0.5 * dp2dPhi) / p,
      dhdPhi = q * dpdPhi - 2 * r * p * drdPhi,
      dra2dPhi = r * j * dp2dPhi + p2 * k * drdPhi,
      mu = -secPhi * drdPhi,
      nu = -secPhi * dra2dPhi,
      zeta = -2 * secPhi * dhdPhi,
      lambda1 = 4 * lambda / pi,
      delta;

  // Slower but accurate bisection method.
  if (lambda > 0.222 * pi || phi < pi / 4 && lambda > 0.175 * pi) {
    x = (h + r * sqrt(a2 * (1 + r2) - h * h)) / (1 + r2);
    if (lambda > pi / 4) return [x, x];
    var x1 = x, x0 = 0.5 * x;
    x = 0.5 * (x0 + x1), i = 50;
    do {
      var g = sqrt(a2 - x * x),
          f = (x * (zeta + mu * g) + nu * asin(x / a)) - lambda1;
      if (!f) break;
      if (f < 0) x0 = x;
      else x1 = x;
      x = 0.5 * (x0 + x1);
    } while (abs(x1 - x0) > epsilon && --i > 0);
  }

  // Newton-Raphson.
  else {
    x = epsilon, i = 25;
    do {
      var x2 = x * x,
          g2 = sqrt(a2 - x2),
          zetaMug = zeta + mu * g2,
          f2 = x * zetaMug + nu * asin(x / a) - lambda1,
          df = zetaMug + (nu - mu * x2) / g2;
      x -= delta = g2 ? f2 / df : 0;
    } while (abs(delta) > epsilon && --i > 0);
  }

  return [x, -h - r * sqrt(a2 - x * x)];
}

function gringortenHexadecantInvert(x, y) {
  var x0 = 0,
      x1 = 1,
      r = 0.5,
      i = 50;

  while (true) {
    var r2 = r * r,
        sinPhi = sqrt(r),
        z = asin(1 / sqrt(1 + r2)),
        v = (1 - r2) + r * (1 + r2) * z,
        p2 = (1 - sinPhi) / v,
        p = sqrt(p2),
        a2 = p2 * (1 + r2),
        h = p * (1 - r2),
        g2 = a2 - x * x,
        g = sqrt(g2),
        y0 = y + h + r * g;
    if (abs(x1 - x0) < epsilon2 || --i === 0 || y0 === 0) break;
    if (y0 > 0) x0 = r;
    else x1 = r;
    r = 0.5 * (x0 + x1);
  }

  if (!i) return null;

  var phi = asin(sinPhi),
      cosPhi = cos(phi),
      secPhi = 1 / cosPhi,
      drdPhi = 2 * sinPhi * cosPhi,
      dvdPhi = (-3 * r + z * (1 + 3 * r2)) * drdPhi,
      dp2dPhi = (-v * cosPhi - (1 - sinPhi) * dvdPhi) / (v * v),
      dpdPhi = 0.5 * dp2dPhi / p,
      dhdPhi = (1 - r2) * dpdPhi - 2 * r * p * drdPhi,
      zeta = -2 * secPhi * dhdPhi,
      mu = -secPhi * drdPhi,
      nu = -secPhi * (r * (1 + r2) * dp2dPhi + p2 * (1 + 3 * r2) * drdPhi);

  return [pi / 4 * (x * (zeta + mu * g) + nu * asin(x / sqrt(a2))), phi];
}

var gringorten = function() {
  return d3Geo.geoProjection(squareRaw(gringortenRaw))
      .scale(239.75);
};

// Returns [sn, cn, dn](u + iv|m).
function ellipticJi(u, v, m) {
  var a, b, c;
  if (!u) {
    b = ellipticJ(v, 1 - m);
    return [
      [0, b[0] / b[1]],
      [1 / b[1], 0],
      [b[2] / b[1], 0]
    ];
  }
  a = ellipticJ(u, m);
  if (!v) return [[a[0], 0], [a[1], 0], [a[2], 0]];
  b = ellipticJ(v, 1 - m);
  c = b[1] * b[1] + m * a[0] * a[0] * b[0] * b[0];
  return [
    [a[0] * b[2] / c, a[1] * a[2] * b[0] * b[1] / c],
    [a[1] * b[1] / c, -a[0] * a[2] * b[0] * b[2] / c],
    [a[2] * b[1] * b[2] / c, -m * a[0] * a[1] * b[0] / c]
  ];
}

// Returns [sn, cn, dn, ph](u|m).
function ellipticJ(u, m) {
  var ai, b, phi, t, twon;
  if (m < epsilon) {
    t = sin(u);
    b = cos(u);
    ai = m * (u - t * b) / 4;
    return [
      t - ai * b,
      b + ai * t,
      1 - m * t * t / 2,
      u - ai
    ];
  }
  if (m >= 1 - epsilon) {
    ai = (1 - m) / 4;
    b = cosh(u);
    t = tanh(u);
    phi = 1 / b;
    twon = b * sinh(u);
    return [
      t + ai * (twon - u) / (b * b),
      phi - ai * t * phi * (twon - u),
      phi + ai * t * phi * (twon + u),
      2 * atan(exp(u)) - halfPi + ai * (twon - u) / b
    ];
  }

  var a = [1, 0, 0, 0, 0, 0, 0, 0, 0],
      c = [sqrt(m), 0, 0, 0, 0, 0, 0, 0, 0],
      i = 0;
  b = sqrt(1 - m);
  twon = 1;

  while (abs(c[i] / a[i]) > epsilon && i < 8) {
    ai = a[i++];
    c[i] = (ai - b) / 2;
    a[i] = (ai + b) / 2;
    b = sqrt(ai * b);
    twon *= 2;
  }

  phi = twon * a[i] * u;
  do {
    t = c[i] * sin(b = phi) / a[i];
    phi = (asin(t) + phi) / 2;
  } while (--i);

  return [sin(phi), t = cos(phi), t / cos(phi - b), phi];
}

// Calculate F(phi+iPsi|m).
// See Abramowitz and Stegun, 17.4.11.
function ellipticFi(phi, psi, m) {
  var r = abs(phi),
      i = abs(psi),
      sinhPsi = sinh(i);
  if (r) {
    var cscPhi = 1 / sin(r),
        cotPhi2 = 1 / (tan(r) * tan(r)),
        b = -(cotPhi2 + m * (sinhPsi * sinhPsi * cscPhi * cscPhi) - 1 + m),
        c = (m - 1) * cotPhi2,
        cotLambda2 = (-b + sqrt(b * b - 4 * c)) / 2;
    return [
      ellipticF(atan(1 / sqrt(cotLambda2)), m) * sign(phi),
      ellipticF(atan(sqrt((cotLambda2 / cotPhi2 - 1) / m)), 1 - m) * sign(psi)
    ];
  }
  return [
    0,
    ellipticF(atan(sinhPsi), 1 - m) * sign(psi)
  ];
}

// Calculate F(phi|m) where m = k² = sin²α.
// See Abramowitz and Stegun, 17.6.7.
function ellipticF(phi, m) {
  if (!m) return phi;
  if (m === 1) return log(tan(phi / 2 + quarterPi));
  var a = 1,
      b = sqrt(1 - m),
      c = sqrt(m);
  for (var i = 0; abs(c) > epsilon; i++) {
    if (phi % pi) {
      var dPhi = atan(b * tan(phi) / a);
      if (dPhi < 0) dPhi += pi;
      phi += dPhi + ~~(phi / pi) * pi;
    } else phi += phi;
    c = (a + b) / 2;
    b = sqrt(a * b);
    c = ((a = c) - b) / 2;
  }
  return phi / (pow(2, i) * a);
}

function guyouRaw(lambda, phi) {
  var k_ = (sqrt2 - 1) / (sqrt2 + 1),
      k = sqrt(1 - k_ * k_),
      K = ellipticF(halfPi, k * k),
      f = -1,
      psi = log(tan(pi / 4 + abs(phi) / 2)),
      r = exp(f * psi) / sqrt(k_),
      at = guyouComplexAtan(r * cos(f * lambda), r * sin(f * lambda)),
      t = ellipticFi(at[0], at[1], k * k);
  return [-t[1], (phi >= 0 ? 1 : -1) * (0.5 * K - t[0])];
}

function guyouComplexAtan(x, y) {
  var x2 = x * x,
      y_1 = y + 1,
      t = 1 - x2 - y * y;
  return [
   0.5 * ((x >= 0 ? halfPi : -halfPi) - atan2(t, 2 * x)),
    -0.25 * log(t * t + 4 * x2) +0.5 * log(y_1 * y_1 + x2)
  ];
}

function guyouComplexDivide(a, b) {
  var denominator = b[0] * b[0] + b[1] * b[1];
  return [
    (a[0] * b[0] + a[1] * b[1]) / denominator,
    (a[1] * b[0] - a[0] * b[1]) / denominator
  ];
}

guyouRaw.invert = function(x, y) {
  var k_ = (sqrt2 - 1) / (sqrt2 + 1),
      k = sqrt(1 - k_ * k_),
      K = ellipticF(halfPi, k * k),
      f = -1,
      j = ellipticJi(0.5 * K - y, -x, k * k),
      tn = guyouComplexDivide(j[0], j[1]),
      lambda = atan2(tn[1], tn[0]) / f;
  return [
    lambda,
    2 * atan(exp(0.5 / f * log(k_ * tn[0] * tn[0] + k_ * tn[1] * tn[1]))) - halfPi
  ];
};

var guyou = function() {
  return d3Geo.geoProjection(squareRaw(guyouRaw))
      .scale(151.496);
};

function hammerRaw(A, B) {
  if (arguments.length < 2) B = A;
  if (B === 1) return d3Geo.geoAzimuthalEqualAreaRaw;
  if (B === Infinity) return hammerQuarticAuthalicRaw;

  function forward(lambda, phi) {
    var coordinates = d3Geo.geoAzimuthalEqualAreaRaw(lambda / B, phi);
    coordinates[0] *= A;
    return coordinates;
  }

  forward.invert = function(x, y) {
    var coordinates = d3Geo.geoAzimuthalEqualAreaRaw.invert(x / A, y);
    coordinates[0] *= B;
    return coordinates;
  };

  return forward;
}

function hammerQuarticAuthalicRaw(lambda, phi) {
  return [
    lambda * cos(phi) / cos(phi /= 2),
    2 * sin(phi)
  ];
}

hammerQuarticAuthalicRaw.invert = function(x, y) {
  var phi = 2 * asin(y / 2);
  return [
    x * cos(phi / 2) / cos(phi),
    phi
  ];
};

var hammer = function() {
  var B = 2,
      m = d3Geo.geoProjectionMutator(hammerRaw),
      p = m(B);

  p.coefficient = function(_) {
    if (!arguments.length) return B;
    return m(B = +_);
  };

  return p
    .scale(169.529);
};

function hammerRetroazimuthalRaw(phi0) {
  var sinPhi0 = sin(phi0),
      cosPhi0 = cos(phi0),
      rotate = hammerRetroazimuthalRotation(phi0);

  rotate.invert = hammerRetroazimuthalRotation(-phi0);

  function forward(lambda, phi) {
    var p = rotate(lambda, phi);
    lambda = p[0], phi = p[1];
    var sinPhi = sin(phi),
        cosPhi = cos(phi),
        cosLambda = cos(lambda),
        z = acos(sinPhi0 * sinPhi + cosPhi0 * cosPhi * cosLambda),
        sinz = sin(z),
        K = abs(sinz) > epsilon ? z / sinz : 1;
    return [
      K * cosPhi0 * sin(lambda),
      (abs(lambda) > halfPi ? K : -K) // rotate for back hemisphere
        * (sinPhi0 * cosPhi - cosPhi0 * sinPhi * cosLambda)
    ];
  }

  forward.invert = function(x, y) {
    var rho = sqrt(x * x + y * y),
        sinz = -sin(rho),
        cosz = cos(rho),
        a = rho * cosz,
        b = -y * sinz,
        c = rho * sinPhi0,
        d = sqrt(a * a + b * b - c * c),
        phi = atan2(a * c + b * d, b * c - a * d),
        lambda = (rho > halfPi ? -1 : 1) * atan2(x * sinz, rho * cos(phi) * cosz + y * sin(phi) * sinz);
    return rotate.invert(lambda, phi);
  };

  return forward;
}

// Latitudinal rotation by phi0.
// Temporary hack until D3 supports arbitrary small-circle clipping origins.
function hammerRetroazimuthalRotation(phi0) {
  var sinPhi0 = sin(phi0),
      cosPhi0 = cos(phi0);

  return function(lambda, phi) {
    var cosPhi = cos(phi),
        x = cos(lambda) * cosPhi,
        y = sin(lambda) * cosPhi,
        z = sin(phi);
    return [
      atan2(y, x * cosPhi0 - z * sinPhi0),
      asin(z * cosPhi0 + x * sinPhi0)
    ];
  };
}

var hammerRetroazimuthal = function() {
  var phi0 = 0,
      m = d3Geo.geoProjectionMutator(hammerRetroazimuthalRaw),
      p = m(phi0),
      rotate_ = p.rotate,
      stream_ = p.stream,
      circle = d3Geo.geoCircle();

  p.parallel = function(_) {
    if (!arguments.length) return phi0 * degrees;
    var r = p.rotate();
    return m(phi0 = _ * radians).rotate(r);
  };

  // Temporary hack; see hammerRetroazimuthalRotation.
  p.rotate = function(_) {
    if (!arguments.length) return (_ = rotate_.call(p), _[1] += phi0 * degrees, _);
    rotate_.call(p, [_[0], _[1] - phi0 * degrees]);
    circle.center([-_[0], -_[1]]);
    return p;
  };

  p.stream = function(stream) {
    stream = stream_(stream);
    stream.sphere = function() {
      stream.polygonStart();
      var epsilon$$1 = 1e-2,
          ring = circle.radius(90 - epsilon$$1)().coordinates[0],
          n = ring.length - 1,
          i = -1,
          p;
      stream.lineStart();
      while (++i < n) stream.point((p = ring[i])[0], p[1]);
      stream.lineEnd();
      ring = circle.radius(90 + epsilon$$1)().coordinates[0];
      n = ring.length - 1;
      stream.lineStart();
      while (--i >= 0) stream.point((p = ring[i])[0], p[1]);
      stream.lineEnd();
      stream.polygonEnd();
    };
    return stream;
  };

  return p
      .scale(79.4187)
      .parallel(45)
      .clipAngle(180 - 1e-3);
};

var healpixParallel = 41 + 48 / 36 + 37 / 3600;
var healpixLambert = cylindricalEqualAreaRaw(0);

function healpixRaw(H) {
  var phi0 = healpixParallel * radians,
      dx = collignonRaw(pi, phi0)[0] - collignonRaw(-pi, phi0)[0],
      y0 = healpixLambert(0, phi0)[1],
      y1 = collignonRaw(0, phi0)[1],
      dy1 = sqrtPi - y1,
      k = tau / H,
      w = 4 / tau,
      h = y0 + (dy1 * dy1 * 4) / tau;

  function forward(lambda, phi) {
    var point,
        phi2 = abs(phi);
    if (phi2 > phi0) {
      var i = min(H - 1, max(0, floor((lambda + pi) / k)));
      lambda += pi * (H - 1) / H - i * k;
      point = collignonRaw(lambda, phi2);
      point[0] = point[0] * tau / dx - tau * (H - 1) / (2 * H) + i * tau / H;
      point[1] = y0 + (point[1] - y1) * 4 * dy1 / tau;
      if (phi < 0) point[1] = -point[1];
    } else {
      point = healpixLambert(lambda, phi);
    }
    point[0] *= w, point[1] /= h;
    return point;
  }

  forward.invert = function(x, y) {
    x /= w, y *= h;
    var y2 = abs(y);
    if (y2 > y0) {
      var i = min(H - 1, max(0, floor((x + pi) / k)));
      x = (x + pi * (H - 1) / H - i * k) * dx / tau;
      var point = collignonRaw.invert(x, 0.25 * (y2 - y0) * tau / dy1 + y1);
      point[0] -= pi * (H - 1) / H - i * k;
      if (y < 0) point[1] = -point[1];
      return point;
    }
    return healpixLambert.invert(x, y);
  };

  return forward;
}

function sphere(step) {
  return {
    type: "Polygon",
    coordinates: [
      d3Array.range(-180, 180 + step / 2, step).map(function(x, i) { return [x, i & 1 ? 90 - 1e-6 : healpixParallel]; })
      .concat(d3Array.range(180, -180 - step / 2, -step).map(function(x, i) { return [x, i & 1 ? -90 + 1e-6 : -healpixParallel]; }))
    ]
  };
}

var healpix = function() {
  var H = 4,
      m = d3Geo.geoProjectionMutator(healpixRaw),
      p = m(H),
      stream_ = p.stream;

  p.lobes = function(_) {
    return arguments.length ? m(H = +_) : H;
  };

  p.stream = function(stream) {
    var rotate = p.rotate(),
        rotateStream = stream_(stream),
        sphereStream = (p.rotate([0, 0]), stream_(stream));
    p.rotate(rotate);
    rotateStream.sphere = function() { d3Geo.geoStream(sphere(180 / H), sphereStream); };
    return rotateStream;
  };

  return p
      .scale(239.75);
};

function hillRaw(K) {
  var L = 1 + K,
      sinBt = sin(1 / L),
      Bt = asin(sinBt),
      A = 2 * sqrt(pi / (B = pi + 4 * Bt * L)),
      B,
      rho0 = 0.5 * A * (L + sqrt(K * (2 + K))),
      K2 = K * K,
      L2 = L * L;

  function forward(lambda, phi) {
    var t = 1 - sin(phi),
        rho,
        omega;
    if (t && t < 2) {
      var theta = halfPi - phi, i = 25, delta;
      do {
        var sinTheta = sin(theta),
            cosTheta = cos(theta),
            Bt_Bt1 = Bt + atan2(sinTheta, L - cosTheta),
            C = 1 + L2 - 2 * L * cosTheta;
        theta -= delta = (theta - K2 * Bt - L * sinTheta + C * Bt_Bt1 -0.5 * t * B) / (2 * L * sinTheta * Bt_Bt1);
      } while (abs(delta) > epsilon2 && --i > 0);
      rho = A * sqrt(C);
      omega = lambda * Bt_Bt1 / pi;
    } else {
      rho = A * (K + t);
      omega = lambda * Bt / pi;
    }
    return [
      rho * sin(omega),
      rho0 - rho * cos(omega)
    ];
  }

  forward.invert = function(x, y) {
    var rho2 = x * x + (y -= rho0) * y,
        cosTheta = (1 + L2 - rho2 / (A * A)) / (2 * L),
        theta = acos(cosTheta),
        sinTheta = sin(theta),
        Bt_Bt1 = Bt + atan2(sinTheta, L - cosTheta);
    return [
      asin(x / sqrt(rho2)) * pi / Bt_Bt1,
      asin(1 - 2 * (theta - K2 * Bt - L * sinTheta + (1 + L2 - 2 * L * cosTheta) * Bt_Bt1) / B)
    ];
  };

  return forward;
}

var hill = function() {
  var K = 1,
      m = d3Geo.geoProjectionMutator(hillRaw),
      p = m(K);