// https://d3js.org/d3-geo-projection/ Version 2.2.0. 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$$1); } } sphereStream.lineEnd(), sphereStream.polygonEnd(); }; return rotateStream; }; return p .scale(87.8076) .center([0, 17.1875]) .clipAngle(180 - 1e-3); }; function hammerRaw(A, B) { if (arguments.length < 2) B = A; if (B === 1) return d3Geo.geoAzimuthalEqualAreaRaw; if (B === Infinity) return hammerQuarticAuthalicRaw; function forward(lambda, phi) { var coordinates = d3Geo.geoAzimuthalEqualAreaRaw(lambda / B, phi); coordinates[0] *= A; return coordinates; } forward.invert = function(x, y) { var coordinates = d3Geo.geoAzimuthalEqualAreaRaw.invert(x / A, y); coordinates[0] *= B; return coordinates; }; return forward; } function hammerQuarticAuthalicRaw(lambda, phi) { return [ lambda * cos(phi) / cos(phi /= 2), 2 * sin(phi) ]; } hammerQuarticAuthalicRaw.invert = function(x, y) { var phi = 2 * asin(y / 2); return [ x * cos(phi / 2) / cos(phi), phi ]; }; var hammer = function() { var B = 2, m = d3Geo.geoProjectionMutator(hammerRaw), p = m(B); p.coefficient = function(_) { if (!arguments.length) return B; return m(B = +_); }; return p .scale(169.529); }; function bertin1953Raw() { var hammer$$1 = hammerRaw(1.68, 2), fu = 1.4, k = 12; return function(lambda, phi) { if (lambda + phi < -fu) { var u = (lambda - phi + 1.6) * (lambda + phi + fu) / 8; lambda += u; phi -= 0.8 * u * sin(phi + pi / 2); } var r = hammer$$1(lambda, phi); var d = (1 - cos(lambda * phi)) / k; if (r[1] < 0) { r[0] *= 1 + d; } if (r[1] > 0) { r[1] *= 1 + d / 1.5 * r[0] * r[0]; } return r; }; } var bertin = function() { var p = d3Geo.geoProjection(bertin1953Raw()); p.rotate([-16.5, -42]); delete p.rotate; return p .scale(176.57) .center([7.93, 0.09]); }; 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 lagrangeRaw(n) { function forward(lambda, phi) { if (abs(abs(phi) - halfPi) < epsilon) return [0, phi < 0 ? -2 : 2]; var sinPhi = sin(phi), v = pow((1 + sinPhi) / (1 - sinPhi), n / 2), c = 0.5 * (v + 1 / v) + cos(lambda *= n); return [ 2 * sin(lambda) / c, (v - 1 / v) / c ]; } forward.invert = function(x, y) { var y0 = abs(y); if (abs(y0 - 2) < epsilon) return x ? null : [0, sign(y) * halfPi]; if (y0 > 2) return null; x /= 2, y /= 2; var x2 = x * x, y2 = y * y, t = 2 * y / (1 + x2 + y2); // tanh(nPhi) t = pow((1 + t) / (1 - t), 1 / n); return [ atan2(2 * x, 1 - x2 - y2) / n, asin((t - 1) / (t + 1)) ]; }; return forward; } var lagrange = function() { var n = 0.5, m = d3Geo.geoProjectionMutator(lagrangeRaw), p = m(n); p.spacing = function(_) { return arguments.length ? m(n = +_) : n; }; return p .scale(124.75); }; function complexAtan(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 complexDivide(a, b) { if (b[1]) a = complexMul(a, [b[0], -b[1]]), b = complexNorm2(b); else b = b[0]; return [ a[0] / b, a[1] / b ]; } function complexMul(a, b) { return [ a[0] * b[0] - a[1] * b[1], a[1] * b[0] + a[0] * b[1] ]; } function complexAdd(a, b) { return [ a[0] + b[0], a[1] + b[1] ]; } function complexSub(a, b) { return [ a[0] - b[0], a[1] - b[1] ]; } function complexNorm2(a) { return a[0] * a[0] + a[1] * a[1]; } function complexNorm(a) { return sqrt(complexNorm2(a)); } function complexLogHypot(a, b) { var _a = abs(a), _b = abs(b); if (a === 0) return log(_b); if (b === 0) return log(_a); if (_a < 3000 && _b < 3000) return log(a * a + b * b) * 0.5; return log(a / cos(atan2(b, a))); } // adapted from https://github.com/infusion/Complex.js function complexPow(a, n) { var b = a[1], arg, loh; a = a[0]; if (a === 0 && b === 0) return [0,0]; if (typeof n === 'number') n = [n,0]; if (!n[1]) { if (b === 0 && a >= 0) { return [pow(a, n[0]), 0]; } else if (a === 0) { switch ((n[1] % 4 + 4) % 4) { case 0: return [pow(b, n[0]), 0]; case 1: return [0, pow(b, n[0])]; case 2: return [-pow(b, n[0]), 0]; case 3: return [0, -pow(b, n[0])]; } } } arg = atan2(b, a); loh = complexLogHypot(a, b); a = exp(n[0] * loh - n[1] * arg); b = n[1] * loh + n[0] * arg; return [a * cos(b), a * sin(b)]; } // w1 = gamma(1/n) * gamma(1 - 2/n) / n / gamma(1 - 1/n) // https://bl.ocks.org/Fil/852557838117687bbd985e4b38ff77d4 var w1 = [-1/2, sqrt(3)/2]; var w1 = [1.7666387502854533, 0]; var m = 0.3 * 0.3; // 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(z) { var k = [0, 0]; // rotate to have s ~= 1 var rot = complexPow(w1, d3Array.scan([0, 1, 2].map(function(i) { return -complexMul(z, complexPow(w1, [i, 0]))[0]; }))); var y = complexMul(rot, z); y = [1 - y[0], - y[1]]; // 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 = [0, 0]; for (var i = F0.length; i--;) F = complexAdd([F0[i],0], complexMul(F, y)); k = complexMul( complexAdd(w1, complexMul([-F[0], -F[1]], complexPow(y, (1-2/3))) ), complexMul(rot, rot) ); // 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 = complexNorm2(z); if (n < m) { var H0 = [ 1, 1/3, 5/27, 10/81, 22/243 //… ]; var z3 = complexPow(z, [3,0]); var h = [0,0]; for (i = H0.length; i--;) h = complexAdd([H0[i],0], complexMul(h, z3)); h = complexMul(h, z); k = complexAdd(complexMul(k, [n / m, 0]), complexMul(h, [1 - n / m, 0])); } return k; } var lagrange1_2 = lagrangeRaw(0.5); function coxRaw(lambda, phi) { var s = lagrange1_2(lambda, phi); var t = sm_1([s[1] / 2, s[0] / 2]); return [t[1], t[0]] } // the Sphere should go *exactly* to the vertices of the triangles // because they are singular points function sphere() { var c = 2 * asin(1 / sqrt(5)) * degrees; return { type: "Polygon", coordinates: [ [ [ 0,90 ], [ -180, -c + epsilon ], [ 0, -90 ], [ 180, -c + epsilon ], [ 0,90 ] ] ] }; } var cox = function() { var p = d3Geo.geoProjection(coxRaw); var stream_ = p.stream; 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(), sphereStream); }; return rotateStream; }; return p .scale(188.682) .center([0, 0]) .translate([ 480, 500 * 2 /3 ]); }; 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 = complexAtan(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])]; } 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 = complexDivide(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 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),