/*
* Geodesic.js
* Transcription of Geodesic.[ch]pp into JavaScript.
*
* See the documentation for the C++ class. The conversion is a literal
* conversion from C++.
*
* The algorithms are derived in
*
* Charles F. F. Karney,
* Algorithms for geodesics, J. Geodesy 87, 43-55 (2013);
* https://doi.org/10.1007/s00190-012-0578-z
* Addenda: https://geographiclib.sourceforge.io/geod-addenda.html
*
* Copyright (c) Charles Karney (2011-2022) <charles@karney.com> and licensed
* under the MIT/X11 License. For more information, see
* https://geographiclib.sourceforge.io/
*/
// Load AFTER Math.js
// To allow swap via [y, x] = [x, y]
/* jshint esversion: 6 */
geodesic.Geodesic = {};
geodesic.GeodesicLine = {};
geodesic.PolygonArea = {};
(function(
/**
* @exports geodesic/Geodesic
* @description Solve geodesic problems via the
* {@link module:geodesic/Geodesic.Geodesic Geodesic} class.
*/
g, l, p, m, c) {
"use strict";
var GEOGRAPHICLIB_GEODESIC_ORDER = 6,
nA1_ = GEOGRAPHICLIB_GEODESIC_ORDER,
nA2_ = GEOGRAPHICLIB_GEODESIC_ORDER,
nA3_ = GEOGRAPHICLIB_GEODESIC_ORDER,
nA3x_ = nA3_,
nC3x_, nC4x_,
maxit1_ = 20,
maxit2_ = maxit1_ + m.digits + 10,
tol0_ = m.epsilon,
tol1_ = 200 * tol0_,
tol2_ = Math.sqrt(tol0_),
tolb_ = tol0_ * tol1_,
xthresh_ = 1000 * tol2_,
CAP_NONE = 0,
CAP_ALL = 0x1F,
OUT_ALL = 0x7F80,
astroid,
A1m1f_coeff, C1f_coeff, C1pf_coeff,
A2m1f_coeff, C2f_coeff,
A3_coeff, C3_coeff, C4_coeff;
// N.B. Number.MIN_VALUE is denormalized; divide by Number.EPSILON to get min
// normalized positive number.
g.tiny_ = Math.sqrt(Number.MIN_VALUE/Number.EPSILON);
g.nC1_ = GEOGRAPHICLIB_GEODESIC_ORDER;
g.nC1p_ = GEOGRAPHICLIB_GEODESIC_ORDER;
g.nC2_ = GEOGRAPHICLIB_GEODESIC_ORDER;
g.nC3_ = GEOGRAPHICLIB_GEODESIC_ORDER;
g.nC4_ = GEOGRAPHICLIB_GEODESIC_ORDER;
nC3x_ = (g.nC3_ * (g.nC3_ - 1)) / 2;
nC4x_ = (g.nC4_ * (g.nC4_ + 1)) / 2;
g.CAP_C1 = 1<<0;
g.CAP_C1p = 1<<1;
g.CAP_C2 = 1<<2;
g.CAP_C3 = 1<<3;
g.CAP_C4 = 1<<4;
g.NONE = 0;
g.ARC = 1<<6;
g.LATITUDE = 1<<7 | CAP_NONE;
g.LONGITUDE = 1<<8 | g.CAP_C3;
g.AZIMUTH = 1<<9 | CAP_NONE;
g.DISTANCE = 1<<10 | g.CAP_C1;
g.STANDARD = g.LATITUDE | g.LONGITUDE | g.AZIMUTH | g.DISTANCE;
g.DISTANCE_IN = 1<<11 | g.CAP_C1 | g.CAP_C1p;
g.REDUCEDLENGTH = 1<<12 | g.CAP_C1 | g.CAP_C2;
g.GEODESICSCALE = 1<<13 | g.CAP_C1 | g.CAP_C2;
g.AREA = 1<<14 | g.CAP_C4;
g.ALL = OUT_ALL| CAP_ALL;
g.LONG_UNROLL = 1<<15;
g.OUT_MASK = OUT_ALL| g.LONG_UNROLL;
g.SinCosSeries = function(sinp, sinx, cosx, c) {
// Evaluate
// y = sinp ? sum(c[i] * sin( 2*i * x), i, 1, n) :
// sum(c[i] * cos((2*i+1) * x), i, 0, n-1)
// using Clenshaw summation. N.B. c[0] is unused for sin series
// Approx operation count = (n + 5) mult and (2 * n + 2) add
var k = c.length, // Point to one beyond last element
n = k - (sinp ? 1 : 0),
ar = 2 * (cosx - sinx) * (cosx + sinx), // 2 * cos(2 * x)
y0 = n & 1 ? c[--k] : 0, y1 = 0; // accumulators for sum
// Now n is even
n = Math.floor(n/2);
while (n--) {
// Unroll loop x 2, so accumulators return to their original role
y1 = ar * y0 - y1 + c[--k];
y0 = ar * y1 - y0 + c[--k];
}
return (sinp ? 2 * sinx * cosx * y0 : // sin(2 * x) * y0
cosx * (y0 - y1)); // cos(x) * (y0 - y1)
};
astroid = function(x, y) {
// Solve k^4+2*k^3-(x^2+y^2-1)*k^2-2*y^2*k-y^2 = 0 for positive
// root k. This solution is adapted from Geocentric::Reverse.
var k,
p = m.sq(x),
q = m.sq(y),
r = (p + q - 1) / 6,
S, r2, r3, disc, u, T3, T, ang, v, uv, w;
if ( !(q === 0 && r <= 0) ) {
// Avoid possible division by zero when r = 0 by multiplying
// equations for s and t by r^3 and r, resp.
S = p * q / 4; // S = r^3 * s
r2 = m.sq(r);
r3 = r * r2;
// The discriminant of the quadratic equation for T3. This is
// zero on the evolute curve p^(1/3)+q^(1/3) = 1
disc = S * (S + 2 * r3);
u = r;
if (disc >= 0) {
T3 = S + r3;
// Pick the sign on the sqrt to maximize abs(T3). This
// minimizes loss of precision due to cancellation. The
// result is unchanged because of the way the T is used
// in definition of u.
T3 += T3 < 0 ? -Math.sqrt(disc) : Math.sqrt(disc); // T3 = (r * t)^3
// N.B. cbrt always returns the real root. cbrt(-8) = -2.
T = m.cbrt(T3); // T = r * t
// T can be zero; but then r2 / T -> 0.
u += T + (T !== 0 ? r2 / T : 0);
} else {
// T is complex, but the way u is defined the result is real.
ang = Math.atan2(Math.sqrt(-disc), -(S + r3));
// There are three possible cube roots. We choose the
// root which avoids cancellation. Note that disc < 0
// implies that r < 0.
u += 2 * r * Math.cos(ang / 3);
}
v = Math.sqrt(m.sq(u) + q); // guaranteed positive
// Avoid loss of accuracy when u < 0.
uv = u < 0 ? q / (v - u) : u + v; // u+v, guaranteed positive
w = (uv - q) / (2 * v); // positive?
// Rearrange expression for k to avoid loss of accuracy due to
// subtraction. Division by 0 not possible because uv > 0, w >= 0.
k = uv / (Math.sqrt(uv + m.sq(w)) + w); // guaranteed positive
} else { // q == 0 && r <= 0
// y = 0 with |x| <= 1. Handle this case directly.
// for y small, positive root is k = abs(y)/sqrt(1-x^2)
k = 0;
}
return k;
};
A1m1f_coeff = [
// (1-eps)*A1-1, polynomial in eps2 of order 3
+1, 4, 64, 0, 256
];
// The scale factor A1-1 = mean value of (d/dsigma)I1 - 1
g.A1m1f = function(eps) {
var p = Math.floor(nA1_/2),
t = m.polyval(p, A1m1f_coeff, 0, m.sq(eps)) / A1m1f_coeff[p + 1];
return (t + eps) / (1 - eps);
};
C1f_coeff = [
// C1[1]/eps^1, polynomial in eps2 of order 2
-1, 6, -16, 32,
// C1[2]/eps^2, polynomial in eps2 of order 2
-9, 64, -128, 2048,
// C1[3]/eps^3, polynomial in eps2 of order 1
+9, -16, 768,
// C1[4]/eps^4, polynomial in eps2 of order 1
+3, -5, 512,
// C1[5]/eps^5, polynomial in eps2 of order 0
-7, 1280,
// C1[6]/eps^6, polynomial in eps2 of order 0
-7, 2048
];
// The coefficients C1[l] in the Fourier expansion of B1
g.C1f = function(eps, c) {
var eps2 = m.sq(eps),
d = eps,
o = 0,
l, p;
for (l = 1; l <= g.nC1_; ++l) { // l is index of C1p[l]
p = Math.floor((g.nC1_ - l) / 2); // order of polynomial in eps^2
c[l] = d * m.polyval(p, C1f_coeff, o, eps2) / C1f_coeff[o + p + 1];
o += p + 2;
d *= eps;
}
};
C1pf_coeff = [
// C1p[1]/eps^1, polynomial in eps2 of order 2
+205, -432, 768, 1536,
// C1p[2]/eps^2, polynomial in eps2 of order 2
+4005, -4736, 3840, 12288,
// C1p[3]/eps^3, polynomial in eps2 of order 1
-225, 116, 384,
// C1p[4]/eps^4, polynomial in eps2 of order 1
-7173, 2695, 7680,
// C1p[5]/eps^5, polynomial in eps2 of order 0
+3467, 7680,
// C1p[6]/eps^6, polynomial in eps2 of order 0
+38081, 61440
];
// The coefficients C1p[l] in the Fourier expansion of B1p
g.C1pf = function(eps, c) {
var eps2 = m.sq(eps),
d = eps,
o = 0,
l, p;
for (l = 1; l <= g.nC1p_; ++l) { // l is index of C1p[l]
p = Math.floor((g.nC1p_ - l) / 2); // order of polynomial in eps^2
c[l] = d * m.polyval(p, C1pf_coeff, o, eps2) / C1pf_coeff[o + p + 1];
o += p + 2;
d *= eps;
}
};
A2m1f_coeff = [
// (eps+1)*A2-1, polynomial in eps2 of order 3
-11, -28, -192, 0, 256
];
// The scale factor A2-1 = mean value of (d/dsigma)I2 - 1
g.A2m1f = function(eps) {
var p = Math.floor(nA2_/2),
t = m.polyval(p, A2m1f_coeff, 0, m.sq(eps)) / A2m1f_coeff[p + 1];
return (t - eps) / (1 + eps);
};
C2f_coeff = [
// C2[1]/eps^1, polynomial in eps2 of order 2
+1, 2, 16, 32,
// C2[2]/eps^2, polynomial in eps2 of order 2
+35, 64, 384, 2048,
// C2[3]/eps^3, polynomial in eps2 of order 1
+15, 80, 768,
// C2[4]/eps^4, polynomial in eps2 of order 1
+7, 35, 512,
// C2[5]/eps^5, polynomial in eps2 of order 0
+63, 1280,
// C2[6]/eps^6, polynomial in eps2 of order 0
+77, 2048
];
// The coefficients C2[l] in the Fourier expansion of B2
g.C2f = function(eps, c) {
var eps2 = m.sq(eps),
d = eps,
o = 0,
l, p;
for (l = 1; l <= g.nC2_; ++l) { // l is index of C2[l]
p = Math.floor((g.nC2_ - l) / 2); // order of polynomial in eps^2
c[l] = d * m.polyval(p, C2f_coeff, o, eps2) / C2f_coeff[o + p + 1];
o += p + 2;
d *= eps;
}
};
/**
* @class
* @property {number} a the equatorial radius (meters).
* @property {number} f the flattening.
* @summary Initialize a Geodesic object for a specific ellipsoid.
* @classdesc Performs geodesic calculations on an ellipsoid of revolution.
* The routines for solving the direct and inverse problems return an
* object with some of the following fields set: lat1, lon1, azi1, lat2,
* lon2, azi2, s12, a12, m12, M12, M21, S12. See {@tutorial 2-interface},
* section "The results".
* @example
* var geodesic = require("geographiclib-geodesic"),
* geod = geodesic.Geodesic.WGS84;
* var inv = geod.Inverse(1,2,3,4);
* console.log("lat1 = " + inv.lat1 + ", lon1 = " + inv.lon1 +
* ", lat2 = " + inv.lat2 + ", lon2 = " + inv.lon2 +
* ",\nazi1 = " + inv.azi1 + ", azi2 = " + inv.azi2 +
* ", s12 = " + inv.s12);
* @param {number} a the equatorial radius of the ellipsoid (meters).
* @param {number} f the flattening of the ellipsoid. Setting f = 0 gives
* a sphere (on which geodesics are great circles). Negative f gives a
* prolate ellipsoid.
* @throws an error if the parameters are illegal.
*/
g.Geodesic = function(a, f) {
this.a = a;
this.f = f;
this._f1 = 1 - this.f;
this._e2 = this.f * (2 - this.f);
this._ep2 = this._e2 / m.sq(this._f1); // e2 / (1 - e2)
this._n = this.f / ( 2 - this.f);
this._b = this.a * this._f1;
// authalic radius squared
this._c2 = (m.sq(this.a) + m.sq(this._b) *
(this._e2 === 0 ? 1 :
(this._e2 > 0 ? m.atanh(Math.sqrt(this._e2)) :
Math.atan(Math.sqrt(-this._e2))) /
Math.sqrt(Math.abs(this._e2))))/2;
// The sig12 threshold for "really short". Using the auxiliary sphere
// solution with dnm computed at (bet1 + bet2) / 2, the relative error in
// the azimuth consistency check is sig12^2 * abs(f) * min(1, 1-f/2) / 2.
// (Error measured for 1/100 < b/a < 100 and abs(f) >= 1/1000. For a given
// f and sig12, the max error occurs for lines near the pole. If the old
// rule for computing dnm = (dn1 + dn2)/2 is used, then the error increases
// by a factor of 2.) Setting this equal to epsilon gives sig12 = etol2.
// Here 0.1 is a safety factor (error decreased by 100) and max(0.001,
// abs(f)) stops etol2 getting too large in the nearly spherical case.
this._etol2 = 0.1 * tol2_ /
Math.sqrt( Math.max(0.001, Math.abs(this.f)) *
Math.min(1, 1 - this.f/2) / 2 );
if (!(isFinite(this.a) && this.a > 0))
throw new Error("Equatorial radius is not positive");
if (!(isFinite(this._b) && this._b > 0))
throw new Error("Polar semi-axis is not positive");
this._A3x = new Array(nA3x_);
this._C3x = new Array(nC3x_);
this._C4x = new Array(nC4x_);
this.A3coeff();
this.C3coeff();
this.C4coeff();
};
A3_coeff = [
// A3, coeff of eps^5, polynomial in n of order 0
-3, 128,
// A3, coeff of eps^4, polynomial in n of order 1
-2, -3, 64,
// A3, coeff of eps^3, polynomial in n of order 2
-1, -3, -1, 16,
// A3, coeff of eps^2, polynomial in n of order 2
+3, -1, -2, 8,
// A3, coeff of eps^1, polynomial in n of order 1
+1, -1, 2,
// A3, coeff of eps^0, polynomial in n of order 0
+1, 1
];
// The scale factor A3 = mean value of (d/dsigma)I3
g.Geodesic.prototype.A3coeff = function() {
var o = 0, k = 0,
j, p;
for (j = nA3_ - 1; j >= 0; --j) { // coeff of eps^j
p = Math.min(nA3_ - j - 1, j); // order of polynomial in n
this._A3x[k++] = m.polyval(p, A3_coeff, o, this._n) /
A3_coeff[o + p + 1];
o += p + 2;
}
};
C3_coeff = [
// C3[1], coeff of eps^5, polynomial in n of order 0
+3, 128,
// C3[1], coeff of eps^4, polynomial in n of order 1
+2, 5, 128,
// C3[1], coeff of eps^3, polynomial in n of order 2
-1, 3, 3, 64,
// C3[1], coeff of eps^2, polynomial in n of order 2
-1, 0, 1, 8,
// C3[1], coeff of eps^1, polynomial in n of order 1
-1, 1, 4,
// C3[2], coeff of eps^5, polynomial in n of order 0
+5, 256,
// C3[2], coeff of eps^4, polynomial in n of order 1
+1, 3, 128,
// C3[2], coeff of eps^3, polynomial in n of order 2
-3, -2, 3, 64,
// C3[2], coeff of eps^2, polynomial in n of order 2
+1, -3, 2, 32,
// C3[3], coeff of eps^5, polynomial in n of order 0
+7, 512,
// C3[3], coeff of eps^4, polynomial in n of order 1
-10, 9, 384,
// C3[3], coeff of eps^3, polynomial in n of order 2
+5, -9, 5, 192,
// C3[4], coeff of eps^5, polynomial in n of order 0
+7, 512,
// C3[4], coeff of eps^4, polynomial in n of order 1
-14, 7, 512,
// C3[5], coeff of eps^5, polynomial in n of order 0
+21, 2560
];
// The coefficients C3[l] in the Fourier expansion of B3
g.Geodesic.prototype.C3coeff = function() {
var o = 0, k = 0,
l, j, p;
for (l = 1; l < g.nC3_; ++l) { // l is index of C3[l]
for (j = g.nC3_ - 1; j >= l; --j) { // coeff of eps^j
p = Math.min(g.nC3_ - j - 1, j); // order of polynomial in n
this._C3x[k++] = m.polyval(p, C3_coeff, o, this._n) /
C3_coeff[o + p + 1];
o += p + 2;
}
}
};
C4_coeff = [
// C4[0], coeff of eps^5, polynomial in n of order 0
+97, 15015,
// C4[0], coeff of eps^4, polynomial in n of order 1
+1088, 156, 45045,
// C4[0], coeff of eps^3, polynomial in n of order 2
-224, -4784, 1573, 45045,
// C4[0], coeff of eps^2, polynomial in n of order 3
-10656, 14144, -4576, -858, 45045,
// C4[0], coeff of eps^1, polynomial in n of order 4
+64, 624, -4576, 6864, -3003, 15015,
// C4[0], coeff of eps^0, polynomial in n of order 5
+100, 208, 572, 3432, -12012, 30030, 45045,
// C4[1], coeff of eps^5, polynomial in n of order 0
+1, 9009,
// C4[1], coeff of eps^4, polynomial in n of order 1
-2944, 468, 135135,
// C4[1], coeff of eps^3, polynomial in n of order 2
+5792, 1040, -1287, 135135,
// C4[1], coeff of eps^2, polynomial in n of order 3
+5952, -11648, 9152, -2574, 135135,
// C4[1], coeff of eps^1, polynomial in n of order 4
-64, -624, 4576, -6864, 3003, 135135,
// C4[2], coeff of eps^5, polynomial in n of order 0
+8, 10725,
// C4[2], coeff of eps^4, polynomial in n of order 1
+1856, -936, 225225,
// C4[2], coeff of eps^3, polynomial in n of order 2
-8448, 4992, -1144, 225225,
// C4[2], coeff of eps^2, polynomial in n of order 3
-1440, 4160, -4576, 1716, 225225,
// C4[3], coeff of eps^5, polynomial in n of order 0
-136, 63063,
// C4[3], coeff of eps^4, polynomial in n of order 1
+1024, -208, 105105,
// C4[3], coeff of eps^3, polynomial in n of order 2
+3584, -3328, 1144, 315315,
// C4[4], coeff of eps^5, polynomial in n of order 0
-128, 135135,
// C4[4], coeff of eps^4, polynomial in n of order 1
-2560, 832, 405405,
// C4[5], coeff of eps^5, polynomial in n of order 0
+128, 99099
];
g.Geodesic.prototype.C4coeff = function() {
var o = 0, k = 0,
l, j, p;
for (l = 0; l < g.nC4_; ++l) { // l is index of C4[l]
for (j = g.nC4_ - 1; j >= l; --j) { // coeff of eps^j
p = g.nC4_ - j - 1; // order of polynomial in n
this._C4x[k++] = m.polyval(p, C4_coeff, o, this._n) /
C4_coeff[o + p + 1];
o += p + 2;
}
}
};
g.Geodesic.prototype.A3f = function(eps) {
// Evaluate A3
return m.polyval(nA3x_ - 1, this._A3x, 0, eps);
};
g.Geodesic.prototype.C3f = function(eps, c) {
// Evaluate C3 coeffs
// Elements c[1] thru c[nC3_ - 1] are set
var mult = 1,
o = 0,
l, p;
for (l = 1; l < g.nC3_; ++l) { // l is index of C3[l]
p = g.nC3_ - l - 1; // order of polynomial in eps
mult *= eps;
c[l] = mult * m.polyval(p, this._C3x, o, eps);
o += p + 1;
}
};
g.Geodesic.prototype.C4f = function(eps, c) {
// Evaluate C4 coeffs
// Elements c[0] thru c[g.nC4_ - 1] are set
var mult = 1,
o = 0,
l, p;
for (l = 0; l < g.nC4_; ++l) { // l is index of C4[l]
p = g.nC4_ - l - 1; // order of polynomial in eps
c[l] = mult * m.polyval(p, this._C4x, o, eps);
o += p + 1;
mult *= eps;
}
};
// return s12b, m12b, m0, M12, M21
g.Geodesic.prototype.Lengths = function(eps, sig12,
ssig1, csig1, dn1, ssig2, csig2, dn2,
cbet1, cbet2, outmask,
C1a, C2a) {
// Return m12b = (reduced length)/_b; also calculate s12b =
// distance/_b, and m0 = coefficient of secular term in
// expression for reduced length.
outmask &= g.OUT_MASK;
var vals = {},
m0x = 0, J12 = 0, A1 = 0, A2 = 0,
B1, B2, l, csig12, t;
if (outmask & (g.DISTANCE | g.REDUCEDLENGTH | g.GEODESICSCALE)) {
A1 = g.A1m1f(eps);
g.C1f(eps, C1a);
if (outmask & (g.REDUCEDLENGTH | g.GEODESICSCALE)) {
A2 = g.A2m1f(eps);
g.C2f(eps, C2a);
m0x = A1 - A2;
A2 = 1 + A2;
}
A1 = 1 + A1;
}
if (outmask & g.DISTANCE) {
B1 = g.SinCosSeries(true, ssig2, csig2, C1a) -
g.SinCosSeries(true, ssig1, csig1, C1a);
// Missing a factor of _b
vals.s12b = A1 * (sig12 + B1);
if (outmask & (g.REDUCEDLENGTH | g.GEODESICSCALE)) {
B2 = g.SinCosSeries(true, ssig2, csig2, C2a) -
g.SinCosSeries(true, ssig1, csig1, C2a);
J12 = m0x * sig12 + (A1 * B1 - A2 * B2);
}
} else if (outmask & (g.REDUCEDLENGTH | g.GEODESICSCALE)) {
// Assume here that nC1_ >= nC2_
for (l = 1; l <= g.nC2_; ++l)
C2a[l] = A1 * C1a[l] - A2 * C2a[l];
J12 = m0x * sig12 + (g.SinCosSeries(true, ssig2, csig2, C2a) -
g.SinCosSeries(true, ssig1, csig1, C2a));
}
if (outmask & g.REDUCEDLENGTH) {
vals.m0 = m0x;
// Missing a factor of _b.
// Add parens around (csig1 * ssig2) and (ssig1 * csig2) to ensure
// accurate cancellation in the case of coincident points.
vals.m12b = dn2 * (csig1 * ssig2) - dn1 * (ssig1 * csig2) -
csig1 * csig2 * J12;
}
if (outmask & g.GEODESICSCALE) {
csig12 = csig1 * csig2 + ssig1 * ssig2;
t = this._ep2 * (cbet1 - cbet2) * (cbet1 + cbet2) / (dn1 + dn2);
vals.M12 = csig12 + (t * ssig2 - csig2 * J12) * ssig1 / dn1;
vals.M21 = csig12 - (t * ssig1 - csig1 * J12) * ssig2 / dn2;
}
return vals;
};
// return sig12, salp1, calp1, salp2, calp2, dnm
g.Geodesic.prototype.InverseStart = function(sbet1, cbet1, dn1,
sbet2, cbet2, dn2,
lam12, slam12, clam12,
C1a, C2a) {
// Return a starting point for Newton's method in salp1 and calp1
// (function value is -1). If Newton's method doesn't need to be
// used, return also salp2 and calp2 and function value is sig12.
// salp2, calp2 only updated if return val >= 0.
var vals = {},
// bet12 = bet2 - bet1 in [0, pi); bet12a = bet2 + bet1 in (-pi, 0]
sbet12 = sbet2 * cbet1 - cbet2 * sbet1,
cbet12 = cbet2 * cbet1 + sbet2 * sbet1,
sbet12a, shortline, omg12, sbetm2, somg12, comg12, t, ssig12, csig12,
x, y, lamscale, betscale, k2, eps, cbet12a, bet12a, m12b, m0, nvals,
k, omg12a, lam12x;
vals.sig12 = -1; // Return value
// Volatile declaration needed to fix inverse cases
// 88.202499451857 0 -88.202499451857 179.981022032992859592
// 89.262080389218 0 -89.262080389218 179.992207982775375662
// 89.333123580033 0 -89.333123580032997687 179.99295812360148422
// which otherwise fail with g++ 4.4.4 x86 -O3
sbet12a = sbet2 * cbet1;
sbet12a += cbet2 * sbet1;
shortline = cbet12 >= 0 && sbet12 < 0.5 && cbet2 * lam12 < 0.5;
if (shortline) {
sbetm2 = m.sq(sbet1 + sbet2);
// sin((bet1+bet2)/2)^2
// = (sbet1 + sbet2)^2 / ((sbet1 + sbet2)^2 + (cbet1 + cbet2)^2)
sbetm2 /= sbetm2 + m.sq(cbet1 + cbet2);
vals.dnm = Math.sqrt(1 + this._ep2 * sbetm2);
omg12 = lam12 / (this._f1 * vals.dnm);
somg12 = Math.sin(omg12); comg12 = Math.cos(omg12);
} else {
somg12 = slam12; comg12 = clam12;
}
vals.salp1 = cbet2 * somg12;
vals.calp1 = comg12 >= 0 ?
sbet12 + cbet2 * sbet1 * m.sq(somg12) / (1 + comg12) :
sbet12a - cbet2 * sbet1 * m.sq(somg12) / (1 - comg12);
ssig12 = m.hypot(vals.salp1, vals.calp1);
csig12 = sbet1 * sbet2 + cbet1 * cbet2 * comg12;
if (shortline && ssig12 < this._etol2) {
// really short lines
vals.salp2 = cbet1 * somg12;
vals.calp2 = sbet12 - cbet1 * sbet2 *
(comg12 >= 0 ? m.sq(somg12) / (1 + comg12) : 1 - comg12);
// norm(vals.salp2, vals.calp2);
t = m.hypot(vals.salp2, vals.calp2); vals.salp2 /= t; vals.calp2 /= t;
// Set return value
vals.sig12 = Math.atan2(ssig12, csig12);
} else if (Math.abs(this._n) > 0.1 || // Skip astroid calc if too eccentric
csig12 >= 0 ||
ssig12 >= 6 * Math.abs(this._n) * Math.PI * m.sq(cbet1)) {
// Nothing to do, zeroth order spherical approximation is OK
} else {
// Scale lam12 and bet2 to x, y coordinate system where antipodal
// point is at origin and singular point is at y = 0, x = -1.
lam12x = Math.atan2(-slam12, -clam12); // lam12 - pi
if (this.f >= 0) { // In fact f == 0 does not get here
// x = dlong, y = dlat
k2 = m.sq(sbet1) * this._ep2;
eps = k2 / (2 * (1 + Math.sqrt(1 + k2)) + k2);
lamscale = this.f * cbet1 * this.A3f(eps) * Math.PI;
betscale = lamscale * cbet1;
x = lam12x / lamscale;
y = sbet12a / betscale;
} else { // f < 0
// x = dlat, y = dlong
cbet12a = cbet2 * cbet1 - sbet2 * sbet1;
bet12a = Math.atan2(sbet12a, cbet12a);
// In the case of lon12 = 180, this repeats a calculation made
// in Inverse.
nvals = this.Lengths(this._n, Math.PI + bet12a,
sbet1, -cbet1, dn1, sbet2, cbet2, dn2,
cbet1, cbet2, g.REDUCEDLENGTH, C1a, C2a);
m12b = nvals.m12b; m0 = nvals.m0;
x = -1 + m12b / (cbet1 * cbet2 * m0 * Math.PI);
betscale = x < -0.01 ? sbet12a / x :
-this.f * m.sq(cbet1) * Math.PI;
lamscale = betscale / cbet1;
y = lam12 / lamscale;
}
if (y > -tol1_ && x > -1 - xthresh_) {
// strip near cut
if (this.f >= 0) {
vals.salp1 = Math.min(1, -x);
vals.calp1 = -Math.sqrt(1 - m.sq(vals.salp1));
} else {
vals.calp1 = Math.max(x > -tol1_ ? 0 : -1, x);
vals.salp1 = Math.sqrt(1 - m.sq(vals.calp1));
}
} else {
// Estimate alp1, by solving the astroid problem.
//
// Could estimate alpha1 = theta + pi/2, directly, i.e.,
// calp1 = y/k; salp1 = -x/(1+k); for f >= 0
// calp1 = x/(1+k); salp1 = -y/k; for f < 0 (need to check)
//
// However, it's better to estimate omg12 from astroid and use
// spherical formula to compute alp1. This reduces the mean number of
// Newton iterations for astroid cases from 2.24 (min 0, max 6) to 2.12
// (min 0 max 5). The changes in the number of iterations are as
// follows:
//
// change percent
// 1 5
// 0 78
// -1 16
// -2 0.6
// -3 0.04
// -4 0.002
//
// The histogram of iterations is (m = number of iterations estimating
// alp1 directly, n = number of iterations estimating via omg12, total
// number of trials = 148605):
//
// iter m n
// 0 148 186
// 1 13046 13845
// 2 93315 102225
// 3 36189 32341
// 4 5396 7
// 5 455 1
// 6 56 0
//
// Because omg12 is near pi, estimate work with omg12a = pi - omg12
k = astroid(x, y);
omg12a = lamscale * ( this.f >= 0 ? -x * k/(1 + k) : -y * (1 + k)/k );
somg12 = Math.sin(omg12a); comg12 = -Math.cos(omg12a);
// Update spherical estimate of alp1 using omg12 instead of
// lam12
vals.salp1 = cbet2 * somg12;
vals.calp1 = sbet12a -
cbet2 * sbet1 * m.sq(somg12) / (1 - comg12);
}
}
// Sanity check on starting guess. Backwards check allows NaN through.
// jshint -W018
if (!(vals.salp1 <= 0)) {
// norm(vals.salp1, vals.calp1);
t = m.hypot(vals.salp1, vals.calp1); vals.salp1 /= t; vals.calp1 /= t;
} else {
vals.salp1 = 1; vals.calp1 = 0;
}
return vals;
};
// return lam12, salp2, calp2, sig12, ssig1, csig1, ssig2, csig2, eps,
// domg12, dlam12,
g.Geodesic.prototype.Lambda12 = function(sbet1, cbet1, dn1,
sbet2, cbet2, dn2,
salp1, calp1, slam120, clam120,
diffp, C1a, C2a, C3a) {
var vals = {},
t, salp0, calp0,
somg1, comg1, somg2, comg2, somg12, comg12, B312, eta, k2, nvals;
if (sbet1 === 0 && calp1 === 0)
// Break degeneracy of equatorial line. This case has already been
// handled.
calp1 = -g.tiny_;
// sin(alp1) * cos(bet1) = sin(alp0)
salp0 = salp1 * cbet1;
calp0 = m.hypot(calp1, salp1 * sbet1); // calp0 > 0
// tan(bet1) = tan(sig1) * cos(alp1)
// tan(omg1) = sin(alp0) * tan(sig1) = tan(omg1)=tan(alp1)*sin(bet1)
vals.ssig1 = sbet1; somg1 = salp0 * sbet1;
vals.csig1 = comg1 = calp1 * cbet1;
// norm(vals.ssig1, vals.csig1);
t = m.hypot(vals.ssig1, vals.csig1); vals.ssig1 /= t; vals.csig1 /= t;
// norm(somg1, comg1); -- don't need to normalize!
// Enforce symmetries in the case abs(bet2) = -bet1. Need to be careful
// about this case, since this can yield singularities in the Newton
// iteration.
// sin(alp2) * cos(bet2) = sin(alp0)
vals.salp2 = cbet2 !== cbet1 ? salp0 / cbet2 : salp1;
// calp2 = sqrt(1 - sq(salp2))
// = sqrt(sq(calp0) - sq(sbet2)) / cbet2
// and subst for calp0 and rearrange to give (choose positive sqrt
// to give alp2 in [0, pi/2]).
vals.calp2 = cbet2 !== cbet1 || Math.abs(sbet2) !== -sbet1 ?
Math.sqrt(m.sq(calp1 * cbet1) + (cbet1 < -sbet1 ?
(cbet2 - cbet1) * (cbet1 + cbet2) :
(sbet1 - sbet2) * (sbet1 + sbet2))) /
cbet2 : Math.abs(calp1);
// tan(bet2) = tan(sig2) * cos(alp2)
// tan(omg2) = sin(alp0) * tan(sig2).
vals.ssig2 = sbet2; somg2 = salp0 * sbet2;
vals.csig2 = comg2 = vals.calp2 * cbet2;
// norm(vals.ssig2, vals.csig2);
t = m.hypot(vals.ssig2, vals.csig2); vals.ssig2 /= t; vals.csig2 /= t;
// norm(somg2, comg2); -- don't need to normalize!
// sig12 = sig2 - sig1, limit to [0, pi]
vals.sig12 = Math.atan2(Math.max(0, vals.csig1 * vals.ssig2 -
vals.ssig1 * vals.csig2),
vals.csig1 * vals.csig2 +
vals.ssig1 * vals.ssig2);
// omg12 = omg2 - omg1, limit to [0, pi]
somg12 = Math.max(0, comg1 * somg2 - somg1 * comg2);
comg12 = comg1 * comg2 + somg1 * somg2;
// eta = omg12 - lam120
eta = Math.atan2(somg12 * clam120 - comg12 * slam120,
comg12 * clam120 + somg12 * slam120);
k2 = m.sq(calp0) * this._ep2;
vals.eps = k2 / (2 * (1 + Math.sqrt(1 + k2)) + k2);
this.C3f(vals.eps, C3a);
B312 = (g.SinCosSeries(true, vals.ssig2, vals.csig2, C3a) -
g.SinCosSeries(true, vals.ssig1, vals.csig1, C3a));
vals.domg12 = -this.f * this.A3f(vals.eps) * salp0 * (vals.sig12 + B312);
vals.lam12 = eta + vals.domg12;
if (diffp) {
if (vals.calp2 === 0)
vals.dlam12 = -2 * this._f1 * dn1 / sbet1;
else {
nvals = this.Lengths(vals.eps, vals.sig12,
vals.ssig1, vals.csig1, dn1,
vals.ssig2, vals.csig2, dn2,
cbet1, cbet2, g.REDUCEDLENGTH, C1a, C2a);
vals.dlam12 = nvals.m12b;
vals.dlam12 *= this._f1 / (vals.calp2 * cbet2);
}
}
return vals;
};
/**
* @summary Solve the inverse geodesic problem.
* @param {number} lat1 the latitude of the first point in degrees.
* @param {number} lon1 the longitude of the first point in degrees.
* @param {number} lat2 the latitude of the second point in degrees.
* @param {number} lon2 the longitude of the second point in degrees.
* @param {bitmask} [outmask = STANDARD] which results to include.
* @return {object} the requested results
* @description The lat1, lon1, lat2, lon2, and a12 fields of the result are
* always set. For details on the outmask parameter, see {@tutorial
* 2-interface}, "The outmask and caps parameters".
*/
g.Geodesic.prototype.Inverse = function(lat1, lon1, lat2, lon2, outmask) {
var r, vals;
if (!outmask) outmask = g.STANDARD;
if (outmask === g.LONG_UNROLL) outmask |= g.STANDARD;
outmask &= g.OUT_MASK;
r = this.InverseInt(lat1, lon1, lat2, lon2, outmask);
vals = r.vals;
if (outmask & g.AZIMUTH) {
vals.azi1 = m.atan2d(r.salp1, r.calp1);
vals.azi2 = m.atan2d(r.salp2, r.calp2);
}
return vals;
};
g.Geodesic.prototype.InverseInt = function(lat1, lon1, lat2, lon2, outmask) {
var vals = {},
lon12, lon12s, lonsign, t, swapp, latsign,
sbet1, cbet1, sbet2, cbet2, s12x, m12x,
dn1, dn2, lam12, slam12, clam12,
sig12, calp1, salp1, calp2, salp2, C1a, C2a, C3a, meridian, nvals,
ssig1, csig1, ssig2, csig2, eps, omg12, dnm,
numit, salp1a, calp1a, salp1b, calp1b,
tripn, tripb, v, dv, dalp1, sdalp1, cdalp1, nsalp1,
lengthmask, salp0, calp0, alp12, k2, A4, C4a, B41, B42,
somg12, comg12, domg12, dbet1, dbet2, salp12, calp12, sdomg12, cdomg12;
// Compute longitude difference (AngDiff does this carefully). Result is
// in [-180, 180] but -180 is only for west-going geodesics. 180 is for
// east-going and meridional geodesics.
vals.lat1 = lat1 = m.LatFix(lat1); vals.lat2 = lat2 = m.LatFix(lat2);
// If really close to the equator, treat as on equator.
lat1 = m.AngRound(lat1);
lat2 = m.AngRound(lat2);
lon12 = m.AngDiff(lon1, lon2); lon12s = lon12.e; lon12 = lon12.d;
if (outmask & g.LONG_UNROLL) {
vals.lon1 = lon1; vals.lon2 = (lon1 + lon12) + lon12s;
} else {
vals.lon1 = m.AngNormalize(lon1); vals.lon2 = m.AngNormalize(lon2);
}
// Make longitude difference positive.
lonsign = m.copysign(1, lon12);
lon12 *= lonsign; lon12s *= lonsign;
lam12 = lon12 * m.degree;
// Calculate sincos of lon12 + error (this applies AngRound internally).
t = m.sincosde(lon12, lon12s); slam12 = t.s; clam12 = t.c;
lon12s = (180 - lon12) - lon12s; // the supplementary longitude difference
// Swap points so that point with higher (abs) latitude is point 1
// If one latitude is a nan, then it becomes lat1.
swapp = Math.abs(lat1) < Math.abs(lat2) || isNaN(lat2) ? -1 : 1;
if (swapp < 0) {
lonsign *= -1;
[lat2, lat1] = [lat1, lat2]; // swap(lat1, lat2);
}
// Make lat1 <= 0
latsign = m.copysign(1, -lat1);
lat1 *= latsign;
lat2 *= latsign;
// Now we have
//
// 0 <= lon12 <= 180
// -90 <= lat1 <= 0
// lat1 <= lat2 <= -lat1
//
// longsign, swapp, latsign register the transformation to bring the
// coordinates to this canonical form. In all cases, 1 means no change was
// made. We make these transformations so that there are few cases to
// check, e.g., on verifying quadrants in atan2. In addition, this
// enforces some symmetries in the results returned.
t = m.sincosd(lat1); sbet1 = this._f1 * t.s; cbet1 = t.c;
// norm(sbet1, cbet1);
t = m.hypot(sbet1, cbet1); sbet1 /= t; cbet1 /= t;
// Ensure cbet1 = +epsilon at poles
cbet1 = Math.max(g.tiny_, cbet1);
t = m.sincosd(lat2); sbet2 = this._f1 * t.s; cbet2 = t.c;
// norm(sbet2, cbet2);
t = m.hypot(sbet2, cbet2); sbet2 /= t; cbet2 /= t;
// Ensure cbet2 = +epsilon at poles
cbet2 = Math.max(g.tiny_, cbet2);
// If cbet1 < -sbet1, then cbet2 - cbet1 is a sensitive measure of the
// |bet1| - |bet2|. Alternatively (cbet1 >= -sbet1), abs(sbet2) + sbet1 is
// a better measure. This logic is used in assigning calp2 in Lambda12.
// Sometimes these quantities vanish and in that case we force bet2 = +/-
// bet1 exactly. An example where is is necessary is the inverse problem
// 48.522876735459 0 -48.52287673545898293 179.599720456223079643
// which failed with Visual Studio 10 (Release and Debug)
if (cbet1 < -sbet1) {
if (cbet2 === cbet1)
sbet2 = m.copysign(sbet1, sbet2);
} else {
if (Math.abs(sbet2) === -sbet1)
cbet2 = cbet1;
}
dn1 = Math.sqrt(1 + this._ep2 * m.sq(sbet1));
dn2 = Math.sqrt(1 + this._ep2 * m.sq(sbet2));
// index zero elements of these arrays are unused
C1a = new Array(g.nC1_ + 1);
C2a = new Array(g.nC2_ + 1);
C3a = new Array(g.nC3_);
meridian = lat1 === -90 || slam12 === 0;
if (meridian) {
// Endpoints are on a single full meridian, so the geodesic might
// lie on a meridian.
calp1 = clam12; salp1 = slam12; // Head to the target longitude
calp2 = 1; salp2 = 0; // At the target we're heading north
// tan(bet) = tan(sig) * cos(alp)
ssig1 = sbet1; csig1 = calp1 * cbet1;
ssig2 = sbet2; csig2 = calp2 * cbet2;
// sig12 = sig2 - sig1
sig12 = Math.atan2(Math.max(0, csig1 * ssig2 - ssig1 * csig2),
csig1 * csig2 + ssig1 * ssig2);
nvals = this.Lengths(this._n, sig12,
ssig1, csig1, dn1, ssig2, csig2, dn2, cbet1, cbet2,
outmask | g.DISTANCE | g.REDUCEDLENGTH,
C1a, C2a);
s12x = nvals.s12b;
m12x = nvals.m12b;
// Ignore m0
if (outmask & g.GEODESICSCALE) {
vals.M12 = nvals.M12;
vals.M21 = nvals.M21;
}
// Add the check for sig12 since zero length geodesics might yield
// m12 < 0. Test case was
//
// echo 20.001 0 20.001 0 | GeodSolve -i
//
// In fact, we will have sig12 > pi/2 for meridional geodesic
// which is not a shortest path.
if (sig12 < 1 || m12x >= 0) {
// Need at least 2, to handle 90 0 90 180
if (sig12 < 3 * g.tiny_ ||
// Prevent negative s12 or m12 for short lines
(sig12 < tol0_ && (s12x < 0 || m12x < 0)))
sig12 = m12x = s12x = 0;
m12x *= this._b;
s12x *= this._b;
vals.a12 = sig12 / m.degree;
} else
// m12 < 0, i.e., prolate and too close to anti-podal
meridian = false;
}
somg12 = 2;
if (!meridian &&
sbet1 === 0 && // and sbet2 == 0
(this.f <= 0 || lon12s >= this.f * 180)) {
// Geodesic runs along equator
calp1 = calp2 = 0; salp1 = salp2 = 1;
s12x = this.a * lam12;
sig12 = omg12 = lam12 / this._f1;
m12x = this._b * Math.sin(sig12);
if (outmask & g.GEODESICSCALE)
vals.M12 = vals.M21 = Math.cos(sig12);
vals.a12 = lon12 / this._f1;
} else if (!meridian) {
// Now point1 and point2 belong within a hemisphere bounded by a
// meridian and geodesic is neither meridional or equatorial.
// Figure a starting point for Newton's method
nvals = this.InverseStart(sbet1, cbet1, dn1, sbet2, cbet2, dn2,
lam12, slam12, clam12, C1a, C2a);
sig12 = nvals.sig12;
salp1 = nvals.salp1;
calp1 = nvals.calp1;
if (sig12 >= 0) {
salp2 = nvals.salp2;
calp2 = nvals.calp2;
// Short lines (InverseStart sets salp2, calp2, dnm)
dnm = nvals.dnm;
s12x = sig12 * this._b * dnm;
m12x = m.sq(dnm) * this._b * Math.sin(sig12 / dnm);
if (outmask & g.GEODESICSCALE)
vals.M12 = vals.M21 = Math.cos(sig12 / dnm);
vals.a12 = sig12 / m.degree;
omg12 = lam12 / (this._f1 * dnm);
} else {
// Newton's method. This is a straightforward solution of f(alp1) =
// lambda12(alp1) - lam12 = 0 with one wrinkle. f(alp) has exactly one
// root in the interval (0, pi) and its derivative is positive at the
// root. Thus f(alp) is positive for alp > alp1 and negative for alp <
// alp1. During the course of the iteration, a range (alp1a, alp1b) is
// maintained which brackets the root and with each evaluation of
// f(alp) the range is shrunk if possible. Newton's method is
// restarted whenever the derivative of f is negative (because the new
// value of alp1 is then further from the solution) or if the new
// estimate of alp1 lies outside (0,pi); in this case, the new starting
// guess is taken to be (alp1a + alp1b) / 2.
numit = 0;
// Bracketing range
salp1a = g.tiny_; calp1a = 1; salp1b = g.tiny_; calp1b = -1;
for (tripn = false, tripb = false; numit < maxit2_; ++numit) {
// the WGS84 test set: mean = 1.47, sd = 1.25, max = 16
// WGS84 and random input: mean = 2.85, sd = 0.60
nvals = this.Lambda12(sbet1, cbet1, dn1, sbet2, cbet2, dn2,
salp1, calp1, slam12, clam12, numit < maxit1_,
C1a, C2a, C3a);
v = nvals.lam12;
salp2 = nvals.salp2;
calp2 = nvals.calp2;
sig12 = nvals.sig12;
ssig1 = nvals.ssig1;
csig1 = nvals.csig1;
ssig2 = nvals.ssig2;
csig2 = nvals.csig2;
eps = nvals.eps;
domg12 = nvals.domg12;
dv = nvals.dlam12;
// Reversed test to allow escape with NaNs
// jshint -W018
if (tripb || !(Math.abs(v) >= (tripn ? 8 : 1) * tol0_))
break;
// Update bracketing values
if (v > 0 && (numit < maxit1_ || calp1/salp1 > calp1b/salp1b)) {
salp1b = salp1; calp1b = calp1;
} else if (v < 0 &&
(numit < maxit1_ || calp1/salp1 < calp1a/salp1a)) {
salp1a = salp1; calp1a = calp1;
}
if (numit < maxit1_ && dv > 0) {
dalp1 = -v/dv;
sdalp1 = Math.sin(dalp1); cdalp1 = Math.cos(dalp1);
nsalp1 = salp1 * cdalp1 + calp1 * sdalp1;
if (nsalp1 > 0 && Math.abs(dalp1) < Math.PI) {
calp1 = calp1 * cdalp1 - salp1 * sdalp1;
salp1 = nsalp1;
// norm(salp1, calp1);
t = m.hypot(salp1, calp1); salp1 /= t; calp1 /= t;
// In some regimes we don't get quadratic convergence because
// slope -> 0. So use convergence conditions based on epsilon
// instead of sqrt(epsilon).
tripn = Math.abs(v) <= 16 * tol0_;
continue;
}
}
// Either dv was not positive or updated value was outside legal
// range. Use the midpoint of the bracket as the next estimate.
// This mechanism is not needed for the WGS84 ellipsoid, but it does
// catch problems with more eccentric ellipsoids. Its efficacy is
// such for the WGS84 test set with the starting guess set to alp1 =
// 90deg:
// the WGS84 test set: mean = 5.21, sd = 3.93, max = 24
// WGS84 and random input: mean = 4.74, sd = 0.99
salp1 = (salp1a + salp1b)/2;
calp1 = (calp1a + calp1b)/2;
// norm(salp1, calp1);
t = m.hypot(salp1, calp1); salp1 /= t; calp1 /= t;
tripn = false;
tripb = (Math.abs(salp1a - salp1) + (calp1a - calp1) < tolb_ ||
Math.abs(salp1 - salp1b) + (calp1 - calp1b) < tolb_);
}
lengthmask = outmask |
(outmask & (g.REDUCEDLENGTH | g.GEODESICSCALE) ?
g.DISTANCE : g.NONE);
nvals = this.Lengths(eps, sig12,
ssig1, csig1, dn1, ssig2, csig2, dn2,
cbet1, cbet2,
lengthmask, C1a, C2a);
s12x = nvals.s12b;
m12x = nvals.m12b;
// Ignore m0
if (outmask & g.GEODESICSCALE) {
vals.M12 = nvals.M12;
vals.M21 = nvals.M21;
}
m12x *= this._b;
s12x *= this._b;
vals.a12 = sig12 / m.degree;
if (outmask & g.AREA) {
// omg12 = lam12 - domg12
sdomg12 = Math.sin(domg12); cdomg12 = Math.cos(domg12);
somg12 = slam12 * cdomg12 - clam12 * sdomg12;
comg12 = clam12 * cdomg12 + slam12 * sdomg12;
}
}
}
if (outmask & g.DISTANCE)
vals.s12 = 0 + s12x; // Convert -0 to 0
if (outmask & g.REDUCEDLENGTH)
vals.m12 = 0 + m12x; // Convert -0 to 0
if (outmask & g.AREA) {
// From Lambda12: sin(alp1) * cos(bet1) = sin(alp0)
salp0 = salp1 * cbet1;
calp0 = m.hypot(calp1, salp1 * sbet1); // calp0 > 0
if (calp0 !== 0 && salp0 !== 0) {
// From Lambda12: tan(bet) = tan(sig) * cos(alp)
ssig1 = sbet1; csig1 = calp1 * cbet1;
ssig2 = sbet2; csig2 = calp2 * cbet2;
k2 = m.sq(calp0) * this._ep2;
eps = k2 / (2 * (1 + Math.sqrt(1 + k2)) + k2);
// Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0).
A4 = m.sq(this.a) * calp0 * salp0 * this._e2;
// norm(ssig1, csig1);
t = m.hypot(ssig1, csig1); ssig1 /= t; csig1 /= t;
// norm(ssig2, csig2);
t = m.hypot(ssig2, csig2); ssig2 /= t; csig2 /= t;
C4a = new Array(g.nC4_);
this.C4f(eps, C4a);
B41 = g.SinCosSeries(false, ssig1, csig1, C4a);
B42 = g.SinCosSeries(false, ssig2, csig2, C4a);
vals.S12 = A4 * (B42 - B41);
} else
// Avoid problems with indeterminate sig1, sig2 on equator
vals.S12 = 0;
if (!meridian && somg12 == 2) {
somg12 = Math.sin(omg12); comg12 = Math.cos(omg12);
}
if (!meridian &&
comg12 > -0.7071 && // Long difference not too big
sbet2 - sbet1 < 1.75) { // Lat difference not too big
// Use tan(Gamma/2) = tan(omg12/2)
// * (tan(bet1/2)+tan(bet2/2))/(1+tan(bet1/2)*tan(bet2/2))
// with tan(x/2) = sin(x)/(1+cos(x))
domg12 = 1 + comg12; dbet1 = 1 + cbet1; dbet2 = 1 + cbet2;
alp12 = 2 * Math.atan2( somg12 * (sbet1*dbet2 + sbet2*dbet1),
domg12 * (sbet1*sbet2 + dbet1*dbet2) );
} else {
// alp12 = alp2 - alp1, used in atan2 so no need to normalize
salp12 = salp2 * calp1 - calp2 * salp1;
calp12 = calp2 * calp1 + salp2 * salp1;
// The right thing appears to happen if alp1 = +/-180 and alp2 = 0, viz
// salp12 = -0 and alp12 = -180. However this depends on the sign
// being attached to 0 correctly. The following ensures the correct
// behavior.
if (salp12 === 0 && calp12 < 0) {
salp12 = g.tiny_ * calp1;
calp12 = -1;
}
alp12 = Math.atan2(salp12, calp12);
}
vals.S12 += this._c2 * alp12;
vals.S12 *= swapp * lonsign * latsign;
// Convert -0 to 0
vals.S12 += 0;
}
// Convert calp, salp to azimuth accounting for lonsign, swapp, latsign.
if (swapp < 0) {
[salp2, salp1] = [salp1, salp2]; // swap(salp1, salp2);
[calp2, calp1] = [calp1, calp2]; // swap(calp1, calp2);
if (outmask & g.GEODESICSCALE) {
[vals.M21, vals.M12] = [vals.M12, vals.M21]; //swap(vals.M12, vals.M21);
}
}
salp1 *= swapp * lonsign; calp1 *= swapp * latsign;
salp2 *= swapp * lonsign; calp2 *= swapp * latsign;
return {vals: vals,
salp1: salp1, calp1: calp1,
salp2: salp2, calp2: calp2};
};
/**
* @summary Solve the general direct geodesic problem.
* @param {number} lat1 the latitude of the first point in degrees.
* @param {number} lon1 the longitude of the first point in degrees.
* @param {number} azi1 the azimuth at the first point in degrees.
* @param {bool} arcmode is the next parameter an arc length?
* @param {number} s12_a12 the (arcmode ? arc length : distance) from the
* first point to the second in (arcmode ? degrees : meters).
* @param {bitmask} [outmask = STANDARD] which results to include.
* @return {object} the requested results.
* @description The lat1, lon1, azi1, and a12 fields of the result are always
* set; s12 is included if arcmode is false. For details on the outmask
* parameter, see {@tutorial 2-interface}, "The outmask and caps
* parameters".
*/
g.Geodesic.prototype.GenDirect = function(lat1, lon1, azi1,
arcmode, s12_a12, outmask) {
var line;
if (!outmask) outmask = g.STANDARD;
else if (outmask === g.LONG_UNROLL) outmask |= g.STANDARD;
// Automatically supply DISTANCE_IN if necessary
if (!arcmode) outmask |= g.DISTANCE_IN;
line = new l.GeodesicLine(this, lat1, lon1, azi1, outmask);
return line.GenPosition(arcmode, s12_a12, outmask);
};
/**
* @summary Solve the direct geodesic problem.
* @param {number} lat1 the latitude of the first point in degrees.
* @param {number} lon1 the longitude of the first point in degrees.
* @param {number} azi1 the azimuth at the first point in degrees.
* @param {number} s12 the distance from the first point to the second in
* meters.
* @param {bitmask} [outmask = STANDARD] which results to include.
* @return {object} the requested results.
* @description The lat1, lon1, azi1, s12, and a12 fields of the result are
* always set. For details on the outmask parameter, see {@tutorial
* 2-interface}, "The outmask and caps parameters".
*/
g.Geodesic.prototype.Direct = function(lat1, lon1, azi1, s12, outmask) {
return this.GenDirect(lat1, lon1, azi1, false, s12, outmask);
};
/**
* @summary Solve the direct geodesic problem with arc length.
* @param {number} lat1 the latitude of the first point in degrees.
* @param {number} lon1 the longitude of the first point in degrees.
* @param {number} azi1 the azimuth at the first point in degrees.
* @param {number} a12 the arc length from the first point to the second in
* degrees.
* @param {bitmask} [outmask = STANDARD] which results to include.
* @return {object} the requested results.
* @description The lat1, lon1, azi1, and a12 fields of the result are
* always set. For details on the outmask parameter, see {@tutorial
* 2-interface}, "The outmask and caps parameters".
*/
g.Geodesic.prototype.ArcDirect = function(lat1, lon1, azi1, a12, outmask) {
return this.GenDirect(lat1, lon1, azi1, true, a12, outmask);
};
/**
* @summary Create a {@link module:geodesic/GeodesicLine.GeodesicLine
* GeodesicLine} object.
* @param {number} lat1 the latitude of the first point in degrees.
* @param {number} lon1 the longitude of the first point in degrees.
* @param {number} azi1 the azimuth at the first point in degrees.
* degrees.
* @param {bitmask} [caps = STANDARD | DISTANCE_IN] which capabilities to
* include.
* @return {object} the
* {@link module:geodesic/GeodesicLine.GeodesicLine
* GeodesicLine} object
* @description For details on the caps parameter, see {@tutorial
* 2-interface}, "The outmask and caps parameters".
*/
g.Geodesic.prototype.Line = function(lat1, lon1, azi1, caps) {
return new l.GeodesicLine(this, lat1, lon1, azi1, caps);
};
/**
* @summary Define a {@link module:geodesic/GeodesicLine.GeodesicLine
* GeodesicLine} in terms of the direct geodesic problem specified in terms
* of distance.
* @param {number} lat1 the latitude of the first point in degrees.
* @param {number} lon1 the longitude of the first point in degrees.
* @param {number} azi1 the azimuth at the first point in degrees.
* degrees.
* @param {number} s12 the distance between point 1 and point 2 (meters); it
* can be negative.
* @param {bitmask} [caps = STANDARD | DISTANCE_IN] which capabilities to
* include.
* @return {object} the
* {@link module:geodesic/GeodesicLine.GeodesicLine
* GeodesicLine} object
* @description This function sets point 3 of the GeodesicLine to correspond
* to point 2 of the direct geodesic problem. For details on the caps
* parameter, see {@tutorial 2-interface}, "The outmask and caps
* parameters".
*/
g.Geodesic.prototype.DirectLine = function(lat1, lon1, azi1, s12, caps) {
return this.GenDirectLine(lat1, lon1, azi1, false, s12, caps);
};
/**
* @summary Define a {@link module:geodesic/GeodesicLine.GeodesicLine
* GeodesicLine} in terms of the direct geodesic problem specified in terms
* of arc length.
* @param {number} lat1 the latitude of the first point in degrees.
* @param {number} lon1 the longitude of the first point in degrees.
* @param {number} azi1 the azimuth at the first point in degrees.
* degrees.
* @param {number} a12 the arc length between point 1 and point 2 (degrees);
* it can be negative.
* @param {bitmask} [caps = STANDARD | DISTANCE_IN] which capabilities to
* include.
* @return {object} the
* {@link module:geodesic/GeodesicLine.GeodesicLine
* GeodesicLine} object
* @description This function sets point 3 of the GeodesicLine to correspond
* to point 2 of the direct geodesic problem. For details on the caps
* parameter, see {@tutorial 2-interface}, "The outmask and caps
* parameters".
*/
g.Geodesic.prototype.ArcDirectLine = function(lat1, lon1, azi1, a12, caps) {
return this.GenDirectLine(lat1, lon1, azi1, true, a12, caps);
};
/**
* @summary Define a {@link module:geodesic/GeodesicLine.GeodesicLine
* GeodesicLine} in terms of the direct geodesic problem specified in terms
* of either distance or arc length.
* @param {number} lat1 the latitude of the first point in degrees.
* @param {number} lon1 the longitude of the first point in degrees.
* @param {number} azi1 the azimuth at the first point in degrees.
* degrees.
* @param {bool} arcmode boolean flag determining the meaning of the
* s12_a12.
* @param {number} s12_a12 if arcmode is false, this is the distance between
* point 1 and point 2 (meters); otherwise it is the arc length between
* point 1 and point 2 (degrees); it can be negative.
* @param {bitmask} [caps = STANDARD | DISTANCE_IN] which capabilities to
* include.
* @return {object} the
* {@link module:geodesic/GeodesicLine.GeodesicLine
* GeodesicLine} object
* @description This function sets point 3 of the GeodesicLine to correspond
* to point 2 of the direct geodesic problem. For details on the caps
* parameter, see {@tutorial 2-interface}, "The outmask and caps
* parameters".
*/
g.Geodesic.prototype.GenDirectLine = function(lat1, lon1, azi1,
arcmode, s12_a12, caps) {
var t;
if (!caps) caps = g.STANDARD | g.DISTANCE_IN;
// Automatically supply DISTANCE_IN if necessary
if (!arcmode) caps |= g.DISTANCE_IN;
t = new l.GeodesicLine(this, lat1, lon1, azi1, caps);
t.GenSetDistance(arcmode, s12_a12);
return t;
};
/**
* @summary Define a {@link module:geodesic/GeodesicLine.GeodesicLine
* GeodesicLine} in terms of the inverse geodesic problem.
* @param {number} lat1 the latitude of the first point in degrees.
* @param {number} lon1 the longitude of the first point in degrees.
* @param {number} lat2 the latitude of the second point in degrees.
* @param {number} lon2 the longitude of the second point in degrees.
* @param {bitmask} [caps = STANDARD | DISTANCE_IN] which capabilities to
* include.
* @return {object} the
* {@link module:geodesic/GeodesicLine.GeodesicLine
* GeodesicLine} object
* @description This function sets point 3 of the GeodesicLine to correspond
* to point 2 of the inverse geodesic problem. For details on the caps
* parameter, see {@tutorial 2-interface}, "The outmask and caps
* parameters".
*/
g.Geodesic.prototype.InverseLine = function(lat1, lon1, lat2, lon2, caps) {
var r, t, azi1;
if (!caps) caps = g.STANDARD | g.DISTANCE_IN;
r = this.InverseInt(lat1, lon1, lat2, lon2, g.ARC);
azi1 = m.atan2d(r.salp1, r.calp1);
// Ensure that a12 can be converted to a distance
if (caps & (g.OUT_MASK & g.DISTANCE_IN)) caps |= g.DISTANCE;
t = new l.GeodesicLine(this, lat1, lon1, azi1, caps, r.salp1, r.calp1);
t.SetArc(r.vals.a12);
return t;
};
/**
* @summary Create a {@link module:geodesic/PolygonArea.PolygonArea
* PolygonArea} object.
* @param {bool} [polyline = false] if true the new PolygonArea object
* describes a polyline instead of a polygon.
* @return {object} the
* {@link module:geodesic/PolygonArea.PolygonArea
* PolygonArea} object
*/
g.Geodesic.prototype.Polygon = function(polyline) {
return new p.PolygonArea(this, polyline);
};
/**
* @summary a {@link module:geodesic/Geodesic.Geodesic Geodesic} object
* initialized for the WGS84 ellipsoid.
* @constant {object}
*/
g.WGS84 = new g.Geodesic(c.WGS84.a, c.WGS84.f);
})(geodesic.Geodesic, geodesic.GeodesicLine,
geodesic.PolygonArea, geodesic.Math, geodesic.Constants);