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Geodesic.cpp
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/**
* \file Geodesic.cpp
* \brief Implementation for GeographicLib::Geodesic class
*
* Copyright (c) Charles Karney (2009-2025) <karney@alum.mit.edu> and licensed
* under the MIT/X11 License. For more information, see
* https://geographiclib.sourceforge.io/
*
* This is a reformulation of the geodesic problem. The notation is as
* follows:
* - at a general point (no suffix or 1 or 2 as suffix)
* - phi = latitude
* - beta = latitude on auxiliary sphere
* - omega = longitude on auxiliary sphere
* - lambda = longitude
* - alpha = azimuth of great circle
* - sigma = arc length along great circle
* - s = distance
* - tau = scaled distance (= sigma at multiples of pi/2)
* - at northwards equator crossing
* - beta = phi = 0
* - omega = lambda = 0
* - alpha = alpha0
* - sigma = s = 0
* - a 12 suffix means a difference, e.g., s12 = s2 - s1.
* - s and c prefixes mean sin and cos
**********************************************************************/
#include <GeographicLib/Geodesic.hpp>
#include <GeographicLib/GeodesicLine.hpp>
#if defined(_MSC_VER)
// Squelch warnings about potentially uninitialized local variables
# pragma warning (disable: 4701)
#endif
namespace GeographicLib {
using namespace std;
Geodesic::Geodesic(real a, real f, bool exact)
: maxit2_(maxit1_ + Math::digits() + 10)
// Underflow guard. We require
// tiny_ * epsilon() > 0
// tiny_ + epsilon() == epsilon()
, tiny_(sqrt(numeric_limits<real>::min()))
, tol0_(numeric_limits<real>::epsilon())
// Increase multiplier in defn of tol1_ from 100 to 200 to fix inverse
// case 52.784459512564 0 -52.784459512563990912 179.634407464943777557
// which otherwise failed for Visual Studio 10 (Release and Debug)
, tol1_(200 * tol0_)
, tol2_(sqrt(tol0_))
, tolb_(tol0_) // Check on bisection interval
, xthresh_(1000 * tol2_)
, _a(a)
, _f(f)
, _exact(exact)
, _f1(1 - _f)
, _e2(_f * (2 - _f))
, _ep2(_e2 / Math::sq(_f1)) // e2 / (1 - e2)
, _n(_f / ( 2 - _f))
, _b(_a * _f1)
, _c2((Math::sq(_a) + Math::sq(_b) *
(_e2 == 0 ? 1 :
Math::eatanhe(real(1), (_f < 0 ? -1 : 1) * sqrt(fabs(_e2))) / _e2))
/ 2) // authalic radius squared
// 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.
, _etol2(real(0.1) * tol2_ /
sqrt( fmax(real(0.001), fabs(_f)) * fmin(real(1), 1 - _f/2) / 2 ))
, _geodexact(_exact ? GeodesicExact(a, f) : GeodesicExact())
{
if (_exact)
_c2 = _geodexact._c2;
else {
if (!(isfinite(_a) && _a > 0))
throw GeographicErr("Equatorial radius is not positive");
if (!(isfinite(_b) && _b > 0))
throw GeographicErr("Polar semi-axis is not positive");
A3coeff();
C3coeff();
C4coeff();
}
}
const Geodesic& Geodesic::WGS84() {
static const Geodesic wgs84(Constants::WGS84_a(), Constants::WGS84_f());
return wgs84;
}
Math::real Geodesic::SinCosSeries(bool sinp,
real sinx, real cosx,
const real c[], int n) {
// 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
c += (n + sinp); // Point to one beyond last element
real
ar = 2 * (cosx - sinx) * (cosx + sinx), // 2 * cos(2 * x)
y0 = n & 1 ? *--c : 0, y1 = 0; // accumulators for sum
// Now n is even
n /= 2;
while (n--) {
// Unroll loop x 2, so accumulators return to their original role
y1 = ar * y0 - y1 + *--c;
y0 = ar * y1 - y0 + *--c;
}
return sinp
? 2 * sinx * cosx * y0 // sin(2 * x) * y0
: cosx * (y0 - y1); // cos(x) * (y0 - y1)
}
GeodesicLine Geodesic::Line(real lat1, real lon1, real azi1,
unsigned caps) const {
return GeodesicLine(*this, lat1, lon1, azi1, caps);
}
Math::real Geodesic::GenDirect(real lat1, real lon1, real azi1,
bool arcmode, real s12_a12, unsigned outmask,
real& lat2, real& lon2, real& azi2,
real& s12, real& m12, real& M12, real& M21,
real& S12) const {
if (_exact)
return _geodexact.GenDirect(lat1, lon1, azi1, arcmode, s12_a12, outmask,
lat2, lon2, azi2,
s12, m12, M12, M21, S12);
// Automatically supply DISTANCE_IN if necessary
if (!arcmode) outmask |= DISTANCE_IN;
return GeodesicLine(*this, lat1, lon1, azi1, outmask)
. // Note the dot!
GenPosition(arcmode, s12_a12, outmask,
lat2, lon2, azi2, s12, m12, M12, M21, S12);
}
GeodesicLine Geodesic::GenDirectLine(real lat1, real lon1, real azi1,
bool arcmode, real s12_a12,
unsigned caps) const {
azi1 = Math::AngNormalize(azi1);
real salp1, calp1;
// Guard against underflow in salp0. Also -0 is converted to +0.
Math::sincosd(Math::AngRound(azi1), salp1, calp1);
// Automatically supply DISTANCE_IN if necessary
if (!arcmode) caps |= DISTANCE_IN;
return GeodesicLine(*this, lat1, lon1, azi1, salp1, calp1,
caps, arcmode, s12_a12);
}
GeodesicLine Geodesic::DirectLine(real lat1, real lon1, real azi1, real s12,
unsigned caps) const {
return GenDirectLine(lat1, lon1, azi1, false, s12, caps);
}
GeodesicLine Geodesic::ArcDirectLine(real lat1, real lon1, real azi1,
real a12, unsigned caps) const {
return GenDirectLine(lat1, lon1, azi1, true, a12, caps);
}
Math::real Geodesic::GenInverse(real lat1, real lon1, real lat2, real lon2,
unsigned outmask, real& s12,
real& salp1, real& calp1,
real& salp2, real& calp2,
real& m12, real& M12, real& M21,
real& S12) const {
if (_exact)
return _geodexact.GenInverse(lat1, lon1, lat2, lon2,
outmask, s12,
salp1, calp1, salp2, calp2,
m12, M12, M21, S12);
// Compute longitude difference (AngDiff does this carefully).
using std::isnan; // Needed for Centos 7, ubuntu 14
real lon12s, lon12 = Math::AngDiff(lon1, lon2, lon12s);
// Make longitude difference positive.
int lonsign = signbit(lon12) ? -1 : 1;
lon12 *= lonsign; lon12s *= lonsign;
real
lam12 = lon12 * Math::degree(),
slam12, clam12;
// Calculate sincos of lon12 + error (this applies AngRound internally).
Math::sincosde(lon12, lon12s, slam12, clam12);
// the supplementary longitude difference
lon12s = (Math::hd - lon12) - lon12s;
// If really close to the equator, treat as on equator.
lat1 = Math::AngRound(Math::LatFix(lat1));
lat2 = Math::AngRound(Math::LatFix(lat2));
// Swap points so that point with higher (abs) latitude is point 1.
// If one latitude is a nan, then it becomes lat1.
int swapp = fabs(lat1) < fabs(lat2) || isnan(lat2) ? -1 : 1;
if (swapp < 0) {
lonsign *= -1;
swap(lat1, lat2);
}
// Make lat1 <= -0
int latsign = signbit(lat1) ? 1 : -1;
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.
real sbet1, cbet1, sbet2, cbet2, s12x, m12x = Math::NaN();
Math::sincosd(lat1, sbet1, cbet1); sbet1 *= _f1;
// Ensure cbet1 = +epsilon at poles; doing the fix on beta means that sig12
// will be <= 2*tiny for two points at the same pole.
Math::norm(sbet1, cbet1); cbet1 = fmax(tiny_, cbet1);
Math::sincosd(lat2, sbet2, cbet2); sbet2 *= _f1;
// Ensure cbet2 = +epsilon at poles
Math::norm(sbet2, cbet2); cbet2 = fmax(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 = copysign(sbet1, sbet2);
} else {
if (fabs(sbet2) == -sbet1)
cbet2 = cbet1;
}
real
dn1 = sqrt(1 + _ep2 * Math::sq(sbet1)),
dn2 = sqrt(1 + _ep2 * Math::sq(sbet2));
real a12, sig12;
// index zero element of this array is unused
real Ca[nC_];
bool meridian = lat1 == -Math::qd || 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
real
// tan(bet) = tan(sig) * cos(alp)
ssig1 = sbet1, csig1 = calp1 * cbet1,
ssig2 = sbet2, csig2 = calp2 * cbet2;
// sig12 = sig2 - sig1
sig12 = atan2(fmax(real(0), csig1 * ssig2 - ssig1 * csig2) + real(0),
csig1 * csig2 + ssig1 * ssig2);
{
real dummy;
Lengths(_n, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2, cbet1, cbet2,
outmask | DISTANCE | REDUCEDLENGTH,
s12x, m12x, dummy, M12, M21, Ca);
}
// 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
if (sig12 < tol2_ || m12x >= 0) {
// Need at least 2, to handle 90 0 90 180
if (sig12 < 3 * tiny_ ||
// Prevent negative s12 or m12 for short lines
(sig12 < tol0_ && (s12x < 0 || m12x < 0)))
sig12 = m12x = s12x = 0;
m12x *= _b;
s12x *= _b;
a12 = sig12 / Math::degree();
} else
// m12 < 0, i.e., prolate and too close to anti-podal
meridian = false;
}
// somg12 == 2 marks that it needs to be calculated
real omg12 = 0, somg12 = 2, comg12 = 0;
if (!meridian &&
sbet1 == 0 && // and sbet2 == 0
(_f <= 0 || lon12s >= _f * Math::hd)) {
// Geodesic runs along equator
calp1 = calp2 = 0; salp1 = salp2 = 1;
s12x = _a * lam12;
sig12 = omg12 = lam12 / _f1;
m12x = _b * sin(sig12);
if (outmask & GEODESICSCALE)
M12 = M21 = cos(sig12);
a12 = lon12 / _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
real dnm;
sig12 = InverseStart(sbet1, cbet1, dn1, sbet2, cbet2, dn2,
lam12, slam12, clam12,
salp1, calp1, salp2, calp2, dnm,
Ca);
if (sig12 >= 0) {
// Short lines (InverseStart sets salp2, calp2, dnm)
s12x = sig12 * _b * dnm;
m12x = Math::sq(dnm) * _b * sin(sig12 / dnm);
if (outmask & GEODESICSCALE)
M12 = M21 = cos(sig12 / dnm);
a12 = sig12 / Math::degree();
omg12 = lam12 / (_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.
//
// initial values to suppress warnings (if loop is executed 0 times)
real ssig1 = 0, csig1 = 0, ssig2 = 0, csig2 = 0, eps = 0, domg12 = 0;
unsigned numit = 0;
// Bracketing range
real salp1a = tiny_, calp1a = 1, salp1b = tiny_, calp1b = -1;
for (bool tripn = false, tripb = false;; ++numit) {
// the WGS84 test set: mean = 1.47, sd = 1.25, max = 16
// WGS84 and random input: mean = 2.85, sd = 0.60
real dv;
real v = Lambda12(sbet1, cbet1, dn1, sbet2, cbet2, dn2, salp1, calp1,
slam12, clam12,
salp2, calp2, sig12, ssig1, csig1, ssig2, csig2,
eps, domg12, numit < maxit1_, dv, Ca);
if (tripb ||
// Reversed test to allow escape with NaNs
!(fabs(v) >= (tripn ? 8 : 1) * tol0_) ||
// Enough bisections to get accurate result
numit == maxit2_)
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) {
real
dalp1 = -v/dv;
// |dalp1| < pi test moved earlier because GEOGRAPHICLIB_PRECISION
// = 5 can result in dalp1 = 10^(10^8). Then sin(dalp1) takes ages
// (because of the need to do accurate range reduction).
if (fabs(dalp1) < Math::pi()) {
real
sdalp1 = sin(dalp1), cdalp1 = cos(dalp1),
nsalp1 = salp1 * cdalp1 + calp1 * sdalp1;
if (nsalp1 > 0) {
calp1 = calp1 * cdalp1 - salp1 * sdalp1;
salp1 = nsalp1;
Math::norm(salp1, calp1);
// In some regimes we don't get quadratic convergence because
// slope -> 0. So use convergence conditions based on epsilon
// instead of sqrt(epsilon).
tripn = fabs(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;
Math::norm(salp1, calp1);
tripn = false;
tripb = (fabs(salp1a - salp1) + (calp1a - calp1) < tolb_ ||
fabs(salp1 - salp1b) + (calp1 - calp1b) < tolb_);
}
{
real dummy;
// Ensure that the reduced length and geodesic scale are computed in
// a "canonical" way, with the I2 integral.
unsigned lengthmask = outmask |
(outmask & (REDUCEDLENGTH | GEODESICSCALE) ? DISTANCE : NONE);
Lengths(eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
cbet1, cbet2, lengthmask, s12x, m12x, dummy, M12, M21, Ca);
}
m12x *= _b;
s12x *= _b;
a12 = sig12 / Math::degree();
if (outmask & AREA) {
// omg12 = lam12 - domg12
real sdomg12 = sin(domg12), cdomg12 = cos(domg12);
somg12 = slam12 * cdomg12 - clam12 * sdomg12;
comg12 = clam12 * cdomg12 + slam12 * sdomg12;
}
}
}
if (outmask & DISTANCE)
s12 = real(0) + s12x; // Convert -0 to 0
if (outmask & REDUCEDLENGTH)
m12 = real(0) + m12x; // Convert -0 to 0
if (outmask & AREA) {
real
// From Lambda12: sin(alp1) * cos(bet1) = sin(alp0)
salp0 = salp1 * cbet1,
calp0 = hypot(calp1, salp1 * sbet1); // calp0 > 0
real alp12;
if (calp0 != 0 && salp0 != 0) {
real
// From Lambda12: tan(bet) = tan(sig) * cos(alp)
ssig1 = sbet1, csig1 = calp1 * cbet1,
ssig2 = sbet2, csig2 = calp2 * cbet2,
k2 = Math::sq(calp0) * _ep2,
eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2),
// Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0).
A4 = Math::sq(_a) * calp0 * salp0 * _e2;
Math::norm(ssig1, csig1);
Math::norm(ssig2, csig2);
C4f(eps, Ca);
real
B41 = SinCosSeries(false, ssig1, csig1, Ca, nC4_),
B42 = SinCosSeries(false, ssig2, csig2, Ca, nC4_);
S12 = A4 * (B42 - B41);
} else
// Avoid problems with indeterminate sig1, sig2 on equator
S12 = 0;
if (!meridian && somg12 == 2) {
somg12 = sin(omg12); comg12 = cos(omg12);
}
if (!meridian &&
// omg12 < 3/4 * pi
comg12 > -real(0.7071) && // Long difference not too big
sbet2 - sbet1 < real(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))
real domg12 = 1 + comg12, dbet1 = 1 + cbet1, dbet2 = 1 + cbet2;
alp12 = 2 * atan2( somg12 * ( sbet1 * dbet2 + sbet2 * dbet1 ),
domg12 * ( sbet1 * sbet2 + dbet1 * dbet2 ) );
} else {
// alp12 = alp2 - alp1, used in atan2 so no need to normalize
real
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 = tiny_ * calp1;
calp12 = -1;
}
alp12 = atan2(salp12, calp12);
}
S12 += _c2 * alp12;
S12 *= swapp * lonsign * latsign;
// Convert -0 to 0
S12 += 0;
}
// Convert calp, salp to azimuth accounting for lonsign, swapp, latsign.
if (swapp < 0) {
swap(salp1, salp2);
swap(calp1, calp2);
if (outmask & GEODESICSCALE)
swap(M12, M21);
}
salp1 *= swapp * lonsign; calp1 *= swapp * latsign;
salp2 *= swapp * lonsign; calp2 *= swapp * latsign;
// Returned value in [0, 180]
return a12;
}
Math::real Geodesic::GenInverse(real lat1, real lon1, real lat2, real lon2,
unsigned outmask,
real& s12, real& azi1, real& azi2,
real& m12, real& M12, real& M21,
real& S12) const {
outmask &= OUT_MASK;
real salp1, calp1, salp2, calp2,
a12 = GenInverse(lat1, lon1, lat2, lon2,
outmask, s12, salp1, calp1, salp2, calp2,
m12, M12, M21, S12);
if (outmask & AZIMUTH) {
azi1 = Math::atan2d(salp1, calp1);
azi2 = Math::atan2d(salp2, calp2);
}
return a12;
}
GeodesicLine Geodesic::InverseLine(real lat1, real lon1,
real lat2, real lon2,
unsigned caps) const {
real t, salp1, calp1, salp2, calp2,
a12 = GenInverse(lat1, lon1, lat2, lon2,
// No need to specify AZIMUTH here
0u, t, salp1, calp1, salp2, calp2,
t, t, t, t),
azi1 = Math::atan2d(salp1, calp1);
// Ensure that a12 can be converted to a distance
if (caps & (OUT_MASK & DISTANCE_IN)) caps |= DISTANCE;
return
GeodesicLine(*this, lat1, lon1, azi1, salp1, calp1, caps, true, a12);
}
void Geodesic::Lengths(real eps, real sig12,
real ssig1, real csig1, real dn1,
real ssig2, real csig2, real dn2,
real cbet1, real cbet2, unsigned outmask,
real& s12b, real& m12b, real& m0,
real& M12, real& M21,
// Scratch area of the right size
real Ca[]) const {
// Return m12b = (reduced length)/_b; also calculate s12b = distance/_b,
// and m0 = coefficient of secular term in expression for reduced length.
outmask &= OUT_MASK;
// outmask & DISTANCE: set s12b
// outmask & REDUCEDLENGTH: set m12b & m0
// outmask & GEODESICSCALE: set M12 & M21
real m0x = 0, J12 = 0, A1 = 0, A2 = 0;
real Cb[nC2_ + 1];
if (outmask & (DISTANCE | REDUCEDLENGTH | GEODESICSCALE)) {
A1 = A1m1f(eps);
C1f(eps, Ca);
if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) {
A2 = A2m1f(eps);
C2f(eps, Cb);
m0x = A1 - A2;
A2 = 1 + A2;
}
A1 = 1 + A1;
}
if (outmask & DISTANCE) {
real B1 = SinCosSeries(true, ssig2, csig2, Ca, nC1_) -
SinCosSeries(true, ssig1, csig1, Ca, nC1_);
// Missing a factor of _b
s12b = A1 * (sig12 + B1);
if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) {
real B2 = SinCosSeries(true, ssig2, csig2, Cb, nC2_) -
SinCosSeries(true, ssig1, csig1, Cb, nC2_);
J12 = m0x * sig12 + (A1 * B1 - A2 * B2);
}
} else if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) {
// Assume here that nC1_ >= nC2_
for (int l = 1; l <= nC2_; ++l)
Cb[l] = A1 * Ca[l] - A2 * Cb[l];
J12 = m0x * sig12 + (SinCosSeries(true, ssig2, csig2, Cb, nC2_) -
SinCosSeries(true, ssig1, csig1, Cb, nC2_));
}
if (outmask & REDUCEDLENGTH) {
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.
m12b = dn2 * (csig1 * ssig2) - dn1 * (ssig1 * csig2) -
csig1 * csig2 * J12;
}
if (outmask & GEODESICSCALE) {
real csig12 = csig1 * csig2 + ssig1 * ssig2;
real t = _ep2 * (cbet1 - cbet2) * (cbet1 + cbet2) / (dn1 + dn2);
M12 = csig12 + (t * ssig2 - csig2 * J12) * ssig1 / dn1;
M21 = csig12 - (t * ssig1 - csig1 * J12) * ssig2 / dn2;
}
}
Math::real Geodesic::Astroid(real x, real 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.
real k;
real
p = Math::sq(x),
q = Math::sq(y),
r = (p + q - 1) / 6;
if ( !(q == 0 && r <= 0) ) {
real
// 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 = Math::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);
real u = r;
if (disc >= 0) {
real 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 ? -sqrt(disc) : sqrt(disc); // T3 = (r * t)^3
// N.B. cbrt always returns the real root. cbrt(-8) = -2.
real T = 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.
real ang = atan2(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 * cos(ang / 3);
}
real
v = sqrt(Math::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 / (sqrt(uv + Math::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;
}
Math::real Geodesic::InverseStart(real sbet1, real cbet1, real dn1,
real sbet2, real cbet2, real dn2,
real lam12, real slam12, real clam12,
real& salp1, real& calp1,
// Only updated if return val >= 0
real& salp2, real& calp2,
// Only updated for short lines
real& dnm,
// Scratch area of the right size
real Ca[]) const {
// 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.
real
sig12 = -1, // Return value
// bet12 = bet2 - bet1 in [0, pi); bet12a = bet2 + bet1 in (-pi, 0]
sbet12 = sbet2 * cbet1 - cbet2 * sbet1,
cbet12 = cbet2 * cbet1 + sbet2 * sbet1;
real sbet12a = sbet2 * cbet1 + cbet2 * sbet1;
bool shortline = cbet12 >= 0 && sbet12 < real(0.5) &&
cbet2 * lam12 < real(0.5);
real somg12, comg12;
if (shortline) {
real sbetm2 = Math::sq(sbet1 + sbet2);
// sin((bet1+bet2)/2)^2
// = (sbet1 + sbet2)^2 / ((sbet1 + sbet2)^2 + (cbet1 + cbet2)^2)
sbetm2 /= sbetm2 + Math::sq(cbet1 + cbet2);
dnm = sqrt(1 + _ep2 * sbetm2);
real omg12 = lam12 / (_f1 * dnm);
somg12 = sin(omg12); comg12 = cos(omg12);
} else {
somg12 = slam12; comg12 = clam12;
}
salp1 = cbet2 * somg12;
calp1 = comg12 >= 0 ?
sbet12 + cbet2 * sbet1 * Math::sq(somg12) / (1 + comg12) :
sbet12a - cbet2 * sbet1 * Math::sq(somg12) / (1 - comg12);
real
ssig12 = hypot(salp1, calp1),
csig12 = sbet1 * sbet2 + cbet1 * cbet2 * comg12;
if (shortline && ssig12 < _etol2) {
// really short lines
salp2 = cbet1 * somg12;
calp2 = sbet12 - cbet1 * sbet2 *
(comg12 >= 0 ? Math::sq(somg12) / (1 + comg12) : 1 - comg12);
Math::norm(salp2, calp2);
// Set return value
sig12 = atan2(ssig12, csig12);
} else if (fabs(_n) > real(0.1) || // Skip astroid calc if too eccentric
csig12 >= 0 ||
ssig12 >= 6 * fabs(_n) * Math::pi() * Math::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.
real x, y, lamscale, betscale;
real lam12x = atan2(-slam12, -clam12); // lam12 - pi
if (_f >= 0) { // In fact f == 0 does not get here
// x = dlong, y = dlat
{
real
k2 = Math::sq(sbet1) * _ep2,
eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2);
lamscale = _f * cbet1 * A3f(eps) * Math::pi();
}
betscale = lamscale * cbet1;
x = lam12x / lamscale;
y = sbet12a / betscale;
} else { // _f < 0
// x = dlat, y = dlong
real
cbet12a = cbet2 * cbet1 - sbet2 * sbet1,
bet12a = atan2(sbet12a, cbet12a);
real m12b, m0, dummy;
// In the case of lon12 = 180, this repeats a calculation made in
// Inverse.
Lengths(_n, Math::pi() + bet12a,
sbet1, -cbet1, dn1, sbet2, cbet2, dn2,
cbet1, cbet2,
REDUCEDLENGTH, dummy, m12b, m0, dummy, dummy, Ca);
x = -1 + m12b / (cbet1 * cbet2 * m0 * Math::pi());
betscale = x < -real(0.01) ? sbet12a / x :
-_f * Math::sq(cbet1) * Math::pi();
lamscale = betscale / cbet1;
y = lam12x / lamscale;
}
if (y > -tol1_ && x > -1 - xthresh_) {
// strip near cut
// Need real(x) here to cast away the volatility of x for min/max
if (_f >= 0) {
salp1 = fmin(real(1), -x); calp1 = - sqrt(1 - Math::sq(salp1));
} else {
calp1 = fmax(real(x > -tol1_ ? 0 : -1), x);
salp1 = sqrt(1 - Math::sq(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
real k = Astroid(x, y);
real
omg12a = lamscale * ( _f >= 0 ? -x * k/(1 + k) : -y * (1 + k)/k );
somg12 = sin(omg12a); comg12 = -cos(omg12a);
// Update spherical estimate of alp1 using omg12 instead of lam12
salp1 = cbet2 * somg12;
calp1 = sbet12a - cbet2 * sbet1 * Math::sq(somg12) / (1 - comg12);
}
}
// Sanity check on starting guess. Backwards check allows NaN through.
if (!(salp1 <= 0))
Math::norm(salp1, calp1);
else {
salp1 = 1; calp1 = 0;
}
return sig12;
}
Math::real Geodesic::Lambda12(real sbet1, real cbet1, real dn1,
real sbet2, real cbet2, real dn2,
real salp1, real calp1,
real slam120, real clam120,
real& salp2, real& calp2,
real& sig12,
real& ssig1, real& csig1,
real& ssig2, real& csig2,
real& eps, real& domg12,
bool diffp, real& dlam12,
// Scratch area of the right size
real Ca[]) const {
if (sbet1 == 0 && calp1 == 0)
// Break degeneracy of equatorial line. This case has already been
// handled.
calp1 = -tiny_;
real
// sin(alp1) * cos(bet1) = sin(alp0)
salp0 = salp1 * cbet1,
calp0 = hypot(calp1, salp1 * sbet1); // calp0 > 0
real somg1, comg1, somg2, comg2, somg12, comg12, lam12;
// tan(bet1) = tan(sig1) * cos(alp1)
// tan(omg1) = sin(alp0) * tan(sig1) = tan(omg1)=tan(alp1)*sin(bet1)
ssig1 = sbet1; somg1 = salp0 * sbet1;
csig1 = comg1 = calp1 * cbet1;
Math::norm(ssig1, csig1);
// Math::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)
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]).
calp2 = cbet2 != cbet1 || fabs(sbet2) != -sbet1 ?
sqrt(Math::sq(calp1 * cbet1) +
(cbet1 < -sbet1 ?
(cbet2 - cbet1) * (cbet1 + cbet2) :
(sbet1 - sbet2) * (sbet1 + sbet2))) / cbet2 :
fabs(calp1);
// tan(bet2) = tan(sig2) * cos(alp2)
// tan(omg2) = sin(alp0) * tan(sig2).
ssig2 = sbet2; somg2 = salp0 * sbet2;
csig2 = comg2 = calp2 * cbet2;
Math::norm(ssig2, csig2);
// Math::norm(somg2, comg2); -- don't need to normalize!
// sig12 = sig2 - sig1, limit to [0, pi]
sig12 = atan2(fmax(real(0), csig1 * ssig2 - ssig1 * csig2) + real(0),
csig1 * csig2 + ssig1 * ssig2);
// omg12 = omg2 - omg1, limit to [0, pi]
somg12 = fmax(real(0), comg1 * somg2 - somg1 * comg2) + real(0);
comg12 = comg1 * comg2 + somg1 * somg2;
// eta = omg12 - lam120
real eta = atan2(somg12 * clam120 - comg12 * slam120,
comg12 * clam120 + somg12 * slam120);
real B312;
real k2 = Math::sq(calp0) * _ep2;
eps = k2 / (2 * (1 + sqrt(1 + k2)) + k2);
C3f(eps, Ca);
B312 = (SinCosSeries(true, ssig2, csig2, Ca, nC3_-1) -
SinCosSeries(true, ssig1, csig1, Ca, nC3_-1));
domg12 = -_f * A3f(eps) * salp0 * (sig12 + B312);
lam12 = eta + domg12;
if (diffp) {
if (calp2 == 0)
dlam12 = - 2 * _f1 * dn1 / sbet1;
else {
real dummy;
Lengths(eps, sig12, ssig1, csig1, dn1, ssig2, csig2, dn2,
cbet1, cbet2, REDUCEDLENGTH,
dummy, dlam12, dummy, dummy, dummy, Ca);
dlam12 *= _f1 / (calp2 * cbet2);
}
}
return lam12;
}
Math::real Geodesic::A3f(real eps) const {
// Evaluate A3
return Math::polyval(nA3_ - 1, _aA3x, eps);
}
void Geodesic::C3f(real eps, real c[]) const {
// Evaluate C3 coeffs
// Elements c[1] thru c[nC3_ - 1] are set
real mult = 1;
int o = 0;
for (int l = 1; l < nC3_; ++l) { // l is index of C3[l]
int m = nC3_ - l - 1; // order of polynomial in eps
mult *= eps;
c[l] = mult * Math::polyval(m, _cC3x + o, eps);
o += m + 1;
}
// Post condition: o == nC3x_
}
void Geodesic::C4f(real eps, real c[]) const {
// Evaluate C4 coeffs
// Elements c[0] thru c[nC4_ - 1] are set
real mult = 1;
int o = 0;
for (int l = 0; l < nC4_; ++l) { // l is index of C4[l]
int m = nC4_ - l - 1; // order of polynomial in eps
c[l] = mult * Math::polyval(m, _cC4x + o, eps);
o += m + 1;
mult *= eps;
}
// Post condition: o == nC4x_
}
// The static const coefficient arrays in the following functions are
// generated by Maxima and give the coefficients of the Taylor expansions for
// the geodesics. The convention on the order of these coefficients is as
// follows:
//
// ascending order in the trigonometric expansion,
// then powers of eps in descending order,
// finally powers of n in descending order.
//
// (For some expansions, only a subset of levels occur.) For each polynomial
// of order n at the lowest level, the (n+1) coefficients of the polynomial
// are followed by a divisor which is applied to the whole polynomial. In
// this way, the coefficients are expressible with no round off error. The
// sizes of the coefficient arrays are:
//
// A1m1f, A2m1f = floor(N/2) + 2
// C1f, C1pf, C2f, A3coeff = (N^2 + 7*N - 2*floor(N/2)) / 4
// C3coeff = (N - 1) * (N^2 + 7*N - 2*floor(N/2)) / 8
// C4coeff = N * (N + 1) * (N + 5) / 6
//
// where N = GEOGRAPHICLIB_GEODESIC_ORDER
// = nA1 = nA2 = nC1 = nC1p = nA3 = nC4
// The scale factor A1-1 = mean value of (d/dsigma)I1 - 1
Math::real Geodesic::A1m1f(real eps) {
// Generated by Maxima on 2015-05-05 18:08:12-04:00
#if GEOGRAPHICLIB_GEODESIC_ORDER/2 == 1
static const real coeff[] = {
// (1-eps)*A1-1, polynomial in eps2 of order 1
1, 0, 4,
};
#elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 2
static const real coeff[] = {
// (1-eps)*A1-1, polynomial in eps2 of order 2
1, 16, 0, 64,
};
#elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 3
static const real coeff[] = {
// (1-eps)*A1-1, polynomial in eps2 of order 3
1, 4, 64, 0, 256,
};
#elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 4
static const real coeff[] = {
// (1-eps)*A1-1, polynomial in eps2 of order 4
25, 64, 256, 4096, 0, 16384,
};
#else
#error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
#endif
static_assert(sizeof(coeff) / sizeof(real) == nA1_/2 + 2,
"Coefficient array size mismatch in A1m1f");
int m = nA1_/2;
real t = Math::polyval(m, coeff, Math::sq(eps)) / coeff[m + 1];
return (t + eps) / (1 - eps);
}
// The coefficients C1[l] in the Fourier expansion of B1
void Geodesic::C1f(real eps, real c[]) {
// Generated by Maxima on 2015-05-05 18:08:12-04:00
#if GEOGRAPHICLIB_GEODESIC_ORDER == 3
static const real coeff[] = {
// C1[1]/eps^1, polynomial in eps2 of order 1
3, -8, 16,
// C1[2]/eps^2, polynomial in eps2 of order 0
-1, 16,
// C1[3]/eps^3, polynomial in eps2 of order 0
-1, 48,
};
#elif GEOGRAPHICLIB_GEODESIC_ORDER == 4
static const real coeff[] = {
// C1[1]/eps^1, polynomial in eps2 of order 1
3, -8, 16,
// C1[2]/eps^2, polynomial in eps2 of order 1
1, -2, 32,
// C1[3]/eps^3, polynomial in eps2 of order 0
-1, 48,
// C1[4]/eps^4, polynomial in eps2 of order 0
-5, 512,
};
#elif GEOGRAPHICLIB_GEODESIC_ORDER == 5
static const real coeff[] = {
// C1[1]/eps^1, polynomial in eps2 of order 2
-1, 6, -16, 32,
// C1[2]/eps^2, polynomial in eps2 of order 1
1, -2, 32,
// C1[3]/eps^3, polynomial in eps2 of order 1
9, -16, 768,
// C1[4]/eps^4, polynomial in eps2 of order 0
-5, 512,
// C1[5]/eps^5, polynomial in eps2 of order 0
-7, 1280,
};
#elif GEOGRAPHICLIB_GEODESIC_ORDER == 6
static const real 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,
};
#elif GEOGRAPHICLIB_GEODESIC_ORDER == 7
static const real coeff[] = {
// C1[1]/eps^1, polynomial in eps2 of order 3
19, -64, 384, -1024, 2048,
// C1[2]/eps^2, polynomial in eps2 of order 2
-9, 64, -128, 2048,
// C1[3]/eps^3, polynomial in eps2 of order 2
-9, 72, -128, 6144,
// C1[4]/eps^4, polynomial in eps2 of order 1
3, -5, 512,
// C1[5]/eps^5, polynomial in eps2 of order 1
35, -56, 10240,
// C1[6]/eps^6, polynomial in eps2 of order 0
-7, 2048,
// C1[7]/eps^7, polynomial in eps2 of order 0
-33, 14336,
};
#elif GEOGRAPHICLIB_GEODESIC_ORDER == 8
static const real coeff[] = {
// C1[1]/eps^1, polynomial in eps2 of order 3
19, -64, 384, -1024, 2048,
// C1[2]/eps^2, polynomial in eps2 of order 3
7, -18, 128, -256, 4096,
// C1[3]/eps^3, polynomial in eps2 of order 2
-9, 72, -128, 6144,
// C1[4]/eps^4, polynomial in eps2 of order 2
-11, 96, -160, 16384,
// C1[5]/eps^5, polynomial in eps2 of order 1
35, -56, 10240,
// C1[6]/eps^6, polynomial in eps2 of order 1
9, -14, 4096,
// C1[7]/eps^7, polynomial in eps2 of order 0
-33, 14336,
// C1[8]/eps^8, polynomial in eps2 of order 0
-429, 262144,
};
#else
#error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
#endif
static_assert(sizeof(coeff) / sizeof(real) ==
(nC1_*nC1_ + 7*nC1_ - 2*(nC1_/2)) / 4,
"Coefficient array size mismatch in C1f");
real
eps2 = Math::sq(eps),
d = eps;
int o = 0;
for (int l = 1; l <= nC1_; ++l) { // l is index of C1p[l]
int m = (nC1_ - l) / 2; // order of polynomial in eps^2
c[l] = d * Math::polyval(m, coeff + o, eps2) / coeff[o + m + 1];
o += m + 2;
d *= eps;
}
// Post condition: o == sizeof(coeff) / sizeof(real)
}
// The coefficients C1p[l] in the Fourier expansion of B1p
void Geodesic::C1pf(real eps, real c[]) {
// Generated by Maxima on 2015-05-05 18:08:12-04:00
#if GEOGRAPHICLIB_GEODESIC_ORDER == 3
static const real coeff[] = {
// C1p[1]/eps^1, polynomial in eps2 of order 1
-9, 16, 32,
// C1p[2]/eps^2, polynomial in eps2 of order 0
5, 16,
// C1p[3]/eps^3, polynomial in eps2 of order 0
29, 96,
};
#elif GEOGRAPHICLIB_GEODESIC_ORDER == 4
static const real coeff[] = {
// C1p[1]/eps^1, polynomial in eps2 of order 1
-9, 16, 32,
// C1p[2]/eps^2, polynomial in eps2 of order 1
-37, 30, 96,
// C1p[3]/eps^3, polynomial in eps2 of order 0
29, 96,
// C1p[4]/eps^4, polynomial in eps2 of order 0
539, 1536,
};
#elif GEOGRAPHICLIB_GEODESIC_ORDER == 5
static const real coeff[] = {
// C1p[1]/eps^1, polynomial in eps2 of order 2
205, -432, 768, 1536,
// C1p[2]/eps^2, polynomial in eps2 of order 1
-37, 30, 96,
// C1p[3]/eps^3, polynomial in eps2 of order 1
-225, 116, 384,
// C1p[4]/eps^4, polynomial in eps2 of order 0
539, 1536,
// C1p[5]/eps^5, polynomial in eps2 of order 0
3467, 7680,
};
#elif GEOGRAPHICLIB_GEODESIC_ORDER == 6
static const real 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,
};
#elif GEOGRAPHICLIB_GEODESIC_ORDER == 7
static const real coeff[] = {
// C1p[1]/eps^1, polynomial in eps2 of order 3
-4879, 9840, -20736, 36864, 73728,
// C1p[2]/eps^2, polynomial in eps2 of order 2
4005, -4736, 3840, 12288,
// C1p[3]/eps^3, polynomial in eps2 of order 2
8703, -7200, 3712, 12288,
// C1p[4]/eps^4, polynomial in eps2 of order 1
-7173, 2695, 7680,
// C1p[5]/eps^5, polynomial in eps2 of order 1
-141115, 41604, 92160,
// C1p[6]/eps^6, polynomial in eps2 of order 0
38081, 61440,
// C1p[7]/eps^7, polynomial in eps2 of order 0
459485, 516096,
};
#elif GEOGRAPHICLIB_GEODESIC_ORDER == 8
static const real coeff[] = {
// C1p[1]/eps^1, polynomial in eps2 of order 3
-4879, 9840, -20736, 36864, 73728,
// C1p[2]/eps^2, polynomial in eps2 of order 3
-86171, 120150, -142080, 115200, 368640,
// C1p[3]/eps^3, polynomial in eps2 of order 2
8703, -7200, 3712, 12288,
// C1p[4]/eps^4, polynomial in eps2 of order 2
1082857, -688608, 258720, 737280,
// C1p[5]/eps^5, polynomial in eps2 of order 1
-141115, 41604, 92160,
// C1p[6]/eps^6, polynomial in eps2 of order 1
-2200311, 533134, 860160,
// C1p[7]/eps^7, polynomial in eps2 of order 0
459485, 516096,
// C1p[8]/eps^8, polynomial in eps2 of order 0
109167851, 82575360,
};
#else
#error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
#endif
static_assert(sizeof(coeff) / sizeof(real) ==
(nC1p_*nC1p_ + 7*nC1p_ - 2*(nC1p_/2)) / 4,
"Coefficient array size mismatch in C1pf");
real
eps2 = Math::sq(eps),
d = eps;
int o = 0;
for (int l = 1; l <= nC1p_; ++l) { // l is index of C1p[l]
int m = (nC1p_ - l) / 2; // order of polynomial in eps^2
c[l] = d * Math::polyval(m, coeff + o, eps2) / coeff[o + m + 1];
o += m + 2;
d *= eps;
}
// Post condition: o == sizeof(coeff) / sizeof(real)
}
// The scale factor A2-1 = mean value of (d/dsigma)I2 - 1
Math::real Geodesic::A2m1f(real eps) {
// Generated by Maxima on 2015-05-29 08:09:47-04:00
#if GEOGRAPHICLIB_GEODESIC_ORDER/2 == 1
static const real coeff[] = {
// (eps+1)*A2-1, polynomial in eps2 of order 1
-3, 0, 4,
}; // count = 3
#elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 2
static const real coeff[] = {
// (eps+1)*A2-1, polynomial in eps2 of order 2
-7, -48, 0, 64,
}; // count = 4
#elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 3
static const real coeff[] = {
// (eps+1)*A2-1, polynomial in eps2 of order 3
-11, -28, -192, 0, 256,
}; // count = 5
#elif GEOGRAPHICLIB_GEODESIC_ORDER/2 == 4
static const real coeff[] = {
// (eps+1)*A2-1, polynomial in eps2 of order 4
-375, -704, -1792, -12288, 0, 16384,
}; // count = 6
#else
#error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
#endif
static_assert(sizeof(coeff) / sizeof(real) == nA2_/2 + 2,
"Coefficient array size mismatch in A2m1f");
int m = nA2_/2;
real t = Math::polyval(m, coeff, Math::sq(eps)) / coeff[m + 1];
return (t - eps) / (1 + eps);
}
// The coefficients C2[l] in the Fourier expansion of B2
void Geodesic::C2f(real eps, real c[]) {
// Generated by Maxima on 2015-05-05 18:08:12-04:00
#if GEOGRAPHICLIB_GEODESIC_ORDER == 3
static const real coeff[] = {
// C2[1]/eps^1, polynomial in eps2 of order 1
1, 8, 16,
// C2[2]/eps^2, polynomial in eps2 of order 0
3, 16,
// C2[3]/eps^3, polynomial in eps2 of order 0
5, 48,
};
#elif GEOGRAPHICLIB_GEODESIC_ORDER == 4
static const real coeff[] = {
// C2[1]/eps^1, polynomial in eps2 of order 1
1, 8, 16,
// C2[2]/eps^2, polynomial in eps2 of order 1
1, 6, 32,
// C2[3]/eps^3, polynomial in eps2 of order 0
5, 48,
// C2[4]/eps^4, polynomial in eps2 of order 0
35, 512,
};
#elif GEOGRAPHICLIB_GEODESIC_ORDER == 5
static const real coeff[] = {
// C2[1]/eps^1, polynomial in eps2 of order 2
1, 2, 16, 32,
// C2[2]/eps^2, polynomial in eps2 of order 1
1, 6, 32,
// C2[3]/eps^3, polynomial in eps2 of order 1
15, 80, 768,
// C2[4]/eps^4, polynomial in eps2 of order 0
35, 512,
// C2[5]/eps^5, polynomial in eps2 of order 0
63, 1280,
};
#elif GEOGRAPHICLIB_GEODESIC_ORDER == 6
static const real 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,
};
#elif GEOGRAPHICLIB_GEODESIC_ORDER == 7
static const real coeff[] = {
// C2[1]/eps^1, polynomial in eps2 of order 3
41, 64, 128, 1024, 2048,
// C2[2]/eps^2, polynomial in eps2 of order 2
35, 64, 384, 2048,
// C2[3]/eps^3, polynomial in eps2 of order 2
69, 120, 640, 6144,
// C2[4]/eps^4, polynomial in eps2 of order 1
7, 35, 512,
// C2[5]/eps^5, polynomial in eps2 of order 1
105, 504, 10240,
// C2[6]/eps^6, polynomial in eps2 of order 0
77, 2048,
// C2[7]/eps^7, polynomial in eps2 of order 0
429, 14336,
};
#elif GEOGRAPHICLIB_GEODESIC_ORDER == 8
static const real coeff[] = {
// C2[1]/eps^1, polynomial in eps2 of order 3
41, 64, 128, 1024, 2048,
// C2[2]/eps^2, polynomial in eps2 of order 3
47, 70, 128, 768, 4096,
// C2[3]/eps^3, polynomial in eps2 of order 2
69, 120, 640, 6144,
// C2[4]/eps^4, polynomial in eps2 of order 2
133, 224, 1120, 16384,
// C2[5]/eps^5, polynomial in eps2 of order 1
105, 504, 10240,
// C2[6]/eps^6, polynomial in eps2 of order 1
33, 154, 4096,
// C2[7]/eps^7, polynomial in eps2 of order 0
429, 14336,
// C2[8]/eps^8, polynomial in eps2 of order 0
6435, 262144,
};
#else
#error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
#endif
static_assert(sizeof(coeff) / sizeof(real) ==
(nC2_*nC2_ + 7*nC2_ - 2*(nC2_/2)) / 4,
"Coefficient array size mismatch in C2f");
real
eps2 = Math::sq(eps),
d = eps;
int o = 0;
for (int l = 1; l <= nC2_; ++l) { // l is index of C2[l]
int m = (nC2_ - l) / 2; // order of polynomial in eps^2
c[l] = d * Math::polyval(m, coeff + o, eps2) / coeff[o + m + 1];
o += m + 2;
d *= eps;
}
// Post condition: o == sizeof(coeff) / sizeof(real)
}
// The scale factor A3 = mean value of (d/dsigma)I3
void Geodesic::A3coeff() {
// Generated by Maxima on 2015-05-05 18:08:13-04:00
#if GEOGRAPHICLIB_GEODESIC_ORDER == 3
static const real coeff[] = {
// A3, coeff of eps^2, polynomial in n of order 0
-1, 4,
// 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,
};
#elif GEOGRAPHICLIB_GEODESIC_ORDER == 4
static const real coeff[] = {
// A3, coeff of eps^3, polynomial in n of order 0
-1, 16,
// A3, coeff of eps^2, polynomial in n of order 1
-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,
};
#elif GEOGRAPHICLIB_GEODESIC_ORDER == 5
static const real coeff[] = {
// A3, coeff of eps^4, polynomial in n of order 0
-3, 64,
// A3, coeff of eps^3, polynomial in n of order 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,
};
#elif GEOGRAPHICLIB_GEODESIC_ORDER == 6
static const real 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,
};
#elif GEOGRAPHICLIB_GEODESIC_ORDER == 7
static const real coeff[] = {
// A3, coeff of eps^6, polynomial in n of order 0
-5, 256,
// A3, coeff of eps^5, polynomial in n of order 1
-5, -3, 128,
// A3, coeff of eps^4, polynomial in n of order 2
-10, -2, -3, 64,
// A3, coeff of eps^3, polynomial in n of order 3
5, -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,
};
#elif GEOGRAPHICLIB_GEODESIC_ORDER == 8
static const real coeff[] = {
// A3, coeff of eps^7, polynomial in n of order 0
-25, 2048,
// A3, coeff of eps^6, polynomial in n of order 1
-15, -20, 1024,
// A3, coeff of eps^5, polynomial in n of order 2
-5, -10, -6, 256,
// A3, coeff of eps^4, polynomial in n of order 3
-5, -20, -4, -6, 128,
// A3, coeff of eps^3, polynomial in n of order 3
5, -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,
};
#else
#error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
#endif
static_assert(sizeof(coeff) / sizeof(real) ==
(nA3_*nA3_ + 7*nA3_ - 2*(nA3_/2)) / 4,
"Coefficient array size mismatch in A3f");
int o = 0, k = 0;
for (int j = nA3_ - 1; j >= 0; --j) { // coeff of eps^j
int m = min(nA3_ - j - 1, j); // order of polynomial in n
_aA3x[k++] = Math::polyval(m, coeff + o, _n) / coeff[o + m + 1];
o += m + 2;
}
// Post condition: o == sizeof(coeff) / sizeof(real) && k == nA3x_
}
// The coefficients C3[l] in the Fourier expansion of B3
void Geodesic::C3coeff() {
// Generated by Maxima on 2015-05-05 18:08:13-04:00
#if GEOGRAPHICLIB_GEODESIC_ORDER == 3
static const real coeff[] = {
// C3[1], coeff of eps^2, polynomial in n of order 0
1, 8,
// C3[1], coeff of eps^1, polynomial in n of order 1
-1, 1, 4,
// C3[2], coeff of eps^2, polynomial in n of order 0
1, 16,
};
#elif GEOGRAPHICLIB_GEODESIC_ORDER == 4
static const real coeff[] = {
// C3[1], coeff of eps^3, polynomial in n of order 0
3, 64,
// C3[1], coeff of eps^2, polynomial in n of order 1
// This is a case where a leading 0 term has been inserted to maintain the
// pattern in the orders of the polynomials.
0, 1, 8,
// C3[1], coeff of eps^1, polynomial in n of order 1
-1, 1, 4,
// C3[2], coeff of eps^3, polynomial in n of order 0
3, 64,
// C3[2], coeff of eps^2, polynomial in n of order 1
-3, 2, 32,
// C3[3], coeff of eps^3, polynomial in n of order 0
5, 192,
};
#elif GEOGRAPHICLIB_GEODESIC_ORDER == 5
static const real coeff[] = {
// C3[1], coeff of eps^4, polynomial in n of order 0
5, 128,
// C3[1], coeff of eps^3, polynomial in n of order 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^4, polynomial in n of order 0
3, 128,
// C3[2], coeff of eps^3, polynomial in n of order 1
-2, 3, 64,
// C3[2], coeff of eps^2, polynomial in n of order 2
1, -3, 2, 32,
// C3[3], coeff of eps^4, polynomial in n of order 0
3, 128,
// C3[3], coeff of eps^3, polynomial in n of order 1
-9, 5, 192,
// C3[4], coeff of eps^4, polynomial in n of order 0
7, 512,
};
#elif GEOGRAPHICLIB_GEODESIC_ORDER == 6
static const real 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,
};
#elif GEOGRAPHICLIB_GEODESIC_ORDER == 7
static const real coeff[] = {
// C3[1], coeff of eps^6, polynomial in n of order 0
21, 1024,
// C3[1], coeff of eps^5, polynomial in n of order 1
11, 12, 512,
// C3[1], coeff of eps^4, polynomial in n of order 2
2, 2, 5, 128,
// C3[1], coeff of eps^3, polynomial in n of order 3
-5, -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^6, polynomial in n of order 0
27, 2048,
// C3[2], coeff of eps^5, polynomial in n of order 1
1, 5, 256,
// C3[2], coeff of eps^4, polynomial in n of order 2
-9, 2, 6, 256,
// C3[2], coeff of eps^3, polynomial in n of order 3
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^6, polynomial in n of order 0
3, 256,
// C3[3], coeff of eps^5, polynomial in n of order 1
-4, 21, 1536,
// C3[3], coeff of eps^4, polynomial in n of order 2
-6, -10, 9, 384,
// C3[3], coeff of eps^3, polynomial in n of order 3
-1, 5, -9, 5, 192,
// C3[4], coeff of eps^6, polynomial in n of order 0
9, 1024,
// C3[4], coeff of eps^5, polynomial in n of order 1
-10, 7, 512,
// C3[4], coeff of eps^4, polynomial in n of order 2
10, -14, 7, 512,
// C3[5], coeff of eps^6, polynomial in n of order 0
9, 1024,
// C3[5], coeff of eps^5, polynomial in n of order 1
-45, 21, 2560,
// C3[6], coeff of eps^6, polynomial in n of order 0
11, 2048,
};
#elif GEOGRAPHICLIB_GEODESIC_ORDER == 8
static const real coeff[] = {
// C3[1], coeff of eps^7, polynomial in n of order 0
243, 16384,
// C3[1], coeff of eps^6, polynomial in n of order 1
10, 21, 1024,
// C3[1], coeff of eps^5, polynomial in n of order 2
3, 11, 12, 512,
// C3[1], coeff of eps^4, polynomial in n of order 3
-2, 2, 2, 5, 128,
// C3[1], coeff of eps^3, polynomial in n of order 3
-5, -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^7, polynomial in n of order 0
187, 16384,
// C3[2], coeff of eps^6, polynomial in n of order 1
69, 108, 8192,
// C3[2], coeff of eps^5, polynomial in n of order 2
-2, 1, 5, 256,
// C3[2], coeff of eps^4, polynomial in n of order 3
-6, -9, 2, 6, 256,
// C3[2], coeff of eps^3, polynomial in n of order 3
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^7, polynomial in n of order 0
139, 16384,
// C3[3], coeff of eps^6, polynomial in n of order 1
-1, 12, 1024,
// C3[3], coeff of eps^5, polynomial in n of order 2
-77, -8, 42, 3072,
// C3[3], coeff of eps^4, polynomial in n of order 3
10, -6, -10, 9, 384,
// C3[3], coeff of eps^3, polynomial in n of order 3
-1, 5, -9, 5, 192,
// C3[4], coeff of eps^7, polynomial in n of order 0
127, 16384,
// C3[4], coeff of eps^6, polynomial in n of order 1
-43, 72, 8192,
// C3[4], coeff of eps^5, polynomial in n of order 2
-7, -40, 28, 2048,
// C3[4], coeff of eps^4, polynomial in n of order 3
-7, 20, -28, 14, 1024,
// C3[5], coeff of eps^7, polynomial in n of order 0
99, 16384,
// C3[5], coeff of eps^6, polynomial in n of order 1
-15, 9, 1024,
// C3[5], coeff of eps^5, polynomial in n of order 2
75, -90, 42, 5120,
// C3[6], coeff of eps^7, polynomial in n of order 0
99, 16384,
// C3[6], coeff of eps^6, polynomial in n of order 1
-99, 44, 8192,
// C3[7], coeff of eps^7, polynomial in n of order 0
429, 114688,
};
#else
#error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
#endif
static_assert(sizeof(coeff) / sizeof(real) ==
((nC3_-1)*(nC3_*nC3_ + 7*nC3_ - 2*(nC3_/2)))/8,
"Coefficient array size mismatch in C3coeff");
int o = 0, k = 0;
for (int l = 1; l < nC3_; ++l) { // l is index of C3[l]
for (int j = nC3_ - 1; j >= l; --j) { // coeff of eps^j
int m = min(nC3_ - j - 1, j); // order of polynomial in n
_cC3x[k++] = Math::polyval(m, coeff + o, _n) / coeff[o + m + 1];
o += m + 2;
}
}
// Post condition: o == sizeof(coeff) / sizeof(real) && k == nC3x_
}
void Geodesic::C4coeff() {
// Generated by Maxima on 2015-05-05 18:08:13-04:00
#if GEOGRAPHICLIB_GEODESIC_ORDER == 3
static const real coeff[] = {
// C4[0], coeff of eps^2, polynomial in n of order 0
-2, 105,
// C4[0], coeff of eps^1, polynomial in n of order 1
16, -7, 35,
// C4[0], coeff of eps^0, polynomial in n of order 2
8, -28, 70, 105,
// C4[1], coeff of eps^2, polynomial in n of order 0
-2, 105,
// C4[1], coeff of eps^1, polynomial in n of order 1
-16, 7, 315,
// C4[2], coeff of eps^2, polynomial in n of order 0
4, 525,
};
#elif GEOGRAPHICLIB_GEODESIC_ORDER == 4
static const real coeff[] = {
// C4[0], coeff of eps^3, polynomial in n of order 0
11, 315,
// C4[0], coeff of eps^2, polynomial in n of order 1
-32, -6, 315,
// C4[0], coeff of eps^1, polynomial in n of order 2
-32, 48, -21, 105,
// C4[0], coeff of eps^0, polynomial in n of order 3
4, 24, -84, 210, 315,
// C4[1], coeff of eps^3, polynomial in n of order 0
-1, 105,
// C4[1], coeff of eps^2, polynomial in n of order 1
64, -18, 945,
// C4[1], coeff of eps^1, polynomial in n of order 2
32, -48, 21, 945,
// C4[2], coeff of eps^3, polynomial in n of order 0
-8, 1575,
// C4[2], coeff of eps^2, polynomial in n of order 1
-32, 12, 1575,
// C4[3], coeff of eps^3, polynomial in n of order 0
8, 2205,
};
#elif GEOGRAPHICLIB_GEODESIC_ORDER == 5
static const real coeff[] = {
// C4[0], coeff of eps^4, polynomial in n of order 0
4, 1155,
// C4[0], coeff of eps^3, polynomial in n of order 1
-368, 121, 3465,
// C4[0], coeff of eps^2, polynomial in n of order 2
1088, -352, -66, 3465,
// C4[0], coeff of eps^1, polynomial in n of order 3
48, -352, 528, -231, 1155,
// C4[0], coeff of eps^0, polynomial in n of order 4
16, 44, 264, -924, 2310, 3465,
// C4[1], coeff of eps^4, polynomial in n of order 0
4, 1155,
// C4[1], coeff of eps^3, polynomial in n of order 1
80, -99, 10395,
// C4[1], coeff of eps^2, polynomial in n of order 2
-896, 704, -198, 10395,
// C4[1], coeff of eps^1, polynomial in n of order 3
-48, 352, -528, 231, 10395,
// C4[2], coeff of eps^4, polynomial in n of order 0
-8, 1925,
// C4[2], coeff of eps^3, polynomial in n of order 1
384, -88, 17325,
// C4[2], coeff of eps^2, polynomial in n of order 2
320, -352, 132, 17325,
// C4[3], coeff of eps^4, polynomial in n of order 0
-16, 8085,
// C4[3], coeff of eps^3, polynomial in n of order 1
-256, 88, 24255,
// C4[4], coeff of eps^4, polynomial in n of order 0
64, 31185,
};
#elif GEOGRAPHICLIB_GEODESIC_ORDER == 6
static const real 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,
};
#elif GEOGRAPHICLIB_GEODESIC_ORDER == 7
static const real coeff[] = {
// C4[0], coeff of eps^6, polynomial in n of order 0
10, 9009,
// C4[0], coeff of eps^5, polynomial in n of order 1
-464, 291, 45045,
// C4[0], coeff of eps^4, polynomial in n of order 2
-4480, 1088, 156, 45045,
// C4[0], coeff of eps^3, polynomial in n of order 3
10736, -224, -4784, 1573, 45045,
// C4[0], coeff of eps^2, polynomial in n of order 4
1664, -10656, 14144, -4576, -858, 45045,
// C4[0], coeff of eps^1, polynomial in n of order 5
16, 64, 624, -4576, 6864, -3003, 15015,
// C4[0], coeff of eps^0, polynomial in n of order 6
56, 100, 208, 572, 3432, -12012, 30030, 45045,
// C4[1], coeff of eps^6, polynomial in n of order 0
10, 9009,
// C4[1], coeff of eps^5, polynomial in n of order 1
112, 15, 135135,
// C4[1], coeff of eps^4, polynomial in n of order 2
3840, -2944, 468, 135135,
// C4[1], coeff of eps^3, polynomial in n of order 3
-10704, 5792, 1040, -1287, 135135,
// C4[1], coeff of eps^2, polynomial in n of order 4
-768, 5952, -11648, 9152, -2574, 135135,
// C4[1], coeff of eps^1, polynomial in n of order 5
-16, -64, -624, 4576, -6864, 3003, 135135,
// C4[2], coeff of eps^6, polynomial in n of order 0
-4, 25025,
// C4[2], coeff of eps^5, polynomial in n of order 1
-1664, 168, 225225,
// C4[2], coeff of eps^4, polynomial in n of order 2
1664, 1856, -936, 225225,
// C4[2], coeff of eps^3, polynomial in n of order 3
6784, -8448, 4992, -1144, 225225,
// C4[2], coeff of eps^2, polynomial in n of order 4
128, -1440, 4160, -4576, 1716, 225225,
// C4[3], coeff of eps^6, polynomial in n of order 0
64, 315315,
// C4[3], coeff of eps^5, polynomial in n of order 1
1792, -680, 315315,
// C4[3], coeff of eps^4, polynomial in n of order 2
-2048, 1024, -208, 105105,
// C4[3], coeff of eps^3, polynomial in n of order 3
-1792, 3584, -3328, 1144, 315315,
// C4[4], coeff of eps^6, polynomial in n of order 0
-512, 405405,
// C4[4], coeff of eps^5, polynomial in n of order 1
2048, -384, 405405,
// C4[4], coeff of eps^4, polynomial in n of order 2
3072, -2560, 832, 405405,
// C4[5], coeff of eps^6, polynomial in n of order 0
-256, 495495,
// C4[5], coeff of eps^5, polynomial in n of order 1
-2048, 640, 495495,
// C4[6], coeff of eps^6, polynomial in n of order 0
512, 585585,
};
#elif GEOGRAPHICLIB_GEODESIC_ORDER == 8
static const real coeff[] = {
// C4[0], coeff of eps^7, polynomial in n of order 0
193, 85085,
// C4[0], coeff of eps^6, polynomial in n of order 1
4192, 850, 765765,
// C4[0], coeff of eps^5, polynomial in n of order 2
20960, -7888, 4947, 765765,
// C4[0], coeff of eps^4, polynomial in n of order 3
12480, -76160, 18496, 2652, 765765,
// C4[0], coeff of eps^3, polynomial in n of order 4
-154048, 182512, -3808, -81328, 26741, 765765,
// C4[0], coeff of eps^2, polynomial in n of order 5
3232, 28288, -181152, 240448, -77792, -14586, 765765,
// C4[0], coeff of eps^1, polynomial in n of order 6
96, 272, 1088, 10608, -77792, 116688, -51051, 255255,
// C4[0], coeff of eps^0, polynomial in n of order 7
588, 952, 1700, 3536, 9724, 58344, -204204, 510510, 765765,
// C4[1], coeff of eps^7, polynomial in n of order 0
349, 2297295,
// C4[1], coeff of eps^6, polynomial in n of order 1
-1472, 510, 459459,
// C4[1], coeff of eps^5, polynomial in n of order 2
-39840, 1904, 255, 2297295,
// C4[1], coeff of eps^4, polynomial in n of order 3
52608, 65280, -50048, 7956, 2297295,
// C4[1], coeff of eps^3, polynomial in n of order 4
103744, -181968, 98464, 17680, -21879, 2297295,
// C4[1], coeff of eps^2, polynomial in n of order 5
-1344, -13056, 101184, -198016, 155584, -43758, 2297295,
// C4[1], coeff of eps^1, polynomial in n of order 6
-96, -272, -1088, -10608, 77792, -116688, 51051, 2297295,
// C4[2], coeff of eps^7, polynomial in n of order 0
464, 1276275,
// C4[2], coeff of eps^6, polynomial in n of order 1
-928, -612, 3828825,
// C4[2], coeff of eps^5, polynomial in n of order 2
64256, -28288, 2856, 3828825,
// C4[2], coeff of eps^4, polynomial in n of order 3
-126528, 28288, 31552, -15912, 3828825,
// C4[2], coeff of eps^3, polynomial in n of order 4
-41472, 115328, -143616, 84864, -19448, 3828825,
// C4[2], coeff of eps^2, polynomial in n of order 5
160, 2176, -24480, 70720, -77792, 29172, 3828825,
// C4[3], coeff of eps^7, polynomial in n of order 0
-16, 97461,
// C4[3], coeff of eps^6, polynomial in n of order 1
-16384, 1088, 5360355,
// C4[3], coeff of eps^5, polynomial in n of order 2
-2560, 30464, -11560, 5360355,
// C4[3], coeff of eps^4, polynomial in n of order 3
35840, -34816, 17408, -3536, 1786785,
// C4[3], coeff of eps^3, polynomial in n of order 4
7168, -30464, 60928, -56576, 19448, 5360355,
// C4[4], coeff of eps^7, polynomial in n of order 0
128, 2297295,
// C4[4], coeff of eps^6, polynomial in n of order 1
26624, -8704, 6891885,
// C4[4], coeff of eps^5, polynomial in n of order 2
-77824, 34816, -6528, 6891885,
// C4[4], coeff of eps^4, polynomial in n of order 3
-32256, 52224, -43520, 14144, 6891885,
// C4[5], coeff of eps^7, polynomial in n of order 0
-6784, 8423415,
// C4[5], coeff of eps^6, polynomial in n of order 1
24576, -4352, 8423415,
// C4[5], coeff of eps^5, polynomial in n of order 2
45056, -34816, 10880, 8423415,
// C4[6], coeff of eps^7, polynomial in n of order 0
-1024, 3318315,
// C4[6], coeff of eps^6, polynomial in n of order 1
-28672, 8704, 9954945,
// C4[7], coeff of eps^7, polynomial in n of order 0
1024, 1640925,
};
#else
#error "Bad value for GEOGRAPHICLIB_GEODESIC_ORDER"
#endif
static_assert(sizeof(coeff) / sizeof(real) ==
(nC4_ * (nC4_ + 1) * (nC4_ + 5)) / 6,
"Coefficient array size mismatch in C4coeff");
int o = 0, k = 0;
for (int l = 0; l < nC4_; ++l) { // l is index of C4[l]
for (int j = nC4_ - 1; j >= l; --j) { // coeff of eps^j
int m = nC4_ - j - 1; // order of polynomial in n
_cC4x[k++] = Math::polyval(m, coeff + o, _n) / coeff[o + m + 1];
o += m + 2;
}
}
// Post condition: o == sizeof(coeff) / sizeof(real) && k == nC4x_
}
} // namespace GeographicLib
real
GeographicLib::Math::real real
Definition GeodSolve.cpp:28
GeodesicLine.hpp
Header for GeographicLib::GeodesicLine class.
Geodesic.hpp
Header for GeographicLib::Geodesic class.
GeographicLib::Constants::WGS84_f
static T WGS84_f()
Definition Constants.hpp:141
GeographicLib::Constants::WGS84_a
static T WGS84_a()
Definition Constants.hpp:135
GeographicLib::GeodesicExact
Exact geodesic calculations.
Definition GeodesicExact.hpp:85
GeographicLib::GeodesicExact::GenDirect
Math::real GenDirect(real lat1, real lon1, real azi1, bool arcmode, real s12_a12, unsigned outmask, real &lat2, real &lon2, real &azi2, real &s12, real &m12, real &M12, real &M21, real &S12) const
Definition GeodesicExact.cpp:402
GeographicLib::GeodesicLine
A geodesic line.
Definition GeodesicLine.hpp:74
GeographicLib::Geodesic
Geodesic calculations
Definition Geodesic.hpp:175
GeographicLib::Geodesic::InverseLine
GeodesicLine InverseLine(real lat1, real lon1, real lat2, real lon2, unsigned caps=ALL) const
Definition Geodesic.cpp:524
GeographicLib::Geodesic::WGS84
static const Geodesic & WGS84()
Definition Geodesic.cpp:94
GeographicLib::Geodesic::ArcDirectLine
GeodesicLine ArcDirectLine(real lat1, real lon1, real azi1, real a12, unsigned caps=ALL) const
Definition Geodesic.cpp:163
GeographicLib::Geodesic::Line
GeodesicLine Line(real lat1, real lon1, real azi1, unsigned caps=ALL) const
Definition Geodesic.cpp:123
GeographicLib::Geodesic::GenDirectLine
GeodesicLine GenDirectLine(real lat1, real lon1, real azi1, bool arcmode, real s12_a12, unsigned caps=ALL) const
Definition Geodesic.cpp:145
GeographicLib::Geodesic::DISTANCE
@ DISTANCE
Definition Geodesic.hpp:294
GeographicLib::Geodesic::REDUCEDLENGTH
@ REDUCEDLENGTH
Definition Geodesic.hpp:311
GeographicLib::Geodesic::AREA
@ AREA
Definition Geodesic.hpp:321
GeographicLib::Geodesic::AZIMUTH
@ AZIMUTH
Definition Geodesic.hpp:289
GeographicLib::Geodesic::NONE
@ NONE
Definition Geodesic.hpp:271
GeographicLib::Geodesic::DISTANCE_IN
@ DISTANCE_IN
Definition Geodesic.hpp:306
GeographicLib::Geodesic::GEODESICSCALE
@ GEODESICSCALE
Definition Geodesic.hpp:316
GeographicLib::Geodesic::GeodesicLine
friend class GeodesicLine
Definition Geodesic.hpp:178
GeographicLib::Geodesic::GenDirect
Math::real GenDirect(real lat1, real lon1, real azi1, bool arcmode, real s12_a12, unsigned outmask, real &lat2, real &lon2, real &azi2, real &s12, real &m12, real &M12, real &M21, real &S12) const
Definition Geodesic.cpp:128
GeographicLib::Geodesic::DirectLine
GeodesicLine DirectLine(real lat1, real lon1, real azi1, real s12, unsigned caps=ALL) const
Definition Geodesic.cpp:158
GeographicLib::Geodesic::Geodesic
Geodesic(real a, real f, bool exact=false)
Definition Geodesic.cpp:41
GeographicLib::GeographicErr
Exception handling for GeographicLib.
Definition Constants.hpp:316
GeographicLib::Math
Mathematical functions needed by GeographicLib.
Definition Math.hpp:82
GeographicLib::Math::degree
static T degree()
Definition Math.hpp:215
GeographicLib::Math::LatFix
static T LatFix(T x)
Definition Math.hpp:315
GeographicLib::Math::sincosd
static void sincosd(T x, T &sinx, T &cosx)
Definition Math.cpp:104
GeographicLib::Math::atan2d
static T atan2d(T y, T x)
Definition Math.cpp:202
GeographicLib::Math::norm
static void norm(T &x, T &y)
Definition Math.hpp:237
GeographicLib::Math::AngRound
static T AngRound(T x)
Definition Math.cpp:95
GeographicLib::Math::sq
static T sq(T x)
Definition Math.hpp:227
GeographicLib::Math::qd
static constexpr int qd
degrees per quarter turn
Definition Math.hpp:150
GeographicLib::Math::AngNormalize
static T AngNormalize(T x)
Definition Math.cpp:69
GeographicLib::Math::sincosde
static void sincosde(T x, T t, T &sinx, T &cosx)
Definition Math.cpp:131
GeographicLib::Math::pi
static T pi()
Definition Math.hpp:205
GeographicLib::Math::NaN
static T NaN()
Definition Math.cpp:283
GeographicLib::Math::polyval
static T polyval(int N, const T p[], T x)
Definition Math.hpp:286
GeographicLib::Math::AngDiff
static T AngDiff(T x, T y, T &e)
Definition Math.cpp:80
GeographicLib::Math::hd
static constexpr int hd
degrees per half turn
Definition Math.hpp:153
GeographicLib::Math::real
double real
Definition Math.hpp:115
GeographicLib
Namespace for GeographicLib.
Definition Accumulator.cpp:12
std
Definition NearestNeighbor.hpp:806
std::swap
void swap(GeographicLib::NearestNeighbor< dist_t, pos_t, distfun_t > &a, GeographicLib::NearestNeighbor< dist_t, pos_t, distfun_t > &b)
Definition NearestNeighbor.hpp:819