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GeodesicLine.cpp
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/**
* \file GeodesicLine.cpp
* \brief Implementation for GeographicLib::GeodesicLine class
*
* Copyright (c) Charles Karney (2009-2023) <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/GeodesicLine.hpp>
namespace GeographicLib {
using namespace std;
void GeodesicLine::LineInit(const Geodesic& g,
real lat1, real lon1,
real azi1, real salp1, real calp1,
unsigned caps) {
tiny_ = g.tiny_;
_lat1 = Math::LatFix(lat1);
_lon1 = lon1;
_azi1 = azi1;
_salp1 = salp1;
_calp1 = calp1;
_a = g._a;
_f = g._f;
_b = g._b;
_c2 = g._c2;
_f1 = g._f1;
// Always allow latitude and azimuth and unrolling of longitude
_caps = caps | LATITUDE | AZIMUTH | LONG_UNROLL;
real cbet1, sbet1;
Math::sincosd(Math::AngRound(_lat1), sbet1, cbet1); sbet1 *= _f1;
// Ensure cbet1 = +epsilon at poles
Math::norm(sbet1, cbet1); cbet1 = fmax(tiny_, cbet1);
_dn1 = sqrt(1 + g._ep2 * Math::sq(sbet1));
// Evaluate alp0 from sin(alp1) * cos(bet1) = sin(alp0),
_salp0 = _salp1 * cbet1; // alp0 in [0, pi/2 - |bet1|]
// Alt: calp0 = hypot(sbet1, calp1 * cbet1). The following
// is slightly better (consider the case salp1 = 0).
_calp0 = hypot(_calp1, _salp1 * sbet1);
// Evaluate sig with tan(bet1) = tan(sig1) * cos(alp1).
// sig = 0 is nearest northward crossing of equator.
// With bet1 = 0, alp1 = pi/2, we have sig1 = 0 (equatorial line).
// With bet1 = pi/2, alp1 = -pi, sig1 = pi/2
// With bet1 = -pi/2, alp1 = 0 , sig1 = -pi/2
// Evaluate omg1 with tan(omg1) = sin(alp0) * tan(sig1).
// With alp0 in (0, pi/2], quadrants for sig and omg coincide.
// No atan2(0,0) ambiguity at poles since cbet1 = +epsilon.
// With alp0 = 0, omg1 = 0 for alp1 = 0, omg1 = pi for alp1 = pi.
_ssig1 = sbet1; _somg1 = _salp0 * sbet1;
_csig1 = _comg1 = sbet1 != 0 || _calp1 != 0 ? cbet1 * _calp1 : 1;
Math::norm(_ssig1, _csig1); // sig1 in (-pi, pi]
// Math::norm(_somg1, _comg1); -- don't need to normalize!
_a13 = _s13 = Math::NaN();
_exact = g._exact;
if (_exact) {
_lineexact.LineInit(g._geodexact, lat1, lon1, azi1, salp1, calp1, caps);
return;
}
_k2 = Math::sq(_calp0) * g._ep2;
real eps = _k2 / (2 * (1 + sqrt(1 + _k2)) + _k2);
if (_caps & CAP_C1) {
_aA1m1 = Geodesic::A1m1f(eps);
Geodesic::C1f(eps, _cC1a);
_bB11 = Geodesic::SinCosSeries(true, _ssig1, _csig1, _cC1a, nC1_);
real s = sin(_bB11), c = cos(_bB11);
// tau1 = sig1 + B11
_stau1 = _ssig1 * c + _csig1 * s;
_ctau1 = _csig1 * c - _ssig1 * s;
// Not necessary because C1pa reverts C1a
// _bB11 = -SinCosSeries(true, _stau1, _ctau1, _cC1pa, nC1p_);
}
if (_caps & CAP_C1p)
Geodesic::C1pf(eps, _cC1pa);
if (_caps & CAP_C2) {
_aA2m1 = Geodesic::A2m1f(eps);
Geodesic::C2f(eps, _cC2a);
_bB21 = Geodesic::SinCosSeries(true, _ssig1, _csig1, _cC2a, nC2_);
}
if (_caps & CAP_C3) {
g.C3f(eps, _cC3a);
_aA3c = -_f * _salp0 * g.A3f(eps);
_bB31 = Geodesic::SinCosSeries(true, _ssig1, _csig1, _cC3a, nC3_-1);
}
if (_caps & CAP_C4) {
g.C4f(eps, _cC4a);
// Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0)
_aA4 = Math::sq(_a) * _calp0 * _salp0 * g._e2;
_bB41 = Geodesic::SinCosSeries(false, _ssig1, _csig1, _cC4a, nC4_);
}
}
GeodesicLine::GeodesicLine(const Geodesic& g,
real lat1, real lon1, real azi1,
unsigned caps) {
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);
LineInit(g, lat1, lon1, azi1, salp1, calp1, caps);
}
GeodesicLine::GeodesicLine(const Geodesic& g,
real lat1, real lon1,
real azi1, real salp1, real calp1,
unsigned caps, bool arcmode, real s13_a13) {
LineInit(g, lat1, lon1, azi1, salp1, calp1, caps);
GenSetDistance(arcmode, s13_a13);
}
Math::real GeodesicLine::GenPosition(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 _lineexact.GenPosition(arcmode, s12_a12, outmask,
lat2, lon2, azi2,
s12, m12, M12, M21, S12);
outmask &= _caps & OUT_MASK;
if (!( Init() && (arcmode || (_caps & (OUT_MASK & DISTANCE_IN))) ))
// Uninitialized or impossible distance calculation requested
return Math::NaN();
// Avoid warning about uninitialized B12.
real sig12, ssig12, csig12, B12 = 0, AB1 = 0;
if (arcmode) {
// Interpret s12_a12 as spherical arc length
sig12 = s12_a12 * Math::degree();
Math::sincosd(s12_a12, ssig12, csig12);
} else {
// Interpret s12_a12 as distance
real
tau12 = s12_a12 / (_b * (1 + _aA1m1)),
s = sin(tau12),
c = cos(tau12);
// tau2 = tau1 + tau12
B12 = - Geodesic::SinCosSeries(true,
_stau1 * c + _ctau1 * s,
_ctau1 * c - _stau1 * s,
_cC1pa, nC1p_);
sig12 = tau12 - (B12 - _bB11);
ssig12 = sin(sig12); csig12 = cos(sig12);
if (fabs(_f) > 0.01) {
// Reverted distance series is inaccurate for |f| > 1/100, so correct
// sig12 with 1 Newton iteration. The following table shows the
// approximate maximum error for a = WGS_a() and various f relative to
// GeodesicExact.
// erri = the error in the inverse solution (nm)
// errd = the error in the direct solution (series only) (nm)
// errda = the error in the direct solution
// (series + 1 Newton) (nm)
//
// f erri errd errda
// -1/5 12e6 1.2e9 69e6
// -1/10 123e3 12e6 765e3
// -1/20 1110 108e3 7155
// -1/50 18.63 200.9 27.12
// -1/100 18.63 23.78 23.37
// -1/150 18.63 21.05 20.26
// 1/150 22.35 24.73 25.83
// 1/100 22.35 25.03 25.31
// 1/50 29.80 231.9 30.44
// 1/20 5376 146e3 10e3
// 1/10 829e3 22e6 1.5e6
// 1/5 157e6 3.8e9 280e6
real
ssig2 = _ssig1 * csig12 + _csig1 * ssig12,
csig2 = _csig1 * csig12 - _ssig1 * ssig12;
B12 = Geodesic::SinCosSeries(true, ssig2, csig2, _cC1a, nC1_);
real serr = (1 + _aA1m1) * (sig12 + (B12 - _bB11)) - s12_a12 / _b;
sig12 = sig12 - serr / sqrt(1 + _k2 * Math::sq(ssig2));
ssig12 = sin(sig12); csig12 = cos(sig12);
// Update B12 below
}
}
real ssig2, csig2, sbet2, cbet2, salp2, calp2;
// sig2 = sig1 + sig12
ssig2 = _ssig1 * csig12 + _csig1 * ssig12;
csig2 = _csig1 * csig12 - _ssig1 * ssig12;
real dn2 = sqrt(1 + _k2 * Math::sq(ssig2));
if (outmask & (DISTANCE | REDUCEDLENGTH | GEODESICSCALE)) {
if (arcmode || fabs(_f) > 0.01)
B12 = Geodesic::SinCosSeries(true, ssig2, csig2, _cC1a, nC1_);
AB1 = (1 + _aA1m1) * (B12 - _bB11);
}
// sin(bet2) = cos(alp0) * sin(sig2)
sbet2 = _calp0 * ssig2;
// Alt: cbet2 = hypot(csig2, salp0 * ssig2);
cbet2 = hypot(_salp0, _calp0 * csig2);
if (cbet2 == 0)
// I.e., salp0 = 0, csig2 = 0. Break the degeneracy in this case
cbet2 = csig2 = tiny_;
// tan(alp0) = cos(sig2)*tan(alp2)
salp2 = _salp0; calp2 = _calp0 * csig2; // No need to normalize
if (outmask & DISTANCE)
s12 = arcmode ? _b * ((1 + _aA1m1) * sig12 + AB1) : s12_a12;
if (outmask & LONGITUDE) {
// tan(omg2) = sin(alp0) * tan(sig2)
real somg2 = _salp0 * ssig2, comg2 = csig2, // No need to normalize
E = copysign(real(1), _salp0); // east-going?
// omg12 = omg2 - omg1
real omg12 = outmask & LONG_UNROLL
? E * (sig12
- (atan2( ssig2, csig2) - atan2( _ssig1, _csig1))
+ (atan2(E * somg2, comg2) - atan2(E * _somg1, _comg1)))
: atan2(somg2 * _comg1 - comg2 * _somg1,
comg2 * _comg1 + somg2 * _somg1);
real lam12 = omg12 + _aA3c *
( sig12 + (Geodesic::SinCosSeries(true, ssig2, csig2, _cC3a, nC3_-1)
- _bB31));
real lon12 = lam12 / Math::degree();
lon2 = outmask & LONG_UNROLL ? _lon1 + lon12 :
Math::AngNormalize(Math::AngNormalize(_lon1) +
Math::AngNormalize(lon12));
}
if (outmask & LATITUDE)
lat2 = Math::atan2d(sbet2, _f1 * cbet2);
if (outmask & AZIMUTH)
azi2 = Math::atan2d(salp2, calp2);
if (outmask & (REDUCEDLENGTH | GEODESICSCALE)) {
real
B22 = Geodesic::SinCosSeries(true, ssig2, csig2, _cC2a, nC2_),
AB2 = (1 + _aA2m1) * (B22 - _bB21),
J12 = (_aA1m1 - _aA2m1) * sig12 + (AB1 - AB2);
if (outmask & REDUCEDLENGTH)
// Add parens around (_csig1 * ssig2) and (_ssig1 * csig2) to ensure
// accurate cancellation in the case of coincident points.
m12 = _b * ((dn2 * (_csig1 * ssig2) - _dn1 * (_ssig1 * csig2))
- _csig1 * csig2 * J12);
if (outmask & GEODESICSCALE) {
real t = _k2 * (ssig2 - _ssig1) * (ssig2 + _ssig1) / (_dn1 + dn2);
M12 = csig12 + (t * ssig2 - csig2 * J12) * _ssig1 / _dn1;
M21 = csig12 - (t * _ssig1 - _csig1 * J12) * ssig2 / dn2;
}
}
if (outmask & AREA) {
real
B42 = Geodesic::SinCosSeries(false, ssig2, csig2, _cC4a, nC4_);
real salp12, calp12;
if (_calp0 == 0 || _salp0 == 0) {
// alp12 = alp2 - alp1, used in atan2 so no need to normalize
salp12 = salp2 * _calp1 - calp2 * _salp1;
calp12 = calp2 * _calp1 + salp2 * _salp1;
// We used to include here some patch up code that purported to deal
// with nearly meridional geodesics properly. However, this turned out
// to be wrong once _salp1 = -0 was allowed (via
// Geodesic::InverseLine). In fact, the calculation of {s,c}alp12
// was already correct (following the IEEE rules for handling signed
// zeros). So the patch up code was unnecessary (as well as
// dangerous).
} else {
// tan(alp) = tan(alp0) * sec(sig)
// tan(alp2-alp1) = (tan(alp2) -tan(alp1)) / (tan(alp2)*tan(alp1)+1)
// = calp0 * salp0 * (csig1-csig2) / (salp0^2 + calp0^2 * csig1*csig2)
// If csig12 > 0, write
// csig1 - csig2 = ssig12 * (csig1 * ssig12 / (1 + csig12) + ssig1)
// else
// csig1 - csig2 = csig1 * (1 - csig12) + ssig12 * ssig1
// No need to normalize
salp12 = _calp0 * _salp0 *
(csig12 <= 0 ? _csig1 * (1 - csig12) + ssig12 * _ssig1 :
ssig12 * (_csig1 * ssig12 / (1 + csig12) + _ssig1));
calp12 = Math::sq(_salp0) + Math::sq(_calp0) * _csig1 * csig2;
}
S12 = _c2 * atan2(salp12, calp12) + _aA4 * (B42 - _bB41);
}
return arcmode ? s12_a12 : sig12 / Math::degree();
}
void GeodesicLine::SetDistance(real s13) {
_s13 = s13;
real t;
// This will set _a13 to NaN if the GeodesicLine doesn't have the
// DISTANCE_IN capability.
_a13 = GenPosition(false, _s13, 0u, t, t, t, t, t, t, t, t);
}
void GeodesicLine::SetArc(real a13) {
_a13 = a13;
// In case the GeodesicLine doesn't have the DISTANCE capability.
_s13 = Math::NaN();
real t;
GenPosition(true, _a13, DISTANCE, t, t, t, _s13, t, t, t, t);
}
void GeodesicLine::GenSetDistance(bool arcmode, real s13_a13) {
arcmode ? SetArc(s13_a13) : SetDistance(s13_a13);
}
} // namespace GeographicLib
real
GeographicLib::Math::real real
Definition GeodSolve.cpp:28
GeodesicLine.hpp
Header for GeographicLib::GeodesicLine class.
GeographicLib::GeodesicLineExact::GenPosition
Math::real GenPosition(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 GeodesicLineExact.cpp:143
GeographicLib::GeodesicLine::SetArc
void SetArc(real a13)
Definition GeodesicLine.cpp:319
GeographicLib::GeodesicLine::Init
bool Init() const
Definition GeodesicLine.hpp:605
GeographicLib::GeodesicLine::GenSetDistance
void GenSetDistance(bool arcmode, real s13_a13)
Definition GeodesicLine.cpp:327
GeographicLib::GeodesicLine::GenPosition
Math::real GenPosition(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 GeodesicLine.cpp:142
GeographicLib::GeodesicLine::SetDistance
void SetDistance(real s13)
Definition GeodesicLine.cpp:311
GeographicLib::GeodesicLine::LATITUDE
@ LATITUDE
Definition GeodesicLine.hpp:137
GeographicLib::GeodesicLine::AZIMUTH
@ AZIMUTH
Definition GeodesicLine.hpp:149
GeographicLib::GeodesicLine::DISTANCE
@ DISTANCE
Definition GeodesicLine.hpp:154
GeographicLib::GeodesicLine::AREA
@ AREA
Definition GeodesicLine.hpp:181
GeographicLib::GeodesicLine::GEODESICSCALE
@ GEODESICSCALE
Definition GeodesicLine.hpp:176
GeographicLib::GeodesicLine::DISTANCE_IN
@ DISTANCE_IN
Definition GeodesicLine.hpp:166
GeographicLib::GeodesicLine::REDUCEDLENGTH
@ REDUCEDLENGTH
Definition GeodesicLine.hpp:171
GeographicLib::GeodesicLine::LONGITUDE
@ LONGITUDE
Definition GeodesicLine.hpp:142
GeographicLib::GeodesicLine::LONG_UNROLL
@ LONG_UNROLL
Definition GeodesicLine.hpp:186
GeographicLib::GeodesicLine::GeodesicLine
GeodesicLine()
Definition GeodesicLine.hpp:251
GeographicLib::Geodesic
Geodesic calculations
Definition Geodesic.hpp:175
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::AngNormalize
static T AngNormalize(T x)
Definition Math.cpp:69
GeographicLib::Math::NaN
static T NaN()
Definition Math.cpp:283
GeographicLib::Math::real
double real
Definition Math.hpp:115
GeographicLib
Namespace for GeographicLib.
Definition Accumulator.cpp:12
std
Definition NearestNeighbor.hpp:806