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LambertConformalConic.cpp
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/**
* \file LambertConformalConic.cpp
* \brief Implementation for GeographicLib::LambertConformalConic class
*
* Copyright (c) Charles Karney (2010-2022) <karney@alum.mit.edu> and licensed
* under the MIT/X11 License. For more information, see
* https://geographiclib.sourceforge.io/
**********************************************************************/
#include <GeographicLib/LambertConformalConic.hpp>
namespace GeographicLib {
using namespace std;
LambertConformalConic::LambertConformalConic(real a, real f,
real stdlat, real k0)
: eps_(numeric_limits<real>::epsilon())
, epsx_(Math::sq(eps_))
, ahypover_(Math::digits() * log(real(numeric_limits<real>::radix)) + 2)
, _a(a)
, _f(f)
, _fm(1 - _f)
, _e2(_f * (2 - _f))
, _es((_f < 0 ? -1 : 1) * sqrt(fabs(_e2)))
{
if (!(isfinite(_a) && _a > 0))
throw GeographicErr("Equatorial radius is not positive");
if (!(isfinite(_f) && _f < 1))
throw GeographicErr("Polar semi-axis is not positive");
if (!(isfinite(k0) && k0 > 0))
throw GeographicErr("Scale is not positive");
if (!(fabs(stdlat) <= Math::qd))
throw GeographicErr("Standard latitude not in [-" + to_string(Math::qd)
+ "d, " + to_string(Math::qd) + "d]");
real sphi, cphi;
Math::sincosd(stdlat, sphi, cphi);
Init(sphi, cphi, sphi, cphi, k0);
}
LambertConformalConic::LambertConformalConic(real a, real f,
real stdlat1, real stdlat2,
real k1)
: eps_(numeric_limits<real>::epsilon())
, epsx_(Math::sq(eps_))
, ahypover_(Math::digits() * log(real(numeric_limits<real>::radix)) + 2)
, _a(a)
, _f(f)
, _fm(1 - _f)
, _e2(_f * (2 - _f))
, _es((_f < 0 ? -1 : 1) * sqrt(fabs(_e2)))
{
if (!(isfinite(_a) && _a > 0))
throw GeographicErr("Equatorial radius is not positive");
if (!(isfinite(_f) && _f < 1))
throw GeographicErr("Polar semi-axis is not positive");
if (!(isfinite(k1) && k1 > 0))
throw GeographicErr("Scale is not positive");
if (!(fabs(stdlat1) <= Math::qd))
throw GeographicErr("Standard latitude 1 not in [-"
+ to_string(Math::qd) + "d, "
+ to_string(Math::qd) + "d]");
if (!(fabs(stdlat2) <= Math::qd))
throw GeographicErr("Standard latitude 2 not in [-"
+ to_string(Math::qd) + "d, "
+ to_string(Math::qd) + "d]");
real sphi1, cphi1, sphi2, cphi2;
Math::sincosd(stdlat1, sphi1, cphi1);
Math::sincosd(stdlat2, sphi2, cphi2);
Init(sphi1, cphi1, sphi2, cphi2, k1);
}
LambertConformalConic::LambertConformalConic(real a, real f,
real sinlat1, real coslat1,
real sinlat2, real coslat2,
real k1)
: eps_(numeric_limits<real>::epsilon())
, epsx_(Math::sq(eps_))
, ahypover_(Math::digits() * log(real(numeric_limits<real>::radix)) + 2)
, _a(a)
, _f(f)
, _fm(1 - _f)
, _e2(_f * (2 - _f))
, _es((_f < 0 ? -1 : 1) * sqrt(fabs(_e2)))
{
if (!(isfinite(_a) && _a > 0))
throw GeographicErr("Equatorial radius is not positive");
if (!(isfinite(_f) && _f < 1))
throw GeographicErr("Polar semi-axis is not positive");
if (!(isfinite(k1) && k1 > 0))
throw GeographicErr("Scale is not positive");
if (signbit(coslat1))
throw GeographicErr("Standard latitude 1 not in [-"
+ to_string(Math::qd) + "d, "
+ to_string(Math::qd) + "d]");
if (signbit(coslat2))
throw GeographicErr("Standard latitude 2 not in [-"
+ to_string(Math::qd) + "d, "
+ to_string(Math::qd) + "d]");
if (!(fabs(sinlat1) <= 1 && coslat1 <= 1) || (coslat1 == 0 && sinlat1 == 0))
throw GeographicErr("Bad sine/cosine of standard latitude 1");
if (!(fabs(sinlat2) <= 1 && coslat2 <= 1) || (coslat2 == 0 && sinlat2 == 0))
throw GeographicErr("Bad sine/cosine of standard latitude 2");
if (coslat1 == 0 || coslat2 == 0)
if (!(coslat1 == coslat2 && sinlat1 == sinlat2))
throw GeographicErr
("Standard latitudes must be equal is either is a pole");
Init(sinlat1, coslat1, sinlat2, coslat2, k1);
}
void LambertConformalConic::Init(real sphi1, real cphi1,
real sphi2, real cphi2, real k1) {
{
real r;
r = hypot(sphi1, cphi1);
sphi1 /= r; cphi1 /= r;
r = hypot(sphi2, cphi2);
sphi2 /= r; cphi2 /= r;
}
bool polar = (cphi1 == 0);
cphi1 = fmax(epsx_, cphi1); // Avoid singularities at poles
cphi2 = fmax(epsx_, cphi2);
// Determine hemisphere of tangent latitude
_sign = sphi1 + sphi2 >= 0 ? 1 : -1;
// Internally work with tangent latitude positive
sphi1 *= _sign; sphi2 *= _sign;
if (sphi1 > sphi2) {
swap(sphi1, sphi2); swap(cphi1, cphi2); // Make phi1 < phi2
}
real
tphi1 = sphi1/cphi1, tphi2 = sphi2/cphi2, tphi0;
//
// Snyder: 15-8: n = (log(m1) - log(m2))/(log(t1)-log(t2))
//
// m = cos(bet) = 1/sec(bet) = 1/sqrt(1+tan(bet)^2)
// bet = parametric lat, tan(bet) = (1-f)*tan(phi)
//
// t = tan(pi/4-chi/2) = 1/(sec(chi) + tan(chi)) = sec(chi) - tan(chi)
// log(t) = -asinh(tan(chi)) = -psi
// chi = conformal lat
// tan(chi) = tan(phi)*cosh(xi) - sinh(xi)*sec(phi)
// xi = eatanhe(sin(phi)), eatanhe(x) = e * atanh(e*x)
//
// n = (log(sec(bet2))-log(sec(bet1)))/(asinh(tan(chi2))-asinh(tan(chi1)))
//
// Let log(sec(bet)) = b(tphi), asinh(tan(chi)) = c(tphi)
// Then n = Db(tphi2, tphi1)/Dc(tphi2, tphi1)
// In limit tphi2 -> tphi1, n -> sphi1
//
real
tbet1 = _fm * tphi1, scbet1 = hyp(tbet1),
tbet2 = _fm * tphi2, scbet2 = hyp(tbet2);
real
scphi1 = 1/cphi1,
xi1 = Math::eatanhe(sphi1, _es), shxi1 = sinh(xi1), chxi1 = hyp(shxi1),
tchi1 = chxi1 * tphi1 - shxi1 * scphi1, scchi1 = hyp(tchi1),
scphi2 = 1/cphi2,
xi2 = Math::eatanhe(sphi2, _es), shxi2 = sinh(xi2), chxi2 = hyp(shxi2),
tchi2 = chxi2 * tphi2 - shxi2 * scphi2, scchi2 = hyp(tchi2),
psi1 = asinh(tchi1);
if (tphi2 - tphi1 != 0) {
// Db(tphi2, tphi1)
real num = Dlog1p(Math::sq(tbet2)/(1 + scbet2),
Math::sq(tbet1)/(1 + scbet1))
* Dhyp(tbet2, tbet1, scbet2, scbet1) * _fm;
// Dc(tphi2, tphi1)
real den = Dasinh(tphi2, tphi1, scphi2, scphi1)
- Deatanhe(sphi2, sphi1) * Dsn(tphi2, tphi1, sphi2, sphi1);
_n = num/den;
if (_n < 1/real(4))
_nc = sqrt((1 - _n) * (1 + _n));
else {
// Compute nc = cos(phi0) = sqrt((1 - n) * (1 + n)), evaluating 1 - n
// carefully. First write
//
// Dc(tphi2, tphi1) * (tphi2 - tphi1)
// = log(tchi2 + scchi2) - log(tchi1 + scchi1)
//
// then den * (1 - n) =
// (log((tchi2 + scchi2)/(2*scbet2)) -
// log((tchi1 + scchi1)/(2*scbet1))) / (tphi2 - tphi1)
// = Dlog1p(a2, a1) * (tchi2+scchi2 + tchi1+scchi1)/(4*scbet1*scbet2)
// * fm * Q
//
// where
// a1 = ( (tchi1 - scbet1) + (scchi1 - scbet1) ) / (2 * scbet1)
// Q = ((scbet2 + scbet1)/fm)/((scchi2 + scchi1)/D(tchi2, tchi1))
// - (tbet2 + tbet1)/(scbet2 + scbet1)
real t;
{
real
// s1 = (scbet1 - scchi1) * (scbet1 + scchi1)
s1 = (tphi1 * (2 * shxi1 * chxi1 * scphi1 - _e2 * tphi1) -
Math::sq(shxi1) * (1 + 2 * Math::sq(tphi1))),
s2 = (tphi2 * (2 * shxi2 * chxi2 * scphi2 - _e2 * tphi2) -
Math::sq(shxi2) * (1 + 2 * Math::sq(tphi2))),
// t1 = scbet1 - tchi1
t1 = tchi1 < 0 ? scbet1 - tchi1 : (s1 + 1)/(scbet1 + tchi1),
t2 = tchi2 < 0 ? scbet2 - tchi2 : (s2 + 1)/(scbet2 + tchi2),
a2 = -(s2 / (scbet2 + scchi2) + t2) / (2 * scbet2),
a1 = -(s1 / (scbet1 + scchi1) + t1) / (2 * scbet1);
t = Dlog1p(a2, a1) / den;
}
// multiply by (tchi2 + scchi2 + tchi1 + scchi1)/(4*scbet1*scbet2) * fm
t *= ( ( (tchi2 >= 0 ? scchi2 + tchi2 : 1/(scchi2 - tchi2)) +
(tchi1 >= 0 ? scchi1 + tchi1 : 1/(scchi1 - tchi1)) ) /
(4 * scbet1 * scbet2) ) * _fm;
// Rewrite
// Q = (1 - (tbet2 + tbet1)/(scbet2 + scbet1)) -
// (1 - ((scbet2 + scbet1)/fm)/((scchi2 + scchi1)/D(tchi2, tchi1)))
// = tbm - tam
// where
real tbm = ( ((tbet1 > 0 ? 1/(scbet1+tbet1) : scbet1 - tbet1) +
(tbet2 > 0 ? 1/(scbet2+tbet2) : scbet2 - tbet2)) /
(scbet1+scbet2) );
// tam = (1 - ((scbet2+scbet1)/fm)/((scchi2+scchi1)/D(tchi2, tchi1)))
//
// Let
// (scbet2 + scbet1)/fm = scphi2 + scphi1 + dbet
// (scchi2 + scchi1)/D(tchi2, tchi1) = scphi2 + scphi1 + dchi
// then
// tam = D(tchi2, tchi1) * (dchi - dbet) / (scchi1 + scchi2)
real
// D(tchi2, tchi1)
dtchi = den / Dasinh(tchi2, tchi1, scchi2, scchi1),
// (scbet2 + scbet1)/fm - (scphi2 + scphi1)
dbet = (_e2/_fm) * ( 1 / (scbet2 + _fm * scphi2) +
1 / (scbet1 + _fm * scphi1) );
// dchi = (scchi2 + scchi1)/D(tchi2, tchi1) - (scphi2 + scphi1)
// Let
// tzet = chxiZ * tphi - shxiZ * scphi
// tchi = tzet + nu
// scchi = sczet + mu
// where
// xiZ = eatanhe(1), shxiZ = sinh(xiZ), chxiZ = cosh(xiZ)
// nu = scphi * (shxiZ - shxi) - tphi * (chxiZ - chxi)
// mu = - scphi * (chxiZ - chxi) + tphi * (shxiZ - shxi)
// then
// dchi = ((mu2 + mu1) - D(nu2, nu1) * (scphi2 + scphi1)) /
// D(tchi2, tchi1)
real
xiZ = Math::eatanhe(real(1), _es),
shxiZ = sinh(xiZ), chxiZ = hyp(shxiZ),
// These are differences not divided differences
// dxiZ1 = xiZ - xi1; dshxiZ1 = shxiZ - shxi; dchxiZ1 = chxiZ - chxi
dxiZ1 = Deatanhe(real(1), sphi1)/(scphi1*(tphi1+scphi1)),
dxiZ2 = Deatanhe(real(1), sphi2)/(scphi2*(tphi2+scphi2)),
dshxiZ1 = Dsinh(xiZ, xi1, shxiZ, shxi1, chxiZ, chxi1) * dxiZ1,
dshxiZ2 = Dsinh(xiZ, xi2, shxiZ, shxi2, chxiZ, chxi2) * dxiZ2,
dchxiZ1 = Dhyp(shxiZ, shxi1, chxiZ, chxi1) * dshxiZ1,
dchxiZ2 = Dhyp(shxiZ, shxi2, chxiZ, chxi2) * dshxiZ2,
// mu1 + mu2
amu12 = (- scphi1 * dchxiZ1 + tphi1 * dshxiZ1
- scphi2 * dchxiZ2 + tphi2 * dshxiZ2),
// D(xi2, xi1)
dxi = Deatanhe(sphi1, sphi2) * Dsn(tphi2, tphi1, sphi2, sphi1),
// D(nu2, nu1)
dnu12 =
( (_f * 4 * scphi2 * dshxiZ2 > _f * scphi1 * dshxiZ1 ?
// Use divided differences
(dshxiZ1 + dshxiZ2)/2 * Dhyp(tphi1, tphi2, scphi1, scphi2)
- ( (scphi1 + scphi2)/2
* Dsinh(xi1, xi2, shxi1, shxi2, chxi1, chxi2) * dxi ) :
// Use ratio of differences
(scphi2 * dshxiZ2 - scphi1 * dshxiZ1)/(tphi2 - tphi1))
+ ( (tphi1 + tphi2)/2 * Dhyp(shxi1, shxi2, chxi1, chxi2)
* Dsinh(xi1, xi2, shxi1, shxi2, chxi1, chxi2) * dxi )
- (dchxiZ1 + dchxiZ2)/2 ),
// dtchi * dchi
dchia = (amu12 - dnu12 * (scphi2 + scphi1)),
tam = (dchia - dtchi * dbet) / (scchi1 + scchi2);
t *= tbm - tam;
_nc = sqrt(fmax(real(0), t) * (1 + _n));
}
{
real r = hypot(_n, _nc);
_n /= r;
_nc /= r;
}
tphi0 = _n / _nc;
} else {
tphi0 = tphi1;
_nc = 1/hyp(tphi0);
_n = tphi0 * _nc;
if (polar)
_nc = 0;
}
_scbet0 = hyp(_fm * tphi0);
real shxi0 = sinh(Math::eatanhe(_n, _es));
_tchi0 = tphi0 * hyp(shxi0) - shxi0 * hyp(tphi0); _scchi0 = hyp(_tchi0);
_psi0 = asinh(_tchi0);
_lat0 = atan(_sign * tphi0) / Math::degree();
_t0nm1 = expm1(- _n * _psi0); // Snyder's t0^n - 1
// a * k1 * m1/t1^n = a * k1 * m2/t2^n = a * k1 * n * (Snyder's F)
// = a * k1 / (scbet1 * exp(-n * psi1))
_scale = _a * k1 / scbet1 *
// exp(n * psi1) = exp(- (1 - n) * psi1) * exp(psi1)
// with (1-n) = nc^2/(1+n) and exp(-psi1) = scchi1 + tchi1
exp( - (Math::sq(_nc)/(1 + _n)) * psi1 )
* (tchi1 >= 0 ? scchi1 + tchi1 : 1 / (scchi1 - tchi1));
// Scale at phi0 = k0 = k1 * (scbet0*exp(-n*psi0))/(scbet1*exp(-n*psi1))
// = k1 * scbet0/scbet1 * exp(n * (psi1 - psi0))
// psi1 - psi0 = Dasinh(tchi1, tchi0) * (tchi1 - tchi0)
_k0 = k1 * (_scbet0/scbet1) *
exp( - (Math::sq(_nc)/(1 + _n)) *
Dasinh(tchi1, _tchi0, scchi1, _scchi0) * (tchi1 - _tchi0))
* (tchi1 >= 0 ? scchi1 + tchi1 : 1 / (scchi1 - tchi1)) /
(_scchi0 + _tchi0);
_nrho0 = polar ? 0 : _a * _k0 / _scbet0;
{
// Figure _drhomax using code at beginning of Forward with lat = -90
real
sphi = -1, cphi = epsx_,
tphi = sphi/cphi,
scphi = 1/cphi, shxi = sinh(Math::eatanhe(sphi, _es)),
tchi = hyp(shxi) * tphi - shxi * scphi, scchi = hyp(tchi),
psi = asinh(tchi),
dpsi = Dasinh(tchi, _tchi0, scchi, _scchi0) * (tchi - _tchi0);
_drhomax = - _scale * (2 * _nc < 1 && dpsi != 0 ?
(exp(Math::sq(_nc)/(1 + _n) * psi ) *
(tchi > 0 ? 1/(scchi + tchi) : (scchi - tchi))
- (_t0nm1 + 1))/(-_n) :
Dexp(-_n * psi, -_n * _psi0) * dpsi);
}
}
const LambertConformalConic& LambertConformalConic::Mercator() {
static const LambertConformalConic mercator(Constants::WGS84_a(),
Constants::WGS84_f(),
real(0), real(1));
return mercator;
}
void LambertConformalConic::Forward(real lon0, real lat, real lon,
real& x, real& y,
real& gamma, real& k) const {
lon = Math::AngDiff(lon0, lon);
// From Snyder, we have
//
// theta = n * lambda
// x = rho * sin(theta)
// = (nrho0 + n * drho) * sin(theta)/n
// y = rho0 - rho * cos(theta)
// = nrho0 * (1-cos(theta))/n - drho * cos(theta)
//
// where nrho0 = n * rho0, drho = rho - rho0
// and drho is evaluated with divided differences
real sphi, cphi;
Math::sincosd(Math::LatFix(lat) * _sign, sphi, cphi);
cphi = fmax(epsx_, cphi);
real
lam = lon * Math::degree(),
tphi = sphi/cphi, scbet = hyp(_fm * tphi),
scphi = 1/cphi, shxi = sinh(Math::eatanhe(sphi, _es)),
tchi = hyp(shxi) * tphi - shxi * scphi, scchi = hyp(tchi),
psi = asinh(tchi),
theta = _n * lam, stheta = sin(theta), ctheta = cos(theta),
dpsi = Dasinh(tchi, _tchi0, scchi, _scchi0) * (tchi - _tchi0),
drho = - _scale * (2 * _nc < 1 && dpsi != 0 ?
(exp(Math::sq(_nc)/(1 + _n) * psi ) *
(tchi > 0 ? 1/(scchi + tchi) : (scchi - tchi))
- (_t0nm1 + 1))/(-_n) :
Dexp(-_n * psi, -_n * _psi0) * dpsi);
x = (_nrho0 + _n * drho) * (_n != 0 ? stheta / _n : lam);
y = _nrho0 *
(_n != 0 ?
(ctheta < 0 ? 1 - ctheta : Math::sq(stheta)/(1 + ctheta)) / _n : 0)
- drho * ctheta;
k = _k0 * (scbet/_scbet0) /
(exp( - (Math::sq(_nc)/(1 + _n)) * dpsi )
* (tchi >= 0 ? scchi + tchi : 1 / (scchi - tchi)) / (_scchi0 + _tchi0));
y *= _sign;
gamma = _sign * theta / Math::degree();
}
void LambertConformalConic::Reverse(real lon0, real x, real y,
real& lat, real& lon,
real& gamma, real& k) const {
// From Snyder, we have
//
// x = rho * sin(theta)
// rho0 - y = rho * cos(theta)
//
// rho = hypot(x, rho0 - y)
// drho = (n*x^2 - 2*y*nrho0 + n*y^2)/(hypot(n*x, nrho0-n*y) + nrho0)
// theta = atan2(n*x, nrho0-n*y)
//
// From drho, obtain t^n-1
// psi = -log(t), so
// dpsi = - Dlog1p(t^n-1, t0^n-1) * drho / scale
y *= _sign;
real
// Guard against 0 * inf in computation of ny
nx = _n * x, ny = _n != 0 ? _n * y : 0, y1 = _nrho0 - ny,
den = hypot(nx, y1) + _nrho0, // 0 implies origin with polar aspect
// isfinite test is to avoid inf/inf
drho = ((den != 0 && isfinite(den))
? (x*nx + y * (ny - 2*_nrho0)) / den
: den);
drho = fmin(drho, _drhomax);
if (_n == 0)
drho = fmax(drho, -_drhomax);
real
tnm1 = _t0nm1 + _n * drho/_scale,
dpsi = (den == 0 ? 0 :
(tnm1 + 1 != 0 ? - Dlog1p(tnm1, _t0nm1) * drho / _scale :
ahypover_));
real tchi;
if (2 * _n <= 1) {
// tchi = sinh(psi)
real
psi = _psi0 + dpsi, tchia = sinh(psi), scchi = hyp(tchia),
dtchi = Dsinh(psi, _psi0, tchia, _tchi0, scchi, _scchi0) * dpsi;
tchi = _tchi0 + dtchi; // Update tchi using divided difference
} else {
// tchi = sinh(-1/n * log(tn))
// = sinh((1-1/n) * log(tn) - log(tn))
// = + sinh((1-1/n) * log(tn)) * cosh(log(tn))
// - cosh((1-1/n) * log(tn)) * sinh(log(tn))
// (1-1/n) = - nc^2/(n*(1+n))
// cosh(log(tn)) = (tn + 1/tn)/2; sinh(log(tn)) = (tn - 1/tn)/2
real
tn = tnm1 + 1 == 0 ? epsx_ : tnm1 + 1,
sh = sinh( -Math::sq(_nc)/(_n * (1 + _n)) *
(2 * tn > 1 ? log1p(tnm1) : log(tn)) );
tchi = sh * (tn + 1/tn)/2 - hyp(sh) * (tnm1 * (tn + 1)/tn)/2;
}
// log(t) = -asinh(tan(chi)) = -psi
gamma = atan2(nx, y1);
real
tphi = Math::tauf(tchi, _es),
scbet = hyp(_fm * tphi), scchi = hyp(tchi),
lam = _n != 0 ? gamma / _n : x / y1;
lat = Math::atand(_sign * tphi);
lon = lam / Math::degree();
lon = Math::AngNormalize(lon + Math::AngNormalize(lon0));
k = _k0 * (scbet/_scbet0) /
(exp(_nc != 0 ? - (Math::sq(_nc)/(1 + _n)) * dpsi : 0)
* (tchi >= 0 ? scchi + tchi : 1 / (scchi - tchi)) / (_scchi0 + _tchi0));
gamma /= _sign * Math::degree();
}
void LambertConformalConic::SetScale(real lat, real k) {
if (!(isfinite(k) && k > 0))
throw GeographicErr("Scale is not positive");
if (!(fabs(lat) <= Math::qd))
throw GeographicErr("Latitude for SetScale not in [-"
+ to_string(Math::qd) + "d, "
+ to_string(Math::qd) + "d]");
if (fabs(lat) == Math::qd && !(_nc == 0 && lat * _n > 0))
throw GeographicErr("Incompatible polar latitude in SetScale");
real x, y, gamma, kold;
Forward(0, lat, 0, x, y, gamma, kold);
k /= kold;
_scale *= k;
_k0 *= k;
}
} // namespace GeographicLib
real
GeographicLib::Math::real real
Definition GeodSolve.cpp:28
LambertConformalConic.hpp
Header for GeographicLib::LambertConformalConic 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::GeographicErr
Exception handling for GeographicLib.
Definition Constants.hpp:316
GeographicLib::LambertConformalConic
Lambert conformal conic projection.
Definition LambertConformalConic.hpp:63
GeographicLib::LambertConformalConic::LambertConformalConic
LambertConformalConic(real a, real f, real stdlat, real k0)
Definition LambertConformalConic.cpp:16
GeographicLib::LambertConformalConic::Reverse
void Reverse(real lon0, real x, real y, real &lat, real &lon, real &gamma, real &k) const
Definition LambertConformalConic.cpp:382
GeographicLib::LambertConformalConic::Forward
void Forward(real lon0, real lat, real lon, real &x, real &y, real &gamma, real &k) const
Definition LambertConformalConic.cpp:340
GeographicLib::LambertConformalConic::SetScale
void SetScale(real lat, real k=real(1))
Definition LambertConformalConic.cpp:450
GeographicLib::LambertConformalConic::Mercator
static const LambertConformalConic & Mercator()
Definition LambertConformalConic.cpp:333
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::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::tauf
static T tauf(T taup, T es)
Definition Math.cpp:241
GeographicLib::Math::AngNormalize
static T AngNormalize(T x)
Definition Math.cpp:69
GeographicLib::Math::atand
static T atand(T x)
Definition Math.cpp:224
GeographicLib::Math::AngDiff
static T AngDiff(T x, T y, T &e)
Definition Math.cpp:80
GeographicLib::Math::eatanhe
static T eatanhe(T x, T es)
Definition Math.cpp:227
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