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Geocentric.cpp
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/**
* \file Geocentric.cpp
* \brief Implementation for GeographicLib::Geocentric class
*
* Copyright (c) Charles Karney (2008-2022) <karney@alum.mit.edu> and licensed
* under the MIT/X11 License. For more information, see
* https://geographiclib.sourceforge.io/
**********************************************************************/
#include <GeographicLib/Geocentric.hpp>
namespace GeographicLib {
using namespace std;
Geocentric::Geocentric(real a, real f)
: _a(a)
, _f(f)
, _e2(_f * (2 - _f))
, _e2m(Math::sq(1 - _f)) // 1 - _e2
, _e2a(fabs(_e2))
, _e4a(Math::sq(_e2))
, _maxrad(2 * _a / numeric_limits<real>::epsilon())
{
if (!(isfinite(_a) && _a > 0))
throw GeographicErr("Equatorial radius is not positive");
if (!(isfinite(_f) && _f < 1))
throw GeographicErr("Polar semiaxis is not positive");
}
const Geocentric& Geocentric::WGS84() {
static const Geocentric wgs84(Constants::WGS84_a(), Constants::WGS84_f());
return wgs84;
}
void Geocentric::IntForward(real lat, real lon, real h,
real& X, real& Y, real& Z,
real M[dim2_]) const {
real sphi, cphi, slam, clam;
Math::sincosd(Math::LatFix(lat), sphi, cphi);
Math::sincosd(lon, slam, clam);
real n = _a/sqrt(1 - _e2 * Math::sq(sphi));
Z = (_e2m * n + h) * sphi;
X = (n + h) * cphi;
Y = X * slam;
X *= clam;
if (M)
Rotation(sphi, cphi, slam, clam, M);
}
void Geocentric::IntReverse(real X, real Y, real Z,
real& lat, real& lon, real& h,
real M[dim2_]) const {
real
R = hypot(X, Y),
slam = R != 0 ? Y / R : 0,
clam = R != 0 ? X / R : 1;
h = hypot(R, Z); // Distance to center of earth
real sphi, cphi;
if (h > _maxrad) {
// We really far away (> 12 million light years); treat the earth as a
// point and h, above, is an acceptable approximation to the height.
// This avoids overflow, e.g., in the computation of disc below. It's
// possible that h has overflowed to inf; but that's OK.
//
// Treat the case X, Y finite, but R overflows to +inf by scaling by 2.
R = hypot(X/2, Y/2);
slam = R != 0 ? (Y/2) / R : 0;
clam = R != 0 ? (X/2) / R : 1;
real H = hypot(Z/2, R);
sphi = (Z/2) / H;
cphi = R / H;
} else if (_e4a == 0) {
// Treat the spherical case. Dealing with underflow in the general case
// with _e2 = 0 is difficult. Origin maps to N pole same as with
// ellipsoid.
real H = hypot(h == 0 ? 1 : Z, R);
sphi = (h == 0 ? 1 : Z) / H;
cphi = R / H;
h -= _a;
} else {
// Treat prolate spheroids by swapping R and Z here and by switching
// the arguments to phi = atan2(...) at the end.
real
p = Math::sq(R / _a),
q = _e2m * Math::sq(Z / _a),
r = (p + q - _e4a) / 6;
if (_f < 0) swap(p, q);
if ( !(_e4a * 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 = _e4a * p * q / 4, // S = r^3 * s
r2 = Math::sq(r),
r3 = r * r2,
disc = S * (2 * r3 + S);
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) + _e4a * q), // guaranteed positive
// Avoid loss of accuracy when u < 0. Underflow doesn't occur in
// e4 * q / (v - u) because u ~ e^4 when q is small and u < 0.
uv = u < 0 ? _e4a * q / (v - u) : u + v, // u+v, guaranteed positive
// Need to guard against w going negative due to roundoff in uv - q.
w = fmax(real(0), _e2a * (uv - q) / (2 * v)),
// 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),
k1 = _f >= 0 ? k : k - _e2,
k2 = _f >= 0 ? k + _e2 : k,
d = k1 * R / k2,
H = hypot(Z/k1, R/k2);
sphi = (Z/k1) / H;
cphi = (R/k2) / H;
h = (1 - _e2m/k1) * hypot(d, Z);
} else { // e4 * q == 0 && r <= 0
// This leads to k = 0 (oblate, equatorial plane) and k + e^2 = 0
// (prolate, rotation axis) and the generation of 0/0 in the general
// formulas for phi and h. using the general formula and division by 0
// in formula for h. So handle this case by taking the limits:
// f > 0: z -> 0, k -> e2 * sqrt(q)/sqrt(e4 - p)
// f < 0: R -> 0, k + e2 -> - e2 * sqrt(q)/sqrt(e4 - p)
real
zz = sqrt((_f >= 0 ? _e4a - p : p) / _e2m),
xx = sqrt( _f < 0 ? _e4a - p : p ),
H = hypot(zz, xx);
sphi = zz / H;
cphi = xx / H;
if (Z < 0) sphi = -sphi; // for tiny negative Z (not for prolate)
h = - _a * (_f >= 0 ? _e2m : 1) * H / _e2a;
}
}
lat = Math::atan2d(sphi, cphi);
lon = Math::atan2d(slam, clam);
if (M)
Rotation(sphi, cphi, slam, clam, M);
}
void Geocentric::Rotation(real sphi, real cphi, real slam, real clam,
real M[dim2_]) {
// This rotation matrix is given by the following quaternion operations
// qrot(lam, [0,0,1]) * qrot(phi, [0,-1,0]) * [1,1,1,1]/2
// or
// qrot(pi/2 + lam, [0,0,1]) * qrot(-pi/2 + phi , [-1,0,0])
// where
// qrot(t,v) = [cos(t/2), sin(t/2)*v[1], sin(t/2)*v[2], sin(t/2)*v[3]]
// Local X axis (east) in geocentric coords
M[0] = -slam; M[3] = clam; M[6] = 0;
// Local Y axis (north) in geocentric coords
M[1] = -clam * sphi; M[4] = -slam * sphi; M[7] = cphi;
// Local Z axis (up) in geocentric coords
M[2] = clam * cphi; M[5] = slam * cphi; M[8] = sphi;
}
} // namespace GeographicLib
Geocentric.hpp
Header for GeographicLib::Geocentric class.
ang
GeographicLib::Angle ang
Definition Geod3Solve.cpp:26
real
GeographicLib::Math::real real
Definition Geod3Solve.cpp:25
GeographicLib::Constants::WGS84_f
static T WGS84_f()
Definition Constants.hpp:118
GeographicLib::Constants::WGS84_a
static T WGS84_a()
Definition Constants.hpp:112
GeographicLib::Geocentric::Geocentric
Geocentric(real a, real f)
Definition Geocentric.cpp:16
GeographicLib::Geocentric::Geocentric
Geocentric()
Definition Geocentric.hpp:118
GeographicLib::Geocentric::WGS84
static const Geocentric & WGS84()
Definition Geocentric.cpp:31
GeographicLib::GeographicErr
Exception handling for GeographicLib.
Definition Constants.hpp:344
GeographicLib::Math
Mathematical functions needed by GeographicLib.
Definition Math.hpp:82
GeographicLib::Math::LatFix
static T LatFix(T x)
Definition Math.hpp:303
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:212
GeographicLib::Math::sq
static T sq(T x)
Definition Math.hpp:209
GeographicLib
Namespace for GeographicLib.
Definition Accumulator.cpp:12
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