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AlbersEqualArea.cpp
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/**
* \file AlbersEqualArea.cpp
* \brief Implementation for GeographicLib::AlbersEqualArea 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/AlbersEqualArea.hpp>
namespace GeographicLib {
using namespace std;
AlbersEqualArea::AlbersEqualArea(real a, real f, real stdlat, real k0)
: eps_(numeric_limits<real>::epsilon())
, epsx_(Math::sq(eps_))
, epsx2_(Math::sq(epsx_))
, tol_(sqrt(eps_))
, tol0_(tol_ * sqrt(sqrt(eps_)))
, _a(a)
, _f(f)
, _fm(1 - _f)
, _e2(_f * (2 - _f))
, _e(sqrt(fabs(_e2)))
, _e2m(1 - _e2)
, _qZ(1 + _e2m * atanhee(real(1)))
, _qx(_qZ / ( 2 * _e2m ))
{
if (!(isfinite(_a) && _a > 0))
throw GeographicErr("Equatorial radius is not positive");
if (!(isfinite(_f) && _f < 1))
throw GeographicErr("Polar semiaxis 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);
}
AlbersEqualArea::AlbersEqualArea(real a, real f, real stdlat1, real stdlat2,
real k1)
: eps_(numeric_limits<real>::epsilon())
, epsx_(Math::sq(eps_))
, epsx2_(Math::sq(epsx_))
, tol_(sqrt(eps_))
, tol0_(tol_ * sqrt(sqrt(eps_)))
, _a(a)
, _f(f)
, _fm(1 - _f)
, _e2(_f * (2 - _f))
, _e(sqrt(fabs(_e2)))
, _e2m(1 - _e2)
, _qZ(1 + _e2m * atanhee(real(1)))
, _qx(_qZ / ( 2 * _e2m ))
{
if (!(isfinite(_a) && _a > 0))
throw GeographicErr("Equatorial radius is not positive");
if (!(isfinite(_f) && _f < 1))
throw GeographicErr("Polar semiaxis 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);
}
AlbersEqualArea::AlbersEqualArea(real a, real f,
real sinlat1, real coslat1,
real sinlat2, real coslat2,
real k1)
: eps_(numeric_limits<real>::epsilon())
, epsx_(Math::sq(eps_))
, epsx2_(Math::sq(epsx_))
, tol_(sqrt(eps_))
, tol0_(tol_ * sqrt(sqrt(eps_)))
, _a(a)
, _f(f)
, _fm(1 - _f)
, _e2(_f * (2 - _f))
, _e(sqrt(fabs(_e2)))
, _e2m(1 - _e2)
, _qZ(1 + _e2m * atanhee(real(1)))
, _qx(_qZ / ( 2 * _e2m ))
{
if (!(isfinite(_a) && _a > 0))
throw GeographicErr("Equatorial radius is not positive");
if (!(isfinite(_f) && _f < 1))
throw GeographicErr("Polar semiaxis 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 && sinlat1 * sinlat2 <= 0)
throw GeographicErr
("Standard latitudes cannot be opposite poles");
Init(sinlat1, coslat1, sinlat2, coslat2, k1);
}
void AlbersEqualArea::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;
// q = (1-e^2)*(sphi/(1-e^2*sphi^2) - atanhee(sphi))
// qZ = q(pi/2) = (1 + (1-e^2)*atanhee(1))
// atanhee(x) = atanh(e*x)/e
// q = sxi * qZ
// dq/dphi = 2*(1-e^2)*cphi/(1-e^2*sphi^2)^2
//
// n = (m1^2-m2^2)/(q2-q1) -> sin(phi0) for phi1, phi2 -> phi0
// C = m1^2 + n*q1 = (m1^2*q2-m2^2*q1)/(q2-q1)
// let
// rho(pi/2)/rho(-pi/2) = (1-s)/(1+s)
// s = n*qZ/C
// = qZ * (m1^2-m2^2)/(m1^2*q2-m2^2*q1)
// = qZ * (scbet2^2 - scbet1^2)/(scbet2^2*q2 - scbet1^2*q1)
// = (scbet2^2 - scbet1^2)/(scbet2^2*sxi2 - scbet1^2*sxi1)
// = (tbet2^2 - tbet1^2)/(scbet2^2*sxi2 - scbet1^2*sxi1)
// 1-s = -((1-sxi2)*scbet2^2 - (1-sxi1)*scbet1^2)/
// (scbet2^2*sxi2 - scbet1^2*sxi1)
//
// Define phi0 to give same value of s, i.e.,
// s = sphi0 * qZ / (m0^2 + sphi0*q0)
// = sphi0 * scbet0^2 / (1/qZ + sphi0 * scbet0^2 * sxi0)
real tphi0, C;
if (polar || tphi1 == tphi2) {
tphi0 = tphi2;
C = 1; // ignored
} else {
real
tbet1 = _fm * tphi1, scbet12 = 1 + Math::sq(tbet1),
tbet2 = _fm * tphi2, scbet22 = 1 + Math::sq(tbet2),
txi1 = txif(tphi1), cxi1 = 1/hyp(txi1), sxi1 = txi1 * cxi1,
txi2 = txif(tphi2), cxi2 = 1/hyp(txi2), sxi2 = txi2 * cxi2,
dtbet2 = _fm * (tbet1 + tbet2),
es1 = 1 - _e2 * Math::sq(sphi1), es2 = 1 - _e2 * Math::sq(sphi2),
/*
dsxi = ( (_e2 * sq(sphi2 + sphi1) + es2 + es1) / (2 * es2 * es1) +
Datanhee(sphi2, sphi1) ) * Dsn(tphi2, tphi1, sphi2, sphi1) /
( 2 * _qx ),
*/
dsxi = ( (1 + _e2 * sphi1 * sphi2) / (es2 * es1) +
Datanhee(sphi2, sphi1) ) * Dsn(tphi2, tphi1, sphi2, sphi1) /
( 2 * _qx ),
den = (sxi2 + sxi1) * dtbet2 + (scbet22 + scbet12) * dsxi,
// s = (sq(tbet2) - sq(tbet1)) / (scbet22*sxi2 - scbet12*sxi1)
s = 2 * dtbet2 / den,
// 1-s = -(sq(scbet2)*(1-sxi2) - sq(scbet1)*(1-sxi1)) /
// (scbet22*sxi2 - scbet12*sxi1)
// Write
// sq(scbet)*(1-sxi) = sq(scbet)*(1-sphi) * (1-sxi)/(1-sphi)
sm1 = -Dsn(tphi2, tphi1, sphi2, sphi1) *
( -( ((sphi2 <= 0 ? (1 - sxi2) / (1 - sphi2) :
Math::sq(cxi2/cphi2) * (1 + sphi2) / (1 + sxi2)) +
(sphi1 <= 0 ? (1 - sxi1) / (1 - sphi1) :
Math::sq(cxi1/cphi1) * (1 + sphi1) / (1 + sxi1))) ) *
(1 + _e2 * (sphi1 + sphi2 + sphi1 * sphi2)) /
(1 + (sphi1 + sphi2 + sphi1 * sphi2)) +
(scbet22 * (sphi2 <= 0 ? 1 - sphi2 :
Math::sq(cphi2) / ( 1 + sphi2)) +
scbet12 * (sphi1 <= 0 ? 1 - sphi1 : Math::sq(cphi1) / ( 1 + sphi1)))
* (_e2 * (1 + sphi1 + sphi2 + _e2 * sphi1 * sphi2)/(es1 * es2)
+_e2m * DDatanhee(sphi1, sphi2) ) / _qZ ) / den;
// C = (scbet22*sxi2 - scbet12*sxi1) / (scbet22 * scbet12 * (sx2 - sx1))
C = den / (2 * scbet12 * scbet22 * dsxi);
tphi0 = (tphi2 + tphi1)/2;
real stol = tol0_ * fmax(real(1), fabs(tphi0));
for (int i = 0;
i < 2*numit0_ ||
GEOGRAPHICLIB_PANIC("Convergence failure in AlbersEqualArea");
++i) {
// Solve (scbet0^2 * sphi0) / (1/qZ + scbet0^2 * sphi0 * sxi0) = s
// for tphi0 by Newton's method on
// v(tphi0) = (scbet0^2 * sphi0) - s * (1/qZ + scbet0^2 * sphi0 * sxi0)
// = 0
// Alt:
// (scbet0^2 * sphi0) / (1/qZ - scbet0^2 * sphi0 * (1-sxi0))
// = s / (1-s)
// w(tphi0) = (1-s) * (scbet0^2 * sphi0)
// - s * (1/qZ - scbet0^2 * sphi0 * (1-sxi0))
// = (1-s) * (scbet0^2 * sphi0)
// - S/qZ * (1 - scbet0^2 * sphi0 * (qZ-q0))
// Now
// qZ-q0 = (1+e2*sphi0)*(1-sphi0)/(1-e2*sphi0^2) +
// (1-e2)*atanhee((1-sphi0)/(1-e2*sphi0))
// In limit sphi0 -> 1, qZ-q0 -> 2*(1-sphi0)/(1-e2), so wrte
// qZ-q0 = 2*(1-sphi0)/(1-e2) + A + B
// A = (1-sphi0)*( (1+e2*sphi0)/(1-e2*sphi0^2) - (1+e2)/(1-e2) )
// = -e2 *(1-sphi0)^2 * (2+(1+e2)*sphi0) / ((1-e2)*(1-e2*sphi0^2))
// B = (1-e2)*atanhee((1-sphi0)/(1-e2*sphi0)) - (1-sphi0)
// = (1-sphi0)*(1-e2)/(1-e2*sphi0)*
// ((atanhee(x)/x-1) - e2*(1-sphi0)/(1-e2))
// x = (1-sphi0)/(1-e2*sphi0), atanhee(x)/x = atanh(e*x)/(e*x)
//
// 1 - scbet0^2 * sphi0 * (qZ-q0)
// = 1 - scbet0^2 * sphi0 * (2*(1-sphi0)/(1-e2) + A + B)
// = D - scbet0^2 * sphi0 * (A + B)
// D = 1 - scbet0^2 * sphi0 * 2*(1-sphi0)/(1-e2)
// = (1-sphi0)*(1-e2*(1+2*sphi0*(1+sphi0)))/((1-e2)*(1+sphi0))
// dD/dsphi0 = -2*(1-e2*sphi0^2*(2*sphi0+3))/((1-e2)*(1+sphi0)^2)
// d(A+B)/dsphi0 = 2*(1-sphi0^2)*e2*(2-e2*(1+sphi0^2))/
// ((1-e2)*(1-e2*sphi0^2)^2)
real
scphi02 = 1 + Math::sq(tphi0), scphi0 = sqrt(scphi02),
// sphi0m = 1-sin(phi0) = 1/( sec(phi0) * (tan(phi0) + sec(phi0)) )
sphi0 = tphi0 / scphi0, sphi0m = 1/(scphi0 * (tphi0 + scphi0)),
// scbet0^2 * sphi0
g = (1 + Math::sq( _fm * tphi0 )) * sphi0,
// dg/dsphi0 = dg/dtphi0 * scphi0^3
dg = _e2m * scphi02 * (1 + 2 * Math::sq(tphi0)) + _e2,
D = sphi0m * (1 - _e2*(1 + 2*sphi0*(1+sphi0))) / (_e2m * (1+sphi0)),
// dD/dsphi0
dD = -2 * (1 - _e2*Math::sq(sphi0) * (2*sphi0+3)) /
(_e2m * Math::sq(1+sphi0)),
A = -_e2 * Math::sq(sphi0m) * (2+(1+_e2)*sphi0) /
(_e2m*(1-_e2*Math::sq(sphi0))),
B = (sphi0m * _e2m / (1 - _e2*sphi0) *
(atanhxm1(_e2 *
Math::sq(sphi0m / (1-_e2*sphi0))) - _e2*sphi0m/_e2m)),
// d(A+B)/dsphi0
dAB = (2 * _e2 * (2 - _e2 * (1 + Math::sq(sphi0))) /
(_e2m * Math::sq(1 - _e2*Math::sq(sphi0)) * scphi02)),
u = sm1 * g - s/_qZ * ( D - g * (A + B) ),
// du/dsphi0
du = sm1 * dg - s/_qZ * (dD - dg * (A + B) - g * dAB),
dtu = -u/du * (scphi0 * scphi02);
tphi0 += dtu;
if (!(fabs(dtu) >= stol))
break;
}
}
_txi0 = txif(tphi0); _scxi0 = hyp(_txi0); _sxi0 = _txi0 / _scxi0;
_n0 = tphi0/hyp(tphi0);
_m02 = 1 / (1 + Math::sq(_fm * tphi0));
_nrho0 = polar ? 0 : _a * sqrt(_m02);
_k0 = sqrt(tphi1 == tphi2 ? 1 : C / (_m02 + _n0 * _qZ * _sxi0)) * k1;
_k2 = Math::sq(_k0);
_lat0 = _sign * atan(tphi0)/Math::degree();
}
const AlbersEqualArea& AlbersEqualArea::CylindricalEqualArea() {
static const AlbersEqualArea
cylindricalequalarea(Constants::WGS84_a(), Constants::WGS84_f(),
real(0), real(1), real(0), real(1), real(1));
return cylindricalequalarea;
}
const AlbersEqualArea& AlbersEqualArea::AzimuthalEqualAreaNorth() {
static const AlbersEqualArea
azimuthalequalareanorth(Constants::WGS84_a(), Constants::WGS84_f(),
real(1), real(0), real(1), real(0), real(1));
return azimuthalequalareanorth;
}
const AlbersEqualArea& AlbersEqualArea::AzimuthalEqualAreaSouth() {
static const AlbersEqualArea
azimuthalequalareasouth(Constants::WGS84_a(), Constants::WGS84_f(),
real(-1), real(0), real(-1), real(0), real(1));
return azimuthalequalareasouth;
}
Math::real AlbersEqualArea::txif(real tphi) const {
// sxi = ( sphi/(1-e2*sphi^2) + atanhee(sphi) ) /
// ( 1/(1-e2) + atanhee(1) )
//
// txi = ( sphi/(1-e2*sphi^2) + atanhee(sphi) ) /
// sqrt( ( (1+e2*sphi)*(1-sphi)/( (1-e2*sphi^2) * (1-e2) ) +
// atanhee((1-sphi)/(1-e2*sphi)) ) *
// ( (1-e2*sphi)*(1+sphi)/( (1-e2*sphi^2) * (1-e2) ) +
// atanhee((1+sphi)/(1+e2*sphi)) ) )
// = ( tphi/(1-e2*sphi^2) + atanhee(sphi, e2)/cphi ) /
// sqrt(
// ( (1+e2*sphi)/( (1-e2*sphi^2) * (1-e2) ) + Datanhee(1, sphi) ) *
// ( (1-e2*sphi)/( (1-e2*sphi^2) * (1-e2) ) + Datanhee(1, -sphi) ) )
//
// This function maintains odd parity
real
cphi = 1 / sqrt(1 + Math::sq(tphi)),
sphi = tphi * cphi,
es1 = _e2 * sphi,
es2m1 = 1 - es1 * sphi, // 1 - e2 * sphi^2
es2m1a = _e2m * es2m1; // (1 - e2 * sphi^2) * (1 - e2)
return ( tphi / es2m1 + atanhee(sphi) / cphi ) /
sqrt( ( (1 + es1) / es2m1a + Datanhee(1, sphi) ) *
( (1 - es1) / es2m1a + Datanhee(1, -sphi) ) );
}
Math::real AlbersEqualArea::tphif(real txi) const {
real
tphi = txi,
stol = tol_ * fmax(real(1), fabs(txi));
// CHECK: min iterations = 1, max iterations = 2; mean = 1.99
for (int i = 0;
i < numit_ ||
GEOGRAPHICLIB_PANIC("Convergence failure in AlbersEqualArea");
++i) {
// dtxi/dtphi = (scxi/scphi)^3 * 2*(1-e^2)/(qZ*(1-e^2*sphi^2)^2)
real
txia = txif(tphi),
tphi2 = Math::sq(tphi),
scphi2 = 1 + tphi2,
scterm = scphi2/(1 + Math::sq(txia)),
dtphi = (txi - txia) * scterm * sqrt(scterm) *
_qx * Math::sq(1 - _e2 * tphi2 / scphi2);
tphi += dtphi;
if (!(fabs(dtphi) >= stol))
break;
}
return tphi;
}
// return atanh(sqrt(x))/sqrt(x) - 1 = x/3 + x^2/5 + x^3/7 + ...
// typical x < e^2 = 2*f
Math::real AlbersEqualArea::atanhxm1(real x) {
real s = 0;
if (fabs(x) < real(0.5)) {
static const real lg2eps_ = -log2(numeric_limits<real>::epsilon() / 2);
int e;
(void) frexp(x, &e);
e = -e;
// x = [0.5,1) * 2^(-e)
// estimate n s.t. x^n/(2*n+1) < x/3 * epsilon/2
// a stronger condition is x^(n-1) < epsilon/2
// taking log2 of both sides, a stronger condition is
// (n-1)*(-e) < -lg2eps or (n-1)*e > lg2eps or n > ceiling(lg2eps/e)+1
int n = x == 0 ? 1 : int(ceil(lg2eps_ / e)) + 1;
while (n--) // iterating from n-1 down to 0
s = x * s + (n ? 1 : 0)/Math::real(2*n + 1);
} else {
real xs = sqrt(fabs(x));
s = (x > 0 ? atanh(xs) : atan(xs)) / xs - 1;
}
return s;
}
// return (Datanhee(1,y) - Datanhee(1,x))/(y-x)
Math::real AlbersEqualArea::DDatanhee(real x, real y) const {
// This function is called with x = sphi1, y = sphi2, phi1 <= phi2, sphi2
// >= 0, abs(sphi1) <= phi2. However for safety's sake we enforce x <= y.
if (y < x) swap(x, y); // ensure that x <= y
real q1 = fabs(_e2),
q2 = fabs(2 * _e / _e2m * (1 - x));
return
x <= 0 || !(fmin(q1, q2) < real(0.75)) ? DDatanhee0(x, y) :
(q1 < q2 ? DDatanhee1(x, y) : DDatanhee2(x, y));
}
// Rearrange difference so that 1 - x is in the denominator, then do a
// straight divided difference.
Math::real AlbersEqualArea::DDatanhee0(real x, real y) const {
return (Datanhee(1, y) - Datanhee(x, y))/(1 - x);
}
// The expansion for e2 small
Math::real AlbersEqualArea::DDatanhee1(real x, real y) const {
// The series in e2 is
// sum( c[l] * e2^l, l, 1, N)
// where
// c[l] = sum( x^i * y^j; i >= 0, j >= 0, i+j < 2*l) / (2*l + 1)
// = ( (x-y) - (1-y) * x^(2*l+1) + (1-x) * y^(2*l+1) ) /
// ( (2*l+1) * (x-y) * (1-y) * (1-x) )
// For x = y = 1, c[l] = l
//
// In the limit x,y -> 1,
//
// DDatanhee -> e2/(1-e2)^2 = sum(l * e2^l, l, 1, inf)
//
// Use if e2 is sufficiently small.
real s = 0;
real z = 1, k = 1, t = 0, c = 0, en = 1;
while (true) {
t = y * t + z; c += t; z *= x;
t = y * t + z; c += t; z *= x;
k += 2; en *= _e2;
// Here en[l] = e2^l, k[l] = 2*l + 1,
// c[l] = sum( x^i * y^j; i >= 0, j >= 0, i+j < 2*l) / (2*l + 1)
// Taylor expansion is
// s = sum( c[l] * e2^l, l, 1, N)
real ds = en * c / k;
s += ds;
if (!(fabs(ds) > fabs(s) * eps_/2))
break; // Iterate until the added term is sufficiently small
}
return s;
}
// The expansion for x (and y) close to 1
Math::real AlbersEqualArea::DDatanhee2(real x, real y) const {
// If x and y are both close to 1, expand in Taylor series in dx = 1-x and
// dy = 1-y:
//
// DDatanhee = sum(C_m * (dx^(m+1) - dy^(m+1)) / (dx - dy), m, 0, inf)
//
// where
//
// C_m = sum( (m+2)!! / (m+2-2*k)!! *
// ((m+1)/2)! / ((m+1)/2-k)! /
// (k! * (2*k-1)!!) *
// e2^((m+1)/2+k),
// k, 0, (m+1)/2) * (-1)^m / ((m+2) * (1-e2)^(m+2))
// for m odd, and
//
// C_m = sum( 2 * (m+1)!! / (m+1-2*k)!! *
// (m/2+1)! / (m/2-k)! /
// (k! * (2*k+1)!!) *
// e2^(m/2+1+k),
// k, 0, m/2)) * (-1)^m / ((m+2) * (1-e2)^(m+2))
// for m even.
//
// Here i!! is the double factorial extended to negative i with
// i!! = (i+2)!!/(i+2).
//
// Note that
// (dx^(m+1) - dy^(m+1)) / (dx - dy) =
// dx^m + dx^(m-1)*dy ... + dx*dy^(m-1) + dy^m
//
// Leading (m = 0) term is e2 / (1 - e2)^2
//
// Magnitude of mth term relative to the leading term scales as
//
// 2*(2*e/(1-e2)*dx)^m
//
// So use series if (2*e/(1-e2)*dx) is sufficiently small
real s, dx = 1 - x, dy = 1 - y, xy = 1, yy = 1, ee = _e2 / Math::sq(_e2m);
s = ee;
for (int m = 1; ; ++m) {
real c = m + 2, t = c;
yy *= dy; // yy = dy^m
xy = dx * xy + yy;
// Now xy = dx^m + dx^(m-1)*dy ... + dx*dy^(m-1) + dy^m
// = (dx^(m+1) - dy^(m+1)) / (dx - dy)
// max value = (m+1) * max(dx,dy)^m
ee /= -_e2m;
if (m % 2 == 0) ee *= _e2;
// Now ee = (-1)^m * e2^(floor(m/2)+1) / (1-e2)^(m+2)
int kmax = (m+1)/2;
for (int k = kmax - 1; k >= 0; --k) {
// max coeff is less than 2^(m+1)
c *= (k + 1) * (2 * (k + m - 2*kmax) + 3);
c /= (kmax - k) * (2 * (kmax - k) + 1);
// Horner sum for inner _e2 series
t = _e2 * t + c;
}
// Straight sum for outer m series
real ds = t * ee * xy / (m + 2);
s = s + ds;
if (!(fabs(ds) > fabs(s) * eps_/2))
break; // Iterate until the added term is sufficiently small
}
return s;
}
void AlbersEqualArea::Forward(real lon0, real lat, real lon,
real& x, real& y, real& gamma, real& k) const {
lon = Math::AngDiff(lon0, lon);
lat *= _sign;
real sphi, cphi;
Math::sincosd(Math::LatFix(lat) * _sign, sphi, cphi);
cphi = fmax(epsx_, cphi);
real
lam = lon * Math::degree(),
tphi = sphi/cphi, txi = txif(tphi), sxi = txi/hyp(txi),
dq = _qZ * Dsn(txi, _txi0, sxi, _sxi0) * (txi - _txi0),
drho = - _a * dq / (sqrt(_m02 - _n0 * dq) + _nrho0 / _a),
theta = _k2 * _n0 * lam, stheta = sin(theta), ctheta = cos(theta),
t = _nrho0 + _n0 * drho;
x = t * (_n0 != 0 ? stheta / _n0 : _k2 * lam) / _k0;
y = (_nrho0 *
(_n0 != 0 ?
(ctheta < 0 ? 1 - ctheta : Math::sq(stheta)/(1 + ctheta)) / _n0 :
0)
- drho * ctheta) / _k0;
k = _k0 * (t != 0 ? t * hyp(_fm * tphi) / _a : 1);
y *= _sign;
gamma = _sign * theta / Math::degree();
}
void AlbersEqualArea::Reverse(real lon0, real x, real y,
real& lat, real& lon,
real& gamma, real& k) const {
y *= _sign;
real
nx = _k0 * _n0 * x, ny = _k0 * _n0 * y, y1 = _nrho0 - ny,
den = hypot(nx, y1) + _nrho0, // 0 implies origin with polar aspect
drho = den != 0 ? (_k0*x*nx - 2*_k0*y*_nrho0 + _k0*y*ny) / den : 0,
// dsxia = scxi0 * dsxi
dsxia = - _scxi0 * (2 * _nrho0 + _n0 * drho) * drho /
(Math::sq(_a) * _qZ),
txi = (_txi0 + dsxia) / sqrt(fmax(1 - dsxia * (2*_txi0 + dsxia), epsx2_)),
tphi = tphif(txi),
theta = atan2(nx, y1),
lam = _n0 != 0 ? theta / (_k2 * _n0) : x / (y1 * _k0);
gamma = _sign * theta / Math::degree();
lat = Math::atand(_sign * tphi);
lon = lam / Math::degree();
lon = Math::AngNormalize(lon + Math::AngNormalize(lon0));
k = _k0 * (den != 0 ? (_nrho0 + _n0 * drho) * hyp(_fm * tphi) / _a : 1);
}
void AlbersEqualArea::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)");
real x, y, gamma, kold;
Forward(0, lat, 0, x, y, gamma, kold);
k /= kold;
_k0 *= k;
_k2 = Math::sq(_k0);
}
} // namespace GeographicLib
AlbersEqualArea.hpp
Header for GeographicLib::AlbersEqualArea class.
real
GeographicLib::Math::real real
Definition Geod3Solve.cpp:25
GEOGRAPHICLIB_PANIC
#define GEOGRAPHICLIB_PANIC(msg)
Definition Math.hpp:67
GeographicLib::AlbersEqualArea::Reverse
void Reverse(real lon0, real x, real y, real &lat, real &lon, real &gamma, real &k) const
Definition AlbersEqualArea.cpp:521
GeographicLib::AlbersEqualArea::AlbersEqualArea
AlbersEqualArea(real a, real f, real stdlat, real k0)
Definition AlbersEqualArea.cpp:16
GeographicLib::AlbersEqualArea::SetScale
void SetScale(real lat, real k=real(1))
Definition AlbersEqualArea.cpp:543
GeographicLib::AlbersEqualArea::CylindricalEqualArea
static const AlbersEqualArea & CylindricalEqualArea()
Definition AlbersEqualArea.cpp:284
GeographicLib::AlbersEqualArea::AzimuthalEqualAreaNorth
static const AlbersEqualArea & AzimuthalEqualAreaNorth()
Definition AlbersEqualArea.cpp:291
GeographicLib::AlbersEqualArea::AzimuthalEqualAreaSouth
static const AlbersEqualArea & AzimuthalEqualAreaSouth()
Definition AlbersEqualArea.cpp:298
GeographicLib::AlbersEqualArea::Forward
void Forward(real lon0, real lat, real lon, real &x, real &y, real &gamma, real &k) const
Definition AlbersEqualArea.cpp:496
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::GeographicErr
Exception handling for GeographicLib.
Definition Constants.hpp:344
GeographicLib::Math
Mathematical functions needed by GeographicLib.
Definition Math.hpp:82
GeographicLib::Math::degree
static T degree()
Definition Math.hpp:197
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::sq
static T sq(T x)
Definition Math.hpp:209
GeographicLib::Math::qd
static constexpr int qd
degrees per quarter turn
Definition Math.hpp:142
GeographicLib::Math::AngNormalize
static T AngNormalize(T x)
Definition Math.cpp:69
GeographicLib::Math::atand
static T atand(T x)
Definition Math.cpp:234
GeographicLib::Math::AngDiff
static T AngDiff(T x, T y, T &e)
Definition Math.cpp:80
GeographicLib::Math::real
double real
Definition Math.hpp:115
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