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DAuxLatitude.cpp
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/**
* \file DAuxLatitude.cpp
* \brief Implementation for the GeographicLib::DAuxLatitude class.
*
* This file is an implementation of the methods described in
* - C. F. F. Karney,
* <a href="https://doi.org/10.1080/00396265.2023.2217604">
* On auxiliary latitudes,</a>
* Survey Review 56(395), 165--180 (2024);
* preprint
* <a href="https://arxiv.org/abs/2212.05818">arXiv:2212.05818</a>.
* .
* Copyright (c) Charles Karney (2022-2024) <karney@alum.mit.edu> and licensed
* under the MIT/X11 License. For more information, see
* https://geographiclib.sourceforge.io/
**********************************************************************/
#include <GeographicLib/DAuxLatitude.hpp>
#include <GeographicLib/EllipticFunction.hpp>
namespace GeographicLib {
using namespace std;
Math::real DAuxLatitude::DRectifying(const AuxAngle& phi1,
const AuxAngle& phi2)
const {
// Stipulate that phi1 and phi2 are in [-90d, 90d]
real x = phi1.radians(), y = phi2.radians();
if (x == y) {
real d;
AuxAngle mu1(base::Rectifying(phi1, &d));
real tphi1 = phi1.tan(), tmu1 = mu1.tan();
return
isfinite(tphi1) ? d * Math::sq(base::sc(tphi1)/base::sc(tmu1)) : 1/d;
} else if (x * y < 0)
return (base::Rectifying(phi2).radians() -
base::Rectifying(phi1).radians()) / (y - x);
else {
AuxAngle bet1(base::Parametric(phi1)), bet2(base::Parametric(phi2));
real dEdbet = DE(bet1, bet2), dbetdphi = DParametric(phi1, phi2);
return base::_b * dEdbet / base::RectifyingRadius(true) * dbetdphi;
}
}
Math::real DAuxLatitude::DParametric(const AuxAngle& phi1,
const AuxAngle& phi2)
const {
real tx = phi1.tan(), ty = phi2.tan(), r;
// DbetaDphi = Datan(fm1*tx, fm1*ty) * fm1 / Datan(tx, ty)
// Datan(x, y) = 1/(1 + x^2), for x = y
// = (atan(y) - atan(x)) / (y-x), for x*y < 0
// = atan( (y-x) / (1 + x*y) ) / (y-x), for x*y > 0
if (!(tx * ty >= 0)) // This includes, e.g., tx = 0, ty = inf
r = (atan(base::_fm1 * ty) - atan(base::_fm1 * tx)) /
(atan(ty) - atan(tx));
else if (tx == ty) { // This includes the case tx = ty = inf
tx *= tx;
if (tx <= 1)
r = base::_fm1 * (1 + tx) / (1 + base::_e2m1 * tx);
else {
tx = 1/tx;
r = base::_fm1 * (1 + tx) / (base::_e2m1 + tx);
}
} else {
if (tx * ty <= 1)
r = atan2(base::_fm1 * (ty - tx), 1 + base::_e2m1 * tx * ty)
/ atan2( ty - tx , 1 + tx * ty);
else {
tx = 1/tx; ty = 1/ty;
r = atan2(base::_fm1 * (ty - tx), base::_e2m1 + tx * ty)
/ atan2( ty - tx , 1 + tx * ty);
}
}
return r;
}
Math::real DAuxLatitude::DE(const AuxAngle& X, const AuxAngle& Y) const {
AuxAngle Xn(X.normalized()), Yn(Y.normalized());
// We assume that X and Y are in [-90d, 90d] and have the same sign
// If not we would include
// if (Xn.y() * Yn.y() < 0)
// return d != 0 ? (E(X) - E(Y)) / d : 1;
// The general formula fails for x = y = 0d and x = y = 90d. Probably this
// is fixable (the formula works for other x = y. But let's also stipulate
// that x != y .
// Make both positive, so we can do the swap a <-> b trick
Xn.y() = fabs(Xn.y()); Yn.y() = fabs(Yn.y());
real k2 = -base::_e12;
bool flip = base::_f < 0;
// Switch prolate to oblate; we then can use the formulas for k2 < 0
if (flip) {
swap(Xn.x(), Xn.y());
swap(Yn.x(), Yn.y());
k2 = base::_e2;
}
real x = Xn.radians(), y = Yn.radians(), d = y - x,
sx = Xn.y(), sy = Yn.y(), cx = Xn.x(), cy = Yn.x();
// See DLMF: Eqs (19.11.2) and (19.11.4) letting
// theta -> x, phi -> -y, psi -> z
//
// (E(y) - E(x)) / d = E(z)/d - k2 * sin(x) * sin(y) * sin(z)/d
// = (E(z)/sin(z) - k2 * sin(x) * sin(y)) * sin(z)/d
// tan(z/2) = (sin(x)*Delta(y) - sin(y)*Delta(x)) / (cos(x) + cos(y))
// = d * Dsin(x,y) * (sin(x) + sin(y))/(cos(x) + cos(y)) /
// (sin(x)*Delta(y) + sin(y)*Delta(x))
// = t = d * Dt
// Delta(x) = sqrt(1 - k2 * sin(x)^2)
// sin(z) = 2*t/(1+t^2); cos(z) = (1-t^2)/(1+t^2)
real Dt = Dsin(x, y) * (sx + sy) /
((cx + cy) * (sx * sqrt(1 - k2 * sy*sy) + sy * sqrt(1 - k2 * sx*sx))),
t = d * Dt, Dsz = 2 * Dt / (1 + t*t),
sz = d * Dsz, cz = (1 - t) * (1 + t) / (1 + t*t),
sz2 = sz*sz, cz2 = cz*cz, dz2 = 1 - k2 * sz2,
// E(z)/sin(z)
Ezbsz = (EllipticFunction::RF(cz2, dz2, 1)
- k2 * sz2 * EllipticFunction::RD(cz2, dz2, 1) / 3);
return (Ezbsz - k2 * sx * sy) * Dsz / (flip ? 1 - base::_f : 1);
}
/// \cond SKIP
Math::real DAuxLatitude::Dsn(real x, real y) {
real sc1 = base::sc(x);
if (x == y) return 1 / (sc1 * (1 + x*x));
real sc2 = base::sc(y), sn1 = base::sn(x), sn2 = base::sn(y);
return x * y > 0 ?
(sn1/sc2 + sn2/sc1) / ((sn1 + sn2) * sc1 * sc2) :
(sn2 - sn1) / (y - x);
}
Math::real DAuxLatitude::Datan(real x, real y) {
real d = y - x, xy = x*y;
return x == y ? 1 / (1 + xy) :
(isinf(xy) && xy > 0 ? 0 :
(2 * xy > -1 ? atan( d / (1 + xy) ) : atan(y) - atan(x)) / d);
}
Math::real DAuxLatitude::Dasinh(real x, real y) {
real d = y - x, xy = x*y, hx = base::sc(x), hy = base::sc(y);
// KF formula for x*y < 0 is asinh(y*hx - x*hy) / (y - x)
// but this has problem if x*y overflows to -inf
return x == y ? 1 / hx :
(isinf(d) ? 0 :
(xy > 0 ? asinh(d * (x*y < 1 ? (x + y) / (x*hy + y*hx) :
(1/x + 1/y) / (hy/y + hx/x))) :
asinh(y) - asinh(x)) / d);
}
Math::real DAuxLatitude::Dh(real x, real y) {
if (isnan(x + y))
return x + y; // N.B. nan for inf-inf
if (isinf(x))
return copysign(1/real(2), x);
if (isinf(y))
return copysign(1/real(2), y);
real sx = base::sn(x), sy = base::sn(y), d = sx*x + sy*y;
if (d / 2 == 0)
return (x + y) / 2; // Handle underflow
if (x * y <= 0)
return (h(y) - h(x)) / (y - x); // Does not include x = y = 0
real scx = base::sc(x), scy = base::sc(y);
return ((x + y) / (2 * d)) *
(Math::sq(sx*sy) + Math::sq(sy/scx) + Math::sq(sx/scy));
}
Math::real DAuxLatitude::Datanhee(real x, real y) const {
// atan(e*sn(tphi))/e:
// Datan(e*sn(x),e*sn(y))*Dsn(x,y)/Datan(x,y)
// asinh(e1*sn(fm1*tphi)):
// Dasinh(e1*sn(fm1*x)), e1*sn(fm1*y)) *
// e1 * Dsn(fm1*x, fm1*y) *fm1 / (e * Datan(x,y))
// = Dasinh(e1*sn(fm1*x)), e1*sn(fm1*y)) *
// Dsn(fm1*x, fm1*y) / Datan(x,y)
return base::_f < 0 ?
Datan(base::_e * base::sn(x), base::_e * base::sn(y)) * Dsn(x, y) :
Dasinh(base::_e1 * base::sn(base::_fm1 * x),
base::_e1 * base::sn(base::_fm1 * y)) *
Dsn(base::_fm1 * x, base::_fm1 * y);
}
/// \endcond
Math::real DAuxLatitude::DIsometric(const AuxAngle& phi1,
const AuxAngle& phi2)
const {
// psi = asinh(tan(phi)) - e^2 * atanhee(tan(phi))
real tphi1 = phi1.tan(), tphi2 = phi2.tan();
return isnan(tphi1) || isnan(tphi2) ? numeric_limits<real>::quiet_NaN() :
(isinf(tphi1) || isinf(tphi2) ? numeric_limits<real>::infinity() :
(Dasinh(tphi1, tphi2) - base::_e2 * Datanhee(tphi1, tphi2)) /
Datan(tphi1, tphi2));
}
Math::real DAuxLatitude::DConvert(int auxin, int auxout,
const AuxAngle& zeta1,
const AuxAngle& zeta2)
const {
int k = base::ind(auxout, auxin);
if (k < 0) return numeric_limits<real>::quiet_NaN();
if (auxin == auxout) return 1;
if ( isnan(base::_c[base::Lmax * (k + 1) - 1]) )
base::fillcoeff(auxin, auxout, k);
AuxAngle zeta1n(zeta1.normalized()), zeta2n(zeta2.normalized());
return 1 + DClenshaw(true, zeta2n.radians() - zeta1n.radians(),
zeta1n.y(), zeta1n.x(), zeta2n.y(), zeta2n.x(),
base::_c + base::Lmax * k, base::Lmax);
}
Math::real DAuxLatitude::DClenshaw(bool sinp, real Delta,
real szeta1, real czeta1,
real szeta2, real czeta2,
const real c[], int K) {
// Evaluate
// (Clenshaw(sinp, szeta2, czeta2, c, K) -
// Clenshaw(sinp, szeta1, czeta1, c, K)) / Delta
// or
// sum(c[k] * (sin( (2*k+2) * zeta2) - sin( (2*k+2) * zeta2)), i, 0, K-1)
// / Delta
// (if !sinp, then change sin->cos here.)
//
// Delta is EITHER 1, giving the plain difference OR (zeta2 - zeta1) in
// radians, giving the divided difference. Other values will give
// nonsense.
//
int k = K;
// suffixes a b denote [1,1], [2,1] elements of matrix/vector
real D2 = Delta * Delta,
czetap = czeta2 * czeta1 - szeta2 * szeta1,
szetap = szeta2 * czeta1 + czeta2 * szeta1,
czetam = czeta2 * czeta1 + szeta2 * szeta1,
// sin(zetam) / Delta
szetamd = (Delta == 1 ? szeta2 * czeta1 - czeta2 * szeta1 :
(Delta != 0 ? sin(Delta) / Delta : 1)),
Xa = 2 * czetap * czetam,
Xb = -2 * szetap * szetamd,
u0a = 0, u0b = 0, u1a = 0, u1b = 0; // accumulators for sum
for (--k; k >= 0; --k) {
// temporary real = X . U0 - U1 + c[k] * I
real ta = Xa * u0a + D2 * Xb * u0b - u1a + c[k],
tb = Xb * u0a + Xa * u0b - u1b;
// U1 = U0; U0 = real
u1a = u0a; u0a = ta;
u1b = u0b; u0b = tb;
}
// P = U0 . F[0] - U1 . F[-1]
// if sinp:
// F[0] = [ sin(2*zeta2) + sin(2*zeta1),
// (sin(2*zeta2) - sin(2*zeta1)) / Delta]
// = 2 * [ szetap * czetam, czetap * szetamd ]
// F[-1] = [0, 0]
// else:
// F[0] = [ cos(2*zeta2) + cos(2*zeta1),
// (cos(2*zeta2) - cos(2*zeta1)) / Delta]
// = 2 * [ czetap * czetam, -szetap * szetamd ]
// F[-1] = [2, 0]
real F0a = (sinp ? szetap : czetap) * czetam,
F0b = (sinp ? czetap : -szetap) * szetamd,
Fm1a = sinp ? 0 : 1; // Fm1b = 0;
// Don't both to compute sum...
// divided difference (or difference if Delta == 1)
return 2 * (F0a * u0b + F0b * u0a - Fm1a * u1b);
}
} // namespace GeographicLib
DAuxLatitude.hpp
Header for the GeographicLib::DAuxLatitude class.
EllipticFunction.hpp
Header for GeographicLib::EllipticFunction class.
real
GeographicLib::Math::real real
Definition Geod3Solve.cpp:25
GeographicLib::AuxAngle
An accurate representation of angles.
Definition AuxAngle.hpp:51
GeographicLib::AuxAngle::y
Math::real y() const
Definition AuxAngle.hpp:74
GeographicLib::AuxAngle::x
Math::real x() const
Definition AuxAngle.hpp:79
GeographicLib::AuxAngle::radians
Math::real radians() const
Definition AuxAngle.hpp:232
GeographicLib::AuxAngle::normalized
AuxAngle normalized() const
Definition AuxAngle.cpp:28
GeographicLib::AuxAngle::tan
Math::real tan() const
Definition AuxAngle.hpp:117
GeographicLib::AuxLatitude::Lmax
static const int Lmax
Definition AuxLatitude.hpp:316
GeographicLib::AuxLatitude::RectifyingRadius
Math::real RectifyingRadius(bool exact=false) const
Definition AuxLatitude.cpp:321
GeographicLib::AuxLatitude::Parametric
AuxAngle Parametric(const AuxAngle &phi, real *diff=nullptr) const
Definition AuxLatitude.cpp:84
GeographicLib::AuxLatitude::Rectifying
AuxAngle Rectifying(const AuxAngle &phi, real *diff=nullptr) const
Definition AuxLatitude.cpp:94
GeographicLib::DAuxLatitude::DParametric
Math::real DParametric(const AuxAngle &phi1, const AuxAngle &phi2) const
Definition DAuxLatitude.cpp:46
GeographicLib::DAuxLatitude::DConvert
Math::real DConvert(int auxin, int auxout, const AuxAngle &zeta1, const AuxAngle &zeta2) const
Definition DAuxLatitude.cpp:191
GeographicLib::DAuxLatitude::DIsometric
Math::real DIsometric(const AuxAngle &phi1, const AuxAngle &phi2) const
Definition DAuxLatitude.cpp:180
GeographicLib::DAuxLatitude::DClenshaw
static Math::real DClenshaw(bool sinp, real Delta, real szeta1, real czeta1, real szeta2, real czeta2, const real c[], int K)
Definition DAuxLatitude.cpp:206
GeographicLib::DAuxLatitude::DRectifying
Math::real DRectifying(const AuxAngle &phi1, const AuxAngle &phi2) const
Definition DAuxLatitude.cpp:25
GeographicLib::EllipticFunction::RD
static real RD(real x, real y, real z)
Definition EllipticFunction.cpp:177
GeographicLib::EllipticFunction::RF
static real RF(real x, real y, real z)
Definition EllipticFunction.cpp:24
GeographicLib::Math::sq
static T sq(T x)
Definition Math.hpp:209
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