/**
* Reverse projection, from Lambert conformal conic to geographic.
*
* @param[in] lon0 central meridian longitude (degrees).
* @param[in] x easting of point (meters).
* @param[in] y northing of point (meters).
* @param[out] lat latitude of point (degrees).
* @param[out] lon longitude of point (degrees).
* @param[out] gamma meridian convergence at point (degrees).
* @param[out] k azimuthal scale of projection at point; the radial
* scale is the 1/\e k.
*
* The latitude origin is given by AlbersEqualArea::LatitudeOrigin(). No
* false easting or northing is added. The value of \e lon returned is in
* the range [−180°, 180°]. The value of \e lat returned is
* in the range [−90°, 90°]. If the input point is outside
* the legal projected space the nearest pole is returned.
**********************************************************************/
public void Reverse(double lon0, double x, double y,
out double lat, out double lon, out double gamma, out double k)
{
y *= _sign;
double
nx = _k0 * _n0 * x, ny = _k0 * _n0 * y, y1 = _nrho0 - ny,
den = GeoMath.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 /
(GeoMath.Square(_a) * _qZ),
txi = (_txi0 + dsxia) / Math.Sqrt(Math.Max(1 - dsxia * (2 * _txi0 + dsxia), epsx2_)),
tphi = tphif(txi),
theta = Math.Atan2(nx, y1),
lam = _n0 != 0 ? theta / (_k2 * _n0) : x / (y1 * _k0);
gamma = _sign * theta / GeoMath.Degree;
lat = GeoMath.Atand(_sign * tphi);
lon = lam / GeoMath.Degree;
lon = GeoMath.AngNormalize(lon + GeoMath.AngNormalize(lon0));
k = _k0 * (den != 0 ? (_nrho0 + _n0 * drho) * hyp(_fm * tphi) / _a : 1);
}
/**
* Reverse projection, from transverse Mercator to geographic.
*
* @param[in] lon0 central meridian of the projection (degrees).
* @param[in] x easting of point (meters).
* @param[in] y northing of point (meters).
* @param[out] lat latitude of point (degrees).
* @param[out] lon longitude of point (degrees).
* @param[out] gamma meridian convergence at point (degrees).
* @param[out] k scale of projection at point.
*
* No false easting or northing is added. The value of \e lon returned is
* in the range [−180°, 180°].
**********************************************************************/
public void Reverse(double lon0, double x, double y,
out double lat, out double lon, out double gamma, out double k)
{
// This undoes the steps in Forward. The wrinkles are: (1) Use of the
// reverted series to express zeta' in terms of zeta. (2) Newton's method
// to solve for phi in terms of tan(phi).
double
xi = y / (_a1 * _k0),
eta = x / (_a1 * _k0);
// Explicitly enforce the parity
int
xisign = (xi < 0) ? -1 : 1,
etasign = (eta < 0) ? -1 : 1;
xi *= xisign;
eta *= etasign;
bool backside = xi > Math.PI / 2;
if (backside)
{
xi = Math.PI - xi;
}
double
c0 = Math.Cos(2 * xi), ch0 = Math.Cosh(2 * eta),
s0 = Math.Sin(2 * xi), sh0 = Math.Sinh(2 * eta);
int n = maxpow_;
Complex a = new Complex(2 * c0 * ch0, -2 * s0 * sh0); // 2 * Math.Cos(2*zeta')
Complex y0 = new Complex((n & 1) != 0 ? -_bet[n] : 0, 0);
Complex y1; // default initializer is 0+i0
Complex z0 = new Complex((n & 1) != 0 ? -2 * n * _bet[n] : 0, 0);
Complex z1;
if ((n & 1) != 0)
{
--n;
}
while (n > 0)
{
y1 = a * y0 - y1 - _bet[n];
z1 = a * z0 - z1 - 2 * n * _bet[n];
--n;
y0 = a * y1 - y0 - _bet[n];
z0 = a * z1 - z0 - 2 * n * _bet[n];
--n;
}
a /= 2; // Math.Cos(2*zeta)
z1 = 1 - z1 + a * z0;
a = new Complex(s0 * ch0, c0 * sh0); // Math.Sin(2*zeta)
y1 = new Complex(xi, eta) + a * y0;
// Convergence and scale for Gauss-Schreiber TM to Gauss-Krueger TM.
gamma = GeoMath.Atan2d(z1.Imaginary, z1.Real);
k = _b1 / Complex.Abs(z1);
// JHS 154 has
//
// phi' = asin(Math.Sin(xi') / Math.Cosh(eta')) (Krueger p 17 (25))
// lam = asin(tanh(eta') / Math.Cos(phi')
// psi = asinh(tan(phi'))
double
xip = y1.Real, etap = y1.Imaginary,
s = Math.Sinh(etap),
c = Math.Max(0, Math.Cos(xip)), // Math.Cos(pi/2) might be negative
r = GeoMath.Hypot(s, c);
if (r != 0)
{
lon = GeoMath.Atan2d(s, c); // Krueger p 17 (25)
// Use Newton's method to solve for tau
double
sxip = Math.Sin(xip),
tau = GeoMath.Tauf(sxip / r, _es);
gamma += GeoMath.Atan2d(sxip * Math.Tanh(etap), c); // Krueger p 19 (31)
lat = GeoMath.Atand(tau);
// Note Math.Cos(phi') * Math.Cosh(eta') = r
k *= Math.Sqrt(_e2m + _e2 / (1 + GeoMath.Square(tau))) *
GeoMath.Hypot(1, tau) * r;
}
else
{
lat = 90;
lon = 0;
k *= _c;
}
lat *= xisign;
if (backside)
{
lon = 180 - lon;
}
lon *= etasign;
lon = GeoMath.AngNormalize(lon + lon0);
if (backside)
{
gamma = 180 - gamma;
}
gamma *= xisign * etasign;
gamma = GeoMath.AngNormalize(gamma);
k *= _k0;
}
