public GeodesicLine(Geodesic g,
double lat1, double lon1, double azi1,
int caps)
{
azi1 = GeoMath.AngNormalize(azi1);
double salp1, calp1;
// Guard against underflow in salp0. Also -0 is converted to +0.
GeoMath.Sincosd(GeoMath.AngRound(azi1), out salp1, out calp1);
LineInit(g, lat1, lon1, azi1, salp1, calp1, caps); //TODO: Check this cast
}
/**
* Reset the origin.
*
* @param[in] lat0 latitude at origin (degrees).
* @param[in] lon0 longitude at origin (degrees).
* @param[in] h0 height above ellipsoid at origin (meters); default 0.
*
* \e lat0 should be in the range [−90°, 90°].
**********************************************************************/
public void Reset(double lat0, double lon0, double h0 = 0)
{
_lat0 = GeoMath.LatFix(lat0);
_lon0 = GeoMath.AngNormalize(lon0);
_h0 = h0;
_earth.Forward(_lat0, _lon0, _h0, out _x0, out _y0, out _z0);
double sphi, cphi, slam, clam;
GeoMath.Sincosd(_lat0, out sphi, out cphi);
GeoMath.Sincosd(_lon0, out slam, out clam);
Geocentric.Rotation(sphi, cphi, slam, clam, _r);
}
/**
* 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);
}
///@}
/** \name The general position function.
**********************************************************************/
///@{
/**
* The general position function. GeodesicLine::Position and
* GeodesicLine::ArcPosition are defined in terms of this function.
*
* @param[in] arcmode boolean flag determining the meaning of the second
* parameter; if \e arcmode is false, then the GeodesicLine object must
* have been constructed with \e caps |= GeodesicLine::DISTANCE_IN.
* @param[in] s12_a12 if \e arcmode is false, this is the distance between
* point 1 and point 2 (meters); otherwise it is the arc length between
* point 1 and point 2 (degrees); it can be negative.
* @param[in] outmask a bitor'ed combination of GeodesicLine::mask values
* specifying which of the following parameters should be set.
* @param[out] lat2 GeodesicMask.LATITUDE of point 2 (degrees).
* @param[out] lon2 GeodesicMask.LONGITUDE of point 2 (degrees); requires that the
* GeodesicLine object was constructed with \e caps |=
* GeodesicLine::GeodesicMask.LONGITUDE.
* @param[out] azi2 (forward) GeodesicMask.AZIMUTH at point 2 (degrees).
* @param[out] s12 distance from point 1 to point 2 (meters); requires
* that the GeodesicLine object was constructed with \e caps |=
* GeodesicLine::DISTANCE.
* @param[out] m12 reduced length of geodesic (meters); requires that the
* GeodesicLine object was constructed with \e caps |=
* GeodesicLine::GeodesicMask.REDUCEDLENGTH.
* @param[out] M12 geodesic scale of point 2 relative to point 1
* (dimensionless); requires that the GeodesicLine object was constructed
* with \e caps |= GeodesicLine::GeodesicMask.GEODESICSCALE.
* @param[out] M21 geodesic scale of point 1 relative to point 2
* (dimensionless); requires that the GeodesicLine object was constructed
* with \e caps |= GeodesicLine::GeodesicMask.GEODESICSCALE.
* @param[out] S12 GeodesicMask.AREA under the geodesic (meters<sup>2</sup>); requires
* that the GeodesicLine object was constructed with \e caps |=
* GeodesicLine::GeodesicMask.AREA.
* @return \e a12 arc length from point 1 to point 2 (degrees).
*
* The GeodesicLine::mask values possible for \e outmask are
* - \e outmask |= GeodesicLine::GeodesicMask.LATITUDE for the GeodesicMask.LATITUDE \e lat2;
* - \e outmask |= GeodesicLine::GeodesicMask.LONGITUDE for the GeodesicMask.LATITUDE \e lon2;
* - \e outmask |= GeodesicLine::GeodesicMask.AZIMUTH for the GeodesicMask.LATITUDE \e azi2;
* - \e outmask |= GeodesicLine::DISTANCE for the distance \e s12;
* - \e outmask |= GeodesicLine::GeodesicMask.REDUCEDLENGTH for the reduced length \e
* m12;
* - \e outmask |= GeodesicLine::GeodesicMask.GEODESICSCALE for the geodesic scales \e
* M12 and \e M21;
* - \e outmask |= GeodesicLine::GeodesicMask.AREA for the GeodesicMask.AREA \e S12;
* - \e outmask |= GeodesicLine::ALL for all of the above;
* - \e outmask |= GeodesicLine::LONG_UNROLL to unroll \e lon2 instead of
* reducing it into the range [−180°, 180°].
* .
* Requesting a value which the GeodesicLine object is not capable of
* computing is not an error; the corresponding argument will not be
* altered. Note, however, that the arc length is always computed and
* returned as the function value.
*
* With the GeodesicLine::LONG_UNROLL bit seout t, the quantity \e lon2 −
* \e lon1 indicates how many times and in what sense the geodesic
* encircles the ellipsoid.
**********************************************************************/
public double GenPosition(bool arcmode, double s12_a12, int outmask,
out double lat2, out double lon2, out double azi2,
out double s12, out double m12, out double M12, out double M21,
out double S12)
{
outmask &= (_caps & GeodesicMask.OUT_MASK);
lat2 = double.NaN;
lon2 = double.NaN;
azi2 = double.NaN;
s12 = double.NaN;
m12 = double.NaN;
M12 = double.NaN;
M21 = double.NaN;
S12 = double.NaN;
if (!(Init() && (arcmode || (_caps & (GeodesicMask.OUT_MASK & GeodesicMask.DISTANCE_IN)) != 0)))
{
// Uninitialized or impossible distance calculation requested
return(double.NaN);
}
// Avoid warning about uninitialized B12.
double sig12, ssig12, csig12, B12 = 0, AB1 = 0;
if (arcmode)
{
// Interpret s12_a12 as spherical arc length
sig12 = s12_a12 * GeoMath.Degree;
GeoMath.Sincosd(s12_a12, out ssig12, out csig12);
}
else
{
// Interpret s12_a12 as distance
double
tau12 = s12_a12 / (_b * (1 + _A1m1)),
s = Math.Sin(tau12),
c = Math.Cos(tau12);
// tau2 = tau1 + tau12
B12 = -Geodesic.SinCosSeries(true,
_stau1 * c + _ctau1 * s,
_ctau1 * c - _stau1 * s,
_C1pa);
sig12 = tau12 - (B12 - _B11);
ssig12 = Math.Sin(sig12); csig12 = Math.Cos(sig12);
if (Math.Abs(_f) > 0.01)
{
// Reverted distance series is inaccurate for |f| > 1/100, so correct
// sig12 with 1 Newton iteration. The following table shows the
// approximate maximum error for a = WGS_a() and various f relative to
// GeodesicExact.
// erri = the error in the inverse solution (nm)
// errd = the error in the direct solution (series only) (nm)
// errda = the error in the direct solution
// (series + 1 Newton) (nm)
//
// f erri errd errda
// -1/5 12e6 1.2e9 69e6
// -1/10 123e3 12e6 765e3
// -1/20 1110 108e3 7155
// -1/50 18.63 200.9 27.12
// -1/100 18.63 23.78 23.37
// -1/150 18.63 21.05 20.26
// 1/150 22.35 24.73 25.83
// 1/100 22.35 25.03 25.31
// 1/50 29.80 231.9 30.44
// 1/20 5376 146e3 10e3
// 1/10 829e3 22e6 1.5e6
// 1/5 157e6 3.8e9 280e6
double
sssig2 = _ssig1 * csig12 + _csig1 * ssig12,
scsig2 = _csig1 * csig12 - _ssig1 * ssig12;
B12 = Geodesic.SinCosSeries(true, sssig2, scsig2, _C1a);
double serr = (1 + _A1m1) * (sig12 + (B12 - _B11)) - s12_a12 / _b;
sig12 = sig12 - serr / Math.Sqrt(1 + _k2 * GeoMath.Square(sssig2));
ssig12 = Math.Sin(sig12); csig12 = Math.Cos(sig12);
// Update B12 below
}
}
double ssig2, csig2, sbet2, cbet2, salp2, calp2;
// sig2 = sig1 + sig12
ssig2 = _ssig1 * csig12 + _csig1 * ssig12;
csig2 = _csig1 * csig12 - _ssig1 * ssig12;
double dn2 = Math.Sqrt(1 + _k2 * GeoMath.Square(ssig2));
if ((outmask & (GeodesicMask.DISTANCE | GeodesicMask.REDUCEDLENGTH | GeodesicMask.GEODESICSCALE)) != 0)
{
if (arcmode || Math.Abs(_f) > 0.01)
{
B12 = Geodesic.SinCosSeries(true, ssig2, csig2, _C1a);
}
AB1 = (1 + _A1m1) * (B12 - _B11);
}
// sin(bet2) = cos(alp0) * sin(sig2)
sbet2 = _calp0 * ssig2;
// Alt: cbet2 = hypot(csig2, salp0 * ssig2);
cbet2 = GeoMath.Hypot(_salp0, _calp0 * csig2);
if (cbet2 == 0)
{
// I.e., salp0 = 0, csig2 = 0. Break the degeneracy in this case
cbet2 = csig2 = Geodesic.tiny_;
}
// tan(alp0) = cos(sig2)*tan(alp2)
salp2 = _salp0; calp2 = _calp0 * csig2; // No need to normalize
if ((outmask & GeodesicMask.DISTANCE) != 0)
{
s12 = arcmode ? _b * ((1 + _A1m1) * sig12 + AB1) : s12_a12;
}
if ((outmask & GeodesicMask.LONGITUDE) != 0)
{
// tan(omg2) = sin(alp0) * tan(sig2)
double somg2 = _salp0 * ssig2, comg2 = csig2, // No need to normalize
E = GeoMath.CopySign(1, _salp0); // east-going?
// omg12 = omg2 - omg1
double omg12 = (outmask & GeodesicMask.LONG_UNROLL) != 0
? E * (sig12
- (Math.Atan2(ssig2, csig2) - Math.Atan2(_ssig1, _csig1))
+ (Math.Atan2(E * somg2, comg2) - Math.Atan2(E * _somg1, _comg1)))
: Math.Atan2(somg2 * _comg1 - comg2 * _somg1,
comg2 * _comg1 + somg2 * _somg1);
double lam12 = omg12 + _A3c *
(sig12 + (Geodesic.SinCosSeries(true, ssig2, csig2, _C3a)
- _B31));
double lon12 = lam12 / GeoMath.Degree;
lon2 = (outmask & GeodesicMask.LONG_UNROLL) != 0 ? _lon1 + lon12 :
GeoMath.AngNormalize(GeoMath.AngNormalize(_lon1) +
GeoMath.AngNormalize(lon12));
}
if ((outmask & GeodesicMask.LATITUDE) != 0)
{
lat2 = GeoMath.Atan2d(sbet2, _f1 * cbet2);
}
if ((outmask & GeodesicMask.AZIMUTH) != 0)
{
azi2 = GeoMath.Atan2d(salp2, calp2);
}
if ((outmask & (GeodesicMask.REDUCEDLENGTH | GeodesicMask.GEODESICSCALE)) != 0)
{
double
B22 = Geodesic.SinCosSeries(true, ssig2, csig2, _C2a),
AB2 = (1 + _A2m1) * (B22 - _B21),
J12 = (_A1m1 - _A2m1) * sig12 + (AB1 - AB2);
if ((outmask & GeodesicMask.REDUCEDLENGTH) != 0)
{
// Add parens around (_csig1 * ssig2) and (_ssig1 * csig2) to ensure
// accurate cancellation in the case of coincident points.
m12 = _b * ((dn2 * (_csig1 * ssig2) - _dn1 * (_ssig1 * csig2))
- _csig1 * csig2 * J12);
}
if ((outmask & GeodesicMask.GEODESICSCALE) != 0)
{
double t = _k2 * (ssig2 - _ssig1) * (ssig2 + _ssig1) / (_dn1 + dn2);
M12 = csig12 + (t * ssig2 - csig2 * J12) * _ssig1 / _dn1;
M21 = csig12 - (t * _ssig1 - _csig1 * J12) * ssig2 / dn2;
}
}
if ((outmask & GeodesicMask.AREA) != 0)
{
double
B42 = Geodesic.SinCosSeries(false, ssig2, csig2, _C4a);
double salp12, calp12;
if (_calp0 == 0 || _salp0 == 0)
{
// alp12 = alp2 - alp1, used in Math.Atan2 so no need to normalize
salp12 = salp2 * _calp1 - calp2 * _salp1;
calp12 = calp2 * _calp1 + salp2 * _salp1;
// We used to include here some patch up code that purported to deal
// with nearly meridional geodesics properly. However, this turned out
// to be wrong once _salp1 = -0 was allowed (via
// Geodesic::InverseLine). In fact, the calculation of {s,c}alp12
// was already correct (following the IEEE rules for handling signed
// zeros). So the patch up code was unnecessary (as well as
// dangerous).
}
else
{
// tan(alp) = tan(alp0) * sec(sig)
// tan(alp2-alp1) = (tan(alp2) -tan(alp1)) / (tan(alp2)*tan(alp1)+1)
// = calp0 * salp0 * (csig1-csig2) / (salp0^2 + calp0^2 * csig1*csig2)
// If csig12 > 0, write
// csig1 - csig2 = ssig12 * (csig1 * ssig12 / (1 + csig12) + ssig1)
// else
// csig1 - csig2 = csig1 * (1 - csig12) + ssig12 * ssig1
// No need to normalize
salp12 = _calp0 * _salp0 *
(csig12 <= 0 ? _csig1 * (1 - csig12) + ssig12 * _ssig1 :
ssig12 * (_csig1 * ssig12 / (1 + csig12) + _ssig1));
calp12 = GeoMath.Square(_salp0) + GeoMath.Square(_calp0) * _csig1 * csig2;
}
S12 = _c2 * Math.Atan2(salp12, calp12) + _A4 * (B42 - _B41);
}
return(arcmode ? s12_a12 : sig12 / GeoMath.Degree);
}
/**
* Forward projection, from geographic to Cassini-Soldner.
*
* @param[in] lat latitude of point (degrees).
* @param[in] lon longitude of point (degrees).
* @param[out] x easting of point (meters).
* @param[out] y northing of point (meters).
* @param[out] azi azimuth of easting direction at point (degrees).
* @param[out] rk reciprocal of azimuthal northing scale at point.
*
* \e lat should be in the range [−90°, 90°]. A call to
* Forward followed by a call to Reverse will return the original (\e lat,
* \e lon) (to within roundoff). The routine does nothing if the origin
* has not been set.
**********************************************************************/
public void Forward(double lat, double lon,
out double x, out double y, out double azi, out double rk)
{
if (!Init())
{
x = double.NaN;
y = double.NaN;
azi = double.NaN;
rk = double.NaN;
}
double e;
double dlon = GeoMath.AngDiff(LongitudeOrigin(), lon, out e);
double sig12, s12, azi1, azi2;
sig12 = _earth.Inverse(lat, -Math.Abs(dlon), lat, Math.Abs(dlon), out s12, out azi1, out azi2);
sig12 *= 0.5;
s12 *= 0.5;
if (s12 == 0)
{
double da = GeoMath.AngDiff(azi1, azi2, out e) / 2;
if (Math.Abs(dlon) <= 90)
{
azi1 = 90 - da;
azi2 = 90 + da;
}
else
{
azi1 = -90 - da;
azi2 = -90 + da;
}
}
if (dlon < 0)
{
azi2 = azi1;
s12 = -s12;
sig12 = -sig12;
}
x = s12;
azi = GeoMath.AngNormalize(azi2);
GeodesicLine perp = _earth.Line(lat, dlon, azi, GeodesicMask.GEODESICSCALE);
double t;
perp.GenPosition(true, -sig12,
GeodesicMask.GEODESICSCALE,
out t, out t, out t, out t, out t, out t, out rk, out t);
double salp0, calp0;
GeoMath.Sincosd(perp.EquatorialAzimuth(), out salp0, out calp0);
double
sbet1 = lat >= 0 ? calp0 : -calp0,
cbet1 = Math.Abs(dlon) <= 90 ? Math.Abs(salp0) : -Math.Abs(salp0),
sbet01 = sbet1 * _cbet0 - cbet1 * _sbet0,
cbet01 = cbet1 * _cbet0 + sbet1 * _sbet0,
sig01 = Math.Atan2(sbet01, cbet01) / GeoMath.Degree;
_meridian.GenPosition(true, sig01,
GeodesicMask.DISTANCE,
out t, out t, out t, out y, out t, out t, out t, out t);
}
/**
* 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;
}
/**
* Forward projection, from geographic to transverse Mercator.
*
* @param[in] lon0 central meridian of the projection (degrees).
* @param[in] lat latitude of point (degrees).
* @param[in] lon longitude of point (degrees).
* @param[out] x easting of point (meters).
* @param[out] y northing of point (meters).
* @param[out] gamma meridian convergence at point (degrees).
* @param[out] k scale of projection at point.
*
* No false easting or northing is added. \e lat should be in the range
* [−90°, 90°].
**********************************************************************/
public void Forward(double lon0, double lat, double lon,
out double x, out double y, out double gamma, out double k)
{
double e;
lat = GeoMath.LatFix(lat);
lon = GeoMath.AngDiff(lon0, lon, out e);
// Explicitly enforce the parity
int
latsign = (lat < 0) ? -1 : 1,
lonsign = (lon < 0) ? -1 : 1;
lon *= lonsign;
lat *= latsign;
bool backside = lon > 90;
if (backside)
{
if (lat == 0)
{
latsign = -1;
}
lon = 180 - lon;
}
double sphi, cphi, slam, clam;
GeoMath.Sincosd(lat, out sphi, out cphi);
GeoMath.Sincosd(lon, out slam, out clam);
// phi = latitude
// phi' = conformal latitude
// psi = isometric latitude
// tau = tan(phi)
// tau' = tan(phi')
// [xi', eta'] = Gauss-Schreiber TM coordinates
// [xi, eta] = Gauss-Krueger TM coordinates
//
// We use
// tan(phi') = Math.Sinh(psi)
// Math.Sin(phi') = tanh(psi)
// Math.Cos(phi') = sech(psi)
// denom^2 = 1-Math.Cos(phi')^2*Math.Sin(lam)^2 = 1-sech(psi)^2*Math.Sin(lam)^2
// Math.Sin(xip) = Math.Sin(phi')/denom = tanh(psi)/denom
// Math.Cos(xip) = Math.Cos(phi')*Math.Cos(lam)/denom = sech(psi)*Math.Cos(lam)/denom
// Math.Cosh(etap) = 1/denom = 1/denom
// Math.Sinh(etap) = Math.Cos(phi')*Math.Sin(lam)/denom = sech(psi)*Math.Sin(lam)/denom
double etap, xip;
if (lat != 90)
{
double
tau = sphi / cphi,
taup = GeoMath.Taupf(tau, _es);
xip = Math.Atan2(taup, clam);
// Used to be
// etap = Math::atanh(Math.Sin(lam) / Math.Cosh(psi));
etap = GeoMath.Asinh(slam / GeoMath.Hypot(taup, clam));
// convergence and scale for Gauss-Schreiber TM (xip, etap) -- gamma0 =
// atan(tan(xip) * tanh(etap)) = atan(tan(lam) * Math.Sin(phi'));
// Math.Sin(phi') = tau'/Math.Sqrt(1 + tau'^2)
// Krueger p 22 (44)
gamma = GeoMath.Atan2d(slam * taup, clam * GeoMath.Hypot(1, taup));
// k0 = Math.Sqrt(1 - _e2 * Math.Sin(phi)^2) * (Math.Cos(phi') / Math.Cos(phi)) * Math.Cosh(etap)
// Note 1/Math.Cos(phi) = Math.Cosh(psip);
// and Math.Cos(phi') * Math.Cosh(etap) = 1/hypot(Math.Sinh(psi), Math.Cos(lam))
//
// This form has cancelling errors. This property is lost if Math.Cosh(psip)
// is replaced by 1/Math.Cos(phi), even though it's using "primary" data (phi
// instead of psip).
k = Math.Sqrt(_e2m + _e2 * GeoMath.Square(cphi)) * GeoMath.Hypot(1, tau)
/ GeoMath.Hypot(taup, clam);
}
else
{
xip = Math.PI / 2;
etap = 0;
gamma = lon;
k = _c;
}
// {xi',eta'} is {northing,easting} for Gauss-Schreiber transverse Mercator
// (for eta' = 0, xi' = bet). {xi,eta} is {northing,easting} for transverse
// Mercator with ant scale on the central meridian (for eta = 0, xip =
// rectifying latitude). Define
//
// zeta = xi + i*eta
// zeta' = xi' + i*eta'
//
// The conversion from conformal to rectifying latitude can be expressed as
// a series in _n:
//
// zeta = zeta' + sum(h[j-1]' * Math.Sin(2 * j * zeta'), j = 1..maxpow_)
//
// where h[j]' = O(_n^j). The reversion of this series gives
//
// zeta' = zeta - sum(h[j-1] * Math.Sin(2 * j * zeta), j = 1..maxpow_)
//
// which is used in Reverse.
//
// Evaluate sums via Clenshaw method. See
// https://en.wikipedia.org/wiki/Clenshaw_algorithm
//
// Let
//
// S = sum(a[k] * phi[k](x), k = 0..n)
// phi[k+1](x) = alpha[k](x) * phi[k](x) + beta[k](x) * phi[k-1](x)
//
// Evaluate S with
//
// b[n+2] = b[n+1] = 0
// b[k] = alpha[k](x) * b[k+1] + beta[k+1](x) * b[k+2] + a[k]
// S = (a[0] + beta[1](x) * b[2]) * phi[0](x) + b[1] * phi[1](x)
//
// Here we have
//
// x = 2 * zeta'
// phi[k](x) = Math.Sin(k * x)
// alpha[k](x) = 2 * Math.Cos(x)
// beta[k](x) = -1
// [ Math.Sin(A+B) - 2*Math.Cos(B)*Math.Sin(A) + Math.Sin(A-B) = 0, A = k*x, B = x ]
// n = maxpow_
// a[k] = _alp[k]
// S = b[1] * Math.Sin(x)
//
// For the derivative we have
//
// x = 2 * zeta'
// phi[k](x) = Math.Cos(k * x)
// alpha[k](x) = 2 * Math.Cos(x)
// beta[k](x) = -1
// [ Math.Cos(A+B) - 2*Math.Cos(B)*Math.Cos(A) + Math.Cos(A-B) = 0, A = k*x, B = x ]
// a[0] = 1; a[k] = 2*k*_alp[k]
// S = (a[0] - b[2]) + b[1] * Math.Cos(x)
//
// Matrix formulation (not used here):
// phi[k](x) = [Math.Sin(k * x); k * Math.Cos(k * x)]
// alpha[k](x) = 2 * [Math.Cos(x), 0; -Math.Sin(x), Math.Cos(x)]
// beta[k](x) = -1 * [1, 0; 0, 1]
// a[k] = _alp[k] * [1, 0; 0, 1]
// b[n+2] = b[n+1] = [0, 0; 0, 0]
// b[k] = alpha[k](x) * b[k+1] + beta[k+1](x) * b[k+2] + a[k]
// N.B., for all k: b[k](1,2) = 0; b[k](1,1) = b[k](2,2)
// S = (a[0] + beta[1](x) * b[2]) * phi[0](x) + b[1] * phi[1](x)
// phi[0](x) = [0; 0]
// phi[1](x) = [Math.Sin(x); Math.Cos(x)]
double
c0 = Math.Cos(2 * xip), ch0 = Math.Cosh(2 * etap),
s0 = Math.Sin(2 * xip), sh0 = Math.Sinh(2 * etap);
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 ? _alp[n] : 0, 0);
Complex y1; // default initializer is 0+i0
Complex z0 = new Complex((n & 1) != 0 ? 2 * n * _alp[n] : 0, 0);
Complex z1;
if ((n & 1) != 0)
{
--n;
}
while (n > 0)
{
y1 = a * y0 - y1 + _alp[n];
z1 = a * z0 - z1 + 2 * n * _alp[n];
--n;
y0 = a * y1 - y0 + _alp[n];
z0 = a * z1 - z0 + 2 * n * _alp[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(xip, etap) + a * y0;
// Fold in change in convergence and scale for Gauss-Schreiber TM to
// Gauss-Krueger TM.
gamma -= GeoMath.Atan2d(z1.Imaginary, z1.Real);
k *= _b1 * Complex.Abs(z1);
double xi = y1.Real, eta = y1.Imaginary;
y = _a1 * _k0 * (backside ? Math.PI - xi : xi) * latsign;
x = _a1 * _k0 * eta * lonsign;
if (backside)
{
gamma = 180 - gamma;
}
gamma *= latsign * lonsign;
gamma = GeoMath.AngNormalize(gamma);
k *= _k0;
}