// The static const coefficient arrays in the following functions are
// generated by Maxima and give the coefficients of the Taylor expansions for
// the geodesics. The convention on the order of these coefficients is as
// follows:
//
// ascending order in the trigonometric expansion,
// then powers of eps in descending order,
// finally powers of n in descending order.
//
// (For some expansions, only a subset of levels occur.) For each polynomial
// of order n at the lowest level, the (n+1) coefficients of the polynomial
// are followed by a divisor which is applied to the whole polynomial. In
// this way, the coefficients are expressible with no round off error. The
// sizes of the coefficient arrays are:
//
// A1m1f, A2m1f = floor(N/2) + 2
// C1f, C1pf, C2f, A3coeff = (N^2 + 7*N - 2*floor(N/2)) / 4
// C3coeff = (N - 1) * (N^2 + 7*N - 2*floor(N/2)) / 8
// C4coeff = N * (N + 1) * (N + 5) / 6
//
// where N = GEOGRAPHICLIB_GEODESIC_ORDER
// = nA1 = nA2 = nC1 = nC1p = nA3 = nC4
// The scale factor A1-1 = mean value of (d/dsigma)I1 - 1
public static double A1m1f(double eps)
{
double[] coeff =
{
// (1-eps)*A1-1, polynomial in eps2 of order 3
1, 4, 64, 0, 256,
};
int m = nA1_ / 2;
double t = GeoMath.PolyVal(m, coeff, 0, GeoMath.Square(eps)) / coeff[m + 1];
return((t + eps) / (1 - eps));
}
/**
* Set the central point of the projection
*
* @param[in] lat0 latitude of center point of projection (degrees).
* @param[in] lon0 longitude of center point of projection (degrees).
*
* \e lat0 should be in the range [−90°, 90°].
**********************************************************************/
public void Reset(double lat0, double lon0)
{
_meridian = _earth.Line(lat0, lon0, 0,
GeodesicMask.LATITUDE | GeodesicMask.LONGITUDE |
GeodesicMask.DISTANCE | GeodesicMask.DISTANCE_IN |
GeodesicMask.AZIMUTH);
double f = _earth.Flattening();
GeoMath.Sincosd(LatitudeOrigin(), out _sbet0, out _cbet0);
_sbet0 *= (1 - f);
GeoMath.Norm(ref _sbet0, ref _cbet0);
}
/**
* 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);
}
// The scale factor A2-1 = mean value of (d/dsigma)I2 - 1
public static double A2m1f(double eps)
{
double[] coeff =
{
// (eps+1)*A2-1, polynomial in eps2 of order 3
-11, -28, -192, 0, 256,
}; // count = 5
int m = nA2_ / 2;
double t = GeoMath.PolyVal(m, coeff, 0, GeoMath.Square(eps)) / coeff[m + 1];
return((t - eps) / (1 + eps));
}
void InitVars(double a, double f)
{
eps_ = GeoMath.Epsilon;
epsx_ = GeoMath.Square(eps_);
epsx2_ = GeoMath.Square(epsx_);
tol_ = GeoMath.Square(eps_);
tol0_ = (tol_ * GeoMath.Square(GeoMath.Square(eps_)));
_a = a;
_f = f;
_fm = 1 - _f;
_e2 = _f * (2 - _f);
_e = GeoMath.Square(Math.Abs(_e2));
_e2m = 1 - _e2;
_qZ = 1 + _e2m * atanhee(1);
_qx = _qZ / (2 * _e2m);
}
/**
* Set the azimuthal scale for the projection.
*
* @param[in] lat (degrees).
* @param[in] k azimuthal scale at latitude \e lat (default 1).
* @exception new GeographicException \e k is not positive.
* @exception new GeographicException if \e lat is not in (−90°,
* 90°).
*
* This allows a "latitude of conformality" to be specified.
**********************************************************************/
public void SetScale(double lat, double k = 1)
{
if (!(GeoMath.IsFinite(k) && k > 0))
{
throw new GeographicException("Scale is not positive");
}
if (!(Math.Abs(lat) < 90))
{
throw new GeographicException("Latitude for SetScale not in (-90d, 90d)");
}
double x, y, gamma, kold;
Forward(0, lat, 0, out x, out y, out gamma, out kold);
k /= kold;
_k0 *= k;
_k2 = GeoMath.Square(_k0);
}
internal void IntForward(double lat, double lon, double h, out double X, out double Y, out double Z,
double[] M)
{
double sphi, cphi, slam, clam;
GeoMath.Sincosd(GeoMath.LatFix(lat), out sphi, out cphi);
GeoMath.Sincosd(lon, out slam, out clam);
double n = _a / Math.Sqrt(1 - _e2 * GeoMath.Square(sphi));
Z = (_e2m * n + h) * sphi;
X = (n + h) * cphi;
Y = X * slam;
X *= clam;
if (M != null)
{
Rotation(sphi, cphi, slam, clam, M);
}
}
/**
* ructor with two standard parallels.
*
* @param[in] a equatorial radius of ellipsoid (meters).
* @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere.
* Negative \e f gives a prolate ellipsoid.
* @param[in] stdlat1 first standard parallel (degrees).
* @param[in] stdlat2 second standard parallel (degrees).
* @param[in] k1 azimuthal scale on the standard parallels.
* @exception new GeographicException if \e a, (1 − \e f) \e a, or \e k1 is
* not positive.
* @exception new GeographicException if \e stdlat1 or \e stdlat2 is not in
* [−90°, 90°], or if \e stdlat1 and \e stdlat2 are
* opposite poles.
**********************************************************************/
public AlbersEqualArea(double a, double f, double stdlat1, double stdlat2, double k1)
{
InitVars(a, f);
eps_ = GeoMath.Epsilon;
epsx_ = GeoMath.Square(eps_);
epsx2_ = GeoMath.Square(epsx_);
tol_ = GeoMath.Square(eps_);
tol0_ = (tol_ * GeoMath.Square(GeoMath.Square(eps_)));
_a = a;
_f = f;
_fm = 1 - _f;
_e2 = _f * (2 - _f);
_e = GeoMath.Square(Math.Abs(_e2));
_e2m = 1 - _e2;
_qZ = 1 + _e2m * atanhee(1);
_qx = _qZ / (2 * _e2m);
if (!(GeoMath.IsFinite(_a) && _a > 0))
{
throw new GeographicException("Equatorial radius is not positive");
}
if (!(GeoMath.IsFinite(_f) && _f < 1))
{
throw new GeographicException("Polar semi-axis is not positive");
}
if (!(GeoMath.IsFinite(k1) && k1 > 0))
{
throw new GeographicException("Scale is not positive");
}
if (!(Math.Abs(stdlat1) <= 90))
{
throw new GeographicException("Standard latitude 1 not in [-90d, 90d]");
}
if (!(Math.Abs(stdlat2) <= 90))
{
throw new GeographicException("Standard latitude 2 not in [-90d, 90d]");
}
double sphi1, cphi1, sphi2, cphi2;
GeoMath.Sincosd(stdlat1, out sphi1, out cphi1);
GeoMath.Sincosd(stdlat2, out sphi2, out cphi2);
Init(sphi1, cphi1, sphi2, cphi2, k1);
}
public void Add(double y)
{
// Here's Shewchuk's solution...
double u; // hold exact sum as [s, t, u]
// Accumulate starting at least significant end
y = GeoMath.Sum(y, _t, out u);
_s = GeoMath.Sum(y, _s, out _t);
// Start is _s, _t decreasing and non-adjacent. Sum is now (s + t + u)
// exactly with s, t, u non-adjacent and in decreasing order (except for
// possible zeros). The following code tries to normalize the result.
// Ideally, we want _s = round(s+t+u) and _u = round(s+t+u - _s). The
// following does an approximate job (and maintains the decreasing
// non-adjacent property). Here are two "failures" using 3-bit floats:
//
// Case 1: _s is not equal to round(s+t+u) -- off by 1 ulp
// [12, -1] - 8 -> [4, 0, -1] -> [4, -1] = 3 should be [3, 0] = 3
//
// Case 2: _s+_t is not as close to s+t+u as it shold be
// [64, 5] + 4 -> [64, 8, 1] -> [64, 8] = 72 (off by 1)
// should be [80, -7] = 73 (exact)
//
// "Fixing" these problems is probably not worth the expense. The
// representation inevitably leads to small errors in the accumulated
// values. The additional errors illustrated here amount to 1 ulp of the
// less significant word during each addition to the Accumulator and an
// additional possible error of 1 ulp in the reported sum.
//
// Incidentally, the "ideal" representation described above is not
// canonical, because _s = round(_s + _t) may not be true. For example,
// with 3-bit floats:
//
// [128, 16] + 1 -> [160, -16] -- 160 = round(145).
// But [160, 0] - 16 -> [128, 16] -- 128 = round(144).
//
if (_s == 0) // This implies t == 0,
{
_s = u; // so result is u
}
else
{
_t += u; // otherwise just accumulate u to t.
}
}
/**
* Constructor for a ellipsoid with
*
* @param[in] a equatorial radius (meters).
* @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere.
* Negative \e f gives a prolate ellipsoid.
* @exception GeographicErr if \e a or (1 − \e f) \e a is not
* positive.
**********************************************************************/
public Geocentric(double a, double f)
{
_a = a;
_f = f;
_e2 = _f * (2 - _f);
_e2m = GeoMath.Square(1 - _f);
_e2a = Math.Abs(_e2);
_e4a = GeoMath.Square(_e2);
_maxrad = 2 * _a / GeoMath.Epsilon;
if (!(GeoMath.IsFinite(_a) && _a > 0))
{
throw new GeographicException("Equatorial radius is not positive");
}
if (!(GeoMath.IsFinite(_f) && _f < 1))
{
throw new GeographicException("Polar semi-axis is not positive");
}
}
/**
* ructor with two standard parallels specified by sines and cosines.
*
* @param[in] a equatorial radius of ellipsoid (meters).
* @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere.
* Negative \e f gives a prolate ellipsoid.
* @param[in] sinlat1 sine of first standard parallel.
* @param[in] coslat1 cosine of first standard parallel.
* @param[in] sinlat2 sine of second standard parallel.
* @param[in] coslat2 cosine of second standard parallel.
* @param[in] k1 azimuthal scale on the standard parallels.
* @exception new GeographicException if \e a, (1 − \e f) \e a, or \e k1 is
* not positive.
* @exception new GeographicException if \e stdlat1 or \e stdlat2 is not in
* [−90°, 90°], or if \e stdlat1 and \e stdlat2 are
* opposite poles.
*
* This allows parallels close to the poles to be specified accurately.
* This routine computes the latitude of origin and the azimuthal scale at
* this latitude. If \e dlat = abs(\e lat2 − \e lat1) ≤ 160°,
* then the error in the latitude of origin is less than 4.5 ×
* 10<sup>−14</sup>d;.
**********************************************************************/
public AlbersEqualArea(double a, double f,
double sinlat1, double coslat1,
double sinlat2, double coslat2,
double k1)
{
InitVars(a, f);
if (!(GeoMath.IsFinite(_a) && _a > 0))
{
throw new GeographicException("Equatorial radius is not positive");
}
if (!(GeoMath.IsFinite(_f) && _f < 1))
{
throw new GeographicException("Polar semi-axis is not positive");
}
if (!(GeoMath.IsFinite(k1) && k1 > 0))
{
throw new GeographicException("Scale is not positive");
}
if (!(coslat1 >= 0))
{
throw new GeographicException("Standard latitude 1 not in [-90d, 90d]");
}
if (!(coslat2 >= 0))
{
throw new GeographicException("Standard latitude 2 not in [-90d, 90d]");
}
if (!(Math.Abs(sinlat1) <= 1 && coslat1 <= 1) || (coslat1 == 0 && sinlat1 == 0))
{
throw new GeographicException("Bad sine/cosine of standard latitude 1");
}
if (!(Math.Abs(sinlat2) <= 1 && coslat2 <= 1) || (coslat2 == 0 && sinlat2 == 0))
{
throw new GeographicException("Bad sine/cosine of standard latitude 2");
}
if (coslat1 == 0 && coslat2 == 0 && sinlat1 * sinlat2 <= 0)
{
throw new GeographicException
("Standard latitudes cannot be opposite poles");
}
Init(sinlat1, coslat1, sinlat2, coslat2, k1);
}
public static double[] GenerateC4(double n)
{
double[] coeff =
{
97, 15015, // C4[0], coeff of eps^5, polynomial in n of order 0
1088, 156, 45045, // C4[0], coeff of eps^4, polynomial in n of order 1
-224, -4784, 1573, 45045, // C4[0], coeff of eps^3, polynomial in n of order 2
-10656, 14144, -4576, -858, 45045, // C4[0], coeff of eps^2, polynomial in n of order 3
64, 624, -4576, 6864, -3003,15015, // C4[0], coeff of eps^1, polynomial in n of order 4
100, 208, 572, 3432, -12012,30030, 45045, // C4[0], coeff of eps^0, polynomial in n of order 5
1, 9009, // C4[1], coeff of eps^5, polynomial in n of order 0
-2944, 468,135135, // C4[1], coeff of eps^4, polynomial in n of order 1
5792, 1040, -1287,135135, // C4[1], coeff of eps^3, polynomial in n of order 2
5952, -11648, 9152, -2574, 135135, // C4[1], coeff of eps^2, polynomial in n of order 3
-64, -624, 4576, -6864, 3003,135135, // C4[1], coeff of eps^1, polynomial in n of order 4
8, 10725, // C4[2], coeff of eps^5, polynomial in n of order 0
1856, -936,225225, // C4[2], coeff of eps^4, polynomial in n of order 1
-8448, 4992, -1144,225225, // C4[2], coeff of eps^3, polynomial in n of order 2
-1440, 4160, -4576, 1716, 225225, // C4[2], coeff of eps^2, polynomial in n of order 3
-136, 63063, // C4[3], coeff of eps^5, polynomial in n of order 0
1024, -208,105105, // C4[3], coeff of eps^4, polynomial in n of order 1
3584, -3328, 1144,315315, // C4[3], coeff of eps^3, polynomial in n of order 2
-128, 135135, // C4[4], coeff of eps^5, polynomial in n of order 0
-2560, 832,405405, // C4[4], coeff of eps^4, polynomial in n of order 1
128, 99099 // C4[5], coeff of eps^5, polynomial in n of order 0
};
var C4x = new double[nC4x_];
int o = 0, k = 0;
for (int l = 0; l < nC4_; ++l) // l is index of C4[l]
{
for (int j = nC4_ - 1; j >= l; --j) // coeff of eps^j
{
int m = nC4_ - j - 1; // order of polynomial in n
C4x[k++] = GeoMath.PolyVal(m, coeff, o, n) / coeff[o + m + 1];
o += m + 2;
}
}
return(C4x);
}
/**
* 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);
}
// return atanh(GeoMath.Square(x))/GeoMath.Square(x) - 1, accurate for small x
static double atanhxm1(double x)
{
double s = 0;
if (Math.Abs(x) < 0.5)
{
double os = -1, y = 1, k = 1;
while (os != s)
{
os = s;
y *= x; // y = x^n
k += 2; // k = 2*n + 1
s += y / k; // sum( x^n/(2*n + 1) )
}
}
else
{
double xs = Math.Sqrt(Math.Abs(x));
s = (x > 0 ? GeoMath.Atanh(xs) : Math.Atan(xs)) / xs - 1;
}
return(s);
}
//TODO: Can make this more efficient by cloning the array once it has been created
public static double[] GenerateA3(double n)
{
double[] coeff =
{
-3,128, // coeff of eps^5, polynomial in n of order 0
-2, -3, 64, // coeff of eps^4, polynomial in n of order 1
-1, -3, -1, 16, // coeff of eps^3, polynomial in n of order 2
3, -1, -2, 8, // coeff of eps^2, polynomial in n of order 2
1, -1, 2, // coeff of eps^1, polynomial in n of order 1
1, 1, // coeff of eps^0, polynomial in n of order 0
};
var A3x = new double[nA3x_];
int o = 0, k = 0;
for (int j = nA3_ - 1; j >= 0; --j) // coeff of eps^j
{
int m = Math.Min(nA3_ - j - 1, j); // order of polynomial in n
A3x[k++] = GeoMath.PolyVal(m, coeff, o, n) / coeff[o + m + 1];
o += m + 2;
}
return(A3x);
}
public static double[] GenerateC3(double n)
{
double[] coeff =
{
3, 128, // C3[1], coeff of eps^5, polynomial in n of order 0
2, 5,128, // C3[1], coeff of eps^4, polynomial in n of order 1
-1, 3, 3, 64, // C3[1], coeff of eps^3, polynomial in n of order 2
-1, 0, 1, 8, // C3[1], coeff of eps^2, polynomial in n of order 2
-1, 1, 4, // C3[1], coeff of eps^1, polynomial in n of order 1
5, 256, // C3[2], coeff of eps^5, polynomial in n of order 0
1, 3,128, // C3[2], coeff of eps^4, polynomial in n of order 1
-3, -2, 3, 64, // C3[2], coeff of eps^3, polynomial in n of order 2
1, -3, 2, 32, // C3[2], coeff of eps^2, polynomial in n of order 2
7, 512, // C3[3], coeff of eps^5, polynomial in n of order 0
-10, 9,384, // C3[3], coeff of eps^4, polynomial in n of order 1
5, -9, 5, 192, // C3[3], coeff of eps^3, polynomial in n of order 2
7, 512, // C3[4], coeff of eps^5, polynomial in n of order 0
-14, 7,512, // C3[4], coeff of eps^4, polynomial in n of order 1
21, 2560 // C3[5], coeff of eps^5, polynomial in n of order 0
};
var C3x = new double[nC3x_];
int o = 0, k = 0;
for (int l = 1; l < nC3_; ++l) // l is index of C3[l]
{
for (int j = nC3_ - 1; j >= l; --j) // coeff of eps^j
{
int m = Math.Min(nC3_ - j - 1, j); // order of polynomial in n
C3x[k++] = GeoMath.PolyVal(m, coeff, o, n) / coeff[o + m + 1];
o += m + 2;
}
}
return(C3x);
}
/**
* 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;
}
internal void IntReverse(double X, double Y, double Z, out double lat, out double lon, out double h,
double[] M)
{
double
R = GeoMath.Hypot(X, Y),
slam = R != 0 ? Y / R : 0,
clam = R != 0 ? X / R : 1;
h = GeoMath.Hypot(R, Z); // Distance to center of earth
double sphi, cphi;
if (h > _maxrad)
{
// We doublely 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 = GeoMath.Hypot(X / 2, Y / 2);
slam = R != 0 ? (Y / 2) / R : 0;
clam = R != 0 ? (X / 2) / R : 1;
double H = GeoMath.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.
double H = GeoMath.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.
double
p = GeoMath.Square(R / _a),
q = _e2m * GeoMath.Square(Z / _a),
r = (p + q - _e4a) / 6;
if (_f < 0)
{
Utility.Swap(ref p, ref q);
}
if (!(_e4a * q == 0 && r <= 0))
{
double
// 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 = GeoMath.Square(r),
r3 = r * r2,
disc = S * (2 * r3 + S);
double u = r;
if (disc >= 0)
{
double 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 ? -Math.Sqrt(disc) : Math.Sqrt(disc); // T3 = (r * t)^3
// N.B. cbrt always returns the double root. cbrt(-8) = -2.
double T = GeoMath.CubeRoot(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 double.
double ang = Math.Atan2(Math.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 * Math.Cos(ang / 3);
}
double
v = Math.Sqrt(GeoMath.Square(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 = Math.Max(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 / (Math.Sqrt(uv + GeoMath.Square(w)) + w),
k1 = _f >= 0 ? k : k - _e2,
k2 = _f >= 0 ? k + _e2 : k,
d = k1 * R / k2,
H = GeoMath.Hypot(Z / k1, R / k2);
sphi = (Z / k1) / H;
cphi = (R / k2) / H;
h = (1 - _e2m / k1) * GeoMath.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 * Math.Sqrt(q)/Math.Sqrt(e4 - p)
// f < 0: R -> 0, k + e2 -> - e2 * Math.Sqrt(q)/Math.Sqrt(e4 - p)
double
zz = Math.Sqrt((_f >= 0 ? _e4a - p : p) / _e2m),
xx = Math.Sqrt(_f < 0 ? _e4a - p : p),
H = GeoMath.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 = GeoMath.Atan2d(sphi, cphi);
lon = GeoMath.Atan2d(slam, clam);
if (M != null)
{
Rotation(sphi, cphi, slam, clam, M);
}
}
/**
* Constructor for a ellipsoid with
*
* @param[in] a equatorial radius (meters).
* @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere.
* Negative \e f gives a prolate ellipsoid.
* @param[in] k0 central scale factor.
* @exception GeographicErr if \e a, (1 − \e f) \e a, or \e k0 is
* not positive.
**********************************************************************/
public TransverseMercator(double a, double f, double k0)
{
_a = a;
_f = f;
_k0 = k0;
_e2 = _f * (2 - _f);
_es = (f < 0 ? -1 : 1) * Math.Sqrt(Math.Abs(_e2));
_e2m = 1 - _e2;
// _c = Math.Sqrt( pow(1 + _e, 1 + _e) * pow(1 - _e, 1 - _e) ) )
// See, for example, Lee (1976), p 100.
_c = Math.Sqrt(_e2m) * Math.Exp(GeoMath.Eatanhe(1, _es));
_n = _f / (2 - _f);
if (!(GeoMath.IsFinite(_a) && _a > 0))
{
throw new GeographicException("Equatorial radius is not positive");
}
if (!(GeoMath.IsFinite(_f) && _f < 1))
{
throw new GeographicException("Polar semi-axis is not positive");
}
if (!(GeoMath.IsFinite(_k0) && _k0 > 0))
{
throw new GeographicException("Scale is not positive");
}
double[] b1coeff =
{
// b1*(n+1), polynomial in n2 of order 3
1, 4, 64, 256, 256,
};
double[] alpcoeff =
{
// alp[1]/n^1, polynomial in n of order 5
31564, -66675, 34440, 47250, -100800, 75600, 151200,
// alp[2]/n^2, polynomial in n of order 4
-1983433, 863232, 748608, -1161216, 524160, 1935360,
// alp[3]/n^3, polynomial in n of order 3
670412, 406647, -533952, 184464, 725760,
// alp[4]/n^4, polynomial in n of order 2
6601661, -7732800, 2230245, 7257600,
// alp[5]/n^5, polynomial in n of order 1
-13675556, 3438171, 7983360,
// alp[6]/n^6, polynomial in n of order 0
212378941, 319334400,
}; // count = 27
double[] betcoeff =
{
// bet[1]/n^1, polynomial in n of order 5
384796, -382725, -6720, 932400, -1612800, 1209600, 2419200,
// bet[2]/n^2, polynomial in n of order 4
-1118711, 1695744, -1174656, 258048, 80640, 3870720,
// bet[3]/n^3, polynomial in n of order 3
22276, -16929, -15984, 12852, 362880,
// bet[4]/n^4, polynomial in n of order 2
-830251, -158400, 197865, 7257600,
// bet[5]/n^5, polynomial in n of order 1
-435388, 453717, 15966720,
// bet[6]/n^6, polynomial in n of order 0
20648693, 638668800,
}; // count = 27
int m = maxpow_ / 2;
_b1 = GeoMath.PolyVal(m, b1coeff, 0, GeoMath.Square(_n)) / (b1coeff[m + 1] * (1 + _n));
// _a1 is the equivalent radius for computing the circumference of
// ellipse.
_a1 = _b1 * _a;
int o = 0;
double d = _n;
for (int l = 1; l <= maxpow_; ++l)
{
m = maxpow_ - l;
_alp[l] = d * GeoMath.PolyVal(m, alpcoeff, o, _n) / alpcoeff[o + m + 1];
_bet[l] = d * GeoMath.PolyVal(m, betcoeff, o, _n) / betcoeff[o + m + 1];
o += m + 2;
d *= _n;
}
// Post condition: o == sizeof(alpcoeff) / sizeof(double) &&
// o == sizeof(betcoeff) / sizeof(double)
}
private void LineInit(Geodesic g,
double lat1, double lon1,
double azi1, double salp1, double calp1,
int caps)
{
_lat1 = GeoMath.LatFix(lat1);
_lon1 = lon1;
_azi1 = azi1;
_salp1 = salp1;
_calp1 = calp1;
_a = g._a;
_f = g._f;
_b = g._b;
_c2 = g._c2;
_f1 = g._f1;
// Always allow latitude and azimuth and unrolling of longitude
_caps = ((int)caps | (int)GeodesicMask.LATITUDE | (int)GeodesicMask.AZIMUTH | (int)GeodesicMask.LONG_UNROLL); //TODO: check this
double cbet1, sbet1;
GeoMath.Sincosd(GeoMath.AngRound(_lat1), out sbet1, out cbet1); sbet1 *= _f1;
// Ensure cbet1 = +epsilon at poles
GeoMath.Norm(ref sbet1, ref cbet1);
cbet1 = Math.Max(Geodesic.tiny_, cbet1);
_dn1 = Math.Sqrt(1 + g._ep2 * GeoMath.Square(sbet1));
// Evaluate alp0 from sin(alp1) * cos(bet1) = sin(alp0),
_salp0 = _salp1 * cbet1; // alp0 in [0, pi/2 - |bet1|]
// Alt: calp0 = hypot(sbet1, calp1 * cbet1). The following
// is slightly better (consider the case salp1 = 0).
_calp0 = GeoMath.Hypot(_calp1, _salp1 * sbet1);
// Evaluate sig with tan(bet1) = tan(sig1) * cos(alp1).
// sig = 0 is nearest northward crossing of equator.
// With bet1 = 0, alp1 = pi/2, we have sig1 = 0 (equatorial line).
// With bet1 = pi/2, alp1 = -pi, sig1 = pi/2
// With bet1 = -pi/2, alp1 = 0 , sig1 = -pi/2
// Evaluate omg1 with tan(omg1) = sin(alp0) * tan(sig1).
// With alp0 in (0, pi/2], quadrants for sig and omg coincide.
// No Math.Atan2(0,0) ambiguity at poles since cbet1 = +epsilon.
// With alp0 = 0, omg1 = 0 for alp1 = 0, omg1 = pi for alp1 = pi.
_ssig1 = sbet1; _somg1 = _salp0 * sbet1;
_csig1 = _comg1 = sbet1 != 0 || _calp1 != 0 ? cbet1 * _calp1 : 1;
GeoMath.Norm(ref _ssig1, ref _csig1); // sig1 in (-pi, pi]
// Math::norm(_somg1, _comg1); -- don't need to normalize!
_k2 = GeoMath.Square(_calp0) * g._ep2;
double eps = _k2 / (2 * (1 + Math.Sqrt(1 + _k2)) + _k2);
if ((_caps & GeodesicMask.CAP_C1) != 0)
{
_C1a = new double[nC1_ + 1];
_A1m1 = GeodesicCoeff.A1m1f(eps);
GeodesicCoeff.C1f(eps, _C1a);
_B11 = Geodesic.SinCosSeries(true, _ssig1, _csig1, _C1a);
double s = Math.Sin(_B11), c = Math.Cos(_B11);
// tau1 = sig1 + B11
_stau1 = _ssig1 * c + _csig1 * s;
_ctau1 = _csig1 * c - _ssig1 * s;
// Not necessary because C1pa reverts C1a
// _B11 = -SinCosSeries(true, _stau1, _ctau1, _C1pa, nC1p_);
}
if ((_caps & GeodesicMask.CAP_C1p) != 0)
{
_C1pa = new double[nC1p_ + 1];
GeodesicCoeff.C1pf(eps, _C1pa);
}
if ((_caps & GeodesicMask.CAP_C2) != 0)
{
_C2a = new double[nC2_ + 1];
_A2m1 = GeodesicCoeff.A2m1f(eps);
GeodesicCoeff.C2f(eps, _C2a);
_B21 = Geodesic.SinCosSeries(true, _ssig1, _csig1, _C2a);
}
if ((_caps & GeodesicMask.CAP_C3) != 0)
{
_C3a = new double[nC3_];
g.C3f(eps, _C3a);
_A3c = -_f *_salp0 *g.A3f(eps);
_B31 = Geodesic.SinCosSeries(true, _ssig1, _csig1, _C3a);
}
if ((_caps & GeodesicMask.CAP_C4) != 0)
{
_C4a = new double[nC4_];
g.C4f(eps, _C4a);
// Multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0)
_A4 = GeoMath.Square(_a) * _calp0 * _salp0 * g._e2;
_B41 = Geodesic.SinCosSeries(false, _ssig1, _csig1, _C4a);
}
_a13 = _s13 = double.NaN;
}
/**
* @return \e a1 the arc length (degrees) between the northward equatorial
* crossing and point 1.
*
* The result lies in (−180°, 180°].
**********************************************************************/
public double EquatorialArc()
{
return(Init() ? GeoMath.Atan2d(_ssig1, _csig1) : double.NaN);
}
/**
* @return \e azi0 the azimuth (degrees) of the geodesic line as it crosses
* the equator in a northward direction.
*
* The result lies in [−90°, 90°].
**********************************************************************/
public double EquatorialAzimuth()
{
return(Init() ? GeoMath.Atan2d(_salp0, _calp0) : double.NaN);
}
///@}
/** \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);
}
static double hyp(double x)
{
return(GeoMath.Hypot(1, x));
}
/**
* 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;
}
void Init(double sphi1, double cphi1, double sphi2, double cphi2, double k1)
{
{
double r;
r = GeoMath.Hypot(sphi1, cphi1);
sphi1 /= r; cphi1 /= r;
r = GeoMath.Hypot(sphi2, cphi2);
sphi2 /= r; cphi2 /= r;
}
bool polar = (cphi1 == 0);
cphi1 = Math.Max(epsx_, cphi1); // Avoid singularities at poles
cphi2 = Math.Max(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)
{
Utility.Swap(ref sphi1, ref sphi2); Utility.Swap(ref cphi1, ref cphi2); // Make phi1 < phi2
}
double
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)
double tphi0, C;
if (polar || tphi1 == tphi2)
{
tphi0 = tphi2;
C = 1; // ignored
}
else
{
double
tbet1 = _fm * tphi1, scbet12 = 1 + GeoMath.Square(tbet1),
tbet2 = _fm * tphi2, scbet22 = 1 + GeoMath.Square(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 * GeoMath.Square(sphi1), es2 = 1 - _e2 * GeoMath.Square(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) :
GeoMath.Square(cxi2 / cphi2) * (1 + sphi2) / (1 + sxi2)) +
(sphi1 <= 0 ? (1 - sxi1) / (1 - sphi1) :
GeoMath.Square(cxi1 / cphi1) * (1 + sphi1) / (1 + sxi1)))) *
(1 + _e2 * (sphi1 + sphi2 + sphi1 * sphi2)) /
(1 + (sphi1 + sphi2 + sphi1 * sphi2)) +
(scbet22 * (sphi2 <= 0 ? 1 - sphi2 :
GeoMath.Square(cphi2) / (1 + sphi2)) +
scbet12 * (sphi1 <= 0 ? 1 - sphi1 : GeoMath.Square(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;
double stol = tol0_ * Math.Max(1, Math.Abs(tphi0));
for (int i = 0; i < 2 * numit0_; ++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)
double
scphi02 = 1 + GeoMath.Square(tphi0), scphi0 = Math.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 + GeoMath.Square(_fm * tphi0)) * sphi0,
// dg/dsphi0 = dg/dtphi0 * scphi0^3
dg = _e2m * scphi02 * (1 + 2 * GeoMath.Square(tphi0)) + _e2,
D = sphi0m * (1 - _e2 * (1 + 2 * sphi0 * (1 + sphi0))) / (_e2m * (1 + sphi0)),
// dD/dsphi0
dD = -2 * (1 - _e2 * GeoMath.Square(sphi0) * (2 * sphi0 + 3)) /
(_e2m * GeoMath.Square(1 + sphi0)),
A = -_e2 *GeoMath.Square(sphi0m) * (2 + (1 + _e2) * sphi0) /
(_e2m * (1 - _e2 * GeoMath.Square(sphi0))),
B = (sphi0m * _e2m / (1 - _e2 * sphi0) *
(atanhxm1(_e2 *
GeoMath.Square(sphi0m / (1 - _e2 * sphi0))) - _e2 * sphi0m / _e2m)),
// d(A+B)/dsphi0
dAB = (2 * _e2 * (2 - _e2 * (1 + GeoMath.Square(sphi0))) /
(_e2m * GeoMath.Square(1 - _e2 * GeoMath.Square(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 (!(Math.Abs(dtu) >= stol))
{
break;
}
}
}
_txi0 = txif(tphi0); _scxi0 = hyp(_txi0); _sxi0 = _txi0 / _scxi0;
_n0 = tphi0 / hyp(tphi0);
_m02 = 1 / (1 + GeoMath.Square(_fm * tphi0));
_nrho0 = polar ? 0 : _a *Math.Sqrt(_m02);
_k0 = Math.Sqrt(tphi1 == tphi2 ? 1 : C / (_m02 + _n0 * _qZ * _sxi0)) * k1;
_k2 = GeoMath.Square(_k0);
_lat0 = _sign * Math.Atan(tphi0) / GeoMath.Degree;
}
