/**
* Reverse projection, from gnomonic to geographic.
* <p>
* @param lat0 latitude of center point of projection (degrees).
* @param lon0 longitude of center point of projection (degrees).
* @param x easting of point (meters).
* @param y northing of point (meters).
* @return {@link GnomonicData} object with the following fields:
* <i>lat0</i>, <i>lon0</i>, <i>lat</i>, <i>lon</i>, <i>x</i>, <i>y</i>,
* <i>azi</i>, <i>rk</i>.
* <p>
* <i>lat0</i> should be in the range [−90°, 90°] and
* <i>lon0</i> should be in the range [−540°, 540°).
* <i>lat</i> will be in the range [−90°, 90°] and <i>lon</i>
* will be in the range [−180°, 180°]. The scale of the
* projection is 1/<i>rk<sup>2</sup></i> in the "radial" direction,
* <i>azi</i> clockwise from true north, and is 1/<i>rk</i> in the direction
* perpendicular to this. Even though all inputs should return a valid
* <i>lat</i> and <i>lon</i>, it's possible that the procedure fails to
* converge for very large <i>x</i> or <i>y</i>; in this case NaNs are
* returned for all the output arguments. A call to Reverse followed by a
* call to Forward will return the original (<i>x</i>, <i>y</i>) (to
* roundoff).
*/
public GnomonicData Reverse(double lat0, double lon0, double x, double y)
{
GnomonicData rev =
new GnomonicData(lat0, lon0, Double.NaN, Double.NaN, x, y, Double.NaN,
Double.NaN);
double azi0 = GeoMath.Atan2d(x, y);
double rho = GeoMath.Hypot(x, y);
double s = _a * Math.Atan(rho / _a);
bool little = rho <= _a;
if (!little)
{
rho = 1 / rho;
}
GeodesicLine line =
_earth.Line(lat0, lon0, azi0, GeodesicMask.LATITUDE
| GeodesicMask.LONGITUDE | GeodesicMask.AZIMUTH
| GeodesicMask.DISTANCE_IN | GeodesicMask.REDUCEDLENGTH
| GeodesicMask.GEODESICSCALE);
int count = numit_, trip = 0;
GeodesicData pos = null;
while (count-- > 0)
{
pos =
line.Position(s, GeodesicMask.LONGITUDE | GeodesicMask.LATITUDE
| GeodesicMask.AZIMUTH | GeodesicMask.DISTANCE_IN
| GeodesicMask.REDUCEDLENGTH
| GeodesicMask.GEODESICSCALE);
if (trip > 0)
{
break;
}
double ds =
little ? ((pos.ReducedLength / pos.GeodesicScale12) - rho) * pos.GeodesicScale12 * pos.GeodesicScale12
: (rho - (pos.GeodesicScale12 / pos.ReducedLength)) * pos.ReducedLength * pos.ReducedLength;
s -= ds;
if (Math.Abs(ds) <= eps_ * _a)
{
trip++;
}
}
if (trip == 0)
{
return(rev);
}
rev.PointLatitude = pos.Latitude2;
rev.PointLongitude = pos.Longitude2;
rev.azi = pos.FinalAzimuth;
rev.rk = pos.GeodesicScale12;
return(rev);
}
/**
* @return <i>azi0</i> the azimuth (degrees) of the geodesic line as it
* crosses the equator in a northward direction.
**********************************************************************/
public double EquatorialAzimuth()
{
return(Init() ?
GeoMath.Atan2d(_salp0, _calp0) : Double.NaN);
}
/**
* @return <i>a1</i> the arc length (degrees) between the northward
* equatorial crossing and point 1.
**********************************************************************/
public double EquatorialArc()
{
return(Init() ?
GeoMath.Atan2d(_ssig1, _csig1) : Double.NaN);
}
/**
* The general position function. {@link #Position(double, int) Position}
* and {@link #ArcPosition(double, int) ArcPosition} are defined in terms of
* this function.
* <p>
* @param arcmode boolean flag determining the meaning of the second
* parameter; if arcmode is false, then the GeodesicLine object must have
* been constructed with <i>caps</i> |= {@link GeodesicMask#DISTANCE_IN}.
* @param s12_a12 if <i>arcmode</i> 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 outmask a bitor'ed combination of {@link GeodesicMask} values
* specifying which results should be returned.
* @return a {@link GeodesicData} object with the requested results.
* <p>
* The {@link GeodesicMask} values possible for <i>outmask</i> are
* <ul>
* <li>
* <i>outmask</i> |= {@link GeodesicMask#LATITUDE} for the latitude
* <i>lat2</i>;
* <li>
* <i>outmask</i> |= {@link GeodesicMask#LONGITUDE} for the latitude
* <i>lon2</i>;
* <li>
* <i>outmask</i> |= {@link GeodesicMask#AZIMUTH} for the latitude
* <i>azi2</i>;
* <li>
* <i>outmask</i> |= {@link GeodesicMask#DISTANCE} for the distance
* <i>s12</i>;
* <li>
* <i>outmask</i> |= {@link GeodesicMask#REDUCEDLENGTH} for the reduced
* length <i>m12</i>;
* <li>
* <i>outmask</i> |= {@link GeodesicMask#GEODESICSCALE} for the geodesic
* scales <i>M12</i> and <i>M21</i>;
* <li>
* <i>outmask</i> |= {@link GeodesicMask#ALL} for all of the above;
* <li>
* <i>outmask</i> |= {@link GeodesicMask#LONG_UNROLL} to unroll <i>lon2</i>
* (instead of reducing it to the range [−180°, 180°]).
* </ul>
* <p>
* Requesting a value which the GeodesicLine object is not capable of
* computing is not an error; Double.NaN is returned instead.
**********************************************************************/
public GeodesicData Position(bool arcmode, double s12_a12,
int outmask)
{
outmask &= _caps & GeodesicMask.OUT_MASK;
GeodesicData r = new GeodesicData();
if (!(Init() &&
(arcmode ||
(_caps & (GeodesicMask.OUT_MASK & GeodesicMask.DISTANCE_IN)) != 0)
))
{
// Uninitialized or impossible distance calculation requested
return(r);
}
r.Latitude1 = _lat1; r.InitialAzimuth = _azi1;
r.Longitude1 = ((outmask & GeodesicMask.LONG_UNROLL) != 0) ? _lon1 :
GeoMath.AngNormalize(_lon1);
// Avoid warning about uninitialized B12.
double sig12, ssig12, csig12, B12 = 0, AB1 = 0;
if (arcmode)
{
// Interpret s12_a12 as spherical arc length
r.ArcLength = s12_a12;
sig12 = Math2.ToRadians(s12_a12);
{
Pair p = GeoMath.SinCosD(s12_a12);
ssig12 = p.First; csig12 = p.Second;
}
}
else
{
// Interpret s12_a12 as distance
r.Distance = s12_a12;
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
}
r.ArcLength = Math2.ToDegrees(sig12);
}
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 && arcmode)
{
r.Distance = _b * ((1 + _A1m1) * sig12 + AB1);
}
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 or west 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 = Math2.ToDegrees(lam12);
r.Longitude2 = ((outmask & GeodesicMask.LONG_UNROLL) != 0) ? _lon1 + lon12 :
GeoMath.AngNormalize(r.Longitude1 + GeoMath.AngNormalize(lon12));
}
if ((outmask & GeodesicMask.LATITUDE) != 0)
{
r.Latitude2 = GeoMath.Atan2d(sbet2, _f1 * cbet2);
}
if ((outmask & GeodesicMask.AZIMUTH) != 0)
{
r.FinalAzimuth = 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.
r.ReducedLength = _b * ((dn2 * (_csig1 * ssig2) - _dn1 * (_ssig1 * csig2))
- _csig1 * csig2 * J12);
}
if ((outmask & GeodesicMask.GEODESICSCALE) != 0)
{
double t = _k2 * (ssig2 - _ssig1) * (ssig2 + _ssig1) / (_dn1 + dn2);
r.GeodesicScale12 = csig12 + (t * ssig2 - csig2 * J12) * _ssig1 / _dn1;
r.GeodesicScale21 = 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 atan2 so no need to normalize
salp12 = salp2 * _calp1 - calp2 * _salp1;
calp12 = calp2 * _calp1 + salp2 * _salp1;
}
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;
}
r.AreaUnderGeodesic = _c2 * Math.Atan2(salp12, calp12) + _A4 * (B42 - _B41);
}
return(r);
}