/// <summary>
/// Converts ‘this’ (geocentric) cartesian (x/y/z) point to (ellipsoidal geodetic) latitude/longitude coordinates on specified datum.
/// Uses Bowring’s (1985) formulation for μm precision in concise form.
/// </summary>
/// <param name="toDatum">Datum to use when converting point.</param>
public LatLonEllipsoidal ToLatLonE(Datum toDatum)
{
var Major = toDatum.Ellipsoid.Major;
var Minor = toDatum.Ellipsoid.Minor;
var Flattening = toDatum.Ellipsoid.Flattening;
var E2 = 2 * Flattening - Flattening * Flattening; // 1st eccentricity squared ≡ (a²-b²)/a²
var ε2 = E2 / (1 - E2); // 2nd eccentricity squared ≡ (a²-b²)/b²
var P = Math.Sqrt(X * X + Y * Y); // distance from minor axis
var R = Math.Sqrt(P * P + Z * Z); // polar radius
// parametric latitude (Bowring eqn 17, replacing tanβ = z·a / p·b)
var Tanβ = (Minor * Z) / (Major * P) * (1 + ε2 * Minor / R);
var Sinβ = Tanβ / Math.Sqrt(1 + Tanβ * Tanβ);
var Cosβ = Sinβ / Tanβ;
// geodetic latitude (Bowring eqn 18: tanφ = z+ε²bsin³β / p−e²cos³β)
var φ = Math.Atan2(Z + ε2 * Minor * Sinβ * Sinβ * Sinβ, P - E2 * Major * Cosβ * Cosβ * Cosβ);
// longitude
var λ = Math.Atan2(Y, X);
// height above ellipsoid (Bowring eqn 7) [not currently used]
var Sinφ = Math.Sin(φ);
var Cosφ = Math.Cos(φ);
var V = Major / Math.Sqrt(1 - E2 * Sinφ * Sinφ); // length of the normal terminated by the minor axis
var H = P * Cosφ + Z * Sinφ - (Major * Major / V);
var point = new LatLonEllipsoidal(φ.ToDegrees(), λ.ToDegrees(), toDatum);
return(point);
}
/// <summary>
/// /Converts ‘this’ lat/lon coordinate to new coordinate system.
/// </summary>
/// <param name="toDatum">Datum this coordinate is to be converted to.</param>
/// <example>
/// var pWGS84 = new LatLon(51.4778, -0.0016, Datum.WGS84);
/// var pOSGB = pWGS84.convertDatum(Datum.OSGB36); // 51.4773°N, 000.0000°E
/// </example>
public LatLonEllipsoidal ConvertDatum(Datum toDatum)
{
LatLonEllipsoidal OldLatLon = this;
double[] Transform = toDatum.Transform;
bool UsingWgs84 = false;
if (OldLatLon.Datum == Datum.WGS84)
{
// converting from WGS 84
Transform = toDatum.Transform;
UsingWgs84 = true;
}
if (toDatum == Datum.WGS84)
{
// converting to WGS 84; use inverse transform (don't overwrite original!)
UsingWgs84 = true;
Transform = new double[7];
for (var p = 0; p < 7; p++)
{
Transform[p] = -OldLatLon.Datum.Transform[p];
}
}
if (!UsingWgs84)
{
// neither this.datum nor toDatum are WGS84: convert this to WGS84 first
OldLatLon = ConvertDatum(Datum.WGS84);
}
var OldCartesian = OldLatLon.ToCartesian(); // convert polar to cartesian...
var NewCartesian = OldCartesian.ApplyTransform(Transform); // ...apply transform...
var NewLatLon = NewCartesian.ToLatLonE(toDatum); // ...and convert cartesian to polar
return(NewLatLon);
}
/// <summary>Converts Ordnance Survey grid reference easting/northing coordinate to latitude/longitude (SW corner of grid square).
///
/// Note formulation implemented here due to Thomas, Redfearn, etc is as published by OS, but is
/// inferior to Krüger as used by e.g. Karney 2011.</summary>
///<param name="gridRef">Grid ref E/N to be converted to lat/long (SW corner of grid square).</param>
///<param name="datum">Datum to convert grid reference into.</param>
///<example>var gridref = new OsGridRef(651409.903, 313177.270);
/// var pWgs84 = OsGridRef.osGridToLatLon(gridref); // 52°39′28.723″N, 001°42′57.787″E
/// to obtain (historical) OSGB36 latitude/longitude point:
/// var pOsgb = OsGridRef.osGridToLatLon(gridref, LatLon.datum.OSGB36); // 52°39′27.253″N, 001°43′04.518″E
///</example>
///<remarks>Currently not accurate enough, owing to a lack of precision in C#'s number handling. Will be ported to use arbitrary-precision maths.</remarks>
public LatLonEllipsoidal ToLatLon(Datum datum)
{
var E = Easting;
var N = Northing;
var a = 6377563.396;
var b = 6356256.909; // Airy 1830 major & minor semi-axes
var F0 = 0.9996012717; // NatGrid scale factor on central meridian
var φ0 = (49.0).ToRadians();
var λ0 = (-2.0).ToRadians(); // NatGrid true origin is 49°N,2°W
var N0 = -100000;
var E0 = 400000; // northing & easting of true origin, metres
var e2 = 1 - b * b / (a * a); // eccentricity squared
var n = (a - b) / (a + b);
var n2 = n * n;
var n3 = n * n * n; // n, n², n³
var φ = φ0;
var M = 0.0;
do
{
φ = (N - N0 - M) / (a * F0) + φ;
var Ma = (1 + n + 1.25 * n2 + 1.25 * n3) * (φ - φ0);
var Mb = (3 * n + 3 * n * n + 2.625 * n3) * Math.Sin(φ - φ0) * Math.Cos(φ + φ0);
var Mc = (1.875 * n2 + 1.875 * n3) * Math.Sin(2 * (φ - φ0)) * Math.Cos(2 * (φ + φ0));
var Md = (35.0 / 24.0) * n3 * Math.Sin(3 * (φ - φ0)) * Math.Cos(3 * (φ + φ0));
M = b * F0 * (Ma - Mb + Mc - Md); // meridional arc
} while (N - N0 - M >= 0.00001); // ie until < 0.01mm
var cosφ = Math.Cos(φ);
var sinφ = Math.Sin(φ);
var ν = a * F0 / Math.Sqrt(1 - e2 * sinφ * sinφ); // nu = transverse radius of curvature
var ρ = a * F0 * (1 - e2) / Math.Pow(1 - e2 * sinφ * sinφ, 1.5); // rho = meridional radius of curvature
var η2 = ν / ρ - 1; // eta = ?
var tanφ = Math.Tan(φ);
var tan2φ = tanφ * tanφ;
var tan4φ = tan2φ * tan2φ;
var tan6φ = tan4φ * tan2φ;
var secφ = 1 / cosφ;
var ν3 = ν * ν * ν;
var ν5 = ν3 * ν * ν;
var ν7 = ν5 * ν * ν;
var VII = tanφ / (2 * ρ * ν);
var VIII = tanφ / (24 * ρ * ν3) * (5 + 3 * tan2φ + η2 - 9 * tan2φ * η2);
var IX = tanφ / (720 * ρ * ν5) * (61 + 90 * tan2φ + 45 * tan4φ);
var X = secφ / ν;
var XI = secφ / (6 * ν3) * (ν / ρ + 2 * tan2φ);
var XII = secφ / (120 * ν5) * (5 + 28 * tan2φ + 24 * tan4φ);
var XIIA = secφ / (5040 * ν7) * (61 + 662 * tan2φ + 1320 * tan4φ + 720 * tan6φ);
var dE = (E - E0);
var dE2 = dE * dE;
var dE3 = dE2 * dE;
var dE4 = dE2 * dE2;
var dE5 = dE3 * dE2;
var dE6 = dE4 * dE2;
var dE7 = dE5 * dE2;
φ = φ - VII * dE2 + VIII * dE4 - IX * dE6;
var λ = λ0 + X * dE - XI * dE3 + XII * dE5 - XIIA * dE7;
var Point = new LatLonEllipsoidal(φ.ToDegrees(), λ.ToDegrees(), Datum.OSGB36);
if (datum != Datum.OSGB36)
{
Point = Point.ConvertDatum(datum);
}
return(Point);
}
public OsGridRef ToGridRef()
{
// if necessary convert to OSGB36 first
if (Datum != Datum.OSGB36)
{
var Point = new LatLonEllipsoidal(Latitude, Longitude, Datum);
Point = Point.ConvertDatum(Datum.OSGB36);
Latitude = Point.Latitude;
Longitude = Point.Longitude;
Datum = Datum.OSGB36;
}
var φ = Latitude.ToRadians();
var λ = Longitude.ToRadians();
var A = 6377563.396;
var B = 6356256.909; // Airy 1830 major & minor semi-axes
var F0 = 0.9996012717; // NatGrid scale factor on central meridian
var φ0 = (49.0).ToRadians();
var λ0 = (-2.0).ToRadians(); // NatGrid true origin is 49°N,2°W
var N0 = -100000;
var E0 = 400000; // northing & easting of true origin, metres
var E2 = 1 - (B * B) / (A * A); // eccentricity squared
var N = (A - B) / (A + B);
var N2 = N * N;
var N3 = N * N * N; // n, n², n³
var Cosφ = Math.Cos(φ);
var Sinφ = Math.Sin(φ);
var ν = A * F0 / Math.Sqrt(1 - E2 * Sinφ * Sinφ); // nu = transverse radius of curvature
var ρ = A * F0 * (1 - E2) / Math.Pow(1 - E2 * Sinφ * Sinφ, 1.5); // rho = meridional radius of curvature
var η2 = ν / ρ - 1; // eta = ?
var Ma = (1 + N + 1.25 * N2 + 1.25 * N3) * (φ - φ0);
var Mb = (3 * N + 3 * N * N + 2.625 * N3) * Math.Sin(φ - φ0) * Math.Cos(φ + φ0);
var Mc = (1.875 * N2 + 1.875 * N3) * Math.Sin(2 * (φ - φ0)) * Math.Cos(2 * (φ + φ0));
var Md = 35.0 / 24.0 * N3 * Math.Sin(3 * (φ - φ0)) * Math.Cos(3 * (φ + φ0));
var M = B * F0 * (Ma - Mb + Mc - Md); // meridional arc
var Cos3φ = Cosφ * Cosφ * Cosφ;
var Cos5φ = Cos3φ * Cosφ * Cosφ;
var Tan2φ = Math.Tan(φ) * Math.Tan(φ);
var Tan4φ = Tan2φ * Tan2φ;
var I = M + N0;
var II = (ν / 2) * Sinφ * Cosφ;
var III = (ν / 24) * Sinφ * Cos3φ * (5 - Tan2φ + 9 * η2);
var IIIA = (ν / 720) * Sinφ * Cos5φ * (61 - 58 * Tan2φ + Tan4φ);
var IV = ν * Cosφ;
var V = (ν / 6) * Cos3φ * (ν / ρ - Tan2φ);
var VI = (ν / 120) * Cos5φ * (5 - 18 * Tan2φ + Tan4φ + 14 * η2 - 58 * Tan2φ * η2);
var Δλ = λ - λ0;
var Δλ2 = Δλ * Δλ;
var Δλ3 = Δλ2 * Δλ;
var Δλ4 = Δλ3 * Δλ;
var Δλ5 = Δλ4 * Δλ;
var Δλ6 = Δλ5 * Δλ;
var North = I + II * Δλ2 + III * Δλ4 + IIIA * Δλ6;
var East = E0 + IV * Δλ + V * Δλ3 + VI * Δλ5;
North = Math.Round(North, 3); //North.toFixed(3)); // round to mm precision
East = Math.Round(East, 3);
return(new OsGridRef(East, North)); // gets truncated to SW corner of 1m grid square
}
