PyGeodetics is a Python library for performing geodetic computations like geodetic inverse and direct problems, conversions between different reference systems like ECEF to ENU, ECEF to geographic etc.
- Convert geodetic coordinates (latitude, longitude, height) to ECEF (Earth-Centered, Earth-Fixed)
- Convert ECEF to geodetic coordinates
- Transform ECEF to local ENU (East-North-Up) and NED (North-East-Down) systems
- Solve geodetic inverse and direct problems
- Distance between two points along the ellipsoid
- Compute radius of curvature and mean radius of the reference ellipsoid
- Support for different reference ellipsoids
pip install pygeodetics- Geodetic to ECEF
- ECEF to Geodetic
- ECEF to ENU
- ECEF to NED
- Geodetic Inverse Problem on the GRS80 ellipsoid
- Geodetic Direct Problem on the GRS80 ellipsoid
- Radius of Curvature for a given Azimuth using Euler's equation.
- Calculate the mean Radius of the International1924 Ellipsoid
- Calculate the distance between two points on the ellipsoid (Vincenty formula)
- Calculate the meridional radius of curvature (M) at a given latitude
- Calculate the normal radius of curvature (N) at a given latitude.
- Use of Mercator Variant C projection
- Use of Transverse Mercator projection
from pygeodetics import Geodetic
# Initialize Geodetic class with WGS84 ellipsoid
geod = Geodetic()
lat = 59.907072474276958 # Latitude in degrees
lon = 10.754482924017791 # Longitude in degrees
h = 63.8281 # Height in meters
X, Y, Z = geod.geod2ecef(lat, lon, h)
print(f"Geodetic to ECEF:\nX: {X:.4f} m\nY: {Y:.4f} m\nZ: {Z:.4f} m")from pygeodetics import Geodetic
X, Y, Z = 3149785.9652, 598260.8822, 5495348.4927
geod = Geodetic()
lat, lon, h = geod.ecef2geod(X, Y, Z, angle_unit='deg')
print(f"ECEF to Geodetic:\nLatitude: {lat:.6f}°\nLongitude: {lon:.6f}°\nHeight: {h:.3f} m")from pygeodetics import Geodetic
X, Y, Z = 3149785.9652, 598260.8822, 5495348.4927
lat0, lon0, h0 = 58.907072, 9.75448, 63.8281
e, n, u = Geodetic().ecef2enu(X, Y, Z, lat0, lon0, h0, radians=False)
print(f"ECEF to ENU:\nEast: {e:.6f} m\nNorth: {n:.6f} m\nUp: {u:.6f} m")from pygeodetics import Geodetic
X, Y, Z = 3149785.9652, 598260.8822, 5495348.4927
lat0, lon0, h0 = 58.907072, 9.75448, 63.8281
n, e, d = Geodetic().ecef2ned(X, Y, Z, lat0, lon0, h0)
print(f"ECEF to NED:\nNorth: {n:.6f} m\nEast: {e:.6f} m\nDown: {d:.6f} m")from pygeodetics import Geodetic
from pygeodetics.Ellipsoid import GRS80
geod = Geodetic(GRS80())
lat1, lon1 = 52.2296756, 21.0122287
lat2, lon2 = 41.8919300, 12.5113300
az1, az2, distance = geod.inverse_problem(lat1, lon1, lat2, lon2, quadrant_correction=False)
print(f"Geodetic Inverse Problem:\nForward Azimuth: {az1:.6f}°\nFinal Azimuth at P2: {az2:.6f}°\nDistance: {distance:.3f} m")from pygeodetics import Geodetic
from pygeodetics.Ellipsoid import GRS80
geod = Geodetic(GRS80())
lat1, lon1 = 52.2296756, 21.0122287
az1 = -147.4628043168
d = 1316208.08334
lat2, lon2, az2 = geod.direct_problem(lat1, lon1, az1, d, quadrant_correction=True)
print(f"Geodetic Direct Problem:\nDestination Latitude: {lat2:.6f}°\nDestination Longitude: {lon2:.6f}°\nFinal Azimuth at destination: {az2:.6f}°")from pygeodetics import Geodetic
lat = 45
azimuth = 30
radius = Geodetic().radius_of_curvature(lat, azimuth, radians=False)
print(f"Radius of Curvature:\n{radius:.3f} meters")from pygeodetics import Geodetic
from pygeodetics.Ellipsoid import International1924
geod = Geodetic(International1924())
mean_radius = geod.get_mean_radius()
print(f"Mean Radius of the Ellipsoid:\n{mean_radius:.3f} meters")from pygeodetics import Geodetic
# Define the coordinates of the first point
lat1 = 52.2296756
lon1 = 21.0122287
# Define the coordinates of the second point
lat2 = 41.8919300
lon2 = 12.5113300
distances = Geodetic().distance_between_two_points(lon1, lat1, lon2, lat2, radians=False)
print(f"Distances between the two points: {distances}")from pygeodetics import Geodetic
# Compute the mean radius of the ellipsoid at a given latitude
lat = 61.456121547 # Latitude in degrees
mradius = Geodetic().mrad(lat)
print(f"Mean Radius of the Ellipsoid at Latitude {lat}°: {mradius:.3f} meters")from pygeodetics import Geodetic
# Compute the normal radius of the ellipsoid at a given latitude
lat = 61.456121547 # Latitude in degrees
mradius = Geodetic().nrad(lat)
print(f"Normal Radius of the Ellipsoid at Latitude {lat}°:\n{mradius:.3f} meters")from pygeodetics import MercatorVariantC
import numpy as np
conv = MercatorVariantC(
a=6378245.0, f=1/298.3,
latSP1=np.deg2rad(42), latFO=np.deg2rad(42), lonFO=np.deg2rad(51),
EFO=0.0, NFO=0.0
)
lon, lat = 53, 53
E, N = conv.geog_to_projected([[lon, lat]], unit="deg").ravel()
rlon, rlat = conv.projected_to_geog([[E, N]]).ravel()
print(f"Easting = {E:.2f} m\nNorthing = {N:.2f} m")
print(f"Reversed lon = {rlon:.8f}°\nReversed lat = {rlat:.8f}°")import numpy as np
from pygeodetics import TransverseMercator
np.set_printoptions(precision=8, suppress=True)
# Projection parameters (radians for origins)
lat_origin = 0.0 # φ0 (rad)
lon_origin = np.radians(9.0) # λ0 (rad), e.g. UTM zone 32
scale_factor = 0.9996 # k0
false_easting = 500000.0 # FE (m)
false_northing = 0.0 # FN (m)
# Ellipsoid (WGS84)
a = 6378137.0
f = 1 / 298.257223563
tm = TransverseMercator(
lat_origin, lon_origin, scale_factor,
false_easting, false_northing, a, f
)
# Input coordinates (lon, lat, h)
coords = np.array([
[3.0, 60.0, 100.0],
[3.2, 61.0, 102.0],
])
# Perform forward projection
easting, northing, height = tm.geog_to_projected(coords, unit="deg")
results = np.vstack((easting, northing, height)).T
print(f"\nProjected Coordinates TM:\n{results}")
# Perform inverse projection
proj_coordinates = np.vstack([easting, northing, height]).T
lon_back, lat_back, height = tm.projected_to_geog(proj_coordinates, unit="deg")
results = np.vstack((lon_back, lat_back, height)).T
print(f"\nGeographic Coordinates TM:\n{results}")This section provides the mathematical foundation for the computations performed in PyGeodetics.
The conversion from geodetic coordinates
where:
-
$N = \frac{a}{\sqrt{1 - e^2 \sin^2\phi}}$ is the prime vertical radius of curvature, -
$a$ is the semi-major axis, -
$b$ is the semi-minor axis, -
$e^2 = \frac{a^2 - b^2}{a^2}$ is the first eccentricity squared, -
$\phi$ is the geodetic latitude, -
$\lambda$ is the geodetic longitude, -
$h$ is the height above the ellipsoid.
PyGeodetics implements three solvers for the inverse problem
-
Compute the longitude:
$$\lambda=\arctan2(Y, X)$$ -
Compute the intermediate values:
$$p=\sqrt{X^2 + Y^2}, \quad \theta = \arctan2(Z,a,; p,b)$$ -
Compute the latitude:
$$\phi=\arctan2!\left(Z + e'^2 b \sin^3\theta,; p - e^2 a \cos^3\theta\right)$$ ,where$e'^2=\frac{a^2 - b^2}{b^2}$ is the second eccentricity squared. -
Compute the height:
$$h=\frac{p}{\cos\phi}-N$$
The geodetic inverse problem calculates the geodesic distance
-
Compute the reduced latitudes (also called parametric latitudes):
$$U_1 = \arctan\left((1 - f) \tan\phi_1\right), \quad U_2 = \arctan\left((1 - f) \tan\phi_2\right)$$ ,where$f = \frac{a - b}{a}$ is the flattening. -
Set
$L = \lambda_2 - \lambda_1$ and initialise$\lambda = L$ . Iterate$\lambda$ (longitude on the auxiliary sphere) together with the angular distance$\sigma$ :$$\sigma = \arctan2!\left(\sqrt{(\cos U_2 \sin\lambda)^2 + (\cos U_1 \sin U_2 - \sin U_1 \cos U_2 \cos\lambda)^2},; \sin U_1 \sin U_2 + \cos U_1 \cos U_2 \cos\lambda\right)$$ The full update equation for
$\lambda$ and the auxiliary quantities$\sin\alpha$ ,$\cos 2\sigma_m$ ,$C$ are given in Section 7. -
Once converged, compute the forward and reverse azimuths:
$$\alpha_1 = \arctan2\left(\cos U_2 \sin\lambda,; \cos U_1 \sin U_2 - \sin U_1 \cos U_2 \cos\lambda\right)$$ $$\alpha_2 = \arctan2\left(\cos U_1 \sin\lambda,; -\sin U_1 \cos U_2 + \cos U_1 \sin U_2 \cos\lambda\right)$$ -
Compute the geodesic distance using the series expansion in
$u^2 = \cos^2\alpha,(a^2 - b^2)/b^2$ :$$s = b,A,(\sigma - \Delta\sigma)$$ with
$A$ ,$B$ and$\Delta\sigma$ defined in Section 7. The simpler one-term form$s \approx b(\sigma - \tfrac{f}{4} \sin\sigma \cos(2\sigma_m + \sigma))$ is not used here — it loses several kilometres of accuracy on continental baselines.
The radius of curvature in the meridian (
The transformation from ECEF to local East-North-Up (ENU) coordinates is given by:
where
PyGeodetics returns the IUGG arithmetic mean radius:
This is one of several conventional definitions (others include the authalic and volumetric mean radii); it is the simple arithmetic mean of the three semi-axes
Vincenty's formula is used to calculate the geodesic distance between two points on an ellipsoid. The method iteratively solves for the distance
-
Reduced Latitude: Compute the reduced latitudes:
$$U_1=\arctan\left((1 - f) \cdot \tan\phi_1\right), \quad U_2=\arctan\left((1 - f) \cdot\tan\phi_2\right)$$ ,where
$f = \frac{a - b}{a}$ is the flattening. -
Longitude Difference: Compute the difference in longitudes:
$$L=\lambda_2 - \lambda_1$$ -
Iterative Solution: Initialize
$\lambda = L$ and iteratively solve for$\lambda$ using:$$\lambda=L + (1 - C) f \sin\alpha \left[\sigma + C \sin\sigma \left(\cos2\sigma_m + C \cos\sigma \left(-1 + 2 \cos^2 2\sigma_m\right)\right)\right]$$ where:
-
$\sigma = \arctan2\left(\sqrt{(\cos U_2 \sin\lambda)^2 + (\cos U_1 \sin U_2 - \sin U_1 \cos U_2 \cos\lambda)^2}, \sin U_1 \sin U_2 + \cos U_1 \cos U_2 \cos\lambda\right)$ , -
$\sin\alpha = \frac{\cos U_1 \cos U_2 \sin\lambda}{\sin\sigma}$ , -
$\cos2\sigma_m = \cos\sigma - \frac{2 \sin U_1 \sin U_2}{\cos^2\alpha}$ , -
$C = \frac{f}{16} \cos^2\alpha \left(4 + f \left(4 - 3 \cos^2\alpha\right)\right)$ .
The iteration stops when
$|\lambda - \lambda_{\text{prev}}| < \text{tolerance}$ . -
-
Geodesic Distance: Compute the geodesic distance:
$$s=b A \left[\sigma - \delta\sigma\right]$$ ,where:-
$u^2=\frac{\cos^2\alpha (a^2 - b^2)}{b^2}$ , -
$A=1 + \frac{u^2}{16384} \left(4096 + u^2 \left(-768 + u^2 (320 - 175 u^2)\right)\right)$ , -
$B=\frac{u^2}{1024} \left(256 + u^2 \left(-128 + u^2 (74 - 47 u^2)\right)\right)$ , -
$\delta\sigma=B \sin\sigma \left[\cos2\sigma_m + \frac{B}{4} \left(\cos\sigma \left(-1 + 2 \cos^2 2\sigma_m\right) - \frac{B}{6} \cos2\sigma_m \left(-3 + 4 \sin^2\sigma\right) \left(-3 + 4 \cos^2 2\sigma_m\right)\right)\right]$ .
-
-
Azimuths: Compute the forward and reverse azimuths:
$$\alpha_1 = \arctan2\left(\cos U_2 \sin\lambda, \cos U_1 \sin U_2 - \sin U_1 \cos U_2 \cos\lambda\right)$$ $$\alpha_2 = \arctan2\left(\cos U_1 \sin\lambda, -\sin U_1 \cos U_2 + \cos U_1 \sin U_2 \cos\lambda\right)$$
Vincenty's formula is highly accurate for most geodetic calculations but may fail to converge for nearly antipodal points.
The meridional radius of curvature, denoted as
where:
-
$a$ is the semi-major axis of the ellipsoid, -
$e^2 = \frac{a^2 - b^2}{a^2}$ is the first eccentricity squared, -
$\phi$ is the geodetic latitude.
This formula accounts for the flattening of the ellipsoid and the variation in curvature with latitude.
The normal radius of curvature, denoted as
where:
-
$a$ is the semi-major axis of the ellipsoid, -
$e^2 = \frac{a^2 - b^2}{a^2}$ is the first eccentricity squared, -
$\phi$ is the geodetic latitude.
The value of
The grid convergence, denoted as
The grid convergence
where:
-
$\Delta\lambda = \lambda - \lambda_0$ is the longitude difference from the central meridian, -
$\epsilon^2 = \frac{e^2}{1 - e^2} \cos^2\phi$ is the second eccentricity squared, -
$e^2 = \frac{a^2 - b^2}{a^2}$ is the first eccentricity squared, -
$a$ is the semi-major axis, -
$b$ is the semi-minor axis, -
$\phi$ is the geodetic latitude, -
$\lambda$ is the geodetic longitude, -
$\lambda_0$ is the central meridian.
The grid convergence
where:
-
$\phi_f$ is the footpoint latitude, computed iteratively, -
$N_f = \frac{a}{\sqrt{1 - e^2 \sin^2\phi_f}}$ is the normal radius of curvature at the footpoint latitude, -
$\epsilon_f^2 = \frac{e^2}{1 - e^2} \cos^2\phi_f$ is the second eccentricity squared at the footpoint latitude, -
$x$ is the easting coordinate (adjusted for false easting), -
$y$ is the northing coordinate.
The scale factor, denoted as
The scale factor
where:
-
$\Delta\lambda = \lambda - \lambda_0$ is the longitude difference from the central meridian, -
$\epsilon^2 = \frac{e^2}{1 - e^2} \cos^2\phi$ is the second eccentricity squared, -
$e^2 = \frac{a^2 - b^2}{a^2}$ is the first eccentricity squared, -
$a$ is the semi-major axis, -
$b$ is the semi-minor axis, -
$\phi$ is the geodetic latitude, -
$\lambda$ is the geodetic longitude, -
$\lambda_0$ is the central meridian.
The scale factor
where:
-
$M_f = \frac{a(1 - e^2)}{(1 - e^2 \sin^2\phi_f)^{3/2}}$ is the meridional radius of curvature at the footpoint latitude, -
$N_f = \frac{a}{\sqrt{1 - e^2 \sin^2\phi_f}}$ is the normal radius of curvature at the footpoint latitude, -
$\phi_f$ is the footpoint latitude, computed iteratively, -
$x$ is the easting coordinate (adjusted for false easting), -
$y$ is the northing coordinate.
For a spherical Earth model, the scale factor
where:
-
$x$ is the easting coordinate (adjusted for false easting), -
$R$ is the radius of the sphere.
This section explains the mathematical foundation for the projections method supported by the library. In this release supported methods are:
- Mercator Variant C
- Transverse Mercator projections.
The Mercator Variant C projection is a cylindrical map projection that preserves angles, making it conformal. This projection is a variant of the Mercator projection, with specific parameters defined for its implementation.
The following table summarizes the key parameters of the Mercator Variant C projection as defined by EPSG:
| Parameter Name | Parameter EPSG Code | Sign Reversal | Description |
|---|---|---|---|
| Latitude of 1st standard parallel | 8823 | No | For a conic projection with two standard parallels, this is the latitude of one of the parallels of intersection of the cone with the ellipsoid. Scale is true along this parallel. |
| Longitude of natural origin | 8802 | No | The longitude of the point from which the values of both the geographical coordinates on the ellipsoid and the grid coordinates on the projection are deemed to increment or decrement for computational purposes. |
| Latitude of false origin | 8821 | No | The latitude of the point which is not the natural origin and at which grid coordinate values false easting and false northing are defined. |
| Easting at false origin | 8826 | No | The easting value assigned to the false origin. |
| Northing at false origin | 8827 | No | The northing value assigned to the false origin. |
The forward projection transforms geographic coordinates
- Constants:
$k_0 = \frac{\cos \phi_1}{\sqrt{1 - e^2 \sin^2 \phi_1}}$ - All logarithms below are natural logarithms (base
$\mathrm{e}$ ). To avoid clashing with the eccentricity$e$ , the natural-log base is denoted$\mathrm{e}$ in this section.
-
Compute the projection constant
$k_0$ :$$k_0 = \frac{\cos \phi_1}{\sqrt{1 - e^2 \sin^2 \phi_1}}$$ where:
-
$\phi_1$ = latitude of the first standard parallel (absolute value, positive), -
$e^2 = \tfrac{a^2 - b^2}{a^2}$ = eccentricity squared, -
$a$ = semi-major axis,$b$ = semi-minor axis.
-
-
Compute the meridional arc at latitude
$\phi$ :$$M(\phi) = a k_0 ,\ln \left[ \tan!\left(\tfrac{\pi}{4} + \tfrac{\phi}{2}\right) \left(\frac{1 - e \sin\phi}{1 + e \sin\phi}\right)^{\tfrac{e}{2}} \right]$$ -
Compute Easting and Northing:
$$E = E_0 + a k_0 (\lambda - \lambda_0)$$ $$N = N_0 - M(\phi_0) + M(\phi)$$ where:
-
$(\lambda_0, \phi_0)$ = longitude/latitude of the false origin, -
$(E_0, N_0)$ = false easting and northing.
-
The inverse projection transforms projected coordinates
-
Longitude:
$$\lambda = \lambda_0 + \frac{E - E_0}{a k_0}$$ -
Latitude (iterative series):
Define:
$$\chi = \frac{\pi}{2} - 2 \arctan !\left( \exp!\left[-\frac{N - N_0 + M(\phi_0)}{a k_0}\right] \right)$$ Then:
$$\phi=\chi+\left(\tfrac{e^2}{2} + \tfrac{5 e^4}{24} + \tfrac{e^6}{12} + \tfrac{13 e^8}{360}\right) \sin(2\chi)+\left(\tfrac{7 e^4}{48} +\tfrac{29 e^6}{240} + \tfrac{811 e^8}{11520}\right) \sin(4\chi)+\left(\tfrac{7 e^6}{120} + \tfrac{81 e^8}{1120}\right) \sin(6\chi)+\left(\tfrac{4279 e^8}{161280}\right) \sin(8\chi)$$
The Transverse Mercator (TM) projection is a conformal map projection widely used for large-scale mapping, such as the Universal Transverse Mercator (UTM) system.
The following table summarizes the key parameters of the Transverse Mercator projection as defined by EPSG:
| Parameter Name | Parameter EPSG Code | Sign Reversal | Description |
|---|---|---|---|
| Latitude of natural origin | 8801 | No | The latitude of the point from which the values of both the geographical coordinates on the ellipsoid and the grid coordinates on the projection are deemed to increment or decrement for computational purposes. |
| Longitude of natural origin | 8802 | No | The longitude of the point from which the values of both the geographical coordinates on the ellipsoid and the grid coordinates on the projection are deemed to increment or decrement for computational purposes. |
| Scale factor at natural origin | 8805 | No | The factor by which the map grid is reduced or enlarged during the projection process, defined by its value at the natural origin. |
| False easting | 8806 | No | The value assigned to the abscissa (east or west) axis of the projection grid at the natural origin to avoid negative coordinates over parts of the mapped area. |
| False northing | 8807 | No | The value assigned to the ordinate (north or south) axis of the projection grid at the natural origin to avoid negative coordinates over parts of the mapped area. |
The forward projection transforms geographic coordinates
-
Calculate constants for the projection:
$$n = \frac{f}{2 - f}$$ $$B = \frac{a}{1 + n} \left(1 + \frac{n^2}{4} + \frac{n^4}{64}\right)$$ $$h_1=\frac{n}{2} - \frac{2}{3}n^2 + \frac{5}{16}n^3 + \frac{41}{180}n^4$$ $$h_2=\frac{13}{48}n^2 - \frac{3}{5}n^3 + \frac{557}{1440}n^4$$ $$h_3=\frac{61}{240}n^3 - \frac{103}{140}n^4$$ $$h_4=\frac{49561}{161280}n^4$$ -
Compute the meridional arc distance from the equator to the projection origin:
If
$\phi_0 = 0$ $$M_0 = 0$$ or if
$\phi_0 = 90^\circ , (\pi/2 , \text{radians})$ $$M_0 = B \cdot \frac{\pi}{2}$$ or if
$\phi_0 = -90^\circ , (-\pi/2 , \text{radians})$ $$M_0 = B \cdot \left(-\frac{\pi}{2}\right)$$ Otherwise:
$$Q_0 = \sinh^{-1}(\tan\phi_0) - e \tanh^{-1}(e \sin\phi_0)$$ $$\beta_0 = \tan^{-1}(\sinh Q_0)$$ $$\xi_{0,0} = \sin^{-1}(\sin\beta_0)$$ $$\xi_0 = \xi_{0,0} + h_1 \sin(2\xi_{0,0}) + h_2 \sin(4\xi_{0,0}) + h_3 \sin(6\xi_{0,0}) + h_4 \sin(8\xi_{0,0})$$ $$M_0 = B \xi_0$$ -
Compute intermediate values for the given latitude
$\phi$ :$$Q = \sinh^{-1}(\tan\phi) - e \tanh^{-1}(e \sin\phi)$$ $$\beta = \tan^{-1}(\sinh Q)$$ $$\eta_0 = \tanh^{-1}(\cos\beta \sin(\lambda - \lambda_0))$$ $$\xi_0 = \sin^{-1}(\sin\beta \cosh\eta_0)$$ $$\xi = \xi_0 + h_1 \sin(2\xi_0) \cosh(2\eta_0) + h_2 \sin(4\xi_0) \cosh(4\eta_0) + h_3 \sin(6\xi_0) \cosh(6\eta_0) + h_4 \sin(8\xi_0) \cosh(8\eta_0)$$ $$\eta = \eta_0 + h_1 \cos(2\xi_0) \sinh(2\eta_0) + h_2 \cos(4\xi_0) \sinh(4\eta_0) + h_3 \cos(6\xi_0) \sinh(6\eta_0) + h_4 \cos(8\xi_0) \sinh(8\eta_0)$$ -
Compute the easting and northing:
$$E = E_0 + k_0 B \eta$$ $$N = N_0 + k_0 \left(B \xi - M_0\right)$$
The inverse projection transforms projected coordinates
-
Compute intermediate values:
$$\eta' = \frac{E - E_0}{k_0 B}, \quad \xi' = \frac{(N - N_0) + k_0 M_0}{k_0 B}$$ -
Iteratively compute
$\xi_0'$ and$\eta_0'$ :$$\xi_0' = \xi' - \left(h_1 \sin(2\xi') \cosh(2\eta') + h_2 \sin(4\xi') \cosh(4\eta') + h_3 \sin(6\xi') \cosh(6\eta') + h_4 \sin(8\xi') \cosh(8\eta')\right)$$ $$\eta_0' = \eta' - \left(h_1 \cos(2\xi') \sinh(2\eta') + h_2 \cos(4\xi') \sinh(4\eta') + h_3 \cos(6\xi') \sinh(6\eta') + h_4 \cos(8\xi') \sinh(8\eta')\right)$$ -
Compute
$\beta'$ and$Q'$ :$$\beta' = \sin^{-1}\left(\frac{\sin\xi_0'}{\cosh\eta_0'}\right)$$ $$Q' = \sinh^{-1}(\tan\beta')$$ -
Iteratively compute latitude
$\phi$ :$$Q'' = Q' + e \tanh^{-1}(e \tanh Q')$$ Repeat until the change in
$Q''$ is insignificant. Then:$$\phi = \tan^{-1}(\sinh Q'')$$ -
Compute longitude
$\lambda$ :$$\lambda = \lambda_0 + \sin^{-1}\left(\frac{\tanh\eta_0'}{\cos\beta'}\right)$$
- These formulas follow EPSG Guidance Note 7-2.
This project is licensed under the MIT License.
