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# -*- coding: utf-8 -*- 

 

u'''A pure Python version of I{Karney}'s C{Exact Transverse Mercator} (ETM) projection. 

 

Classes L{Etm}, L{ETMError} and L{ExactTransverseMercator}, transcoded from I{Karney}'s 

C++ class U{TransverseMercatorExact<https://GeographicLib.SourceForge.io/html/ 

classGeographicLib_1_1TransverseMercatorExact.html>} (abbreviated as C{TMExact} below). 

 

Class L{ExactTransverseMercator} provides C{Exact Transverse Mercator} projections 

while instances of class L{Etm} represent ETM C{(easting, nothing)} locations. 

 

Following is a copy of I{Karney}'s U{TransverseMercatorExact.hpp 

<https://GeographicLib.SourceForge.io/html/TransverseMercatorExact_8hpp_source.html>} 

file C{Header}. 

 

Copyright (C) U{Charles Karney<mailto:Charles@Karney.com>} (2008-2021) and licensed 

under the MIT/X11 License. For more information, see the U{GeographicLib<https:// 

GeographicLib.SourceForge.io>} documentation. 

 

The method entails using the U{Thompson Transverse Mercator<https://WikiPedia.org/ 

wiki/Transverse_Mercator_projection>} as an intermediate projection. The projections 

from the intermediate coordinates to C{phi, lam} and C{x, y} are given by elliptic 

functions. The inverse of these projections are found by Newton's method with a 

suitable starting guess. 

 

The relevant section of L.P. Lee's paper U{Conformal Projections Based On Jacobian 

Elliptic Functions<https://DOI.org/10.3138/X687-1574-4325-WM62>} in part V, pp 

67-101. The C++ implementation and notation closely follow Lee, with the following 

exceptions:: 

 

Lee here Description 

 

x/a xi Northing (unit Earth) 

 

y/a eta Easting (unit Earth) 

 

s/a sigma xi + i * eta 

 

y x Easting 

 

x y Northing 

 

k e Eccentricity 

 

k^2 mu Elliptic function parameter 

 

k'^2 mv Elliptic function complementary parameter 

 

m k Scale 

 

zeta zeta Complex longitude = Mercator = chi in paper 

 

s sigma Complex GK = zeta in paper 

 

Minor alterations have been made in some of Lee's expressions in an attempt to 

control round-off. For example, C{atanh(sin(phi))} is replaced by C{asinh(tan(phi))} 

which maintains accuracy near C{phi = pi/2}. Such changes are noted in the code. 

''' 

# make sure int/int division yields float quotient, see .basics 

from __future__ import division as _; del _ # PYCHOK semicolon 

 

from pygeodesy.basics import copysign0, isnear0, neg, neg_, _xinstanceof 

from pygeodesy.datums import _ellipsoidal_datum, _WGS84 

from pygeodesy.elliptic import Elliptic, EllipticError 

from pygeodesy.errors import _incompatible 

from pygeodesy.fmath import cbrt, hypot, hypot1, hypot2 

from pygeodesy.fsums import Fsum, fsum1_ 

from pygeodesy.interns import EPS, EPS02, NN, PI_2, PI_4, \ 

_COMMASPACE_, _convergence_, _easting_, \ 

_1_EPS, _EPSmin, _lat_, _lon_, _no_, \ 

_northing_, _scale_, _0_0, _0_1, _0_5, \ 

_1_0, _2_0, _3_0, _4_0, _90_0, _180_0 

from pygeodesy.karney import _diff182, _fix90, _norm180 

from pygeodesy.lazily import _ALL_LAZY 

from pygeodesy.named import _NamedBase, _NamedTuple, _update_all 

from pygeodesy.props import deprecated_method, deprecated_property_RO, \ 

Property_RO, property_RO, property_doc_ 

# _update_all # from .named 

from pygeodesy.streprs import pairs, unstr 

from pygeodesy.units import Degrees, Easting, Float, Lat, Lon, Northing, \ 

Scalar, Scalar_ 

from pygeodesy.utily import atand, atan2d, sincos2 

from pygeodesy.utm import _cmlon, _K0_UTM, _LLEB, _parseUTM5, _toBand, _toXtm8, \ 

_to7zBlldfn, Utm, UTMError 

 

from math import asinh, atan2, degrees, radians, sinh, sqrt, tan 

 

__all__ = _ALL_LAZY.etm 

__version__ = '22.04.27' 

 

_OVERFLOW = _1_EPS**2 # about 2e+31 

_TOL_10 = _0_1 * EPS 

_TAYTOL = pow(EPS, 0.6) 

_TAYTOL2 = _2_0 * _TAYTOL 

_TRIPS = 21 # 7 might be sufficient 

 

 

class EasNorExact4Tuple(_NamedTuple): 

'''4-Tuple C{(easting, northing, convergence, scale)} in 

C{meter}, C{meter}, C{degrees} and C{scalar}. 

''' 

_Names_ = (_easting_, _northing_, _convergence_, _scale_) 

_Units_ = ( Easting, Northing, Degrees, Scalar) 

 

 

class ETMError(UTMError): 

'''Exact Transverse Mercator (ETM) parse, projection or other L{Etm} issue. 

''' 

pass 

 

 

class Etm(Utm): 

'''Exact Transverse Mercator (ETM) coordinate, a sub-class of L{Utm}, 

a Universal Transverse Mercator (UTM) coordinate using the 

L{ExactTransverseMercator} projection for highest accuracy. 

 

@note: Conversion of (geodetic) lat- and longitudes to/from L{Etm} 

coordinates is 3-4 times slower than to/from L{Utm}. 

 

@see: Karney's U{Detailed Description<https://GeographicLib.SourceForge.io/ 

html/classGeographicLib_1_1TransverseMercatorExact.html#details>}. 

''' 

_Error = ETMError # see utm.UTMError 

_exactTM = None 

 

def __init__(self, zone, hemisphere, easting, northing, band=NN, # PYCHOK expected 

datum=_WGS84, falsed=True, 

convergence=None, scale=None, name=NN): 

'''New L{Etm} coordinate. 

 

@arg zone: Longitudinal UTM zone (C{int}, 1..60) or zone with/-out 

I{latitudinal} Band letter (C{str}, '01C'|..|'60X'). 

@arg hemisphere: Northern or southern hemisphere (C{str}, C{'N[orth]'} 

or C{'S[outh]'}). 

@arg easting: Easting, see B{C{falsed}} (C{meter}). 

@arg northing: Northing, see B{C{falsed}} (C{meter}). 

@kwarg band: Optional, I{latitudinal} band (C{str}, 'C'|..|'X'). 

@kwarg datum: Optional, this coordinate's datum (L{Datum}, L{Ellipsoid}, 

L{Ellipsoid2} or L{a_f2Tuple}). 

@kwarg falsed: If C{True}, both B{C{easting}} and B{C{northing}} are 

C{falsed} (C{bool}). 

@kwarg convergence: Optional meridian convergence, bearing off grid North, 

clockwise from true North (C{degrees}) or C{None}. 

@kwarg scale: Optional grid scale factor (C{scalar}) or C{None}. 

@kwarg name: Optional name (C{str}). 

 

@raise ETMError: Invalid B{C{zone}}, B{C{hemishere}} or B{C{band}} 

or near-spherical B{C{datum}} or C{ellipsoid}. 

 

@raise TypeError: Invalid or near-spherical B{C{datum}}. 

 

@example: 

 

>>> import pygeodesy 

>>> u = pygeodesy.Etm(31, 'N', 448251, 5411932) 

''' 

Utm.__init__(self, zone, hemisphere, easting, northing, 

band=band, datum=datum, falsed=falsed, 

convergence=convergence, scale=scale, 

name=name) 

self.exactTM = self.datum.exactTM # ExactTransverseMercator(datum=self.datum) 

 

@property_doc_(''' the ETM projection (L{ExactTransverseMercator}).''') 

def exactTM(self): 

'''Get the ETM projection (L{ExactTransverseMercator}). 

''' 

return self._exactTM 

 

@exactTM.setter # PYCHOK setter! 

def exactTM(self, exactTM): 

'''Set the ETM projection (L{ExactTransverseMercator}). 

 

@raise ETMError: The B{C{exacTM}}'s datum incompatible 

with this ETM coordinate's C{datum}. 

''' 

_xinstanceof(ExactTransverseMercator, exactTM=exactTM) 

 

E = self.datum.ellipsoid 

if E != exactTM.ellipsoid: # may be None 

raise ETMError(repr(exactTM), txt=_incompatible(repr(E))) 

self._exactTM = exactTM 

self._scale0 = exactTM.k0 

 

def parse(self, strETM, name=NN): 

'''Parse a string to a similar L{Etm} instance. 

 

@arg strETM: The ETM coordinate (C{str}), 

see function L{parseETM5}. 

@kwarg name: Optional instance name (C{str}), 

overriding this name. 

 

@return: The instance (L{Etm}). 

 

@raise ETMError: Invalid B{C{strETM}}. 

 

@see: Function L{pygeodesy.parseUPS5}, L{pygeodesy.parseUTM5} 

and L{pygeodesy.parseUTMUPS5}. 

''' 

return parseETM5(strETM, datum=self.datum, Etm=self.classof, 

name=name or self.name) 

 

@deprecated_method 

def parseETM(self, strETM): # PYCHOK no cover 

'''DEPRECATED, use method L{Etm.parse}. 

''' 

return self.parse(strETM) 

 

def toLatLon(self, LatLon=None, unfalse=True, **unused): # PYCHOK expected 

'''Convert this ETM coordinate to an (ellipsoidal) geodetic point. 

 

@kwarg LatLon: Optional, ellipsoidal class to return the geodetic 

point (C{LatLon}) or C{None}. 

@kwarg unfalse: Unfalse B{C{easting}} and B{C{northing}} if 

C{falsed} (C{bool}). 

 

@return: This ETM coordinate as (B{C{LatLon}}) or a 

L{LatLonDatum5Tuple}C{(lat, lon, datum, convergence, 

scale)} if B{C{LatLon}} is C{None}. 

 

@raise EllipticError: No convergence transforming to lat- and 

longitude. 

 

@raise ETMError: This ETM coordinate's C{exacTM} and C{datum} 

incompatible. 

 

@raise TypeError: Invalid or non-ellipsoidal B{C{LatLon}}. 

 

@example: 

 

>>> from pygeodesy import ellipsoidalVincenty as eV, Etm 

>>> u = Etm(31, 'N', 448251.795, 5411932.678) 

>>> ll = u.toLatLon(eV.LatLon) # 48°51′29.52″N, 002°17′40.20″E 

''' 

if not self._latlon or self._latlon._toLLEB_args != (unfalse, self.exactTM): 

self._toLLEB(unfalse=unfalse) 

return self._latlon5(LatLon) 

 

def _toLLEB(self, unfalse=True, **unused): # PYCHOK signature 

'''(INTERNAL) Compute (ellipsoidal) lat- and longitude. 

''' 

xTM, d = self.exactTM, self.datum 

# double check that this and exactTM's ellipsoid match 

if xTM._E != d.ellipsoid: # PYCHOK no cover 

t = repr(d.ellipsoid) 

raise ETMError(repr(xTM._E), txt=_incompatible(t)) 

 

e, n = self.eastingnorthing2(falsed=not unfalse) 

lon0 = _cmlon(self.zone) if bool(unfalse) == self.falsed else None 

lat, lon, g, k = xTM.reverse(e, n, lon0=lon0) 

 

ll = _LLEB(lat, lon, datum=d, name=self.name) # utm._LLEB 

ll._convergence = g 

ll._scale = k 

self._latlon5args(ll, _toBand, unfalse, xTM) 

 

def toUtm(self): # PYCHOK signature 

'''Copy this ETM to a UTM coordinate. 

 

@return: The UTM coordinate (L{Utm}). 

''' 

return self._xcopy2(Utm) 

 

 

class ExactTransverseMercator(_NamedBase): 

'''A Python version of Karney's U{TransverseMercatorExact 

<https://GeographicLib.SourceForge.io/html/TransverseMercatorExact_8cpp_source.html>} 

C++ class, a numerically exact transverse Mercator projection, here referred to as 

C{TMExact}. 

''' 

_datum = None # Datum 

_E = None # Ellipsoid 

_extendp = True # use extended domain 

# _iteration = None # ._sigmaInv and ._zetaInv 

_k0 = _1_0 # central scale factor 

_lon0 = _0_0 # central meridian 

_mu = _0_0 # ._E.e2, 1st eccentricity squared 

_mv = _1_0 # _1_0 - ._mu 

 

def __init__(self, datum=_WGS84, lon0=0, k0=_K0_UTM, extendp=True, name=NN): 

'''New L{ExactTransverseMercator} projection. 

 

@kwarg datum: The non-spherical datum or ellipsoid (L{Datum}, 

L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}). 

@kwarg lon0: The central meridian (C{degrees180}). 

@kwarg k0: The central scale factor (C{float}). 

@kwarg extendp: Use the I{extended} or I{standard} domain (C{bool}). 

@kwarg name: Optional name for the projection (C{str}). 

 

@raise ETMError: Invalid B{C{lon0}} or B{C{k0}} or near-spherical 

B{C{datum}} or C{ellipsoid}. 

 

@raise TypeError: Invalid or near-spherical B{C{datum}}. 

 

@see: U{Constructor TransverseMercatorExact<https://GeographicLib.SourceForge.io/ 

html/classGeographicLib_1_1TransverseMercatorExact.html} for more details, 

especially on B{X{extendp}}. 

 

@note: For all 255.5K U{TMcoords.dat<https://Zenodo.org/record/32470>} tests (with 

C{0 <= lat <= 84} and C{0 <= lon}) the maximum error is C{5.2e-08 .forward} 

(or 52 nano-meter) easting and northing and C{3.8e-13 .reverse} (or 0.38 

pico-degrees) lat- and longitude (with Python 3.7.3+, 2.7.16+, PyPy6 3.5.3 

and PyPy6 2.7.13, all in 64-bit on macOS 10.13.6 High Sierra C{x86_64} and 

12.2 Monterey C{arm64} and C{"arm64_x86_64"}). 

''' 

if not extendp: 

self._extendp = False 

if name: 

self.name = name 

if lon0: 

self.lon0 = lon0 

 

self.datum = datum # invokes ._reset 

self.k0 = k0 

 

@property_doc_(''' the datum (L{Datum}).''') 

def datum(self): 

'''Get the datum (L{Datum}) or C{None}. 

''' 

return self._datum 

 

@datum.setter # PYCHOK setter! 

def datum(self, datum): 

'''Set the datum and ellipsoid (L{Datum}, L{Ellipsoid}, 

L{Ellipsoid2} or L{a_f2Tuple}). 

 

@raise ETMError: Near-spherical B{C{datum}} or C{ellipsoid}. 

 

@raise TypeError: Invalid or near-spherical B{C{datum}}. 

''' 

d = _ellipsoidal_datum(datum, name=self.name, raiser=True) 

self._reset(datum) 

self._datum = d 

 

@Property_RO 

def _1_e_90(self): # PYCHOK no cover 

'''(INTERNAL) Get and cache C{(1 - _e) * 90}. 

''' 

return (_1_0 - self._E.e) * _90_0 

 

@property_RO 

def ellipsoid(self): 

'''Get the ellipsoid (L{Ellipsoid}) or C{None}. 

''' 

return self._E 

 

@Property_RO 

def _e_PI_2(self): 

'''(INTERNAL) Get and cache C{_e * PI / 2}. 

''' 

return self._E.e * PI_2 

 

@Property_RO 

def _e_PI_4_(self): 

'''(INTERNAL) Get and cache C{- _e * PI / 4}. 

''' 

return -self._E.e * PI_4 

 

@Property_RO 

def _1_e_PI_2(self): 

'''(INTERNAL) Get and cache C{(1 - _e) * PI / 2}. 

''' 

return (_1_0 - self._E.e) * PI_2 

 

@Property_RO 

def _1_2e_PI_2(self): 

'''(INTERNAL) Get and cache C{(1 - 2 * _e) * PI / 2}. 

''' 

return (_1_0 - _2_0 * self._E.e) * PI_2 

 

@property_RO 

def equatoradius(self): 

'''Get the C{ellipsoid}'s equatorial radius, semi-axis (C{meter}). 

''' 

return self._E.a 

 

@Property_RO 

def _e_TAYTOL(self): 

'''(INTERNAL) Get and cache C{e * TAYTOL}. 

''' 

return self._E.e * _TAYTOL 

 

@Property_RO 

def _Eu(self): 

'''(INTERNAL) Get and cache C{Elliptic(_mu)}. 

''' 

return Elliptic(self._mu) 

 

@Property_RO 

def _Eu_cE_4(self): 

'''(INTERNAL) Get and cache C{_Eu.cE / 4}. 

''' 

return self._Eu.cE / _4_0 

 

@Property_RO 

def _Eu_cK_cE(self): 

'''(INTERNAL) Get and cache C{_Eu.cK / _Eu.cE}. 

''' 

return self._Eu.cK / self._Eu.cE 

 

@Property_RO 

def _Eu_2cK_PI(self): 

'''(INTERNAL) Get and cache C{_Eu.cK * 2 / PI}. 

''' 

return self._Eu.cK / PI_2 

 

@Property_RO 

def _Ev(self): 

'''(INTERNAL) Get and cache C{Elliptic(_mv)}. 

''' 

return Elliptic(self._mv) 

 

@Property_RO 

def _Ev_3cKE_4(self): 

'''(INTERNAL) Get and cache C{_Ev.cKE * 3 / 4}. 

''' 

return self._Ev.cKE * 0.75 

 

@Property_RO 

def _Ev_5cKE_4(self): 

'''(INTERNAL) Get and cache C{_Ev.cKE * 5 / 4}. 

''' 

return self._Ev.cKE * 1.25 

 

@Property_RO 

def extendp(self): 

'''Get using the I{extended} or I{standard} domain (C{bool}). 

''' 

return self._extendp 

 

@property_RO 

def flattening(self): 

'''Get the C{ellipsoid}'s flattening (C{float}). 

''' 

return self._E.f 

 

def forward(self, lat, lon, lon0=None): # MCCABE 13 

'''Forward projection, from geographic to transverse Mercator. 

 

@arg lat: Latitude of point (C{degrees}). 

@arg lon: Longitude of point (C{degrees}). 

@kwarg lon0: Central meridian of the projection (C{degrees}), 

overriding the default. 

 

@return: L{EasNorExact4Tuple}C{(easting, northing, 

convergence, scale)} in C{meter}, C{meter}, 

C{degrees} and C{scalar}. 

 

@see: C{void TMExact::Forward(real lon0, real lat, real lon, 

real &x, real &y, 

real &gamma, real &k)}. 

 

@raise EllipticError: No convergence transforming to ETM 

east- and northing. 

''' 

lat = _fix90(lat) 

lon, _ = _diff182((self.lon0 if lon0 is None else lon0), lon) 

# Explicitly enforce the parity 

backside = _lat = _lon = False 

if not self.extendp: 

if lat < 0: 

_lat, lat = True, -lat 

if lon < 0: 

_lon, lon = True, -lon 

if lon > _90_0: 

lon = _180_0 - lon 

if lat == 0: 

_lat = True 

backside = True 

 

# u,v = coordinates for the Thompson TM, Lee 54 

if lat == _90_0: 

u = self._Eu.cK 

v = self._iteration = 0 

elif lat == 0 and lon == self._1_e_90: # PYCHOK no cover 

u = self._iteration = 0 

v = self._Ev.cK 

else: # tau = tan(phi), taup = sinh(psi) 

tau, lam = tan(radians(lat)), radians(lon) 

u, v = self._zetaInv(self._E.es_taupf(tau), lam) 

 

sncndn6 = self._sncndn6(u, v) 

xi, eta, _ = self._sigma3(v, *sncndn6) 

if backside: 

xi = self._xi_backside(xi) 

y = xi * self._k0_a 

x = eta * self._k0_a 

 

g, k = self._zetaScaled(sncndn6, ll=False) \ 

if lat != _90_0 else (lon, self._k0) 

if backside: 

g = _180_0 - g 

if _lat: 

y, g = neg_(y, g) 

if _lon: 

x, g = neg_(x, g) 

 

return EasNorExact4Tuple(x, y, g, k, iteration=self._iteration) 

 

def _Inv3(self, psi, dlam, _3_mv_e, _e_TAYTOL): # (xi, deta, _3_mv, _TAYTOL2) 

'''(INTERNAL) Partial C{zetaInv0} or C{sigmaInv0}. 

''' 

h = hypot(psi, dlam) 

r = cbrt(h * _3_mv_e) 

# atan2(dlam-psi, psi+dlam) + 45d gives arg(zeta - zeta0) in 

# range [-135, 225). Subtracting 180 (multiplier is negative) 

# makes range [-315, 45). Multiplying by 1/3 (for cube root) 

# gives range [-105, 15). In particular the range [-90, 180] 

# in zeta space maps to [-90, 0] in w space as required. 

a = atan2(dlam - psi, psi + dlam) / _3_0 - PI_4 

s, c = sincos2(a) 

u = r * c 

v = r * s + self._Ev.cK 

# Error using this guess is about 0.068 * rad^(5/3) 

return u, v, (h < _e_TAYTOL) 

 

@property_RO 

def iteration(self): 

'''Get the most recent C{ExactTransverseMercator.forward} 

or C{ExactTransverseMercator.reverse} iteration number 

(C{int}) or C{None} if not available/applicable. 

''' 

return self._iteration 

 

@property_doc_(''' the central scale factor (C{float}).''') 

def k0(self): 

'''Get the central scale factor (C{float}), aka I{C{scale0}}. 

''' 

return self._k0 # aka scale0 

 

@k0.setter # PYCHOK setter! 

def k0(self, k0): 

'''Set the central scale factor (C{float}), aka I{C{scale0}}. 

 

@raise ETMError: Invalid B{C{k0}}. 

''' 

self._k0 = Scalar_(k0=k0, Error=ETMError, low=_TOL_10, high=_1_0) 

ExactTransverseMercator._k0_a._update(self) # redo ._k0_a 

 

@Property_RO 

def _k0_a(self): 

'''(INTERNAL) Get and cache C{k0 * equatoradius}. 

''' 

return self.k0 * self.equatoradius 

 

@property_doc_(''' the central meridian (C{degrees180}).''') 

def lon0(self): 

'''Get the central meridian (C{degrees180}). 

''' 

return self._lon0 

 

@lon0.setter # PYCHOK setter! 

def lon0(self, lon0): 

'''Set the central meridian (C{degrees180}). 

 

@raise ETMError: Invalid B{C{lon0}}. 

''' 

self._lon0 = _norm180(Float(lon0=lon0, Error=ETMError)) 

 

@deprecated_property_RO 

def majoradius(self): # PYCHOK no cover 

'''DEPRECATED, use property C{equatoradius}.''' 

return self.equatoradius 

 

@Property_RO 

def _mu_2_1(self): # PYCHOK no cover 

'''(INTERNAL) Get and cache C{_mu / 2 + 1}. 

''' 

return (self._mu + _2_0) * _0_5 

 

@Property_RO 

def _3_mv(self): 

'''(INTERNAL) Get and cache C{3 / _mv}. 

''' 

return _3_0 / self._mv 

 

@Property_RO 

def _3_mv_e(self): 

'''(INTERNAL) Get and cache C{3 / (_mv * _e)}. 

''' 

return _3_0 / (self._mv * self._E.e) 

 

def _Newton2(self, taup, lam, zeta=True): # or (xi, eta, False) 

'''(INTERNAL) Invert C{zeta} or C{sigma} using Newton's method. 

 

@return: 2-Tuple C{(u, v)}. 

 

@raise EllipticError: No convergence. 

''' 

if zeta: 

psi = asinh(taup) 

u, v, trip = self._zetaInv0(psi, lam) 

else: 

u, v, trip = self._sigmaInv0(taup, lam) 

if trip: 

i = 0 

else: 

sca1, tol2 = _1_0, _TOL_10 

if zeta: # /= chokes PyChecker 

sca1 = sca1 / hypot1(taup) 

tol2 = tol2 / max(psi**2, _1_0) 

U_fsum2_ = Fsum(u).fsum2_ 

V_fsum2_ = Fsum(v).fsum2_ 

 

_sncndn6 = self._sncndn6 

_zeta3 = self._zeta3 if zeta else self._sigma3 # PYCHOK attr 

_zetaDwd = self._zetaDwd if zeta else self._sigmaDwd # PYCHOK attr 

# min iterations 2, max 6 or 7, mean 3.9 or 4.0 

for i in range(1, _TRIPS): # GEOGRAPHICLIB_PANIC 

sncndn6 = _sncndn6(u, v) 

dw, dv = _zetaDwd( *sncndn6) 

T, L, _ = _zeta3(v, *sncndn6) 

T = (taup - T) * sca1 

L -= lam 

u, du = U_fsum2_(T * dw, L * dv) 

v, dv = V_fsum2_(T * dv, -L * dw) 

if trip: 

break 

trip = hypot2(du, dv) < tol2 

 

else: # PYCHOK no cover 

self._iteration = _TRIPS 

w = self._zetaInv if zeta else self._sigmaInv # PYCHOK attr 

t = unstr(w.__name__, taup, lam) 

raise EllipticError(_no_(_convergence_), tol2, txt=t) 

 

self._iteration = i 

return u, v 

 

def _reset(self, datum): 

'''(INTERNAL) Set the ellipsoid and elliptic moduli. 

 

@arg datum: Ellipsoidal datum (C{Datum}). 

 

@raise ETMError: Near-spherical B{C{datum}} or C{ellipsoid}. 

''' 

E = datum.ellipsoid 

mu = E.e2 # .eccentricity1st2 

mv = _1_0 - mu 

if isnear0(E.e) or isnear0(mu, eps0=EPS02) \ 

or isnear0(mv, eps0=EPS02): 

raise ETMError(e=E.e, e2=mu, txt=repr(datum)) 

 

if self._datum or self._E: 

_i = ExactTransverseMercator.iteration._uname 

_update_all(self, _i) 

 

self._E = E 

self._mu = mu 

self._mv = mv 

 

def reverse(self, x, y, lon0=None): 

'''Reverse projection, from Transverse Mercator to geographic. 

 

@arg x: Easting of point (C{meters}). 

@arg y: Northing of point (C{meters}). 

@kwarg lon0: Central meridian of the projection (C{degrees}), 

overriding the default. 

 

@return: L{LatLonExact4Tuple}C{(lat, lon, convergence, scale)} 

in C{degrees}, C{degrees180}, C{degrees} and C{scalar}. 

 

@see: C{void TMExact::Reverse(real lon0, real x, real y, 

real &lat, real &lon, 

real &gamma, real &k)} 

 

@raise EllipticError: No convergence transforming to lat- and 

longitude. 

''' 

# undoes the steps in .forward. 

xi = y / self._k0_a 

eta = x / self._k0_a 

backside = _lat = _lon = False 

if not self.extendp: # enforce the parity 

if y < 0: 

_lat, xi = True, -xi 

if x < 0: 

_lon, eta = True, -eta 

if xi > self._Eu.cE: # PYCHOK no cover 

xi = self._xi_backside(xi) 

backside = True 

 

# u,v = coordinates for the Thompson TM, Lee 54 

if xi or eta != self._Ev.cKE: 

u, v = self._sigmaInv(xi, eta) 

else: # PYCHOK no cover 

u, v = _0_0, self._Ev.cK 

self._iteration = 0 

 

if v or u != self._Eu.cK: 

g, k, lat, lon = self._zetaScaled(self._sncndn6(u, v)) 

else: # PYCHOK no cover 

g, k, lat, lon = _0_0, self._k0, _90_0, _0_0 

 

if backside: # PYCHOK no cover 

lon, g = (_180_0 - lon), (_180_0 - g) 

if _lat: 

lat, g = neg_(lat, g) 

if _lon: 

lon, g = neg_(lon, g) 

 

lon += self.lon0 if lon0 is None else _norm180(lon0) 

return LatLonExact4Tuple(lat, _norm180(lon), g, k, # _norm180(lat) 

iteration=self._iteration) 

 

def _scaled(self, tau, d2, snu, cnu, dnu, snv, cnv, dnv): 

'''(INTERNAL) C{scaled}. 

 

@note: Argument B{C{d2}} is C{_mu * cnu**2 + _mv * cnv**2} 

from C{._sigma3} or C{._zeta3}. 

 

@return: 2-Tuple C{(convergence, scale)}. 

 

@see: C{void TMExact::Scale(real tau, real /*lam*/, 

real snu, real cnu, real dnu, 

real snv, real cnv, real dnv, 

real &gamma, real &k)}. 

''' 

mu, mv = self._mu, self._mv 

cnudnv = cnu * dnv 

# Lee 55.12 -- negated for our sign convention. g gives 

# the bearing (clockwise from true north) of grid north 

g = atan2d(mv * cnv * snv * snu, cnudnv * dnu) 

# Lee 55.13 with nu given by Lee 9.1 -- in sqrt change 

# the numerator from (1 - snu^2 * dnv^2) to (_mv * snv^2 

# + cnu^2 * dnv^2) to maintain accuracy near phi = 90 

# and change the denomintor from (dnu^2 + dnv^2 - 1) to 

# (_mu * cnu^2 + _mv * cnv^2) to maintain accuracy near 

# phi = 0, lam = 90 * (1 - e). Similarly rewrite sqrt in 

# 9.1 as _mv + _mu * c^2 instead of 1 - _mu * sin(phi)^2 

q2 = (mv * snv**2 + cnudnv**2) / d2 

# originally: sec2 = 1 + tau**2 # sec(phi)^2 

# k = sqrt(mv + mu / sec2) * sqrt(sec2) * sqrt(q2) 

# = sqrt(mv * sec2 + mu) * sqrt(q2) 

# = sqrt(mv + mv * tau**2 + mu) * sqrt(q2) 

k = sqrt(fsum1_(mu, mv, mv * tau**2)) * sqrt(q2) 

return g, (k * self._k0) 

 

def _sigma3(self, v, snu, cnu, dnu, snv, cnv, dnv): 

'''(INTERNAL) C{sigma}. 

 

@return: 3-Tuple C{(xi, eta, d2)}. 

 

@see: C{void TMExact::sigma(real /*u*/, real snu, real cnu, real dnu, 

real v, real snv, real cnv, real dnv, 

real &xi, real &eta)}. 

 

@raise EllipticError: No convergence. 

''' 

# Lee 55.4 writing 

# dnu^2 + dnv^2 - 1 = _mu * cnu^2 + _mv * cnv^2 

d2 = self._mu * cnu**2 + self._mv * cnv**2 

xi = self._Eu.fE(snu, cnu, dnu) - self._mu * snu * cnu * dnu / d2 

v -= self._Ev.fE(snv, cnv, dnv) - self._mv * snv * cnv * dnv / d2 

return xi, v, d2 

 

def _sigmaDwd(self, snu, cnu, dnu, snv, cnv, dnv): 

'''(INTERNAL) C{sigmaDwd}. 

 

@return: 2-Tuple C{(du, dv)}. 

 

@see: C{void TMExact::dwdsigma(real /*u*/, real snu, real cnu, real dnu, 

real /*v*/, real snv, real cnv, real dnv, 

real &du, real &dv)}. 

''' 

snuv = snu * snv 

# Reciprocal of 55.9: dw / ds = dn(w)^2/_mv, 

# expanding complex dn(w) using A+S 16.21.4 

d = self._mv * (cnv**2 + self._mu * snuv**2)**2 

r = cnv * dnu * dnv 

i = cnu * snuv * self._mu 

du = (r**2 - i**2) / d 

dv = neg(_2_0 * i * r / d) 

return du, dv 

 

def _sigmaInv(self, xi, eta): 

'''(INTERNAL) Invert C{sigma} using Newton's method. 

 

@return: 2-Tuple C{(u, v)}. 

 

@see: C{void TMExact::sigmainv(real xi, real eta, 

real &u, real &v)}. 

 

@raise EllipticError: No convergence. 

''' 

return self._Newton2(xi, eta, zeta=False) 

 

def _sigmaInv0(self, xi, eta): 

'''(INTERNAL) Starting point for C{sigmaInv}. 

 

@return: 3-Tuple C{(u, v, trip)}. 

 

@see: C{bool TMExact::sigmainv0(real xi, real eta, 

real &u, real &v)}. 

''' 

d = eta - self._Ev.cKE 

if eta > self._Ev_5cKE_4 or (xi < d and xi < -self._Eu_cE_4): 

# sigma as a simple pole at 

# w = w0 = Eu.K() + i * Ev.K() 

# and sigma is approximated by 

# sigma = (Eu.E() + i * Ev.KE()) + 1 / (w - w0) 

x = xi - self._Eu.cE # y = d 

h = hypot2(x, d) 

u = self._Eu.cK + x / h 

v = self._Ev.cK - d / h 

 

elif (eta > self._Ev_3cKE_4 and xi < self._Eu_cE_4) or d > 0: 

# At w = w0 = i * Ev.K(), we have 

# sigma = sigma0 = i * Ev.KE() 

# sigma' = sigma'' = 0 

# including the next term in the Taylor series gives: 

# sigma = sigma0 - _mv / 3 * (w - w0)^3 

# When inverting this, we map arg(w - w0) = [-pi/2, -pi/6] 

# to arg(sigma - sigma0) = [-pi/2, pi/2] mapping arg = 

# [-pi/2, -pi/6] to [-pi/2, pi/2] 

return self._Inv3(xi, d, self._3_mv, _TAYTOL2) 

 

else: # use w = sigma * Eu.K/Eu.E (correct in limit _e -> 0) 

r = self._Eu_cK_cE 

u = r * xi 

v = r * eta 

 

return u, v, False 

 

def _sncndn6(self, u, v): 

'''(INTERNAL) Get 6-tuple C{(snu, cnu, dnu, snv, cnv, dnv)}. 

''' 

# snu, cnu, dnu = self._Eu.sncndn(u) 

# snv, cnv, dnv = self._Ev.sncndn(v) 

return self._Eu.sncndn(u) + self._Ev.sncndn(v) 

 

def toStr(self, **kwds): 

'''Return a C{str} representation. 

 

@arg kwds: Optional, overriding keyword arguments. 

''' 

d = dict(datum=self.datum.name, lon0=self.lon0, 

k0=self.k0, extendp=self.extendp) 

if self.name: 

d.update(name=self.name) 

return _COMMASPACE_.join(pairs(d, **kwds)) 

 

def _xi_backside(self, xi): 

'''(INTERNAL) Return B{C{xi}} for the I{backside}. 

''' 

return _2_0 * self._Eu.cE - xi 

 

def _zeta3(self, unused, snu, cnu, dnu, snv, cnv, dnv): # _sigma3 signature 

'''(INTERNAL) C{zeta}. 

 

@return: 3-Tuple C{(taup, lambda, d2)}. 

 

@see: C{void TMExact::zeta(real /*u*/, real snu, real cnu, real dnu, 

real /*v*/, real snv, real cnv, real dnv, 

real &taup, real &lam)} 

''' 

e, cnu2, lam = self._E.e, cnu**2, True 

# Overflow value like atan(overflow) = pi/2 

t1 = t2 = copysign0(_OVERFLOW, snu) 

# Lee 54.17 but write 

# atanh(snu * dnv) = asinh(snu * dnv / sqrt(cnu^2 + _mv * snu^2 * snv^2)) 

# atanh(_e * snu / dnv) = asinh(_e * snu / sqrt(_mu * cnu^2 + _mv * cnv^2)) 

d1 = cnu2 + self._mv * (snu * snv)**2 

if d1 > EPS02: # _EPSmin 

t1 = snu * dnv / sqrt(d1) 

elif d1 > -_EPSmin: # PYCHOK no cover 

lam = False # d1 near 0 

d2 = self._mu * cnu2 + self._mv * cnv**2 

if d2 > EPS02: # _EPSmin 

t2 = sinh(e * asinh(e * snu / sqrt(d2))) 

elif d2 > -_EPSmin: # PYCHOK no cover 

lam = False # d2 near 0 

# psi = asinh(t1) - asinh(t2) 

# taup = sinh(psi) 

taup = t1 * hypot1(t2) - t2 * hypot1(t1) 

lam = (atan2(dnu * snv, cnu * cnv) - 

atan2(cnu * snv * e, dnu * cnv) * e) if lam else _0_0 

return taup, lam, d2 

 

def _zetaDwd(self, snu, cnu, dnu, snv, cnv, dnv): 

'''(INTERNAL) C{zetaDwd}. 

 

@return: 2-Tuple C{(du, dv)}. 

 

@see: C{void TMExact::dwdzeta(real /*u*/, real snu, real cnu, real dnu, 

real /*v*/, real snv, real cnv, real dnv, 

real &du, real &dv)}. 

''' 

cnu2 = cnu**2 * self._mu 

cnv2 = cnv**2 

dnuv = dnu * dnv 

dnuv2 = dnuv**2 

snuv = snu * snv 

snuv2 = snuv**2 * self._mu 

# Lee 54.21 but write (see A+S 16.21.4) 

# (1 - dnu^2 * snv^2) = (cnv^2 + _mu * snu^2 * snv^2) 

d = self._mv * (cnv2 + snuv2)**2 # max(d, EPS02)? 

du = cnu * dnuv * (cnv2 - snuv2) / d 

dv = cnv * snuv * (cnu2 + dnuv2) / d 

return du, neg(dv) 

 

def _zetaInv(self, taup, lam): 

'''(INTERNAL) Invert C{zeta} using Newton's method. 

 

@return: 2-Tuple C{(u, v)}. 

 

@see: C{void TMExact::zetainv(real taup, real lam, 

real &u, real &v)}. 

 

@raise EllipticError: No convergence. 

''' 

return self._Newton2(taup, lam, True) 

 

def _zetaInv0(self, psi, lam): 

'''(INTERNAL) Starting point for C{zetaInv}. 

 

@return: 3-Tuple C{(u, v, trip)}. 

 

@see: C{bool TMExact::zetainv0(real psi, real lam, # radians 

real &u, real &v)}. 

''' 

if lam > self._1_2e_PI_2: 

d = lam - self._1_e_PI_2 

if psi < d and psi < self._e_PI_4_: # PYCHOK no cover 

# N.B. this branch is normally *not* taken because psi < 0 is 

# converted psi > 0 by .forward. There's a log singularity at 

# w = w0 = Eu.K() + i * Ev.K(), corresponding to the south pole, 

# where we have, approximately 

# psi = _e + i * pi/2 - _e * atanh(cos(i * (w - w0)/(1 + _mu/2))) 

# Inverting this gives: 

e = self._E.e # eccentricity 

s, c = sincos2((PI_2 - lam) / e) 

h, r = sinh(_1_0 - psi / e), self._mu_2_1 

u = self._Eu.cK - r * asinh(s / hypot(c, h)) 

v = self._Ev.cK - r * atan2(c, h) 

return u, v, False 

 

elif psi < self._e_PI_2: 

# At w = w0 = i * Ev.K(), we have 

# zeta = zeta0 = i * (1 - _e) * pi/2 

# zeta' = zeta'' = 0 

# including the next term in the Taylor series gives: 

# zeta = zeta0 - (_mv * _e) / 3 * (w - w0)^3 

# When inverting this, we map arg(w - w0) = [-90, 0] 

# to arg(zeta - zeta0) = [-90, 180] 

return self._Inv3(psi, d, self._3_mv_e, self._e_TAYTOL) 

 

# Use spherical TM, Lee 12.6 -- writing C{atanh(sin(lam) / 

# cosh(psi)) = asinh(sin(lam) / hypot(cos(lam), sinh(psi)))}. 

# This takes care of the log singularity at C{zeta = Eu.K()}, 

# corresponding to the north pole. 

s, c = sincos2(lam) 

h, r = sinh(psi), self._Eu_2cK_PI 

# But scale to put 90, 0 on the right place 

u = r * atan2(h, c) 

v = r * asinh(s / hypot(h, c)) 

return u, v, False 

 

def _zetaScaled(self, sncndn6, ll=True): 

'''(INTERNAL) Recompute (T, L) from (u, v) to improve accuracy of Scale. 

 

@arg sncndn6: 6-Tuple C{(snu, cnu, dnu, snv, cnv, dnv)}. 

 

@return: 2-Tuple C{(g, k)} if not C{B{ll}} else 

4-tuple C{(g, k, lat, lon)}. 

''' 

t, lam, d2 = self._zeta3(None, *sncndn6) 

tau = self._E.es_tauf(t) 

g_k = self._scaled(tau, d2, *sncndn6) 

if ll: 

g_k += atand(tau), degrees(lam) 

return g_k # or (g, k, lat, lon) 

 

 

class LatLonExact4Tuple(_NamedTuple): 

'''4-Tuple C{(lat, lon, convergence, scale)} in C{degrees180}, 

C{degrees180}, C{degrees} and C{scalar}. 

''' 

_Names_ = (_lat_, _lon_, _convergence_, _scale_) 

_Units_ = ( Lat, Lon, Degrees, Scalar) 

 

 

def parseETM5(strUTM, datum=_WGS84, Etm=Etm, falsed=True, name=NN): 

'''Parse a string representing a UTM coordinate, consisting 

of C{"zone[band] hemisphere easting northing"}. 

 

@arg strUTM: A UTM coordinate (C{str}). 

@kwarg datum: Optional datum to use (L{Datum}, L{Ellipsoid}, 

L{Ellipsoid2} or L{a_f2Tuple}). 

@kwarg Etm: Optional class to return the UTM coordinate 

(L{Etm}) or C{None}. 

@kwarg falsed: Both easting and northing are C{falsed} (C{bool}). 

@kwarg name: Optional B{C{Etm}} name (C{str}). 

 

@return: The UTM coordinate (B{C{Etm}}) or if B{C{Etm}} is 

C{None}, a L{UtmUps5Tuple}C{(zone, hemipole, easting, 

northing, band)}. The C{hemipole} is the hemisphere 

C{'N'|'S'}. 

 

@raise ETMError: Invalid B{C{strUTM}}. 

 

@raise TypeError: Invalid or near-spherical B{C{datum}}. 

 

@example: 

 

>>> u = parseETM5('31 N 448251 5411932') 

>>> u.toRepr() # [Z:31, H:N, E:448251, N:5411932] 

>>> u = parseETM5('31 N 448251.8 5411932.7') 

>>> u.toStr() # 31 N 448252 5411933 

''' 

r = _parseUTM5(strUTM, datum, Etm, falsed, Error=ETMError, name=name) 

return r 

 

 

def toEtm8(latlon, lon=None, datum=None, Etm=Etm, falsed=True, name=NN, 

zone=None, **cmoff): 

'''Convert a lat-/longitude point to an ETM coordinate. 

 

@arg latlon: Latitude (C{degrees}) or an (ellipsoidal) 

geodetic C{LatLon} point. 

@kwarg lon: Optional longitude (C{degrees}) or C{None}. 

@kwarg datum: Optional datum for this ETM coordinate, 

overriding B{C{latlon}}'s datum (L{Datum}, 

L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}). 

@kwarg Etm: Optional class to return the ETM coordinate 

(L{Etm}) or C{None}. 

@kwarg falsed: False both easting and northing (C{bool}). 

@kwarg name: Optional B{C{Utm}} name (C{str}). 

@kwarg zone: Optional UTM zone to enforce (C{int} or C{str}). 

@kwarg cmoff: DEPRECATED, use B{C{falsed}}. Offset longitude 

from the zone's central meridian (C{bool}). 

 

@return: The ETM coordinate (B{C{Etm}}) or a 

L{UtmUps8Tuple}C{(zone, hemipole, easting, northing, 

band, datum, convergence, scale)} if B{C{Etm}} is 

C{None} or not B{C{falsed}}. The C{hemipole} is the 

C{'N'|'S'} hemisphere. 

 

@raise EllipticError: No convergence transforming to ETM east- 

and northing. 

 

@raise ETMError: Invalid B{C{zone}} or near-spherical or 

incompatible B{C{datum}} or C{ellipsoid}. 

 

@raise RangeError: If B{C{lat}} outside the valid UTM bands or 

if B{C{lat}} or B{C{lon}} outside the valid 

range and L{pygeodesy.rangerrors} set to C{True}. 

 

@raise TypeError: Invalid or near-spherical B{C{datum}} or 

B{C{latlon}} not ellipsoidal. 

 

@raise ValueError: The B{C{lon}} value is missing or B{C{latlon}} 

is invalid. 

''' 

z, B, lat, lon, d, f, name = _to7zBlldfn(latlon, lon, datum, 

falsed, name, zone, 

ETMError, **cmoff) 

lon0 = _cmlon(z) if f else None 

x, y, g, k = d.exactTM.forward(lat, lon, lon0=lon0) 

 

return _toXtm8(Etm, z, lat, x, y, B, d, g, k, f, 

name, latlon, d.exactTM, Error=ETMError) 

 

# **) MIT License 

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