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# -*- coding: utf-8 -*- 

 

u'''Classes L{ETMError} and L{Etm}, a pure Python implementation of 

I{Charles Karney's} C++ class U{TransverseMercatorExact 

<https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1TransverseMercatorExact.html>}, 

abbreviated as C{TMExact} below. 

 

Python class L{ExactTransverseMercator} implements the C{Exact Transverse 

Mercator} (ETM) projection. Instances of class L{Etm} represent ETM 

C{easting, nothing} locations. 

 

Following is a copy of 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>} (2008-2017) 

and licensed under the MIT/X11 License. For more information, see the 

U{GeographicLib<https://GeographicLib.SourceForge.io/>} documentation. 

 

The method entails using the C{Thompson Transverse Mercator} 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>} 

is 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 

from __future__ import division 

division = 1 / 2 # double check int division, see .datum.py, .utily.py 

if not division: 

raise ImportError('%s 1/2 == %d' % ('division', division)) 

del division 

 

from pygeodesy.datum import Datum, Datums 

from pygeodesy.elliptic import Elliptic, EllipticError, _TRIPS 

from pygeodesy.fmath import cbrt, EPS, Fsum, hypot, hypot1 

from pygeodesy.lazily import _ALL_LAZY 

from pygeodesy.named import EasNorExact4Tuple, LatLonExact4Tuple, \ 

_NamedBase, _xnamed 

from pygeodesy.utily import PI_2, PI_4, property_RO, sincos2, \ 

_TypeError 

from pygeodesy.utm import _cmlon, _K0, _parseUTM5, Utm, UTMError, \ 

_toXtm8, _to7zBlldfn 

from pygeodesy.utmupsBase import _LLEB 

 

from math import asinh, atan, atan2, copysign, degrees, \ 

fmod, radians, sinh, sqrt, tan 

 

__all__ = _ALL_LAZY.etm 

__version__ = '20.01.22' 

 

_OVERFLOW = 1.0 / EPS**2 

_TOL = EPS 

_TOL_10 = 0.1 * _TOL 

_TAYTOL = pow(_TOL, 0.6) 

_TAYTOL2 = 2.0 * _TAYTOL 

 

try: 

from geographiclib.geomath import Math 

 

_diff182 = Math.AngDiff # returns 2-tuple 

_fix90 = Math.LatFix 

_sum2 = Math.sum # PYCHOK for testEtm.py 

_wrap180 = Math.AngNormalize 

 

del Math 

 

except ImportError: # no geographiclib 

 

from pygeodesy.fmath import NAN as _NAN 

 

def _diff182(deg0, deg): # mimick Math.AngDiff 

'''Compute C{deg - deg0}, reduced to C{[-180,180]} accurately. 

''' 

d, t = _sum2(_wrap180(-deg0), _wrap180(deg)) 

d = _wrap180(d) 

if d == 180 and t > 0: 

d = -180 

return _sum2(d, t) 

 

def _fix90(deg): # mimick Math.LatFix 

'''Replace angles outside [-90,90] by NaN. 

''' 

return _NAN if abs(deg) > 90 else deg 

 

def _sum2(u, v): # mimick Math::sum, actually sum2 

'''Error free transformation of a C{sum}. 

 

@return: 2-Tuple C{(sum_u_plus_v, residual)}. 

 

@note: The C{residual} can be the same as one 

of the first two arguments. 

''' 

s = u + v 

r = s - v 

t = s - r 

r -= u 

t -= v 

t = -(r + t) 

# u + v = s + t 

# = round(u + v) + t 

return s, t 

 

def _wrap180(deg): # mimick Math.AngNormalize 

'''Reduce angle to (-180,180] 

''' 

# with Python 2.7.16 and 3.7.3 on macOS 10.13.6 

# fmod( 0, 360) == 0.0 

# fmod( 360, 360) == 0.0 

# fmod(-0, 360) == 0.0 

# fmod(-0.0, 360) == -0.0 

# fmod(-360, 360) == -0.0 

# however, using the % operator ... 

# 0 % 360 == 0 

# 360 % 360 == 0 

# 360.0 % 360 == 0.0 

# -0 % 360 == 0 

# -360 % 360 == 0 

# -0.0 % 360 == 0.0 

# -360.0 % 360 == 0.0 

 

# On Windows 32-bit with Python 2.7, math.fmod(-0.0, 360) 

# == +0.0. This fixes this bug. See also Math::AngNormalize 

# in the C++ library. Math::sincosd has a similar fix. 

d = fmod(deg, 360) if deg else deg 

return (d + 360) if d <= -180 else ( 

d if d <= 180 else (d - 360)) 

 

 

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 L{Etm} coordinates to and from (geodetic) 

lat- and longitude is 3-4 times slower than L{Utm}. 

 

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

html/classGeographicLib_1_1TransverseMercatorExact.html#details>}. 

''' 

_Error = ETMError 

_exactTM = None 

 

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

datum=Datums.WGS84, falsed=True, 

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

'''New L{Etm} coordinate. 

 

@param zone: Longitudinal UTM zone (C{int}, 1..60) or zone 

with/-out (latitudinal) Band letter (C{str}, 

'01C'..'60X'). 

@param hemisphere: Northern or southern hemisphere (C{str}, 

C{'N[orth]'} or C{'S[outh]'}). 

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

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

@keyword band: Optional, (latitudinal) band (C{str}, 'C'..'X'). 

@keyword datum: Optional, this coordinate's datum (L{Datum}). 

@keyword falsed: Both B{C{easting}} and B{C{northing}} are 

falsed (C{bool}). 

@keyword convergence: Optional meridian convergence, bearing 

off grid North, clockwise from true 

North (C{degrees}) or C{None}. 

@keyword scale: Optional grid scale factor (C{scalar}) or 

C{None}. 

@keyword name: Optional name (C{str}). 

 

@raise EllipticError: No convergence. 

 

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

B{C{band}}. 

 

@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 

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}). 

''' 

_TypeError(ExactTransverseMercator, exactTM=exactTM) 

 

E = self.datum.ellipsoid 

if exactTM._E != E or exactTM.majoradius != E.a \ 

or exactTM.flattening != E.f: 

raise ETMError('%r vs %r' % (exactTM, E)) 

self._exactTM = exactTM 

self._scale0 = exactTM.k0 

 

def parseETM(self, strETM): 

'''Parse a string to a ETM coordinate. 

 

@return: The coordinate (L{Etm}). 

 

@see: Function L{parseETM5} in this module L{etm}. 

''' 

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

 

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

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

 

@keyword LatLon: Optional, ellipsoidal (sub-)class to return 

the point (C{LatLon}) or C{None}. 

@keyword unfalse: Unfalse B{C{easting}} and B{C{northing}} 

if 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. 

 

@raise TypeError: If B{C{LatLon}} is not ellipsoidal. 

 

@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 

''' 

xTM, d = self.exactTM, self.datum 

# double check that this and exactTM's ellipsoids stil match 

if xTM._E != d.ellipsoid: 

raise ETMError('%r vs %r' % (xTM._E, d.ellipsoid)) 

 

if self._latlon and self._latlon_args == (xTM, unfalse): 

return self._latlon5(LatLon) 

 

f = not unfalse 

e, n = self.to2en(falsed=f) 

# f = unfalse == self.falsed 

# == unfalse and self.falsed or (not unfalse and not self.falsed) 

# == unfalse if self.falsed else not unfalse 

# == unfalse if self.falsed else f 

if self.falsed: 

f = unfalse 

lon0 = _cmlon(self.zone) if f else None 

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

 

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

ll._convergence = g 

ll._scale = k 

 

self._latlon_to(ll, xTM, unfalse) 

return self._latlon5(LatLon) 

 

def _latlon_to(self, ll, xTM, unfalse): 

'''(INTERNAL) See C{.toLatLon}, C{toEtm8}, C{_toXtm8}. 

''' 

self._latlon, self._latlon_args = ll, (xTM, unfalse) 

 

def toUtm(self): # PYCHOK signature 

'''Coopy this ETM to a UTM coordinate. 

 

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

''' 

return self._xnamed(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, 

referred to as C{TMExact} here. 

 

@see: C{TMExact(real a, real f, real k0, bool extendp)}. 

''' 

_a = 0 # major radius 

_datum = None # Datum 

_e = 0 # eccentricity 

_E = None # Ellipsoid 

_extendp = False 

_f = 0 # flattening 

_k0 = 1 # central scale factor 

_k0_a = 0 

_lon0 = 0 # central meridian 

_trips_ = _TRIPS 

 

def __init__(self, datum=Datums.WGS84, lon0=0, k0=_K0, extendp=True, name=''): 

'''New L{ExactTransverseMercator} projection. 

 

@keyword datum: The datum and ellipsoid to use (C{Datum}). 

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

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

@keyword extendp: Use the extended domain (C{bool}). 

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

 

@raise EllipticError: No convergence. 

 

@raise ETMError: Invalid B{C{k0}}. 

 

@raise TypeError: Invalid B{C{datum}}. 

 

@note: The maximum error for all 255.5K U{TMcoords.dat 

<https://Zenodo.org/record/32470>} tests (with 

C{0 <= lat <= 84} and C{0 <= lon}) 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). 

''' 

if extendp: 

self._extendp = bool(extendp) 

if name: 

self.name = name 

 

self.datum = datum 

self.lon0 = lon0 

self.k0 = k0 

 

@property 

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}). 

 

@raise EllipticError: No convergence. 

 

@raise TypeError: Invalid B{C{datum}}. 

''' 

_TypeError(Datum, datum=datum) 

 

E = datum.ellipsoid 

self._reset(E.e, E.e2) 

self._a = E.a 

self._f = E.f # flattening = (a - b) / a 

 

self._datum = datum 

self._E = E 

 

@property_RO 

def extendp(self): 

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

''' 

return self._extendp 

 

@property_RO 

def flattening(self): 

'''Get the flattening (C{float}). 

''' 

return self._f 

 

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

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

 

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

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

@keyword lon0: Central meridian of the projection (C{degrees}). 

 

@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. 

''' 

lat = _fix90(lat) 

lon, _ = _diff182((self._lon0 if lon0 is None else lon0), lon) 

# Explicitly enforce the parity 

_lat = _lon = backside = False 

if not self.extendp: 

if lat < 0: 

_lat, lat = True, -lat 

if lon < 0: 

_lon, lon = True, -lon 

if lon > 90: 

backside = True 

if lat == 0: 

_lat = True 

lon = 180 - lon 

 

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

if lat == 90: 

u, v = self._Eu_K, 0 

elif lat == 0 and lon == self._1_e_90: 

u, v = 0, self._Ev_K 

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

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

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

 

snu, cnu, dnu = self._Eu.sncndn(u) 

snv, cnv, dnv = self._Ev.sncndn(v) 

xi, eta, _ = self._sigma3(v, snu, cnu, dnu, snv, cnv, dnv) 

if backside: 

xi = 2 * self._Eu_E - xi 

y = xi * self._k0_a 

x = eta * self._k0_a 

 

if lat == 90: 

g, k = lon, self._k0 

else: # Recompute (T, L) from (u, v) to improve accuracy of Scale 

tau, lam, d = self._zeta3( snu, cnu, dnu, snv, cnv, dnv) 

tau = self._E.es_tauf(tau) 

g, k = self._scaled(tau, d, snu, cnu, dnu, snv, cnv, dnv) 

 

if backside: 

g = 180 - g 

if _lat: 

y, g = -y, -g 

if _lon: 

x, g = -x, -g 

return EasNorExact4Tuple(x, y, g, k) 

 

@property 

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 EllipticError: Invalid B{C{k0}}. 

''' 

self._k0 = float(k0) 

if not 0 < self._k0 <= 1: 

raise ETMError('%s invalid: %r' % ('k0', k0)) 

self._k0_a = self._k0 * self._a 

 

@property 

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}). 

''' 

self._lon0 = _wrap180(lon0) 

 

@property_RO 

def majoradius(self): 

'''Get the major (equatorial) radius, semi-axis (C{float}). 

''' 

return self._a 

 

def _reset(self, e, e2): 

'''(INTERNAL) Get elliptic functions and pre-compute 

frequently used values. 

 

@raise EllipticError: No convergence. 

''' 

# assert e2 == e**2 

self._e = e # eccentricity = sqrt(f * (2 - f)) 

 

self._e_PI_2 = e * PI_2 

self._e_PI_4 = e * PI_4 

self._e_taytol_ = e * _TAYTOL 

 

self._1_e_90 = (1 - e) * 90 

self._1_e_PI_2 = (1 - e) * PI_2 

self._1_e2_PI_2 = (1 - e * 2) * PI_2 

 

self._mu = e2 # eccentricity**2 = f * (2 - f) = 1 - (b / a)**2 

self._mu_2_1 = (e2 + 2) * 0.5 

 

self._Eu = Elliptic(self._mu) 

self._Eu_E = self._Eu.cE # constant 

self._Eu_E_1_4 = self._Eu_E * 0.25 

 

self._Eu_K = self._Eu.cK # constant 

 

self._mv = 1 - e2 # 1 - eccentricity**2 = 1 - e2 

self._3_mv = 3.0 / self._mv 

self._3_mv_e = self._3_mv / e 

 

self._Ev = Elliptic(self._mv) 

self._Ev_E = self._Ev.cE # constant 

 

self._Ev_K = self._Ev.cK # constant 

self._Ev_KE = self._Ev.cKE # constant 

self._Ev_KE_3_4 = self._Ev_KE * 0.75 

self._Ev_KE_5_4 = self._Ev_KE * 1.25 

 

def reverse(self, x, y, lon0=None): 

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

 

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

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

@keyword lon0: Central meridian of the projection (C{degrees}). 

 

@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. 

''' 

# undoes the steps in .forward. 

xi = y / self._k0_a 

eta = x / self._k0_a 

_lat = _lon = backside = 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_E: 

backside = True 

xi = 2 * self._Eu_E - xi 

 

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

if xi == 0 and eta == self._Ev_KE: 

u, v = 0, self._Ev_K 

else: 

u, v = self._sigmaInv(xi, eta) 

 

if v != 0 or u != self._Eu_K: 

snu, cnu, dnu = self._Eu.sncndn(u) 

snv, cnv, dnv = self._Ev.sncndn(v) 

tau, lam, d = self._zeta3( snu, cnu, dnu, snv, cnv, dnv) 

tau = self._E.es_tauf(tau) 

lat, lon = degrees(atan(tau)), degrees(lam) 

g, k = self._scaled(tau, d, snu, cnu, dnu, snv, cnv, dnv) 

else: 

lat, lon = 90, 0 

g, k = 0, self._k0 

 

if backside: 

lon, g = 180 - lon, 180 - g 

if _lat: 

lat, g = -lat, -g 

if _lon: 

lon, g = -lon, -g 

 

lat = _wrap180(lat) 

lon = _wrap180(lon + (self._lon0 if lon0 is None else _wrap180(lon0))) 

return LatLonExact4Tuple(lat, lon, g, k) 

 

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

''' 

@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 = atan2(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 term in 9.1 as 

# 

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

sec2 = 1 + tau**2 # sec(phi)^2 

q2 = (mv * snv**2 + cnudnv**2) / d2 

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

return degrees(g), k * self._k0 

 

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

''' 

@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 

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

return xi, eta, d2 

 

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

''' 

@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 = 2 * r * i / d 

return du, dv 

 

def _sigmaInv(self, xi, eta): 

'''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. 

''' 

u, v, trip = self._sigmaInv0(xi, eta) 

if not trip: 

U, V = Fsum(u), Fsum(v) 

# min iterations = 2, max = 7, mean = 3.9 

for _ in range(self._trips_): # GEOGRAPHICLIB_PANIC 

snu, cnu, dnu = self._Eu.sncndn(u) 

snv, cnv, dnv = self._Ev.sncndn(v) 

X, E, _ = self._sigma3(v, snu, cnu, dnu, snv, cnv, dnv) 

dw, dv = self._sigmaDwd( snu, cnu, dnu, snv, cnv, dnv) 

X = xi - X 

E -= eta 

u, du = U.fsum2_(X * dw, E * dv) 

v, dv = V.fsum2_(X * dv, -E * dw) 

if trip: 

break 

trip = (du**2 + dv**2) < _TOL_10 

else: 

raise EllipticError('no %s convergence' % ('sigmaInv',)) 

return u, v 

 

def _sigmaInv0(self, xi, eta): 

'''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)}. 

''' 

trip = False 

if eta > self._Ev_KE_5_4 or xi < min(- self._Eu_E_1_4, 

eta - self._Ev_KE): 

# 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_E 

y = eta - self._Ev_KE 

d = x**2 + y**2 

u = self._Eu_K + x / d 

v = self._Ev_K - y / d 

 

elif eta > self._Ev_KE or (eta > self._Ev_KE_3_4 and 

xi < self._Eu_E_1_4): 

# 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] 

 

d = eta - self._Ev_KE 

r = hypot(xi, d) 

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

trip = r < _TAYTOL2 

# Map the range [-90, 180] in sigma space to [-90, 0] in 

# w space. See discussion in zetainv0 on the cut for ang. 

r = cbrt(r * self._3_mv) 

a = atan2(d - xi, xi + d) / 3.0 - PI_4 

s, c = sincos2(a) 

u = r * c 

v = r * s + self._Ev_K 

 

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

r = self._Eu_K / self._Eu_E 

u = xi * r 

v = eta * r 

return u, v, trip 

 

def toStr(self, **kwds): 

'''Return a C{str} representation. 

 

@param kwds: Optional, keyword arguments. 

''' 

d = dict(datum=self.datum.name, lon0=self.lon0, k0=self.k0, extendp=self.extendp) 

if self.name: 

d['name'] = self.name 

if kwds: 

d.update(kwds) 

return ', '.join('%s=%s' % t for t in sorted(d.items())) 

 

def _zeta3(self, snu, cnu, dnu, snv, cnv, dnv): 

''' 

@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 = self._e 

# 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 = cnu**2 + self._mv * (snu * snv)**2 

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

# Overflow value s.t. atan(overflow) = pi/2 

t1 = t2 = copysign(_OVERFLOW, snu) 

if d1 > 0: 

t1 = snu * dnv / sqrt(d1) 

if d2 > 0: 

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

# psi = asinh(t1) - asinh(t2) 

# taup = sinh(psi) 

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

lam = (atan2( dnu * snv, cnu * cnv) - e * 

atan2(e * cnu * snv, dnu * cnv)) if (d1 > 0 and 

d2 > 0) else 0 

return taup, lam, d2 

 

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

''' 

@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 

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

# (see A+S 16.21.4) 

d = self._mv * (cnv2 + snuv2)**2 

du = cnu * dnuv * (cnv2 - snuv2) / d 

dv = -cnv * snuv * (cnu2 + dnuv2) / d 

return du, dv 

 

def _zetaInv(self, taup, lam): 

'''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. 

''' 

psi = asinh(taup) 

sca = 1.0 / hypot1(taup) 

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

if not trip: 

stol2 = _TOL_10 / max(psi, 1.0)**2 

U, V = Fsum(u), Fsum(v) 

# min iterations = 2, max = 6, mean = 4.0 

for _ in range(self._trips_): # GEOGRAPHICLIB_PANIC 

snu, cnu, dnu = self._Eu.sncndn(u) 

snv, cnv, dnv = self._Ev.sncndn(v) 

T, L, _ = self._zeta3( snu, cnu, dnu, snv, cnv, dnv) 

dw, dv = self._zetaDwd(snu, cnu, dnu, snv, cnv, dnv) 

T = (taup - T) * sca 

L -= lam 

u, du = U.fsum2_(T * dw, L * dv) 

v, dv = V.fsum2_(T * dv, -L * dw) 

if trip: 

break 

trip = (du**2 + dv**2) < stol2 

else: 

raise EllipticError('no %s convergence' % ('zetaInv',)) 

return u, v 

 

def _zetaInv0(self, psi, lam): 

'''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)}. 

''' 

trip = False 

if (psi < -self._e_PI_4 and lam > self._1_e2_PI_2 

and psi < lam - self._1_e_PI_2): 

# 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: 

h = sinh(1 - psi / self._e) 

a = (PI_2 - lam) / self._e 

s, c = sincos2(a) 

u = self._Eu_K - asinh(s / hypot(c, h)) * self._mu_2_1 

v = self._Ev_K - atan2(c, h) * self._mu_2_1 

 

elif (psi < self._e_PI_2 and lam > self._1_e2_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] 

 

d = lam - self._1_e_PI_2 

r = hypot(psi, d) 

# Error using this guess is about 0.21 * (rad/e)^(5/3) 

trip = r < self._e_taytol_ 

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

# in range [-135, 225). Subtracting 180 (since 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. 

r = cbrt(r * self._3_mv_e) 

a = atan2(d - psi, psi + d) / 3.0 - PI_4 

s, c = sincos2(a) 

u = r * c 

v = r * s + self._Ev_K 

 

else: 

# 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_K / PI_2 

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

u = r * atan2(h, c) 

v = r * asinh(s / hypot(c, h)) 

return u, v, trip 

 

 

def parseETM5(strUTM, datum=Datums.WGS84, Etm=Etm, falsed=True, name=''): 

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

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

 

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

@keyword datum: Optional datum to use (L{Datum}). 

@keyword Etm: Optional (sub-)class to return the UTM 

coordinate (L{Etm}) or C{None}. 

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

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

 

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

L{UtmUps5Tuple}C{(zone, hemipole, 

easting, northing, band)} if B{C{Etm}} is 

C{None}. The C{hemipole} is the hemisphere 

C{'N'|'S'}. 

 

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

 

@example: 

 

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

>>> u.toStr2() # [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, ETMError) 

if Etm is not None: 

z, h, e, n, B = r 

r = Etm(z, h, e, n, band=B, datum=datum, falsed=falsed) 

return _xnamed(r, name) 

 

 

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

zone=None, **cmoff): 

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

 

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

geodetic C{LatLon} point. 

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

@keyword datum: Optional datum for this ETM coordinate, 

overriding B{C{latlon}}'s datum (C{Datum}). 

@keyword Etm: Optional (sub-)class to return the ETM 

coordinate (L{Etm}) or C{None}. 

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

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

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

@keyword 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. 

 

@raise ETMError: Invalid B{C{zone}}. 

 

@raise TypeError: If B{C{latlon}} is not ellipsoidal. 

 

@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{rangerrors} set to C{True}. 

 

@raise ValueError: If B{C{lon}} value is missing or if 

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) 

 

t = z, lat, x, y, B, d, g, k, f 

return _toXtm8(Etm, t, name, latlon, d.exactTM) 

 

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