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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.basics import EPS, NN, PI_2, PI_4, property_doc_, \ property_RO, _xinstanceof from pygeodesy.datum import Datum, Datums from pygeodesy.elliptic import Elliptic, EllipticError, _TRIPS from pygeodesy.errors import _incompatible from pygeodesy.fmath import cbrt, Fsum, fsum_, hypot, hypot1, hypot2 from pygeodesy.karney import _diff182, _fix90, _norm180 from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY from pygeodesy.named import _NamedBase, _NamedTuple, _xnamed from pygeodesy.streprs import pairs, unstr from pygeodesy.units import Lon, Scalar_ from pygeodesy.utily import sincos2 from pygeodesy.utm import _cmlon, _K0, _parseUTM5, Utm, UTMError, \ _toXtm8, _to7zBlldfn from pygeodesy.utmupsBase import _LLEB from math import asinh, atan, atan2, copysign, degrees, radians, \ sinh, sqrt, tan __all__ = _ALL_LAZY.etm + _ALL_DOCS('EasNorExact4Tuple', 'LatLonExact4Tuple') __version__ = '20.05.14' _OVERFLOW = 1.0 / EPS**2 _TOL = EPS _TOL_10 = 0.1 * _TOL _TAYTOL = pow(_TOL, 0.6) _TAYTOL2 = 2.0 * _TAYTOL 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') 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=NN, # PYCHOK expected datum=Datums.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 (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, (latitudinal) band (C{str}, 'C'..'X'). @kwarg datum: Optional, this coordinate's datum (L{Datum}). @kwarg falsed: Both B{C{easting}} and B{C{northing}} are 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}}. @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}). ''' _xinstanceof(ExactTransverseMercator, exactTM=exactTM) E = self.datum.ellipsoid if exactTM._E != E or exactTM.majoradius != E.a \ or exactTM.flattening != E.f: raise ETMError(repr(exactTM), txt=_incompatible(repr(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. @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 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: t = repr(d.ellipsoid) raise ETMError(repr(xTM._E), txt=_incompatible(t)) 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 = True _f = 0 # flattening _k0 = 1 # central scale factor _k0_a = 0 _lon0 = 0 # central meridian _trips_ = _TRIPS # see ._reset() below: # _e_PI_2 = _e * PI_2 # _e_PI_4 = _e * PI_4 # _e_TAYTOL = _e * _TAYTOL # _1_e_90 = (1 - _e) * 90 # _1_e_PI_2 = (1 - _e) * PI_2 # _1_e2_PI_2 = (1 - _e * 2) * PI_2 # _mu = _e**2 # _mu_2_1 = (_e**2 + 2) * 0.5 # _Eu = Elliptic(_mu) # _Eu_cE_1_4 = _Eu.cE * 0.25 # _Eu_cK_cE = _Eu.cK / _Eu.cE # _Eu_cK_PI_2 = _Eu.cK / PI_2 # _mv = 1 - _mu # _3_mv = 3.0 / _mv # _3_mv_e = _3_mv / _e # _Ev = Elliptic(_mv) # _Ev_cKE_3_4 = _Ev.cKE * 0.75 # _Ev_cKE_5_4 = _Ev.cKE * 1.25 def __init__(self, datum=Datums.WGS84, lon0=0, k0=_K0, extendp=True, name=NN): '''New L{ExactTransverseMercator} projection. @kwarg datum: The datum and ellipsoid to use (C{Datum}). @kwarg lon0: The central meridian (C{degrees180}). @kwarg k0: The central scale factor (C{float}). @kwarg extendp: Use the extended domain (C{bool}). @kwarg name: Optional name for the projection (C{str}). @raise EllipticError: No convergence. @raise ETMError: Invalid B{C{k0}}. @raise TypeError: Invalid B{C{datum}}. @raise ValueError: Invalid B{C{lon0}} or B{C{k0}}. @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 not extendp: self._extendp = False if name: self.name = name self.datum = datum self.lon0 = lon0 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}). @raise EllipticError: No convergence. @raise TypeError: Invalid B{C{datum}}. ''' _xinstanceof(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. @arg lat: Latitude of point (C{degrees}). @arg lon: Longitude of point (C{degrees}). @kwarg 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 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: 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.cK, 0 elif lat == 0 and lon == self._1_e_90: u, v = 0, 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 = 2 * self._Eu.cE - xi y = xi * self._k0_a x = eta * self._k0_a if lat == 90: g, k = lon, self._k0 else: g, k = self._zetaScaled(sncndn6, ll=False) if backside: g = 180 - g if _lat: y, g = -y, -g if _lon: x, g = -x, -g return EasNorExact4Tuple(x, y, g, k) @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, name='k0', Error=ETMError, low=_TOL_10, high=1.0) # if not self._k0 > 0: # raise Scalar_.Error_(Scalar_, k0, name='k0', Error=ETMError) self._k0_a = self._k0 * self._a @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 ValueError: Invalid B{C{lon0}}. ''' self._lon0 = _norm180(Lon(lon0, name='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 some frequently used values. @arg e: Eccentricity (C{float}). @arg e2: Eccentricity squared (C{float}). @raise EllipticError: No convergence. ''' # assert e2 == e**2 self._e = e 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 self._mu_2_1 = (e2 + 2) * 0.5 self._Eu = Elliptic(self._mu) self._Eu_cE_1_4 = self._Eu.cE * 0.25 self._Eu_cK_cE = self._Eu.cK / self._Eu.cE self._Eu_cK_PI_2 = self._Eu.cK / PI_2 self._mv = 1 - e2 self._3_mv = 3.0 / self._mv self._3_mv_e = self._3_mv / e self._Ev = Elliptic(self._mv) self._Ev_cKE_3_4 = self._Ev.cKE * 0.75 self._Ev_cKE_5_4 = self._Ev.cKE * 1.25 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}). @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 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: xi = 2 * self._Eu.cE - xi backside = True # u,v = coordinates for the Thompson TM, Lee 54 if xi != 0 or eta != self._Ev.cKE: u, v = self._sigmaInv(xi, eta) else: u, v = 0, self._Ev.cK if v != 0 or u != self._Eu.cK: g, k, lat, lon = self._zetaScaled(self._sncndn6(u, v)) else: g, k, lat, lon = 0, self._k0, 90, 0 if backside: lon, g = (180 - lon), (180 - g) if _lat: lat, g = -lat, -g if _lon: lon, g = -lon, -g lon += self._lon0 if lon0 is None else _norm180(lon0) return LatLonExact4Tuple(_norm180(lat), _norm180(lon), g, k) 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 = 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 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 + mv * tau**2 + mu) * sqrt(q2) k = sqrt(fsum_(mu, mv, mv * tau**2)) * sqrt(q2) return degrees(g), k * self._k0 def _sigma3(self, v, snu, cnu, dnu, snv, cnv, dnv): # PYCHOK unused '''(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 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): '''(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 = 2 * r * i / 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. ''' 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 sncndn6 = self._sncndn6(u, v) X, E, _ = self._sigma3(v, *sncndn6) dw, dv = self._sigmaDwd( *sncndn6) 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 = hypot2(du, dv) < _TOL_10 else: t = unstr(self._sigmaInv.__name__, xi, eta) raise EllipticError('no convergence', txt=t) return u, v 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)}. ''' trip = False if eta > self._Ev_cKE_5_4 or xi < min(- self._Eu_cE_1_4, eta - self._Ev.cKE): # 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 = eta - self._Ev.cKE d = hypot2(x, y) u = self._Eu.cK + x / d v = self._Ev.cK - y / d elif eta > self._Ev.cKE or (xi < self._Eu_cE_1_4 and eta > self._Ev_cKE_3_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.cKE 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.cK else: # use w = sigma * Eu.K/Eu.E (correct in the limit _e -> 0) u = xi * self._Eu_cK_cE v = eta * self._Eu_cK_cE return u, v, trip 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(name=self.name) if self.name else {} d = dict(datum=self.datum.name, lon0=self.lon0, k0=self.k0, extendp=self.extendp, **d) return ', '.join(pairs(d, **kwds)) def _zeta3(self, snu, cnu, dnu, snv, cnv, dnv): '''(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 = 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) lam = 0 if d2 > 0: t2 = sinh(e * asinh(e * snu / sqrt(d2))) if d1 > 0: lam = atan2(dnu * snv , cnu * cnv) - \ atan2(cnu * snv * e, dnu * cnv) * e # psi = asinh(t1) - asinh(t2) # taup = sinh(psi) taup = t1 * hypot1(t2) - t2 * hypot1(t1) 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 # (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): '''(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. ''' psi = asinh(taup) sca = 1.0 / hypot1(taup) u, v, trip = self._zetaInv0(psi, lam) if not trip: stol2 = _TOL_10 / max(psi**2, 1.0) U, V = Fsum(u), Fsum(v) # min iterations = 2, max = 6, mean = 4.0 for _ in range(self._trips_): # GEOGRAPHICLIB_PANIC sncndn6 = self._sncndn6(u, v) T, L, _ = self._zeta3( *sncndn6) dw, dv = self._zetaDwd(*sncndn6) 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 = hypot2(du, dv) < stol2 else: t = unstr(self._zetaInv.__name__, taup, lam) raise EllipticError('no convergence', txt=t) return u, v 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)}. ''' 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.cK - asinh(s / hypot(c, h)) * self._mu_2_1 v = self._Ev.cK - 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.cK 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_cK_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 _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 B{C{ll}} is C{False} else 4-tuple C{(g, k, lat, lon)}. ''' t, lam, d2 = self._zeta3( *sncndn6) tau = self._E.es_tauf(t) r = self._scaled(tau, d2, *sncndn6) if ll: r += degrees(atan(tau)), degrees(lam) return r class LatLonExact4Tuple(_NamedTuple): '''4-Tuple C{(lat, lon, convergence, scale)} in C{degrees180}, C{degrees}, C{degrees} and C{scalar}. ''' _Names_ = ('lat', 'lon', 'convergence', 'scale') def parseETM5(strUTM, datum=Datums.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}). @kwarg Etm: Optional class to return the UTM coordinate (L{Etm}) or C{None}. @kwarg falsed: Both easting and northing are 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}}. @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, datum, Etm, falsed, Error=ETMError) return _xnamed(r, name) 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 (C{Datum}). @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. @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) return _toXtm8(Etm, z, lat, x, y, B, d, g, k, f, name, latlon, d.exactTM, Error=ETMError) # **) MIT License # # Copyright (C) 2016-2020 -- mrJean1 at Gmail -- All Rights Reserved. # # Permission is hereby granted, free of charge, to any person obtaining a # copy of this software and associated documentation files (the "Software"), # to deal in the Software without restriction, including without limitation # the rights to use, copy, modify, merge, publish, distribute, sublicense, # and/or sell copies of the Software, and to permit persons to whom the # Software is furnished to do so, subject to the following conditions: # # The above copyright notice and this permission notice shall be included # in all copies or substantial portions of the Software. # # THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS # OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, # FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL # THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR # OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, # ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR # OTHER DEALINGS IN THE SOFTWARE.