Coverage for pygeodesy/etm.py: 92%
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2# -*- coding: utf-8 -*-
4u'''A pure Python version of I{Karney}'s C{Exact Transverse Mercator} (ETM) projection.
6Classes L{Etm}, L{ETMError} and L{ExactTransverseMercator}, transcoded from I{Karney}'s
7C++ class U{TransverseMercatorExact<https://GeographicLib.SourceForge.io/C++/doc/
8classGeographicLib_1_1TransverseMercatorExact.html>}, abbreviated as C{TMExact} below.
10Class L{ExactTransverseMercator} provides C{Exact Transverse Mercator} projections while
11instances of class L{Etm} represent ETM C{(easting, northing)} locations. See also
12I{Karney}'s utility U{TransverseMercatorProj<https://GeographicLib.SourceForge.io/C++/doc/
13TransverseMercatorProj.1.html>} and use C{"python[3] -m pygeodesy.etm ..."} to compare
14the results.
16Following is a copy of I{Karney}'s U{TransverseMercatorExact.hpp
17<https://GeographicLib.SourceForge.io/C++/doc/TransverseMercatorExact_8hpp_source.html>}
18file C{Header}.
20Copyright (C) U{Charles Karney<mailto:Karney@Alum.MIT.edu>} (2008-2023) and licensed
21under the MIT/X11 License. For more information, see the U{GeographicLib<https://
22GeographicLib.SourceForge.io>} documentation.
24The method entails using the U{Thompson Transverse Mercator<https://WikiPedia.org/
25wiki/Transverse_Mercator_projection>} as an intermediate projection. The projections
26from the intermediate coordinates to C{phi, lam} and C{x, y} are given by elliptic
27functions. The inverse of these projections are found by Newton's method with a
28suitable starting guess.
30The relevant section of L.P. Lee's paper U{Conformal Projections Based On Jacobian
31Elliptic Functions<https://DOI.org/10.3138/X687-1574-4325-WM62>} in part V, pp
3267-101. The C++ implementation and notation closely follow Lee, with the following
33exceptions::
35 Lee here Description
37 x/a xi Northing (unit Earth)
39 y/a eta Easting (unit Earth)
41 s/a sigma xi + i * eta
43 y x Easting
45 x y Northing
47 k e Eccentricity
49 k^2 mu Elliptic function parameter
51 k'^2 mv Elliptic function complementary parameter
53 m k Scale
55 zeta zeta Complex longitude = Mercator = chi in paper
57 s sigma Complex GK = zeta in paper
59Minor alterations have been made in some of Lee's expressions in an attempt to
60control round-off. For example, C{atanh(sin(phi))} is replaced by C{asinh(tan(phi))}
61which maintains accuracy near C{phi = pi/2}. Such changes are noted in the code.
62'''
63# make sure int/int division yields float quotient, see .basics
64from __future__ import division as _; del _ # PYCHOK semicolon
66from pygeodesy.basics import map1, neg, neg_, _xinstanceof
67from pygeodesy.constants import EPS, EPS02, PI_2, PI_4, _K0_UTM, \
68 _1_EPS, _0_0, _0_1, _0_5, _1_0, _2_0, \
69 _3_0, _4_0, _90_0, isnear0, isnear90
70from pygeodesy.datums import _ellipsoidal_datum, _WGS84, _EWGS84
71# from pygeodesy.ellipsoids import _EWGS84 # from .datums
72from pygeodesy.elliptic import _ALL_LAZY, Elliptic
73# from pygeodesy.errors import _incompatible # from .named
74# from pygeodesy.fsums import Fsum # from .fmath
75from pygeodesy.fmath import cbrt, hypot, hypot1, hypot2, Fsum
76from pygeodesy.interns import _COMMASPACE_, _DASH_, _near_, _SPACE_, \
77 _spherical_
78from pygeodesy.karney import _copyBit, _diff182, _fix90, _norm2, _norm180, \
79 _tand, _unsigned2
80# from pygeodesy.lazily import _ALL_LAZY # from .elliptic
81from pygeodesy.named import callername, _incompatible, _NamedBase
82from pygeodesy.namedTuples import Forward4Tuple, Reverse4Tuple
83from pygeodesy.props import deprecated_method, deprecated_property_RO, \
84 Property_RO, property_RO, _update_all, \
85 property_doc_
86from pygeodesy.streprs import Fmt, pairs, unstr
87from pygeodesy.units import Degrees, Scalar_
88from pygeodesy.utily import atan1d, atan2d, _loneg, sincos2
89from pygeodesy.utm import _cmlon, _LLEB, _parseUTM5, _toBand, _toXtm8, \
90 _to7zBlldfn, Utm, UTMError
92from math import asinh, atan2, degrees, radians, sinh, sqrt
94__all__ = _ALL_LAZY.etm
95__version__ = '24.05.31'
97_OVERFLOW = _1_EPS**2 # about 2e+31
98_TAYTOL = pow(EPS, 0.6)
99_TAYTOL2 = _TAYTOL * _2_0
100_TOL_10 = EPS * _0_1
101_TRIPS = 21 # C++ 10
104class ETMError(UTMError):
105 '''Exact Transverse Mercator (ETM) parse, projection or other
106 L{Etm} issue or L{ExactTransverseMercator} conversion failure.
107 '''
108 pass
111class Etm(Utm):
112 '''Exact Transverse Mercator (ETM) coordinate, a sub-class of L{Utm},
113 a Universal Transverse Mercator (UTM) coordinate using the
114 L{ExactTransverseMercator} projection for highest accuracy.
116 @note: Conversion of (geodetic) lat- and longitudes to/from L{Etm}
117 coordinates is 3-4 times slower than to/from L{Utm}.
119 @see: Karney's U{Detailed Description<https://GeographicLib.SourceForge.io/
120 C++/doc/classGeographicLib_1_1TransverseMercatorExact.html#details>}.
121 '''
122 _Error = ETMError # see utm.UTMError
123 _exactTM = None
125 __init__ = Utm.__init__
126 '''New L{Etm} Exact Transverse Mercator coordinate, raising L{ETMError}s.
128 @see: L{Utm.__init__} for more information.
129 '''
131 @property_doc_(''' the ETM projection (L{ExactTransverseMercator}).''')
132 def exactTM(self):
133 '''Get the ETM projection (L{ExactTransverseMercator}).
134 '''
135 if self._exactTM is None:
136 self.exactTM = self.datum.exactTM # ExactTransverseMercator(datum=self.datum)
137 return self._exactTM
139 @exactTM.setter # PYCHOK setter!
140 def exactTM(self, exactTM):
141 '''Set the ETM projection (L{ExactTransverseMercator}).
143 @raise ETMError: The B{C{exacTM}}'s datum incompatible
144 with this ETM coordinate's C{datum}.
145 '''
146 _xinstanceof(ExactTransverseMercator, exactTM=exactTM)
148 E = self.datum.ellipsoid
149 if E != exactTM.ellipsoid: # may be None
150 raise ETMError(repr(exactTM), txt=_incompatible(repr(E)))
151 self._exactTM = exactTM
152 self._scale0 = exactTM.k0
154 def parse(self, strETM, **name):
155 '''Parse a string to a similar L{Etm} instance.
157 @arg strETM: The ETM coordinate (C{str}), see function L{parseETM5}.
158 @kwarg name: Optional C{B{name}=NN} (C{str}), overriding this name.
160 @return: The instance (L{Etm}).
162 @raise ETMError: Invalid B{C{strETM}}.
164 @see: Function L{pygeodesy.parseUPS5}, L{pygeodesy.parseUTM5} and
165 L{pygeodesy.parseUTMUPS5}.
166 '''
167 return parseETM5(strETM, datum=self.datum, Etm=self.classof,
168 name=self._name__(name))
170 @deprecated_method
171 def parseETM(self, strETM): # PYCHOK no cover
172 '''DEPRECATED, use method L{Etm.parse}.
173 '''
174 return self.parse(strETM)
176 def toLatLon(self, LatLon=None, unfalse=True, **unused): # PYCHOK expected
177 '''Convert this ETM coordinate to an (ellipsoidal) geodetic point.
179 @kwarg LatLon: Optional, ellipsoidal class to return the geodetic
180 point (C{LatLon}) or C{None}.
181 @kwarg unfalse: Unfalse B{C{easting}} and B{C{northing}} if
182 C{falsed} (C{bool}).
184 @return: This ETM coordinate as (B{C{LatLon}}) or a
185 L{LatLonDatum5Tuple}C{(lat, lon, datum, gamma,
186 scale)} if B{C{LatLon}} is C{None}.
188 @raise ETMError: This ETM coordinate's C{exacTM} and this C{datum}
189 incompatible or no convergence transforming to
190 lat- and longitude.
192 @raise TypeError: Invalid or non-ellipsoidal B{C{LatLon}}.
193 '''
194 if not self._latlon or self._latlon._toLLEB_args != (unfalse, self.exactTM):
195 self._toLLEB(unfalse=unfalse)
196 return self._latlon5(LatLon)
198 def _toLLEB(self, unfalse=True, **unused): # PYCHOK signature
199 '''(INTERNAL) Compute (ellipsoidal) lat- and longitude.
200 '''
201 xTM, d = self.exactTM, self.datum
202 # double check that this and exactTM's ellipsoid match
203 if xTM._E != d.ellipsoid: # PYCHOK no cover
204 t = repr(d.ellipsoid)
205 raise ETMError(repr(xTM._E), txt=_incompatible(t))
207 e, n = self.eastingnorthing2(falsed=not unfalse)
208 lon0 = _cmlon(self.zone) if bool(unfalse) == self.falsed else None
209 lat, lon, g, k = xTM.reverse(e, n, lon0=lon0)
211 ll = _LLEB(lat, lon, datum=d, name=self.name) # utm._LLEB
212 self._latlon5args(ll, g, k, _toBand, unfalse, xTM)
214 def toUtm(self): # PYCHOK signature
215 '''Copy this ETM to a UTM coordinate.
217 @return: The UTM coordinate (L{Utm}).
218 '''
219 return self._xcopy2(Utm)
222class ExactTransverseMercator(_NamedBase):
223 '''Pure Python version of Karney's C++ class U{TransverseMercatorExact
224 <https://GeographicLib.SourceForge.io/C++/doc/TransverseMercatorExact_8cpp_source.html>},
225 a numerically exact transverse Mercator projection, further referred to as C{TMExact}.
226 '''
227 _datum = _WGS84 # Datum
228 _E = _EWGS84 # Ellipsoid
229 _extendp = False # use extended domain
230# _iteration = None # ._sigmaInv2 and ._zetaInv2
231 _k0 = _K0_UTM # central scale factor
232 _lat0 = _0_0 # central parallel
233 _lon0 = _0_0 # central meridian
234 _mu = _EWGS84.e2 # 1st eccentricity squared
235 _mv = _EWGS84.e21 # 1 - ._mu
236 _raiser = False # throw Error
237 _sigmaC = None # most recent _sigmaInv04 case C{int}
238 _zetaC = None # most recent _zetaInv04 case C{int}
240 def __init__(self, datum=_WGS84, lon0=0, k0=_K0_UTM, extendp=False, raiser=False, **name):
241 '''New L{ExactTransverseMercator} projection.
243 @kwarg datum: The I{non-spherical} datum or ellipsoid (L{Datum},
244 L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}).
245 @kwarg lon0: Central meridian, default (C{degrees180}).
246 @kwarg k0: Central scale factor (C{float}).
247 @kwarg extendp: Use the I{extended} domain (C{bool}), I{standard} otherwise.
248 @kwarg raiser: If C{True}, throw an L{ETMError} for convergence failures (C{bool}).
249 @kwarg name: Optional C{B{name}=NN} for the projection (C{str}).
251 @raise ETMError: Near-spherical B{C{datum}} or C{ellipsoid} or invalid B{C{lon0}}
252 or B{C{k0}}.
254 @see: U{Constructor TransverseMercatorExact<https://GeographicLib.SourceForge.io/
255 C++/doc/classGeographicLib_1_1TransverseMercatorExact.html>} for more details,
256 especially on B{X{extendp}}.
258 @note: For all 255.5K U{TMcoords.dat<https://Zenodo.org/record/32470>} tests (with
259 C{0 <= lat <= 84} and C{0 <= lon}) the maximum error is C{5.2e-08 .forward}
260 (or 52 nano-meter) easting and northing and C{3.8e-13 .reverse} (or 0.38
261 pico-degrees) lat- and longitude (with Python 3.7.3+, 2.7.16+, PyPy6 3.5.3
262 and PyPy6 2.7.13, all in 64-bit on macOS 10.13.6 High Sierra C{x86_64} and
263 12.2 Monterey C{arm64} and C{"arm64_x86_64"}).
264 '''
265 if extendp:
266 self._extendp = True
267 if name:
268 self.name = name
269 if raiser:
270 self.raiser = True
272 TM = ExactTransverseMercator
273 if datum not in (TM._datum, TM._E, None):
274 self.datum = datum # invokes ._resets
275 if lon0 or lon0 != TM._lon0:
276 self.lon0 = lon0
277 if k0 is not TM._k0:
278 self.k0 = k0
280 @property_doc_(''' the datum (L{Datum}).''')
281 def datum(self):
282 '''Get the datum (L{Datum}) or C{None}.
283 '''
284 return self._datum
286 @datum.setter # PYCHOK setter!
287 def datum(self, datum):
288 '''Set the datum and ellipsoid (L{Datum}, L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}).
290 @raise ETMError: Near-spherical B{C{datum}} or C{ellipsoid}.
291 '''
292 d = _ellipsoidal_datum(datum, name=self.name) # raiser=_datum_)
293 self._resets(d)
294 self._datum = d
296 @Property_RO
297 def _e(self):
298 '''(INTERNAL) Get and cache C{_e}.
299 '''
300 return self._E.e
302 @Property_RO
303 def _1_e_90(self): # PYCHOK no cover
304 '''(INTERNAL) Get and cache C{(1 - _e) * 90}.
305 '''
306 return (_1_0 - self._e) * _90_0
308 @property_RO
309 def ellipsoid(self):
310 '''Get the ellipsoid (L{Ellipsoid}).
311 '''
312 return self._E
314 @Property_RO
315 def _e_PI_2(self):
316 '''(INTERNAL) Get and cache C{_e * PI / 2}.
317 '''
318 return self._e * PI_2
320 @Property_RO
321 def _e_PI_4_(self):
322 '''(INTERNAL) Get and cache C{-_e * PI / 4}.
323 '''
324 return -self._e * PI_4
326 @Property_RO
327 def _1_e_PI_2(self):
328 '''(INTERNAL) Get and cache C{(1 - _e) * PI / 2}.
329 '''
330 return (_1_0 - self._e) * PI_2
332 @Property_RO
333 def _1_2e_PI_2(self):
334 '''(INTERNAL) Get and cache C{(1 - 2 * _e) * PI / 2}.
335 '''
336 return (_1_0 - self._e * _2_0) * PI_2
338 @property_RO
339 def equatoradius(self):
340 '''Get the C{ellipsoid}'s equatorial radius, semi-axis (C{meter}).
341 '''
342 return self._E.a
344 a = equatoradius
346 @Property_RO
347 def _e_TAYTOL(self):
348 '''(INTERNAL) Get and cache C{e * TAYTOL}.
349 '''
350 return self._e * _TAYTOL
352 @Property_RO
353 def _Eu(self):
354 '''(INTERNAL) Get and cache C{Elliptic(_mu)}.
355 '''
356 return Elliptic(self._mu)
358 @Property_RO
359 def _Eu_cE(self):
360 '''(INTERNAL) Get and cache C{_Eu.cE}.
361 '''
362 return self._Eu.cE
364 def _Eu_2cE_(self, xi):
365 '''(INTERNAL) Return C{_Eu.cE * 2 - B{xi}}.
366 '''
367 return self._Eu_cE * _2_0 - xi
369 @Property_RO
370 def _Eu_cE_4(self):
371 '''(INTERNAL) Get and cache C{_Eu.cE / 4}.
372 '''
373 return self._Eu_cE / _4_0
375 @Property_RO
376 def _Eu_cK(self):
377 '''(INTERNAL) Get and cache C{_Eu.cK}.
378 '''
379 return self._Eu.cK
381 @Property_RO
382 def _Eu_cK_cE(self):
383 '''(INTERNAL) Get and cache C{_Eu.cK / _Eu.cE}.
384 '''
385 return self._Eu_cK / self._Eu_cE
387 @Property_RO
388 def _Eu_2cK_PI(self):
389 '''(INTERNAL) Get and cache C{_Eu.cK * 2 / PI}.
390 '''
391 return self._Eu_cK / PI_2
393 @Property_RO
394 def _Ev(self):
395 '''(INTERNAL) Get and cache C{Elliptic(_mv)}.
396 '''
397 return Elliptic(self._mv)
399 @Property_RO
400 def _Ev_cK(self):
401 '''(INTERNAL) Get and cache C{_Ev.cK}.
402 '''
403 return self._Ev.cK
405 @Property_RO
406 def _Ev_cKE(self):
407 '''(INTERNAL) Get and cache C{_Ev.cKE}.
408 '''
409 return self._Ev.cKE
411 @Property_RO
412 def _Ev_3cKE_4(self):
413 '''(INTERNAL) Get and cache C{_Ev.cKE * 3 / 4}.
414 '''
415 return self._Ev_cKE * 0.75 # _0_75
417 @Property_RO
418 def _Ev_5cKE_4(self):
419 '''(INTERNAL) Get and cache C{_Ev.cKE * 5 / 4}.
420 '''
421 return self._Ev_cKE * 1.25 # _1_25
423 @Property_RO
424 def extendp(self):
425 '''Get the domain (C{bool}), I{extended} or I{standard}.
426 '''
427 return self._extendp
429 @property_RO
430 def flattening(self):
431 '''Get the C{ellipsoid}'s flattening (C{scalar}).
432 '''
433 return self._E.f
435 f = flattening
437 def forward(self, lat, lon, lon0=None, **name): # MCCABE 13
438 '''Forward projection, from geographic to transverse Mercator.
440 @arg lat: Latitude of point (C{degrees}).
441 @arg lon: Longitude of point (C{degrees}).
442 @kwarg lon0: Central meridian (C{degrees180}), overriding
443 the default if not C{None}.
444 @kwarg name: Optional C{B{name}=NN} (C{str}).
446 @return: L{Forward4Tuple}C{(easting, northing, gamma, scale)}.
448 @see: C{void TMExact::Forward(real lon0, real lat, real lon,
449 real &x, real &y,
450 real &gamma, real &k)}.
452 @raise ETMError: No convergence, thrown iff property
453 C{B{raiser}=True}.
454 '''
455 lat = _fix90(lat - self._lat0)
456 lon, _ = _diff182((self.lon0 if lon0 is None else lon0), lon)
457 if self.extendp:
458 backside = _lat = _lon = False
459 else: # enforce the parity
460 lat, _lat = _unsigned2(lat)
461 lon, _lon = _unsigned2(lon)
462 backside = lon > 90
463 if backside: # PYCHOK no cover
464 lon = _loneg(lon)
465 if lat == 0:
466 _lat = True
468 # u, v = coordinates for the Thompson TM, Lee 54
469 if lat == 90: # isnear90(lat)
470 u = self._Eu_cK
471 v = self._iteration = self._zetaC = 0
472 elif lat == 0 and lon == self._1_e_90: # PYCHOK no cover
473 u = self._iteration = self._zetaC = 0
474 v = self._Ev_cK
475 else: # tau = tan(phi), taup = sinh(psi)
476 tau, lam = _tand(lat), radians(lon)
477 u, v = self._zetaInv2(self._E.es_taupf(tau), lam)
479 sncndn6 = self._sncndn6(u, v)
480 y, x, _ = self._sigma3(v, *sncndn6)
481 g, k = (lon, self.k0) if isnear90(lat) else \
482 self._zetaScaled(sncndn6, ll=False)
484 if backside:
485 y, g = self._Eu_2cE_(y), _loneg(g)
486 y *= self._k0_a
487 x *= self._k0_a
488 if _lat:
489 y, g = neg_(y, g)
490 if _lon:
491 x, g = neg_(x, g)
492 return Forward4Tuple(x, y, g, k, iteration=self._iteration,
493 name=self._name__(name))
495 def _Inv03(self, psi, dlam, _3_mv_e): # (xi, deta, _3_mv)
496 '''(INTERNAL) Partial C{_zetaInv04} or C{_sigmaInv04}, Case 2
497 '''
498 # atan2(dlam-psi, psi+dlam) + 45d gives arg(zeta - zeta0) in
499 # range [-135, 225). Subtracting 180 (multiplier is negative)
500 # makes range [-315, 45). Multiplying by 1/3 (for cube root)
501 # gives range [-105, 15). In particular the range [-90, 180]
502 # in zeta space maps to [-90, 0] in w space as required.
503 a = atan2(dlam - psi, psi + dlam) / _3_0 - PI_4
504 s, c = sincos2(a)
505 h = hypot(psi, dlam)
506 r = cbrt(h * _3_mv_e)
507 u = r * c
508 v = r * s + self._Ev_cK
509 # Error using this guess is about 0.068 * rad^(5/3)
510 return u, v, h
512 @property_RO
513 def iteration(self):
514 '''Get the most recent C{ExactTransverseMercator.forward}
515 or C{ExactTransverseMercator.reverse} iteration number
516 (C{int}) or C{None} if not available/applicable.
517 '''
518 return self._iteration
520 @property_doc_(''' the central scale factor (C{float}).''')
521 def k0(self):
522 '''Get the central scale factor (C{float}), aka I{C{scale0}}.
523 '''
524 return self._k0 # aka scale0
526 @k0.setter # PYCHOK setter!
527 def k0(self, k0):
528 '''Set the central scale factor (C{float}), aka I{C{scale0}}.
530 @raise ETMError: Invalid B{C{k0}}.
531 '''
532 k0 = Scalar_(k0=k0, Error=ETMError, low=_TOL_10, high=_1_0)
533 if self._k0 != k0:
534 ExactTransverseMercator._k0_a._update(self) # redo ._k0_a
535 self._k0 = k0
537 @Property_RO
538 def _k0_a(self):
539 '''(INTERNAL) Get and cache C{k0 * equatoradius}.
540 '''
541 return self.k0 * self.equatoradius
543 @property_doc_(''' the central meridian (C{degrees180}).''')
544 def lon0(self):
545 '''Get the central meridian (C{degrees180}).
546 '''
547 return self._lon0
549 @lon0.setter # PYCHOK setter!
550 def lon0(self, lon0):
551 '''Set the central meridian (C{degrees180}).
553 @raise ETMError: Invalid B{C{lon0}}.
554 '''
555 self._lon0 = _norm180(Degrees(lon0=lon0, Error=ETMError))
557 @deprecated_property_RO
558 def majoradius(self): # PYCHOK no cover
559 '''DEPRECATED, use property C{equatoradius}.'''
560 return self.equatoradius
562 @Property_RO
563 def _1_mu_2(self):
564 '''(INTERNAL) Get and cache C{_mu / 2 + 1}.
565 '''
566 return self._mu * _0_5 + _1_0
568 @Property_RO
569 def _3_mv(self):
570 '''(INTERNAL) Get and cache C{3 / _mv}.
571 '''
572 return _3_0 / self._mv
574 @Property_RO
575 def _3_mv_e(self):
576 '''(INTERNAL) Get and cache C{3 / (_mv * _e)}.
577 '''
578 return _3_0 / (self._mv * self._e)
580 def _Newton2(self, taup, lam, u, v, C, *psi): # or (xi, eta, u, v)
581 '''(INTERNAL) Invert C{_zetaInv2} or C{_sigmaInv2} using Newton's method.
583 @return: 2-Tuple C{(u, v)}.
585 @raise ETMError: No convergence.
586 '''
587 sca1, tol2 = _1_0, _TOL_10
588 if psi: # _zetaInv2
589 sca1 = sca1 / hypot1(taup) # /= chokes PyChecker
590 tol2 = tol2 / max(psi[0], _1_0)**2
592 _zeta3 = self._zeta3
593 _zetaDwd2 = self._zetaDwd2
594 else: # _sigmaInv2
595 _zeta3 = self._sigma3
596 _zetaDwd2 = self._sigmaDwd2
598 d2, r = tol2, self.raiser
599 _U_2 = Fsum(u).fsum2f_
600 _V_2 = Fsum(v).fsum2f_
601 # min iterations 2, max 6 or 7, mean 3.9 or 4.0
602 _hy2 = hypot2
603 for i in range(1, _TRIPS): # GEOGRAPHICLIB_PANIC
604 sncndn6 = self._sncndn6(u, v)
605 du, dv = _zetaDwd2(*sncndn6)
606 T, L, _ = _zeta3(v, *sncndn6)
607 T = (taup - T) * sca1
608 L -= lam
609 u, dU = _U_2(T * du, L * dv)
610 v, dV = _V_2(T * dv, -L * du)
611 if d2 < tol2:
612 r = False
613 break
614 d2 = _hy2(dU, dV)
616 self._iteration = i
617 if r: # PYCHOK no cover
618 n = callername(up=2, underOK=True)
619 t = unstr(n, taup, lam, u, v, C=C)
620 raise ETMError(Fmt.no_convergence(d2, tol2), txt=t)
621 return u, v
623 @property_doc_(''' raise an L{ETMError} for convergence failures (C{bool}).''')
624 def raiser(self):
625 '''Get the error setting (C{bool}).
626 '''
627 return self._raiser
629 @raiser.setter # PYCHOK setter!
630 def raiser(self, raiser):
631 '''Set the error setting (C{bool}), if C{True} throw an L{ETMError}
632 for convergence failures.
633 '''
634 self._raiser = bool(raiser)
636 def reset(self, lat0, lon0):
637 '''Set the central parallel and meridian.
639 @arg lat0: Latitude of the central parallel (C{degrees90}).
640 @arg lon0: Longitude of the central parallel (C{degrees180}).
642 @return: 2-Tuple C{(lat0, lon0)} of the previous central
643 parallel and meridian.
645 @raise ETMError: Invalid B{C{lat0}} or B{C{lon0}}.
646 '''
647 t = self._lat0, self.lon0
648 self._lat0 = _fix90(Degrees(lat0=lat0, Error=ETMError))
649 self. lon0 = lon0
650 return t
652 def _resets(self, datum):
653 '''(INTERNAL) Set the ellipsoid and elliptic moduli.
655 @arg datum: Ellipsoidal datum (C{Datum}).
657 @raise ETMError: Near-spherical B{C{datum}} or C{ellipsoid}.
658 '''
659 E = datum.ellipsoid
660 mu = E.e2 # .eccentricity1st2
661 mv = E.e21 # _1_0 - mu
662 if isnear0(E.e) or isnear0(mu, eps0=EPS02) \
663 or isnear0(mv, eps0=EPS02): # or sqrt(mu) != E.e
664 raise ETMError(ellipsoid=E, txt=_near_(_spherical_))
666 if self._datum or self._E:
667 _i = ExactTransverseMercator.iteration._uname
668 _update_all(self, _i, '_sigmaC', '_zetaC') # _under
670 self._E = E
671 self._mu = mu
672 self._mv = mv
674 def reverse(self, x, y, lon0=None, **name):
675 '''Reverse projection, from Transverse Mercator to geographic.
677 @arg x: Easting of point (C{meters}).
678 @arg y: Northing of point (C{meters}).
679 @kwarg lon0: Optional central meridian (C{degrees180}),
680 overriding the default (C{iff not None}).
681 @kwarg name: Optional C{B{name}=NN} (C{str}).
683 @return: L{Reverse4Tuple}C{(lat, lon, gamma, scale)}.
685 @see: C{void TMExact::Reverse(real lon0, real x, real y,
686 real &lat, real &lon,
687 real &gamma, real &k)}
689 @raise ETMError: No convergence, thrown iff property
690 C{B{raiser}=True}.
691 '''
692 # undoes the steps in .forward.
693 xi = y / self._k0_a
694 eta = x / self._k0_a
695 if self.extendp:
696 backside = _lat = _lon = False
697 else: # enforce the parity
698 eta, _lon = _unsigned2(eta)
699 xi, _lat = _unsigned2(xi)
700 backside = xi > self._Eu_cE
701 if backside: # PYCHOK no cover
702 xi = self._Eu_2cE_(xi)
704 # u, v = coordinates for the Thompson TM, Lee 54
705 if xi or eta != self._Ev_cKE:
706 u, v = self._sigmaInv2(xi, eta)
707 else: # PYCHOK no cover
708 u = self._iteration = self._sigmaC = 0
709 v = self._Ev_cK
711 if v or u != self._Eu_cK:
712 g, k, lat, lon = self._zetaScaled(self._sncndn6(u, v))
713 else: # PYCHOK no cover
714 g, k, lat, lon = _0_0, self.k0, _90_0, _0_0
716 if backside: # PYCHOK no cover
717 lon, g = _loneg(lon), _loneg(g)
718 if _lat:
719 lat, g = neg_(lat, g)
720 if _lon:
721 lon, g = neg_(lon, g)
722 lat += self._lat0
723 lon += self._lon0 if lon0 is None else _norm180(lon0)
724 return Reverse4Tuple(lat, _norm180(lon), g, k, # _fix90(lat)
725 iteration=self._iteration,
726 name=self._name__(name))
728 def _scaled2(self, tau, d2, snu, cnu, dnu, snv, cnv, dnv):
729 '''(INTERNAL) C{scaled}.
731 @note: Argument B{C{d2}} is C{_mu * cnu**2 + _mv * cnv**2}
732 from C{._zeta3}.
734 @return: 2-Tuple C{(convergence, scale)}.
736 @see: C{void TMExact::Scale(real tau, real /*lam*/,
737 real snu, real cnu, real dnu,
738 real snv, real cnv, real dnv,
739 real &gamma, real &k)}.
740 '''
741 mu, mv = self._mu, self._mv
742 cnudnv = cnu * dnv
743 # Lee 55.12 -- negated for our sign convention. g gives
744 # the bearing (clockwise from true north) of grid north
745 g = atan2d(mv * cnv * snv * snu, cnudnv * dnu)
746 # Lee 55.13 with nu given by Lee 9.1 -- in sqrt change
747 # the numerator from (1 - snu^2 * dnv^2) to (_mv * snv^2
748 # + cnu^2 * dnv^2) to maintain accuracy near phi = 90
749 # and change the denomintor from (dnu^2 + dnv^2 - 1) to
750 # (_mu * cnu^2 + _mv * cnv^2) to maintain accuracy near
751 # phi = 0, lam = 90 * (1 - e). Similarly rewrite sqrt in
752 # 9.1 as _mv + _mu * c^2 instead of 1 - _mu * sin(phi)^2
753 if d2 > 0:
754 # originally: sec2 = 1 + tau**2 # sec(phi)^2
755 # d2 = (mu * cnu**2 + mv * cnv**2)
756 # q2 = (mv * snv**2 + cnudnv**2) / d2
757 # k = sqrt(mv + mu / sec2) * sqrt(sec2) * sqrt(q2)
758 # = sqrt(mv * sec2 + mu) * sqrt(q2)
759 # = sqrt(mv + mv * tau**2 + mu) * sqrt(q2)
760 k, q2 = _0_0, (mv * snv**2 + cnudnv**2)
761 if q2 > 0:
762 k2 = (tau**2 + _1_0) * mv + mu
763 if k2 > 0:
764 k = sqrt(k2) * sqrt(q2 / d2) * self.k0
765 else:
766 k = _OVERFLOW
767 return g, k
769 def _sigma3(self, v, snu, cnu, dnu, snv, cnv, dnv):
770 '''(INTERNAL) C{sigma}.
772 @return: 3-Tuple C{(xi, eta, d2)}.
774 @see: C{void TMExact::sigma(real /*u*/, real snu, real cnu, real dnu,
775 real v, real snv, real cnv, real dnv,
776 real &xi, real &eta)}.
778 @raise ETMError: No convergence.
779 '''
780 mu = self._mu * cnu
781 mv = self._mv * cnv
782 # Lee 55.4 writing
783 # dnu^2 + dnv^2 - 1 = _mu * cnu^2 + _mv * cnv^2
784 d2 = cnu * mu + cnv * mv
785 mu *= snu * dnu
786 mv *= snv * dnv
787 if d2 > 0: # /= chokes PyChecker
788 mu = mu / d2
789 mv = mv / d2
790 else:
791 mu, mv = map1(_overflow, mu, mv)
792 xi = self._Eu.fE(snu, cnu, dnu) - mu
793 v -= self._Ev.fE(snv, cnv, dnv) - mv
794 return xi, v, d2
796 def _sigmaDwd2(self, snu, cnu, dnu, snv, cnv, dnv):
797 '''(INTERNAL) C{sigmaDwd}.
799 @return: 2-Tuple C{(du, dv)}.
801 @see: C{void TMExact::dwdsigma(real /*u*/, real snu, real cnu, real dnu,
802 real /*v*/, real snv, real cnv, real dnv,
803 real &du, real &dv)}.
804 '''
805 mu = self._mu
806 snuv = snu * snv
807 # Reciprocal of 55.9: dw / ds = dn(w)^2/_mv,
808 # expanding complex dn(w) using A+S 16.21.4
809 d = (cnv**2 + snuv**2 * mu)**2 * self._mv
810 r = cnv * dnu * dnv
811 i = cnu * snuv * mu
812 du = (r**2 - i**2) / d # (r + i) * (r - i) / d
813 dv = neg(r * i * _2_0 / d)
814 return du, dv
816 def _sigmaInv2(self, xi, eta):
817 '''(INTERNAL) Invert C{sigma} using Newton's method.
819 @return: 2-Tuple C{(u, v)}.
821 @see: C{void TMExact::sigmainv(real xi, real eta,
822 real &u, real &v)}.
824 @raise ETMError: No convergence.
825 '''
826 u, v, t, self._sigmaC = self._sigmaInv04(xi, eta)
827 if not t:
828 u, v = self._Newton2(xi, eta, u, v, self._sigmaC)
829 return u, v
831 def _sigmaInv04(self, xi, eta):
832 '''(INTERNAL) Starting point for C{sigmaInv}.
834 @return: 4-Tuple C{(u, v, trip, Case)}.
836 @see: C{bool TMExact::sigmainv0(real xi, real eta,
837 real &u, real &v)}.
838 '''
839 t = False
840 d = eta - self._Ev_cKE
841 if eta > self._Ev_5cKE_4 or (xi < d and xi < -self._Eu_cE_4):
842 # sigma as a simple pole at
843 # w = w0 = Eu.K() + i * Ev.K()
844 # and sigma is approximated by
845 # sigma = (Eu.E() + i * Ev.KE()) + 1 / (w - w0)
846 u, v = _norm2(xi - self._Eu_cE, -d)
847 u += self._Eu_cK
848 v += self._Ev_cK
849 C = 1
851 elif (eta > self._Ev_3cKE_4 and xi < self._Eu_cE_4) or d > 0:
852 # At w = w0 = i * Ev.K(), we have
853 # sigma = sigma0 = i * Ev.KE()
854 # sigma' = sigma'' = 0
855 # including the next term in the Taylor series gives:
856 # sigma = sigma0 - _mv / 3 * (w - w0)^3
857 # When inverting this, we map arg(w - w0) = [-pi/2, -pi/6]
858 # to arg(sigma - sigma0) = [-pi/2, pi/2] mapping arg =
859 # [-pi/2, -pi/6] to [-pi/2, pi/2]
860 u, v, h = self._Inv03(xi, d, self._3_mv)
861 t = h < _TAYTOL2
862 C = 2
864 else: # use w = sigma * Eu.K/Eu.E (correct in limit _e -> 0)
865 u = v = self._Eu_cK_cE
866 u *= xi
867 v *= eta
868 C = 3
870 return u, v, t, C
872 def _sncndn6(self, u, v):
873 '''(INTERNAL) Get 6-tuple C{(snu, cnu, dnu, snv, cnv, dnv)}.
874 '''
875 # snu, cnu, dnu = self._Eu.sncndn(u)
876 # snv, cnv, dnv = self._Ev.sncndn(v)
877 return self._Eu.sncndn(u) + self._Ev.sncndn(v)
879 def toStr(self, joined=_COMMASPACE_, **kwds): # PYCHOK signature
880 '''Return a C{str} representation.
882 @kwarg joined: Separator to join the attribute strings
883 (C{str} or C{None} or C{NN} for non-joined).
884 @kwarg kwds: Optional, overriding keyword arguments.
885 '''
886 d = dict(datum=self.datum.name, lon0=self.lon0,
887 k0=self.k0, extendp=self.extendp)
888 if self.name:
889 d.update(name=self.name)
890 t = pairs(d, **kwds)
891 return joined.join(t) if joined else t
893 def _zeta3(self, unused, snu, cnu, dnu, snv, cnv, dnv): # _sigma3 signature
894 '''(INTERNAL) C{zeta}.
896 @return: 3-Tuple C{(taup, lambda, d2)}.
898 @see: C{void TMExact::zeta(real /*u*/, real snu, real cnu, real dnu,
899 real /*v*/, real snv, real cnv, real dnv,
900 real &taup, real &lam)}
901 '''
902 e, cnu2, mv = self._e, cnu**2, self._mv
903 # Overflow value like atan(overflow) = pi/2
904 t1 = t2 = _overflow(snu)
905 # Lee 54.17 but write
906 # atanh(snu * dnv) = asinh(snu * dnv / sqrt(cnu^2 + _mv * snu^2 * snv^2))
907 # atanh(_e * snu / dnv) = asinh(_e * snu / sqrt(_mu * cnu^2 + _mv * cnv^2))
908 d1 = cnu2 + mv * (snu * snv)**2
909 if d1 > EPS02: # _EPSmin
910 t1 = snu * dnv / sqrt(d1)
911 else:
912 d1 = 0
913 d2 = self._mu * cnu2 + mv * cnv**2
914 if d2 > EPS02: # _EPSmin
915 t2 = sinh(e * asinh(e * snu / sqrt(d2)))
916 else:
917 d2 = 0
918 # psi = asinh(t1) - asinh(t2)
919 # taup = sinh(psi)
920 taup = t1 * hypot1(t2) - t2 * hypot1(t1)
921 lam = (atan2(dnu * snv, cnu * cnv) -
922 atan2(cnu * snv * e, dnu * cnv) * e) if d1 and d2 else _0_0
923 return taup, lam, d2
925 def _zetaDwd2(self, snu, cnu, dnu, snv, cnv, dnv):
926 '''(INTERNAL) C{zetaDwd}.
928 @return: 2-Tuple C{(du, dv)}.
930 @see: C{void TMExact::dwdzeta(real /*u*/, real snu, real cnu, real dnu,
931 real /*v*/, real snv, real cnv, real dnv,
932 real &du, real &dv)}.
933 '''
934 cnu2 = cnu**2 * self._mu
935 cnv2 = cnv**2
936 dnuv = dnu * dnv
937 dnuv2 = dnuv**2
938 snuv = snu * snv
939 snuv2 = snuv**2 * self._mu
940 # Lee 54.21 but write (see A+S 16.21.4)
941 # (1 - dnu^2 * snv^2) = (cnv^2 + _mu * snu^2 * snv^2)
942 d = self._mv * (cnv2 + snuv2)**2 # max(d, EPS02)?
943 du = cnu * dnuv * (cnv2 - snuv2) / d
944 dv = cnv * snuv * (cnu2 + dnuv2) / d
945 return du, neg(dv)
947 def _zetaInv2(self, taup, lam):
948 '''(INTERNAL) Invert C{zeta} using Newton's method.
950 @return: 2-Tuple C{(u, v)}.
952 @see: C{void TMExact::zetainv(real taup, real lam,
953 real &u, real &v)}.
955 @raise ETMError: No convergence.
956 '''
957 psi = asinh(taup)
958 u, v, t, self._zetaC = self._zetaInv04(psi, lam)
959 if not t:
960 u, v = self._Newton2(taup, lam, u, v, self._zetaC, psi)
961 return u, v
963 def _zetaInv04(self, psi, lam):
964 '''(INTERNAL) Starting point for C{zetaInv}.
966 @return: 4-Tuple C{(u, v, trip, Case)}.
968 @see: C{bool TMExact::zetainv0(real psi, real lam, # radians
969 real &u, real &v)}.
970 '''
971 if lam > self._1_2e_PI_2:
972 d = lam - self._1_e_PI_2
973 if psi < d and psi < self._e_PI_4_: # PYCHOK no cover
974 # N.B. this branch is normally *not* taken because psi < 0
975 # is converted psi > 0 by .forward. There's a log singularity
976 # at w = w0 = Eu.K() + i * Ev.K(), corresponding to the south
977 # pole, where we have, approximately
978 # psi = _e + i * pi/2 - _e * atanh(cos(i * (w - w0)/(1 + _mu/2)))
979 # Inverting this gives:
980 e = self._e # eccentricity
981 s, c = sincos2((PI_2 - lam) / e)
982 h, r = sinh(_1_0 - psi / e), self._1_mu_2
983 u = self._Eu_cK - r * asinh(s / hypot(c, h))
984 v = self._Ev_cK - r * atan2(c, h)
985 return u, v, False, 1
987 elif psi < self._e_PI_2:
988 # At w = w0 = i * Ev.K(), we have
989 # zeta = zeta0 = i * (1 - _e) * pi/2
990 # zeta' = zeta'' = 0
991 # including the next term in the Taylor series gives:
992 # zeta = zeta0 - (_mv * _e) / 3 * (w - w0)^3
993 # When inverting this, we map arg(w - w0) = [-90, 0]
994 # to arg(zeta - zeta0) = [-90, 180]
995 u, v, h = self._Inv03(psi, d, self._3_mv_e)
996 return u, v, (h < self._e_TAYTOL), 2
998 # Use spherical TM, Lee 12.6 -- writing C{atanh(sin(lam) /
999 # cosh(psi)) = asinh(sin(lam) / hypot(cos(lam), sinh(psi)))}.
1000 # This takes care of the log singularity at C{zeta = Eu.K()},
1001 # corresponding to the north pole.
1002 s, c = sincos2(lam)
1003 h, r = sinh(psi), self._Eu_2cK_PI
1004 # But scale to put 90, 0 on the right place
1005 u = r * atan2(h, c)
1006 v = r * asinh(s / hypot(h, c))
1007 return u, v, False, 3
1009 def _zetaScaled(self, sncndn6, ll=True):
1010 '''(INTERNAL) Recompute (T, L) from (u, v) to improve accuracy of Scale.
1012 @arg sncndn6: 6-Tuple C{(snu, cnu, dnu, snv, cnv, dnv)}.
1014 @return: 2-Tuple C{(g, k)} if not C{B{ll}} else
1015 4-tuple C{(g, k, lat, lon)}.
1016 '''
1017 t, lam, d2 = self._zeta3(None, *sncndn6)
1018 tau = self._E.es_tauf(t)
1019 g_k = self._scaled2(tau, d2, *sncndn6)
1020 if ll:
1021 g_k += atan1d(tau), degrees(lam)
1022 return g_k # or (g, k, lat, lon)
1025def _overflow(x):
1026 '''(INTERNAL) Like C{copysign0(OVERFLOW, B{x})}.
1027 '''
1028 return _copyBit(_OVERFLOW, x)
1031def parseETM5(strUTM, datum=_WGS84, Etm=Etm, falsed=True, **name):
1032 '''Parse a string representing a UTM coordinate, consisting
1033 of C{"zone[band] hemisphere easting northing"}.
1035 @arg strUTM: A UTM coordinate (C{str}).
1036 @kwarg datum: Optional datum to use (L{Datum}, L{Ellipsoid},
1037 L{Ellipsoid2} or L{a_f2Tuple}).
1038 @kwarg Etm: Optional class to return the UTM coordinate
1039 (L{Etm}) or C{None}.
1040 @kwarg falsed: Both easting and northing are C{falsed} (C{bool}).
1041 @kwarg name: Optional B{C{Etm}} C{B{name}=NN} (C{str}).
1043 @return: The UTM coordinate (B{C{Etm}}) or if B{C{Etm}} is
1044 C{None}, a L{UtmUps5Tuple}C{(zone, hemipole, easting,
1045 northing, band)}. The C{hemipole} is the hemisphere
1046 C{'N'|'S'}.
1048 @raise ETMError: Invalid B{C{strUTM}}.
1050 @raise TypeError: Invalid or near-spherical B{C{datum}}.
1051 '''
1052 r = _parseUTM5(strUTM, datum, Etm, falsed, Error=ETMError, **name)
1053 return r
1056def toEtm8(latlon, lon=None, datum=None, Etm=Etm, falsed=True,
1057 strict=True, zone=None, **name_cmoff):
1058 '''Convert a geodetic lat-/longitude to an ETM coordinate.
1060 @arg latlon: Latitude (C{degrees}) or an (ellipsoidal)
1061 geodetic C{LatLon} instance.
1062 @kwarg lon: Optional longitude (C{degrees}), required
1063 if B{C{latlon}} is in C{degrees}.
1064 @kwarg datum: Optional datum for the ETM coordinate,
1065 overriding B{C{latlon}}'s datum (L{Datum},
1066 L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}).
1067 @kwarg Etm: Optional class to return the ETM coordinate
1068 (L{Etm}) or C{None}.
1069 @kwarg falsed: False both easting and northing (C{bool}).
1070 @kwarg strict: Restrict B{C{lat}} to UTM ranges (C{bool}).
1071 @kwarg zone: Optional UTM zone to enforce (C{int} or C{str}).
1072 @kwarg name_cmoff: Optional B{C{Etm}} C{B{name}=NN} (C{str})
1073 and DEPRECATED keyword argument C{B{cmoff}=True}
1074 to offset the longitude from the zone's central
1075 meridian (C{bool}), use B{C{falsed}} instead.
1077 @return: The ETM coordinate as an B{C{Etm}} instance or a
1078 L{UtmUps8Tuple}C{(zone, hemipole, easting, northing,
1079 band, datum, gamma, scale)} if B{C{Etm}} is C{None}
1080 or not B{C{falsed}}. The C{hemipole} is the C{'N'|'S'}
1081 hemisphere.
1083 @raise ETMError: No convergence transforming to ETM easting
1084 and northing.
1086 @raise ETMError: Invalid B{C{zone}} or near-spherical or
1087 incompatible B{C{datum}} or C{ellipsoid}.
1089 @raise RangeError: If B{C{lat}} outside the valid UTM bands or
1090 if B{C{lat}} or B{C{lon}} outside the valid
1091 range and L{pygeodesy.rangerrors} set to C{True}.
1093 @raise TypeError: Invalid or near-spherical B{C{datum}} or
1094 B{C{latlon}} not ellipsoidal.
1096 @raise ValueError: The B{C{lon}} value is missing or B{C{latlon}}
1097 is invalid.
1098 '''
1099 z, B, lat, lon, d, f, n = _to7zBlldfn(latlon, lon, datum,
1100 falsed, zone, strict,
1101 ETMError, **name_cmoff)
1102 lon0 = _cmlon(z) if f else None
1103 x, y, g, k = d.exactTM.forward(lat, lon, lon0=lon0)
1105 return _toXtm8(Etm, z, lat, x, y, B, d, g, k, f,
1106 n, latlon, d.exactTM, Error=ETMError)
1109if __name__ == '__main__': # MCCABE 13
1111 from pygeodesy import fstr, KTransverseMercator, printf
1112 from pygeodesy.internals import _usage
1113 from sys import argv, exit as _exit
1115 # mimick some of I{Karney}'s utility C{TransverseMercatorProj}
1116 _f = _r = _s = _t = False
1117 _p = -6
1118 _as = argv[1:]
1119 while _as and _as[0].startswith(_DASH_):
1120 _a = _as.pop(0)
1121 if len(_a) < 2:
1122 _exit('%s: option %r invalid' % (_usage(*argv), _a))
1123 elif '-forward'.startswith(_a):
1124 _f, _r = True, False
1125 elif '-reverse'.startswith(_a):
1126 _f, _r = False, True
1127 elif '-precision'.startswith(_a):
1128 _p = int(_as.pop(0))
1129 elif '-series'.startswith(_a):
1130 _s, _t = True, False
1131 elif _a == '-t':
1132 _s, _t = False, True
1133 elif '-help'.startswith(_a):
1134 _exit(_usage(argv[0], '[-s | -t ]',
1135 '[-p[recision] <ndigits>',
1136 '[-f[orward] <lat> <lon>',
1137 '|-r[everse] <easting> <northing>',
1138 '|<lat> <lon>]',
1139 '|-h[elp]'))
1140 else:
1141 _exit('%s: option %r not supported' % (_usage(*argv), _a))
1142 if len(_as) > 1:
1143 f2 = map1(float, *_as[:2])
1144 else:
1145 _exit('%s ...: incomplete' % (_usage(*argv),))
1147 if _s: # -series
1148 tm = KTransverseMercator()
1149 else:
1150 tm = ExactTransverseMercator(extendp=_t)
1152 if _f:
1153 t = tm.forward(*f2)
1154 elif _r:
1155 t = tm.reverse(*f2)
1156 else:
1157 t = tm.forward(*f2)
1158 printf('%s: %s', tm.classname, fstr(t, prec=_p, sep=_SPACE_))
1159 t = tm.reverse(t.easting, t.northing)
1160 printf('%s: %s', tm.classname, fstr(t, prec=_p, sep=_SPACE_))
1163# % python3 -m pygeodesy.etm -p 12 33.33 44.44
1164# ExactTransverseMercator: 4276926.11480390653 4727193.767015309073 28.375536563148 1.233325101778
1165# ExactTransverseMercator: 33.33 44.44 28.375536563148 1.233325101778
1167# % python3 -m pygeodesy.etm -s -p 12 33.33 44.44
1168# KTransverseMercator: 4276926.114803904667 4727193.767015310004 28.375536563148 1.233325101778
1169# KTransverseMercator: 33.33 44.44 28.375536563148 1.233325101778
1171# % echo 33.33 44.44 | .../bin/TransverseMercatorProj
1172# 4276926.114804 4727193.767015 28.375536563148 1.233325101778
1174# **) MIT License
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1183# Software is furnished to do so, subject to the following conditions:
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