Coverage for pygeodesy/karney.py: 93%
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2# -*- coding: utf-8 -*-
4u'''Wrapper around several C{geomath.Math} functions from I{Karney}'s Python package U{geographiclib
5<https://PyPI.org/project/geographiclib>}, provided that package is installed.
7Methods of the I{wrapped} L{Geodesic<geodesicw.Geodesic>} and L{GeodesicLine<geodesicw.GeodesicLine>}
8classes return a L{GDict} instance offering access to the C{dict} items either by C{key} or by
9C{attribute} name.
11With env variable C{PYGEODESY_GEOGRAPHICLIB} left undefined or set to C{"2"}, modules L{geodesicw},
12L{geodesicx} and this module will use U{GeographicLib 2.0+<https://GeographicLib.SourceForge.io/C++/doc/>}
13and newer transcoding, otherwise C{1.52} or older. Set C{PYGEODESY_GEOGRAPHICLIB=2.4} to default to the
14C{Jacobi amplitude} instead of C{Bulirsch}' function in methods L{ExactTransverseMercator.forward
15<pygeodesy.ExactTransverseMercator.forward>} and L{reverse <pygeodesy.ExactTransverseMercator.reverse>}.
17Karney-based functionality
18==========================
201. The following classes and functions in C{pygeodesy}
22 - L{AlbersEqualArea}, L{AlbersEqualArea2}, L{AlbersEqualArea4},
23 L{AlbersEqualAreaCylindrical}, L{AlbersEqualAreaNorth}, L{AlbersEqualAreaSouth} --
24 U{AlbersEqualArea<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1AlbersEqualArea.html>}
26 - L{AuxAngle}, L{AuxDST}, L{AuxDLat}, L{AuxLat} -- U{AuxAngle
27 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1AuxAngle.html>},
28 U{DST<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1DST.html>},
29 U{DAuxLatitude<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1DAuxLatitude.html>},
30 U{AuxLatitude<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1AuxLatitude.html>} in I{GeographicLib 2.2+}
32 - L{CassiniSoldner} -- U{CassiniSoldner<https://GeographicLib.SourceForge.io/C++/doc/
33 classGeographicLib_1_1CassiniSoldner.html>}
35 - L{EcefKarney} -- U{Geocentric<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Geocentric.html>}
37 - L{Elliptic} -- U{EllipticFunction<https://GeographicLib.SourceForge.io/C++/doc/
38 classGeographicLib_1_1EllipticFunction.html>}
40 - L{EquidistantExact}, L{EquidistantGeodSolve}, L{EquidistantKarney} -- U{AzimuthalEquidistant
41 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1AzimuthalEquidistant.html>}
43 - L{Etm}, L{ExactTransverseMercator} -- U{TransverseMercatorExact
44 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1TransverseMercatorExact.html>}
46 - L{Geodesic}, L{GeodesicLine} -- I{wrapped} U{geodesic.Geodesic<https://PyPI.org/project/geographiclib>},
47 I{wrapped} U{geodesicline.GeodesicLine<https://PyPI.org/project/geographiclib>}
49 - L{GeodesicAreaExact}, L{PolygonArea} -- U{PolygonArea<https://GeographicLib.SourceForge.io/C++/doc/
50 classGeographicLib_1_1PolygonAreaT.html>}
52 - L{GeodesicExact}, L{GeodesicLineExact} -- U{GeodesicExact<https://GeographicLib.SourceForge.io/C++/doc/
53 classGeographicLib_1_1GeodesicExact.html>}, U{GeodesicLineExact<https://GeographicLib.SourceForge.io/C++/doc/
54 classGeographicLib_1_1GeodesicLineExact.html>}
56 - L{GeoidKarney} -- U{Geoid<https://GeographicLib.SourceForge.io/C++/doc/geoid.html>}
58 - L{Georef} -- U{Georef<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Georef.html>}
60 - L{GnomonicExact}, L{GnomonicGeodSolve}, L{GnomonicKarney} -- U{Gnomonic
61 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Gnomonic.html>}
63 - L{Intersector} -- U{Intersect
64 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Intersect.html>} from I{GeographicLib 2.3+}
66 - L{JacobiConformal} -- U{JacobiConformal
67 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1experimental_1_1JacobiConformal.html>}
69 - L{KTransverseMercator} - U{TransverseMercator
70 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1TransverseMercator.html>}
72 - L{LocalCartesian}, L{Ltp} -- U{LocalCartesian<https://GeographicLib.SourceForge.io/C++/doc/
73 classGeographicLib_1_1LocalCartesian.html>}
75 - L{Osgr} -- U{OSGB<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1OSGB.html>}
77 - L{rhumb.aux_}, L{RhumbAux}, L{RhumbLineAux} -- U{Rhumb
78 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Rhumb.html>} and U{RhumbLine
79 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1RhumbLine.html>} from I{GeographicLib 2.2+}
81 - L{rhumb.ekx}, L{Rhumb}, L{RhumbLine} -- U{Rhumb
82 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Rhumb.html>},
83 U{RhumbLine<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1RhumbLine.html>},
84 U{TransverseMercator<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1TransverseMercator.html>}
85 from I{GeographicLib 2.0}
87 - L{Ups} -- U{PolarStereographic<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1PolarStereographic.html>}
89 - L{Utm} -- U{TransverseMercator<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1TransverseMercator.html>}
91 - L{UtmUps}, L{Epsg} -- U{UTMUPS<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1UTMUPS.html>}
93 - L{atan1d}, L{atan2d}, L{sincos2}, L{sincos2d}, L{tand} -- U{geomath.Math
94 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Math.html>}
96are I{transcoded} from C++ classes in I{Karney}'s U{GeographicLib<https://GeographicLib.SourceForge.io/C++/doc/annotated.html>}.
982. These C{pygeodesy} modules and classes
100 - L{ellipsoidalGeodSolve}, L{ellipsoidalKarney}, L{geodesici}, L{geodsolve}, L{karney}, L{rhumb.solve}
101 - L{EquidistantKarney}, L{FrechetKarney}, L{GeodesicSolve}, L{GeodesicLineSolve}, L{GnomonicGeodSolve},
102 L{GnomonicKarney}, L{HeightIDWkarney}, L{Intersectool}
104are or use I{wrappers} around I{Karney}'s Python U{geographiclib<https://PyPI.org/project/geographiclib>} or
105C++ utility U{GeodSolve<https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>},
106U{IntersectTool<https://GeographicLib.SourceForge.io/C++/doc/IntersectTool.1.html>} or
107U{RhumbSolve<https://GeographicLib.SourceForge.io/C++/doc/RhumbSolve.1.html>}.
1093. All C{pygeodesy} functions and methods to compute I{ellipsoidal} intersections, nearest points and trilaterations
111 - L{ellipsoidalExact.intersection3}, L{ellipsoidalExact.intersections2}, L{ellipsoidalExact.nearestOn},
112 L{ellipsoidalExact.LatLon.intersection3}, L{ellipsoidalExact.LatLon.intersections2},
113 L{ellipsoidalExact.LatLon.nearestOn}, L{ellipsoidalExact.LatLon.trilaterate5}
115 - L{ellipsoidalKarney.intersection3}, L{ellipsoidalKarney.intersections2}, L{ellipsoidalKarney.nearestOn},
116 L{ellipsoidalKarney.LatLon.intersection3}, L{ellipsoidalKarney.LatLon.intersections2},
117 L{ellipsoidalKarney.LatLon.nearestOn}, L{ellipsoidalKarney.LatLon.trilaterate5}
119 - L{ellipsoidalVincenty.intersection3}, L{ellipsoidalVincenty.intersections2}, L{ellipsoidalVincenty.nearestOn},
120 L{ellipsoidalVincenty.LatLon.intersection3}, L{ellipsoidalVincenty.LatLon.intersections2},
121 L{ellipsoidalVincenty.LatLon.nearestOn}, L{ellipsoidalVincenty.LatLon.trilaterate5}
123 - L{RhumbLineAux.Intersection} and L{RhumbLine.Intersection}
125are implementations of I{Karney}'s iterative solution posted under U{The B{ellipsoidal} case
126<https://GIS.StackExchange.com/questions/48937/calculating-intersection-of-two-circles>} and in paper U{Geodesics
127on an ellipsoid of revolution<https://ArXiv.org/pdf/1102.1215.pdf>} (pp 20-21, section B{14. MARITIME BOUNDARIES}).
1294. The C{pygeodesy} methods to compute I{ellipsoidal} intersections and nearest points
131 - L{RhumbLineAux.Intersecant2}, L{RhumbLineAux.PlumbTo}, L{RhumbLine.Intersecant2} and L{RhumbLine.PlumbTo}
133are I{transcoded} of I{Karney}'s iterative C++ function U{rhumb-intercept
134<https://SourceForge.net/p/geographiclib/discussion/1026620/thread/2ddc295e/>}.
1365. Spherical functions
138 - L{pygeodesy.excessKarney_}, L{sphericalTrigonometry.areaOf}
140in C{pygeodesy} are based on I{Karney}'s post U{Area of a spherical polygon
141<https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>}, 3rd Answer.
142'''
143# make sure int/int division yields float quotient, see .basics
144from __future__ import division as _; del _ # PYCHOK semicolon
146from pygeodesy.basics import _copysign, isint, neg, unsigned0, \
147 _xgeographiclib, _zip
148from pygeodesy.constants import NAN, _isfinite as _math_isfinite, _0_0, \
149 _1_16th, _1_0, _2_0, _180_0, _N_180_0, _360_0
150from pygeodesy.errors import GeodesicError, _ValueError, _xkwds
151from pygeodesy.fmath import cbrt, fremainder, norm2 # Fhorner, Fsum
152from pygeodesy.internals import _getenv, _popen2, _PYGEODESY, _version_info
153from pygeodesy.interns import NN, _a12_, _area_, _azi1_, _azi2_, _azi12_, \
154 _composite_, _lat1_, _lat2_, _lon1_, _lon2_, \
155 _m12_, _M12_, _M21_, _number_, _s12_, _S12_, \
156 _SPACE_, _UNDER_, _X_, _1_, _2_, _BAR_ # PYCHOK used!
157from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY, _ALL_MODS as _MODS
158from pygeodesy.named import ADict, _NamedBase, _NamedTuple, notImplemented, _Pass
159from pygeodesy.props import deprecated_method, Property_RO, property_RO, \
160 property_ROnce
161from pygeodesy.units import Azimuth as _Azi, Degrees as _Deg, Lat, Lon, \
162 Meter as _M, Meter2 as _M2, Number_
163from pygeodesy.utily import atan2d, sincos2d, tand, _unrollon, fabs
165# from math import fabs # from .utily
167__all__ = _ALL_LAZY.karney
168__version__ = '25.04.08'
170_2_4_ = '2.4'
171_K_2_0 = _getenv(_PYGEODESY(_xgeographiclib, 1), _2_)
172_K_2_4 = _K_2_0 == _2_4_
173_K_2_0 = _K_2_4 or (_K_2_0 == _2_)
174_perimeter_ = 'perimeter'
177class _GTuple(_NamedTuple): # in .testNamedTuples
178 '''(INTERNAL) Helper.
179 '''
180 def toGDict(self, **updates): # NO name=NN
181 '''Convert this C{*Tuple} to a L{GDict}.
183 @kwarg updates: Optional items to apply (C{name=value} pairs)
184 '''
185 r = GDict(_zip(self._Names_, self)) # strict=True
186 if updates:
187 r.update(updates)
188 if self._iteration is not None:
189 r._iteration = self._iteration
190 return r
193class _Lat(Lat):
194 '''(INTERNAL) Latitude B{C{lat}}.
195 '''
196 def __init__(self, *lat, **Error_name):
197 kwds = _xkwds(Error_name, clip=0, Error=GeodesicError)
198 Lat.__new__(_Lat, *lat, **kwds)
201class _Lon(Lon):
202 '''(INTERNAL) Longitude B{C{lon}}.
203 '''
204 def __init__(self, *lon, **Error_name):
205 kwds = _xkwds(Error_name, clip=0, Error=GeodesicError)
206 Lon.__new__(_Lon, *lon, **kwds)
209class Area3Tuple(_NamedTuple): # in .geodesicx.gxarea
210 '''3-Tuple C{(number, perimeter, area)} with the C{number}
211 of points of the polygon or polyline, the C{perimeter} in
212 C{meter} and the C{area} in C{meter} I{squared}.
213 '''
214 _Names_ = (_number_, _perimeter_, _area_)
215 _Units_ = ( Number_, _M, _M2)
218class Caps(object):
219 '''(INTERNAL) Overriden by C{Caps} below.
220 '''
221 EMPTY = 0 # formerly aka NONE
222 _CAP_1 = 1 << 0 # for goedesici/-w
223 _CAP_1p = 1 << 1 # for goedesici/-w
224 _CAP_2 = 1 << 2
225 _CAP_3 = 1 << 3 # for goedesici/-w
226 _CAP_4 = 1 << 4 # for goedesicw
227 _CAP_ALL = 0x1F
228# _CAP_MASK = _CAP_ALL
229 LATITUDE = 1 << 7 # compute latitude C{lat2}
230 LONGITUDE = 1 << 8 | _CAP_3 # compute longitude C{lon2}
231 AZIMUTH = 1 << 9 # azimuths C{azi1} and C{azi2}
232 DISTANCE = 1 << 10 | _CAP_1 # compute distance C{s12}
233 DISTANCE_IN = 1 << 11 | _CAP_1 | _CAP_1p # allow distance C{s12} in Direct
234 REDUCEDLENGTH = 1 << 12 | _CAP_1 | _CAP_2 # compute reduced length C{m12}
235 GEODESICSCALE = 1 << 13 | _CAP_1 | _CAP_2 # compute geodesic scales C{M12} and C{M21}
236 AREA = 1 << 14 | _CAP_4 # compute area C{S12}
238 STANDARD = AZIMUTH | DISTANCE | LATITUDE | LONGITUDE
240 _DIRECT3 = AZIMUTH | LATITUDE | LONGITUDE # for goedesicw only
241 _INVERSE3 = AZIMUTH | DISTANCE # for goedesicw only
242 _STD = STANDARD # for goedesicw only
243 _STD_LINE = _STD | DISTANCE_IN # for goedesici/-w
245 LINE_CAPS = _STD_LINE | REDUCEDLENGTH | GEODESICSCALE # .geodesici only
246 LONG_UNROLL = 1 << 15 # unroll C{lon2} in .Direct and .Position
247# = 1 << 16 # unused
248 LINE_OFF = 1 << 17 # Line without updates from parent geodesic or rhumb
249 REVERSE2 = 1 << 18 # reverse C{azi2}
250 ALL = 0x7F80 | _CAP_ALL # without LONG_UNROLL, LINE_OFF, REVERSE2 and _DEBUG_*
252 LATITUDE_LONGITUDE = LATITUDE | LONGITUDE
253 LATITUDE_LONGITUDE_AREA = LATITUDE | LONGITUDE | AREA
255 AZIMUTH_DISTANCE = AZIMUTH | DISTANCE
256 AZIMUTH_DISTANCE_AREA = AZIMUTH | DISTANCE | AREA
258 _SALP_CALPs_ = 1 << 19 # (INTERNAL) GeodesicExact._GenInverse
260 _DEBUG_AREA = 1 << 20 # (INTERNAL) include Line details
261 _DEBUG_DIRECT = 1 << 21 # (INTERNAL) include Direct details
262 _DEBUG_INVERSE = 1 << 22 # (INTERNAL) include Inverse details
263 _DEBUG_LINE = 1 << 23 # (INTERNAL) include Line details
264 _DEBUG_ALL = _DEBUG_AREA | _DEBUG_DIRECT | _DEBUG_INVERSE | \
265 _DEBUG_LINE | _SALP_CALPs_
267 _OUT_ALL = ALL # see geographiclib.geodesiccapabilities.py
268 _OUT_MASK = ALL | LONG_UNROLL | REVERSE2 | _DEBUG_ALL
270 _AZIMUTH_LATITUDE_LONGITUDE = AZIMUTH | LATITUDE | LONGITUDE
271 _AZIMUTH_LATITUDE_LONG_UNROLL = AZIMUTH | LATITUDE | LONG_UNROLL
272 _DEBUG_DIRECT_LINE = _DEBUG_DIRECT | _DEBUG_LINE
273# _DISTANCE_IN_OUT = DISTANCE_IN & _OUT_MASK # == DISTANCE_IN in .gx, .gxline
274 _REDUCEDLENGTH_GEODESICSCALE = REDUCEDLENGTH | GEODESICSCALE
275# _REDUCEDLENGTH_GEODESICSCALE_DISTANCE = REDUCEDLENGTH | GEODESICSCALE | DISTANCE
277 def items(self):
278 '''Yield all I{public} C{Caps} as 2-tuple C{(NAME, mask)}.
279 '''
280 for n, C in Caps.__class__.__dict__.items():
281 if isint(C) and not n.startswith(_UNDER_) \
282 and n.replace(_UNDER_, NN).isupper():
283 yield n, C
285 def toStr(self, Cask, sep=_BAR_):
286 '''Return C{Caps} or an C{outmask} as C{str} or tuple of C{str}s.
287 '''
288 s = (n for n, C in sorted(Caps.items())
289 if C and (Cask & C) == C) # and int1s(C) <= 3
290 return sep.join(s) if sep else tuple(s)
292Caps = Caps() # PYCHOK singleton
293'''I{Enum}-style masks to be bit-C{or}'ed to specify geodesic or
294rhumb capabilities (C{caps}) and results (C{outmask}).
296C{AREA} - compute area C{S12},
298C{AZIMUTH} - include azimuths C{azi1} and C{azi2},
300C{DISTANCE} - compute distance C{s12} and angular distance C{a12},
302C{DISTANCE_IN} - allow distance C{s12} in C{.Direct},
304C{EMPTY} - nothing, formerly aka C{NONE},
306C{GEODESICSCALE} - compute geodesic scales C{M12} and C{M21},
308C{LATITUDE} - compute latitude C{lat2},
310C{LINE_OFF} - Line without updates from parent geodesic or rhumb.
312C{LONGITUDE} - compute longitude C{lon2},
314C{LONG_UNROLL} - unroll C{lon2} in C{.Direct},
316C{REDUCEDLENGTH} - compute reduced length C{m12},
318C{REVERSE2} - reverse C{azi2},
320and C{ALL} - all of the above.
322C{STANDARD} = C{AZIMUTH | DISTANCE | DISTANCE_IN | LATITUDE | LONGITUDE}'''
324_key2Caps = dict(a12 =Caps.DISTANCE, # in GDict._unCaps
325 azi2=Caps.AZIMUTH,
326 lat2=Caps.LATITUDE,
327 lon2=Caps.LONGITUDE,
328 m12 =Caps.REDUCEDLENGTH,
329 M12 =Caps.GEODESICSCALE,
330 M21 =Caps.GEODESICSCALE,
331 s12 =Caps.DISTANCE,
332 S12 =Caps.AREA)
335class _CapsBase(_NamedBase): # in .auxilats, .geodesicx.gxbases
336 '''(INTERNAL) Base class for C{Geodesic*}, C{Geodesic*Exact}, C{Intersectool} and C{Rhumb*Base}.
337 '''
338 ALL = Caps.ALL
339 AREA = Caps.AREA
340 AZIMUTH = Caps.AZIMUTH
341 DISTANCE = Caps.DISTANCE
342 DISTANCE_IN = Caps.DISTANCE_IN
343 EMPTY = Caps.EMPTY # aka NONE
344 GEODESICSCALE = Caps.GEODESICSCALE
345 LATITUDE = Caps.LATITUDE
346 LINE_CAPS = Caps.LINE_CAPS
347 LINE_OFF = Caps.LINE_OFF
348 LONGITUDE = Caps.LONGITUDE
349 LONG_UNROLL = Caps.LONG_UNROLL
350 REDUCEDLENGTH = Caps.REDUCEDLENGTH
351 STANDARD = Caps.STANDARD
352 _STD_LINE = Caps._STD_LINE # for geodesici
354 _caps = 0 # None
355 _debug = 0 # or Caps._DEBUG_...
357 @Property_RO
358 def caps(self):
359 '''Get the capabilities (bit-or'ed C{Caps}).
360 '''
361 return self._caps
363 def caps_(self, caps):
364 '''Check the available capabilities.
366 @arg caps: Bit-or'ed combination of L{Caps} values
367 for all capabilities to be checked.
369 @return: C{True} if I{all} B{C{caps}} are available,
370 C{False} otherwise (C{bool}).
371 '''
372 caps &= Caps._OUT_ALL
373 return (self.caps & caps) == caps
375 @property
376 def debug(self):
377 '''Get the C{debug} option (C{bool}).
378 '''
379 return bool(self._debug)
381 @debug.setter # PYCHOK setter!
382 def debug(self, debug):
383 '''Set the C{debug} option (C{bool}) to include
384 more details in L{GDict} results.
385 '''
386 self._debug = Caps._DEBUG_ALL if debug else 0
388 def _iter2tion(self, r, iter=None, **unused):
389 '''(INTERNAL) Copy C{C{s}.iter} into C{B{r}._iteration}.
390 '''
391 if iter is not None:
392 self._iteration = r._iteration = iter
393 return r
396class Direct9Tuple(_GTuple):
397 '''9-Tuple C{(a12, lat2, lon2, azi2, s12, m12, M12, M21, S12)} with arc
398 length C{a12}, angles C{lat2}, C{lon2} and azimuth C{azi2} in C{degrees},
399 distance C{s12} and reduced length C{m12} in C{meter} and area C{S12} in
400 C{meter} I{squared}.
401 '''
402 _Names_ = (_a12_, _lat2_, _lon2_, _azi2_, _s12_, _m12_, _M12_, _M21_, _S12_)
403 _Units_ = (_Azi, _Lat, _Lon, _Azi, _M, _Pass, _Pass, _Pass, _M2)
406class GDict(ADict): # XXX _NamedDict
407 '''A C{dict} with both C{key} I{and} C{attribute} access to the C{dict} items.
409 Results of all C{geodesic} and C{rhumb} methods (with capitalized named) are
410 returned as L{GDict} instances, see for example L{GeodesicExact} and L{RhumbAux}.
411 '''
412 def toDirect9Tuple(self, dflt=NAN):
413 '''Convert this L{GDict} result to a 9-tuple, like I{Karney}'s method
414 C{geographiclib.geodesic.Geodesic._GenDirect}.
416 @kwarg dflt: Default value for missing items (C{any}).
418 @return: L{Direct9Tuple}C{(a12, lat2, lon2, azi2,
419 s12, m12, M12, M21, S12)}
420 '''
421 return self._toTuple(Direct9Tuple, dflt)
423 def toGeodSolve12Tuple(self, dflt=NAN): # PYCHOK 12 args
424 '''Convert this L{GDict} result to a 12-Tuple, compatible with I{Karney}'s
425 U{GeodSolve<https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>}
426 result.
428 @kwarg dflt: Default value for missing items (C{any}).
430 @return: L{GeodSolve12Tuple}C{(lat1, lon1, azi1, lat2, lon2, azi2,
431 s12, a12, m12, M12, M21, S12)}.
432 '''
433 return self._toTuple(_MODS.geodsolve.GeodSolve12Tuple, dflt)
435 def toInverse10Tuple(self, dflt=NAN):
436 '''Convert this L{GDict} result to a 10-tuple, like I{Karney}'s
437 method C{geographiclib.geodesic.Geodesic._GenInverse}.
439 @kwarg dflt: Default value for missing items (C{any}).
441 @return: L{Inverse10Tuple}C{(a12, s12, salp1, calp1,
442 salp2, calp2, m12, M12, M21, S12)}.
443 '''
444 return self._toTuple(Inverse10Tuple, dflt)
446 def _toNAN(self, outmask): # .GeodesicLineExact._GenPosition
447 '''(INTERNAL) Convert this C{GDict} to all C{NAN}s.
448 '''
449 d = dict((n, NAN) for n in GeodSolve12Tuple._Names_)
450 return self.set_(**d)._unCaps(outmask)
452 @deprecated_method
453 def toRhumb7Tuple(self, dflt=NAN): # PYCHOK no cover
454 '''DEPRECATED on 23.12.07, use method C{toRhumb8Tuple}.
456 @return: A I{DEPRECATED} L{Rhumb7Tuple}.
457 '''
458 return self._toTuple(_MODS.deprecated.classes.Rhumb7Tuple, dflt)
460 def toRhumb8Tuple(self, dflt=NAN):
461 '''Convert this L{GDict} result to a 8-tuple.
463 @kwarg dflt: Default value for missing items (C{any}).
465 @return: L{Rhumb8Tuple}C{(lat1, lon1, lat2, lon2,
466 azi12, s12, S12, a12)}.
467 '''
468 return self._toTuple(Rhumb8Tuple, dflt)
470 def toRhumbSolve7Tuple(self, dflt=NAN):
471 '''Convert this L{GDict} result to a 8-tuple.
473 @kwarg dflt: Default value for missing items (C{any}).
475 @return: L{RhumbSolve7Tuple}C{(lat1, lon1, lat2, lon2,
476 azi12, s12, S12)}.
477 '''
478 return self._toTuple(_MODS.rhumb.solve.RhumbSolve7Tuple, dflt)
480 def _toTuple(self, nTuple, dflt):
481 '''(INTERNAL) Convert this C{GDict} to an B{C{nTuple}}.
482 '''
483 _g = getattr
484 t = tuple(_g(self, n, dflt) for n in nTuple._Names_)
485 return nTuple(t, iteration=self._iteration)
487 def _2X(self, gl, _2X=_X_): # .Intersectool, .Intersector
488 '''(INTERNAL) Rename C{-2} attr to C{-X} or C{-M}.
489 '''
490 X = GDict(self)
491 for n in (_lat2_, _lon2_, _azi2_, _s12_, _a12_):
492 if n in X: # X._X = X._2
493 X[n[:-1] + _2X] = X.pop(n)
494 v = getattr(gl, n, X)
495 if v is not X: # X._2 = gl._2
496 X[n] = v
497 return X
499 def _unCaps(self, outmask): # in .geodsolve
500 '''(INTERNAL) Remove superfluous items.
501 '''
502 for k, C in _key2Caps.items():
503 if k in self and (outmask & C) != C:
504 self.pop(k) # delattr(self, k)
505 return self
508class GeodSolve12Tuple(_GTuple):
509 '''12-Tuple C{(lat1, lon1, azi1, lat2, lon2, azi2, s12, a12, m12, M12, M21, S12)} with
510 angles C{lat1}, C{lon1}, C{azi1}, C{lat2}, C{lon2} and C{azi2} and arc C{a12} all in
511 C{degrees}, initial C{azi1} and final C{azi2} forward azimuths, distance C{s12} and
512 reduced length C{m12} in C{meter}, area C{S12} in C{meter} I{squared} and geodesic
513 scale factors C{M12} and C{M21}, both C{scalar}, see U{GeodSolve
514 <https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>}.
515 '''
516 # from GeodSolve --help option -f ... lat1 lon1 azi1 lat2 lon2 azi2 s12 a12 m12 M12 M21 S12
517 _Names_ = (_lat1_, _lon1_, _azi1_, _lat2_, _lon2_, _azi2_, _s12_, _a12_, _m12_, _M12_, _M21_, _S12_)
518 _Units_ = (_Lat, _Lon, _Azi, _Lat, _Lon, _Azi, _M, _Deg, _Pass, _Pass, _Pass, _M2)
521class Inverse10Tuple(_GTuple):
522 '''10-Tuple C{(a12, s12, salp1, calp1, salp2, calp2, m12, M12, M21, S12)} with arc length
523 C{a12} in C{degrees}, distance C{s12} and reduced length C{m12} in C{meter}, area
524 C{S12} in C{meter} I{squared} and the sines C{salp1}, C{salp2} and cosines C{calp1},
525 C{calp2} of the initial C{1} and final C{2} (forward) azimuths.
526 '''
527 _Names_ = (_a12_, _s12_, 'salp1', 'calp1', 'salp2', 'calp2', _m12_, _M12_, _M21_, _S12_)
528 _Units_ = (_Azi, _M, _Pass, _Pass, _Pass, _Pass, _Pass, _Pass, _Pass, _M2)
530 def toGDict(self, **updates):
531 '''Convert this C{Inverse10Tuple} to a L{GDict}.
533 @kwarg updates: Optional items to apply (C{nam=value} pairs)
534 '''
535 return _GTuple.toGDict(self, azi1=atan2d(self.salp1, self.calp1), # PYCHOK namedTuple
536 azi2=atan2d(self.salp2, self.calp2), # PYCHOK namedTuple
537 **updates) # PYCHOK indent
540class _kWrapped(object): # in .geodesicw
541 ''''(INTERNAL) Wrapper for some of I{Karney}'s U{geographiclib
542 <https://PyPI.org/project/geographiclib>} classes.
543 '''
545 @property_ROnce
546 def geographiclib(self):
547 '''Lazily import C{geographiclib}, provided the U{geographiclib
548 <https://PyPI.org/project/geographiclib>} package is installed,
549 otherwise raise a C{LazyImportError}.
550 '''
551 g = _xgeographiclib(self.__class__.__module__, 1, 49)
552 from geographiclib.geodesic import Geodesic
553 g.Geodesic = Geodesic
554 from geographiclib.geodesicline import GeodesicLine
555 g.GeodesicLine = GeodesicLine
556 from geographiclib.geomath import Math
557 g.Math = Math
558 return g
560 @property_ROnce
561 def Math(self):
562 '''Wrap the C{geomath.Math} class, provided the U{geographiclib
563 <https://PyPI.org/project/geographiclib>} package is installed,
564 otherwise C{None}.
565 '''
566 try:
567 g = self.geographiclib
568 M = g.Math
569 if _version_info(g) < (2,):
570 if _K_2_0:
571 M = None
572# elif not _K_2_0: # XXX set 2.0?
573# _K_2_0 = True
574 except (AttributeError, ImportError):
575 M = None
576 return M
578 @property_RO
579 def Math_K_2(self):
580 return (_2_4_ if _K_2_4 else
581 (_2_ if _K_2_0 else _1_)) if self.Math else NN
583_wrapped = _kWrapped() # PYCHOK singleton, .datum, .test/base.py
586class Rhumb8Tuple(_GTuple):
587 '''8-Tuple C{(lat1, lon1, lat2, lon2, azi12, s12, S12, a12)} with lat- C{lat1},
588 C{lat2} and longitudes C{lon1}, C{lon2} of both points, the azimuth of the
589 rhumb line C{azi12}, the distance C{s12}, the area C{S12} under the rhumb
590 line and the angular distance C{a12} between both points.
591 '''
592 _Names_ = (_lat1_, _lon1_, _lat2_, _lon2_, _azi12_, _s12_, _S12_, _a12_)
593 _Units_ = ( Lat, Lon, Lat, Lon, _Azi, _M, _M2, _Deg)
595 def toDirect9Tuple(self, dflt=NAN, **a12_azi1_azi2_m12_M12_M21):
596 '''Convert this L{Rhumb8Tuple} result to a 9-tuple, like I{Karney}'s
597 method C{geographiclib.geodesic.Geodesic._GenDirect}.
599 @kwarg dflt: Default value for missing items (C{any}).
600 @kwarg a12_azi1_azi2_m12_M12_M21: Optional keyword arguments
601 to specify or override L{Inverse10Tuple} items.
603 @return: L{Direct9Tuple}C{(a12, lat2, lon2, azi2, s12,
604 m12, M12, M21, S12)}
605 '''
606 d = dict(azi1=self.azi12, M12=_1_0, m12=self.s12, # PYCHOK attr
607 azi2=self.azi12, M21=_1_0) # PYCHOK attr
608 if a12_azi1_azi2_m12_M12_M21:
609 d.update(a12_azi1_azi2_m12_M12_M21)
610 return self._toTuple(Direct9Tuple, dflt, d)
612 def toInverse10Tuple(self, dflt=NAN, **a12_m12_M12_M21_salp1_calp1_salp2_calp2):
613 '''Convert this L{Rhumb8Tuple} to a 10-tuple, like I{Karney}'s
614 method C{geographiclib.geodesic.Geodesic._GenInverse}.
616 @kwarg dflt: Default value for missing items (C{any}).
617 @kwarg a12_m12_M12_M21_salp1_calp1_salp2_calp2: Optional keyword
618 arguments to specify or override L{Inverse10Tuple} items.
620 @return: L{Inverse10Tuple}C{(a12, s12, salp1, calp1, salp2, calp2,
621 m12, M12, M21, S12)}.
622 '''
623 s, c = sincos2d(self.azi12) # PYCHOK attr
624 d = dict(salp1=s, calp1=c, M12=_1_0, m12=self.s12, # PYCHOK attr
625 salp2=s, calp2=c, M21=_1_0)
626 if a12_m12_M12_M21_salp1_calp1_salp2_calp2:
627 d.update(a12_m12_M12_M21_salp1_calp1_salp2_calp2)
628 return self._toTuple(Inverse10Tuple, dflt, d)
630 def _toTuple(self, nTuple, dflt, updates={}):
631 '''(INTERNAL) Convert this C{Rhumb8Tuple} to an B{C{nTuple}}.
632 '''
633 _g = self.toGDict(**updates).get
634 t = tuple(_g(n, dflt) for n in nTuple._Names_)
635 return nTuple(t, name=self.name)
637 @deprecated_method
638 def _to7Tuple(self):
639 '''DEPRECATED, do not use!'''
640 return _MODS.deprecated.classes.Rhumb7Tuple(self[:-1])
643class _Xables(object):
644 '''(INTERNAL) Get I{Karney}'s executable paths from/and env vars.
645 '''
646 bin_ = '/opt/local/bin/' # '/opt/local/Cellar/geographiclib/2.3/bin/' # HomeBrew on macOS
647 ENV = NN
649 def GeoConvert(self, *dir_):
650 return self._path(self.GeoConvert, *dir_)
652 def GeodSolve(self, *dir_):
653 return self._path(self.GeodSolve, *dir_)
655 def IntersectTool(self, *dir_):
656 return self._path(self.IntersectTool, *dir_)
658 def RhumbSolve(self, *dir_):
659 return self._path(self.RhumbSolve, *dir_)
661 def name_version(self, path, base=True):
662 # return C{(path + ' ' + version)} of an executable
663 if path:
664 try:
665 r, s = _popen2((path, '--version'))
666 if base:
667 path = _MODS.os.path.basename(path)
668 r = _SPACE_(path, r.split()[-1])
669 else:
670 r = _MODS.streprs.Fmt.PARENSPACED(r, s)
671 return r
672 except (IndexError, IOError, OSError):
673 pass
674 return NN
676 def _path(self, which, *dir_):
677 self.ENV = E = _PYGEODESY(which)
678 return _getenv(E, NN) or \
679 (NN(dir_[0], which.__name__) if dir_ else E)
681 def X_not(self, path):
682 return 'env %s=%r not executable' % (self.ENV, path)
684 def X_OK(self, path): # is C{path} an executable?
685 os = _MODS.os # import os
686 return os.access(path, os.X_OK) if path else False
688_Xables = _Xables() # PYCHOK singleton
691def _around(x): # in .utily.sincos2d
692 '''I{Coarsen} a scalar by rounding small values to underflow to C{0.0}.
694 @return: Coarsened value (C{float}).
696 @see: I{Karney}'s U{geomath.Math.AngRound<https://SourceForge.net/p/
697 geographiclib/code/ci/release/tree/python/geographiclib/geomath.py>}
698 '''
699 try:
700 return _wrapped.Math.AngRound(x)
701 except AttributeError:
702 if x:
703 y = _1_16th - fabs(x)
704 if y > 0: # fabs(x) < _1_16th
705 x = _copysign(_1_16th - y, x)
706 else:
707 x = _0_0 # -0 to 0
708 return x
711def _atan2d(y, x):
712 '''Return C{atan2(B{y}, B{x})} in C{degrees}.
713 '''
714 try:
715 return _wrapped.Math.atan2d(y, x)
716 except AttributeError:
717 return atan2d(y, x)
720def _cbrt(x):
721 '''Return C{cubic root(B{x})}.
722 '''
723 try:
724 return _wrapped.Math.cbrt(x)
725 except AttributeError:
726 return cbrt(x)
729def _copyBit(x, y):
730 '''Like C{copysign0(B{x}, B{y})}, with C{B{x} > 0}.
731 '''
732 return (-x) if _signBit(y) else x
735def _2cos2x(cx, sx): # in .auxDST, .auxLat, .gxbases
736 '''Return M{2 * cos(2 * x)} from cos(x) and sin(x).
737 '''
738 r = cx - sx
739 if r:
740 r *= (cx + sx) * _2_0
741 return r
744def _diff182(deg0, deg, K_2_0=False):
745 '''Compute C{deg - deg0}, reduced to C{[-180,180]} accurately.
747 @return: 2-Tuple C{(delta_angle, residual)} in C{degrees}.
748 '''
749 try:
750 return _wrapped.Math.AngDiff(deg0, deg)
751 except AttributeError:
752 if K_2_0 or _K_2_0: # geographiclib 2.0
753 _r, _360 = fremainder, _360_0
754 d, t = _sum2(_r(-deg0, _360),
755 _r( deg, _360))
756 d, t = _sum2(_r( d, _360), t)
757 if d in (_0_0, _180_0, _N_180_0):
758 d = _copysign(d, -t if t else (deg - deg0))
759 else:
760 _n = _norm180
761 d, t = _sum2(_n(-deg0), _n(deg))
762 d = _n(d)
763 if t > 0 and d == _180_0:
764 d = _N_180_0
765 d, t = _sum2(d, t)
766 return d, t
769def _fix90(deg): # mimick Math.LatFix
770 '''Replace angle in C{degrees} outside [-90,90] by NAN.
772 @return: Angle C{degrees} or NAN.
773 '''
774 try:
775 return _wrapped.Math.LatFix(deg)
776 except AttributeError:
777 return NAN if fabs(deg) > 90 else deg
780def _isfinite(x): # mimick geomath.Math.isfinite
781 '''Check finiteness of C{x}.
783 @return: C{True} if finite.
784 '''
785 try:
786 return _wrapped.Math.isfinite(x)
787 except AttributeError:
788 return _math_isfinite(x) # and fabs(x) <= _MAX
791def _llz2gl(gl, **llz2): # see .geodesici._llz2G
792 '''(INTERNAL) Set C{gl.lat2, .lon2, .azi2} from C{llz2}.
793 '''
794 if llz2:
795 for n in (_lat2_, _lon2_, _azi2_): # _lat1_, _lon1_, _azi1_
796 v = llz2.get(n, None)
797 if v is not None:
798 setattr(gl, n, v)
799 return gl
802def _norm2(x, y): # mimick geomath.Math.norm
803 '''Normalize C{B{x}} and C{B{y}}.
805 @return: 2-Tuple of C{(B{x}, B{y})}, normalized.
806 '''
807 try:
808 return _wrapped.Math.norm(x, y)
809 except AttributeError:
810 return norm2(x, y)
813def _norm180(deg): # mimick geomath.Math.AngNormalize
814 '''Reduce angle in C{degrees} to (-180,180].
816 @return: Reduced angle C{degrees}.
817 '''
818 try:
819 return _wrapped.Math.AngNormalize(deg)
820 except AttributeError:
821 d = fremainder(deg, _360_0)
822 if d in (_180_0, _N_180_0):
823 d = _copysign(_180_0, deg) if _K_2_0 else _180_0
824 return d
827def _polygon(geodesic, points, closed, line, wrap):
828 '''(INTERNAL) Compute the area or perimeter of a polygon,
829 using a L{GeodesicExact}, L{GeodesicSolve} or (if the
830 C{geographiclib} package is installed) a C{Geodesic}
831 or C{geodesicw.Geodesic} instance.
832 '''
833 if not wrap: # capability LONG_UNROLL can't be off
834 notImplemented(None, wrap=wrap, up=3)
836 if _MODS.booleans.isBoolean(points):
837 # recursive call for each boolean clip
839 def _a_p(clip, *args, **unused):
840 return _polygon(geodesic, clip, *args)
842 if not closed: # closed only
843 raise _ValueError(closed=closed, points=_composite_)
845 return points._sum1(_a_p, closed, line, wrap)
847 gP = geodesic.Polygon(line)
848 _A = gP.AddPoint
850 Ps = _MODS.iters.PointsIter(points, loop=1, wrap=wrap) # base=LatLonEllipsoidalBase(0, 0)
851 p1 = p0 = Ps[0]
853 # note, lon deltas are unrolled, by default
854 _A(p1.lat, p1.lon)
855 for p2 in Ps.iterate(closed=closed):
856 if wrap and not Ps.looped:
857 p2 = _unrollon(p1, p2)
858 _A(p2.lat, p2.lon)
859 p1 = p2
860 if closed and line and p1 != p0:
861 _A(p0.lat, p0.lon)
863 # gP.Compute returns (number_of_points, perimeter, signed area)
864 return gP.Compute(False, True)[1 if line else 2]
867try:
868 from math import fma as _fma # since 3.13
870 def _poly_fma(x, s, *cs):
871 for c in cs:
872 s = _fma(s, x, c)
873 return s
875except ImportError: # Python 3.12-
877 def _poly_fma(x, s, *cs): # PYCHOK redef
878 t = _0_0
879 for c in cs:
880 s, t, _ = _sum3(s * x, t * x, c)
881 return s + t
883# def _poly_fma(x, s, *cs):
884# S = Fhorner(x, *cs, incx=False)
885# S += s
886# return float(S)
888def _polynomial(x, cs, i, j): # PYCHOK shared
889 '''(INTERNAL) Like C++ C{GeographicLib.Math.hpp.polyval} but with a
890 signature and cascaded summation different from C{karney._sum3}.
892 @return: M{sum(x**(j - k - 1) * cs[k] for k in range(i, j)}
893 '''
894 if (i + 1) < j <= len(cs): # load _Rtuple._tuple
895 try:
896 r = _wrapped.Math.polyval(j - i - 1, cs, i, x)
897 except AttributeError: # no .Math
898 r = _poly_fma(x, _0_0, *cs[i:j])
899 else:
900 r = cs[i]
901 return float(r)
904def _remainder(x, y):
905 '''Remainder of C{x / y}.
907 @return: Remainder in the range M{[-y / 2, y / 2]}, preserving signed 0.0.
908 '''
909 try:
910 return _wrapped.Math.remainder(x, y)
911 except AttributeError:
912 return fremainder(x, y)
915if _K_2_0:
916 from math import cos as _cos, sin as _sin
918 def _sincos2(rad):
919 return _sin(rad), _cos(rad)
921 _signBit = _MODS.basics.signBit
922else:
923 _sincos2 = _MODS.utily.sincos2 # PYCHOK shared
925 def _signBit(x):
926 '''(INTERNAL) GeographicLib 1.52-.
927 '''
928 return x < 0
931def _sincos2d(deg):
932 '''Return sine and cosine of an angle in C{degrees}.
934 @return: 2-Tuple C{(sin(B{deg}), cos(B{deg}))}.
935 '''
936 try:
937 return _wrapped.Math.sincosd(deg)
938 except AttributeError:
939 return sincos2d(deg)
942def _sincos2de(deg, t):
943 '''Return sine and cosine of a corrected angle in C{degrees}.
945 @return: 2-Tuple C{(sin(B{deg}), cos(B{deg}))}.
946 '''
947 try:
948 return _wrapped.Math.sincosde(deg, t)
949 except AttributeError:
950 return sincos2d(deg, adeg=t)
953def _sum2(a, b): # mimick geomath.Math.sum, actually sum2
954 '''Error-free summation like C{geomath.Math.sum}.
956 @return: 2-Tuple C{(B{a} + B{b}, residual)}.
958 @note: The C{residual} can be the same as B{C{a}} or B{C{b}}.
960 @see: U{TwoSum<https://accurate-algorithms.readthedocs.io/en/latest/ch04summation.html>}
961 and I{Knuth}'s U{Algorithm 3.1<https://www.TUHH.De/ti3/paper/rump/OgRuOi05.pdf>}.
962 '''
963 try:
964 return _wrapped.Math.sum(a, b)
965 except AttributeError:
966 # if Algorithm_3_1:
967 s = a + b
968 r = s - b
969 t = s - r
970 # elif C_CPP: # Math::sum C/C++
971 # r -= a; t -= b; t += r; t = -t
972 # else:
973 t = (a - r) + (b - t)
974 # assert fabs(s) >= fabs(t)
975 return s, t
978def _sum3(s, t, *xs):
979 '''Accumulate any B{C{xs}} into a previous C{_sum2(s, t)}.
981 @return: 3-Tuple C{(s, t, n)} where C{s} is the sum of B{s}, B{t} and all
982 B{xs}, C{t} the residual and C{n} the number of zero C{xs}.
984 @see: I{Karney's} C++ U{Accumulator<https://GeographicLib.SourceForge.io/
985 C++/doc/Accumulator_8hpp_source.html>} comments for more details and
986 function C{_sum2} above.
988 @note: Not "error-free", see C{pygeodesy.test/testKarney.py}.
989 '''
990 z = 0
991 for x in xs:
992 if x:
993 t, r = _sum2(t, x) # start at the least-
994 if s:
995 s, t = _sum2(s, t) # -significant end
996 if s:
997 t += r # accumulate r into t
998 else:
999 # assert t == 0 # s == 0 implies t == 0
1000 s = unsigned0(r) # result is r, t = 0
1001 else:
1002 s, t = unsigned0(t), r
1003 else:
1004 z += 1
1005 # assert fabs(s) >= fabs(t)
1006 return s, t, z
1009def _tand(x):
1010 '''Return C{tan(B{x})} in C{degrees}.
1011 '''
1012 try:
1013 return _wrapped.Math.tand(x)
1014 except AttributeError:
1015 return tand(x) # Error=None
1018def _unroll2(lon1, lon2, wrap=False): # see .ellipsoidalBaseDI._intersects2
1019 '''Unroll B{C{lon2 - lon1}} like C{geodesic.Geodesic.Inverse}.
1021 @return: 2-Tuple C{(B{lon2} - B{lon1}, B{lon2})} with B{C{lon2}}
1022 unrolled if C{B{wrap} is True}, normalized otherwise.
1023 '''
1024 if wrap:
1025 d, t = _diff182(lon1, lon2)
1026 lon2, t, _ = _sum3(d, t, lon1) # (lon1 + d) + t
1027 lon2 += t
1028 else:
1029 lon2 = _norm180(lon2)
1030 return (lon2 - lon1), lon2
1033def _unsigned2(x):
1034 '''(INTERNAL) Unsign B{C{x}}.
1035 '''
1036 return (neg(x), True) if _signBit(x) else (x, False)
1039__all__ += _ALL_DOCS(_CapsBase)
1041# **) MIT License
1042#
1043# Copyright (C) 2016-2025 -- mrJean1 at Gmail -- All Rights Reserved.
1044#
1045# Permission is hereby granted, free of charge, to any person obtaining a
1046# copy of this software and associated documentation files (the "Software"),
1047# to deal in the Software without restriction, including without limitation
1048# the rights to use, copy, modify, merge, publish, distribute, sublicense,
1049# and/or sell copies of the Software, and to permit persons to whom the
1050# Software is furnished to do so, subject to the following conditions:
1051#
1052# The above copyright notice and this permission notice shall be included
1053# in all copies or substantial portions of the Software.
1054#
1055# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
1056# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
1057# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
1058# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
1059# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
1060# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
1061# OTHER DEALINGS IN THE SOFTWARE.