Coverage for pygeodesy/formy.py: 98%
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« prev ^ index » next coverage.py v7.6.1, created at 2024-12-12 17:20 -0500
2# -*- coding: utf-8 -*-
4u'''Formulary of basic geodesy functions and approximations.
5'''
6# make sure int/int division yields float quotient, see .basics
7from __future__ import division as _; del _ # PYCHOK semicolon
9# from pygeodesy.basics import _args_kwds_count2, _copysign # from .constants
10# from pygeodesy.cartesianBase import CartesianBase # _MODS
11from pygeodesy.constants import EPS, EPS0, EPS1, PI, PI2, PI3, PI_2, R_M, \
12 _0_0s, float0_, isnon0, remainder, _umod_PI2, \
13 _0_0, _0_125, _0_25, _0_5, _1_0, _2_0, _4_0, \
14 _32_0, _90_0, _180_0, _360_0, _copysign
15from pygeodesy.datums import Datum, Ellipsoid, _ellipsoidal_datum, \
16 _mean_radius, _spherical_datum, _WGS84, _EWGS84
17# from pygeodesy.ellipsoids import Ellipsoid, _EWGS84 # from .datums
18from pygeodesy.errors import IntersectionError, LimitError, limiterrors, \
19 _TypeError, _ValueError, _xattr, _xError, \
20 _xcallable,_xkwds, _xkwds_pop2
21from pygeodesy.fmath import euclid, hypot, hypot2, sqrt0
22from pygeodesy.fsums import fsumf_, Fmt, unstr
23# from pygeodesy.internals import _DUNDER_nameof # from .named
24from pygeodesy.interns import _delta_, _distant_, _inside_, _SPACE_, _too_
25from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS
26from pygeodesy.named import _name__, _name2__, _NamedTuple, _xnamed, \
27 _DUNDER_nameof
28from pygeodesy.namedTuples import Bearing2Tuple, Distance4Tuple, LatLon2Tuple, \
29 Intersection3Tuple, PhiLam2Tuple
30# from pygeodesy.streprs import Fmt, unstr # from .fsums
31# from pygeodesy.triaxials import _hartzell3 # _MODS
32from pygeodesy.units import _isDegrees, _isHeight, _isRadius, Bearing, Degrees_, \
33 Distance, Distance_, Height, Lamd, Lat, Lon, Meter_, \
34 Phid, Radians, Radians_, Radius, Radius_, Scalar, _100km
35from pygeodesy.utily import acos1, atan2, atan2b, degrees2m, _loneg, m2degrees, \
36 tan_2, sincos2, sincos2_, _Wrap
37# from pygeodesy.vector3d import _otherV3d # _MODS
38# from pygeodesy.vector3dBase import _xyz_y_z3 # _MODS
39# from pygeodesy import ellipsoidalExact, ellipsoidalKarney, vector3d, \
40# sphericalNvector, sphericalTrigonometry # _MODS
42from contextlib import contextmanager
43from math import asin, atan, cos, degrees, fabs, radians, sin, sqrt # pow
45__all__ = _ALL_LAZY.formy
46__version__ = '24.12.06'
48_RADIANS2 = (PI / _180_0)**2 # degrees- to radians-squared
49_ratio_ = 'ratio'
50_xline_ = 'xline'
53def angle2chord(rad, radius=R_M):
54 '''Get the chord length of a (central) angle or I{angular} distance.
56 @arg rad: Central angle (C{radians}).
57 @kwarg radius: Mean earth radius (C{meter}, conventionally), datum (L{Datum}) or ellipsoid
58 (L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}) to use or C{None}.
60 @return: Chord length (C{meter}, same units as B{C{radius}} or if C{B{radius} is None}, C{radians}).
62 @see: Function L{chord2angle}, method L{intermediateChordTo<sphericalNvector.LatLon.intermediateChordTo>} and
63 U{great-circle-distance<https://WikiPedia.org/wiki/Great-circle_distance#Relation_between_central_angle_and_chord_length>}.
64 '''
65 d = _isDegrees(rad, iscalar=False)
66 r = sin((radians(rad) if d else rad) / _2_0) * _2_0
67 return (degrees(r) if d else r) if radius is None else (_mean_radius(radius) * r)
70def _anti2(a, b, n_2, n, n2):
71 '''(INTERNAL) Helper for C{antipode} and C{antipode_}.
72 '''
73 r = remainder(a, n) if fabs(a) > n_2 else a
74 if r == a:
75 r = -r
76 b += n
77 if fabs(b) > n:
78 b = remainder(b, n2)
79 return float0_(r, b)
82def antipode(lat, lon, **name):
83 '''Return the antipode, the point diametrically opposite to a given
84 point in C{degrees}.
86 @arg lat: Latitude (C{degrees}).
87 @arg lon: Longitude (C{degrees}).
88 @kwarg name: Optional C{B{name}=NN} (C{str}).
90 @return: A L{LatLon2Tuple}C{(lat, lon)}.
92 @see: Functions L{antipode_} and L{normal} and U{Geosphere
93 <https://CRAN.R-Project.org/web/packages/geosphere/geosphere.pdf>}.
94 '''
95 return LatLon2Tuple(*_anti2(lat, lon, _90_0, _180_0, _360_0), **name)
98def antipode_(phi, lam, **name):
99 '''Return the antipode, the point diametrically opposite to a given
100 point in C{radians}.
102 @arg phi: Latitude (C{radians}).
103 @arg lam: Longitude (C{radians}).
104 @kwarg name: Optional C{B{name}=NN} (C{str}).
106 @return: A L{PhiLam2Tuple}C{(phi, lam)}.
108 @see: Functions L{antipode} and L{normal_} and U{Geosphere
109 <https://CRAN.R-Project.org/web/packages/geosphere/geosphere.pdf>}.
110 '''
111 return PhiLam2Tuple(*_anti2(phi, lam, PI_2, PI, PI2), **name)
114def bearing(lat1, lon1, lat2, lon2, **final_wrap):
115 '''Compute the initial or final bearing (forward or reverse azimuth)
116 between two (spherical) points.
118 @arg lat1: Start latitude (C{degrees}).
119 @arg lon1: Start longitude (C{degrees}).
120 @arg lat2: End latitude (C{degrees}).
121 @arg lon2: End longitude (C{degrees}).
122 @kwarg final_wrap: Optional keyword arguments for function
123 L{pygeodesy.bearing_}.
125 @return: Initial or final bearing (compass C{degrees360}) or zero if
126 both points coincide.
127 '''
128 r = bearing_(Phid(lat1=lat1), Lamd(lon1=lon1),
129 Phid(lat2=lat2), Lamd(lon2=lon2), **final_wrap)
130 return degrees(r)
133def bearing_(phi1, lam1, phi2, lam2, final=False, wrap=False):
134 '''Compute the initial or final bearing (forward or reverse azimuth) between
135 two (spherical) points.
137 @arg phi1: Start latitude (C{radians}).
138 @arg lam1: Start longitude (C{radians}).
139 @arg phi2: End latitude (C{radians}).
140 @arg lam2: End longitude (C{radians}).
141 @kwarg final: If C{True}, return the final, otherwise the initial bearing
142 (C{bool}).
143 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{phi2}} and
144 B{C{lam2}} (C{bool}).
146 @return: Initial or final bearing (compass C{radiansPI2}) or zero if both
147 are coincident.
149 @see: U{Bearing<https://www.Movable-Type.co.UK/scripts/latlong.html>}, U{Course
150 between two points<https://www.EdWilliams.org/avform147.htm#Crs>} and
151 U{Bearing Between Two Points<https://web.Archive.org/web/20020630205931/
152 https://MathForum.org/library/drmath/view/55417.html>}.
153 '''
154 db, phi2, lam2 = _Wrap.philam3(lam1, phi2, lam2, wrap)
155 if final: # swap plus PI
156 phi1, lam1, phi2, lam2, db = phi2, lam2, phi1, lam1, -db
157 r = PI3
158 else:
159 r = PI2
160 sa1, ca1, sa2, ca2, sdb, cdb = sincos2_(phi1, phi2, db)
162 x = ca1 * sa2 - sa1 * ca2 * cdb
163 y = sdb * ca2
164 return _umod_PI2(atan2(y, x) + r) # .utily.wrapPI2
167def _bearingTo2(p1, p2, wrap=False): # for points.ispolar, sphericalTrigonometry.areaOf
168 '''(INTERNAL) Compute initial and final bearing.
169 '''
170 try: # for LatLon_ and ellipsoidal LatLon
171 return p1.bearingTo2(p2, wrap=wrap)
172 except AttributeError:
173 pass
174 # XXX spherical version, OK for ellipsoidal ispolar?
175 t = p1.philam + p2.philam
176 return Bearing2Tuple(degrees(bearing_(*t, final=False, wrap=wrap)),
177 degrees(bearing_(*t, final=True, wrap=wrap)),
178 name__=_bearingTo2)
181def chord2angle(chord, radius=R_M):
182 '''Get the (central) angle from a chord length or distance.
184 @arg chord: Length or distance (C{meter}, same units as B{C{radius}}).
185 @kwarg radius: Mean earth radius (C{meter}, conventionally), datum (L{Datum}) or
186 ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}) to use.
188 @return: Angle (C{radians} with sign of B{C{chord}}) or C{0} if C{B{radius}=0}.
190 @note: The angle will exceed C{PI} if C{B{chord} > B{radius} * 2}.
192 @see: Function L{angle2chord}.
193 '''
194 m = _mean_radius(radius)
195 r = fabs(chord / (m * _2_0)) if m > 0 else _0_0
196 if r:
197 i = int(r)
198 if i > 0:
199 r -= i
200 i *= PI
201 r = asin(r) + i
202 return _copysign(r * _2_0, chord)
205def compassAngle(lat1, lon1, lat2, lon2, adjust=True, wrap=False):
206 '''Return the angle from North for the direction vector M{(lon2 - lon1,
207 lat2 - lat1)} between two points.
209 Suitable only for short, not near-polar vectors up to a few hundred
210 Km or Miles. Use function L{pygeodesy.bearing} for longer vectors.
212 @arg lat1: From latitude (C{degrees}).
213 @arg lon1: From longitude (C{degrees}).
214 @arg lat2: To latitude (C{degrees}).
215 @arg lon2: To longitude (C{degrees}).
216 @kwarg adjust: Adjust the longitudinal delta by the cosine of the
217 mean latitude (C{bool}).
218 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
219 and B{C{lon2}} (C{bool}).
221 @return: Compass angle from North (C{degrees360}).
223 @note: Courtesy of Martin Schultz.
225 @see: U{Local, flat earth approximation
226 <https://www.EdWilliams.org/avform.htm#flat>}.
227 '''
228 d_lon, lat2, lon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap)
229 if adjust: # scale delta lon
230 d_lon *= _scale_deg(lat1, lat2)
231 return atan2b(d_lon, lat2 - lat1)
234def cosineAndoyerLambert(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):
235 '''Compute the distance between two (ellipsoidal) points using the U{Andoyer-Lambert
236 <https://books.google.com/books?id=x2UiAQAAIAAJ>} correction of the U{Law of
237 Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula.
239 @arg lat1: Start latitude (C{degrees}).
240 @arg lon1: Start longitude (C{degrees}).
241 @arg lat2: End latitude (C{degrees}).
242 @arg lon2: End longitude (C{degrees}).
243 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
244 L{a_f2Tuple}) to use.
245 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} and
246 B{C{lon2}} (C{bool}).
248 @return: Distance (C{meter}, same units as the B{C{datum}}'s ellipsoid axes or
249 C{radians} if C{B{datum} is None}).
251 @raise TypeError: Invalid B{C{datum}}.
253 @see: Functions L{cosineAndoyerLambert_}, L{cosineForsytheAndoyerLambert},
254 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny},
255 L{flatPolar}, L{haversine}, L{thomas} and L{vincentys} and method
256 L{Ellipsoid.distance2}.
257 '''
258 return _dE(cosineAndoyerLambert_, datum, wrap, lat1, lon1, lat2, lon2)
261def cosineAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84):
262 '''Compute the I{angular} distance between two (ellipsoidal) points using the U{Andoyer-Lambert
263 <https://books.google.com/books?id=x2UiAQAAIAAJ>} correction of the U{Law of
264 Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula.
266 @arg phi2: End latitude (C{radians}).
267 @arg phi1: Start latitude (C{radians}).
268 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
269 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
270 L{a_f2Tuple}) to use.
272 @return: Angular distance (C{radians}).
274 @raise TypeError: Invalid B{C{datum}}.
276 @see: Functions L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert_},
277 L{cosineLaw_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_},
278 L{haversine_}, L{thomas_} and L{vincentys_} and U{Geodesy-PHP
279 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/Distance/
280 AndoyerLambert.php>}.
281 '''
282 s2, c2, s1, c1, r, c21 = _sincosa6(phi2, phi1, lam21)
283 if isnon0(c1) and isnon0(c2):
284 E = _ellipsoidal(datum, cosineAndoyerLambert_)
285 if E.f: # ellipsoidal
286 r2 = atan2(E.b_a * s2, c2)
287 r1 = atan2(E.b_a * s1, c1)
288 s2, c2, s1, c1 = sincos2_(r2, r1)
289 r = acos1(s1 * s2 + c1 * c2 * c21)
290 if r:
291 sr, _, sr_2, cr_2 = sincos2_(r, r * _0_5)
292 if isnon0(sr_2) and isnon0(cr_2):
293 s = (sr + r) * ((s1 - s2) / sr_2)**2
294 c = (sr - r) * ((s1 + s2) / cr_2)**2
295 r += (c - s) * E.f * _0_125
296 return r
299def cosineForsytheAndoyerLambert(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):
300 '''Compute the distance between two (ellipsoidal) points using the U{Forsythe-Andoyer-Lambert
301 <https://www2.UNB.Ca/gge/Pubs/TR77.pdf>} correction of the U{Law of Cosines
302 <https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula.
304 @arg lat1: Start latitude (C{degrees}).
305 @arg lon1: Start longitude (C{degrees}).
306 @arg lat2: End latitude (C{degrees}).
307 @arg lon2: End longitude (C{degrees}).
308 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
309 L{a_f2Tuple}) to use.
310 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} and
311 B{C{lon2}} (C{bool}).
313 @return: Distance (C{meter}, same units as the B{C{datum}}'s ellipsoid axes or
314 C{radians} if C{B{datum} is None}).
316 @raise TypeError: Invalid B{C{datum}}.
318 @see: Functions L{cosineForsytheAndoyerLambert_}, L{cosineAndoyerLambert},
319 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny},
320 L{flatPolar}, L{haversine}, L{thomas} and L{vincentys} and method
321 L{Ellipsoid.distance2}.
322 '''
323 return _dE(cosineForsytheAndoyerLambert_, datum, wrap, lat1, lon1, lat2, lon2)
326def cosineForsytheAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84):
327 '''Compute the I{angular} distance between two (ellipsoidal) points using the
328 U{Forsythe-Andoyer-Lambert<https://www2.UNB.Ca/gge/Pubs/TR77.pdf>} correction of
329 the U{Law of Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>}
330 formula.
332 @arg phi2: End latitude (C{radians}).
333 @arg phi1: Start latitude (C{radians}).
334 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
335 @kwarg datum: Datum (L{Datum}) or ellipsoid to use (L{Ellipsoid},
336 L{Ellipsoid2} or L{a_f2Tuple}).
338 @return: Angular distance (C{radians}).
340 @raise TypeError: Invalid B{C{datum}}.
342 @see: Functions L{cosineForsytheAndoyerLambert}, L{cosineAndoyerLambert_},
343 L{cosineLaw_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_},
344 L{haversine_}, L{thomas_} and L{vincentys_} and U{Geodesy-PHP
345 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/
346 Distance/ForsytheCorrection.php>}.
347 '''
348 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21)
349 if r and isnon0(c1) and isnon0(c2):
350 E = _ellipsoidal(datum, cosineForsytheAndoyerLambert_)
351 if E.f: # ellipsoidal
352 sr, cr, s2r, _ = sincos2_(r, r * 2)
353 if isnon0(sr) and fabs(cr) < EPS1:
354 s = (s1 + s2)**2 / (1 + cr)
355 t = (s1 - s2)**2 / (1 - cr)
356 x = s + t
357 y = s - t
359 s = 8 * r**2 / sr
360 a = 64 * r + s * cr * 2 # 16 * r**2 / tan(r)
361 d = 48 * sr + s # 8 * r**2 / tan(r)
362 b = -2 * d
363 e = 30 * s2r
364 c = fsumf_(30 * r, e * _0_5, s * cr) # 8 * r**2 / tan(r)
365 t = fsumf_( a * x, e * y**2, b * y, -c * x**2, d * x * y)
367 r += fsumf_(-r * x, 3 * y * sr, t * E.f / _32_0) * E.f * _0_25
368 return r
371def cosineLaw(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
372 '''Compute the distance between two points using the U{spherical Law of Cosines
373 <https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula.
375 @arg lat1: Start latitude (C{degrees}).
376 @arg lon1: Start longitude (C{degrees}).
377 @arg lat2: End latitude (C{degrees}).
378 @arg lon2: End longitude (C{degrees}).
379 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
380 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
381 L{a_f2Tuple}) to use.
382 @kwarg wrap: If C{True}, wrap or I{normalize} and B{C{lat2}}
383 and B{C{lon2}} (C{bool}).
385 @return: Distance (C{meter}, same units as B{C{radius}} or the
386 ellipsoid or datum axes).
388 @raise TypeError: Invalid B{C{radius}}.
390 @see: Functions L{cosineLaw_}, L{cosineAndoyerLambert},
391 L{cosineForsytheAndoyerLambert}, L{equirectangular}, L{euclidean},
392 L{flatLocal}/L{hubeny}, L{flatPolar}, L{haversine}, L{thomas} and
393 L{vincentys} and method L{Ellipsoid.distance2}.
395 @note: See note at function L{vincentys_}.
396 '''
397 return _dS(cosineLaw_, radius, wrap, lat1, lon1, lat2, lon2)
400def cosineLaw_(phi2, phi1, lam21):
401 '''Compute the I{angular} distance between two points using the U{spherical Law of
402 Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula.
404 @arg phi2: End latitude (C{radians}).
405 @arg phi1: Start latitude (C{radians}).
406 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
408 @return: Angular distance (C{radians}).
410 @see: Functions L{cosineLaw}, L{cosineAndoyerLambert_},
411 L{cosineForsytheAndoyerLambert_}, L{euclidean_},
412 L{flatLocal_}/L{hubeny_}, L{flatPolar_}, L{haversine_},
413 L{thomas_} and L{vincentys_}.
415 @note: See note at function L{vincentys_}.
416 '''
417 return _sincosa6(phi2, phi1, lam21)[4]
420def _d3(wrap, lat1, lon1, lat2, lon2):
421 '''(INTERNAL) Helper for _dE, _dS and _eA.
422 '''
423 if wrap:
424 d_lon, lat2, _ = _Wrap.latlon3(lon1, lat2, lon2, wrap)
425 return radians(lat2), Phid(lat1=lat1), radians(d_lon)
426 else: # for backward compaibility
427 return Phid(lat2=lat2), Phid(lat1=lat1), Phid(d_lon=lon2 - lon1)
430def _dE(func_, earth, *wrap_lls):
431 '''(INTERNAL) Helper for ellipsoidal distances.
432 '''
433 E = _ellipsoidal(earth, func_)
434 r = func_(*_d3(*wrap_lls), datum=E)
435 return r * E.a
438def _dS(func_, radius, *wrap_lls, **adjust):
439 '''(INTERNAL) Helper for spherical distances.
440 '''
441 r = func_(*_d3(*wrap_lls), **adjust)
442 if radius is not R_M:
443 _, lat1, _, lat2, _ = wrap_lls
444 radius = _mean_radius(radius, lat1, lat2)
445 return r * radius
448def _eA(excess_, radius, *wrap_lls):
449 '''(INTERNAL) Helper for spherical excess or area.
450 '''
451 r = excess_(*_d3(*wrap_lls))
452 if radius:
453 _, lat1, _, lat2, _ = wrap_lls
454 r *= _mean_radius(radius, lat1, lat2)**2
455 return r
458def _ellipsoidal(earth, where):
459 '''(INTERNAL) Helper for distances.
460 '''
461 return _EWGS84 if earth in (_WGS84, _EWGS84) else (
462 earth if isinstance(earth, Ellipsoid) else
463 (earth if isinstance(earth, Datum) else # PYCHOK indent
464 _ellipsoidal_datum(earth, name__=where)).ellipsoid)
467def equirectangular(lat1, lon1, lat2, lon2, radius=R_M, **adjust_limit_wrap):
468 '''Approximate the distance between two points using the U{Equirectangular Approximation
469 / Projection<https://www.Movable-Type.co.UK/scripts/latlong.html#equirectangular>}.
471 @arg lat1: Start latitude (C{degrees}).
472 @arg lon1: Start longitude (C{degrees}).
473 @arg lat2: End latitude (C{degrees}).
474 @arg lon2: End longitude (C{degrees}).
475 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum}) or ellipsoid
476 (L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}).
477 @kwarg adjust_limit_wrap: Optional keyword arguments for function L{equirectangular4}.
479 @return: Distance (C{meter}, same units as B{C{radius}} or the ellipsoid or datum axes).
481 @raise TypeError: Invalid B{C{radius}}.
483 @see: Function L{equirectangular4} for more details, the available B{C{options}},
484 errors, restrictions and other, approximate or accurate distance functions.
485 '''
486 d = sqrt(equirectangular4(Lat(lat1=lat1), Lon(lon1=lon1),
487 Lat(lat2=lat2), Lon(lon2=lon2),
488 **adjust_limit_wrap).distance2) # PYCHOK 4 vs 2-3
489 return degrees2m(d, radius=_mean_radius(radius, lat1, lat2))
492def _equirectangular(lat1, lon1, lat2, lon2, **adjust_limit_wrap):
493 '''(INTERNAL) Helper for classes L{frechet._FrechetMeterRadians} and
494 L{hausdorff._HausdorffMeterRedians}.
495 '''
496 return equirectangular4(lat1, lon1, lat2, lon2, **adjust_limit_wrap).distance2 * _RADIANS2
499def equirectangular4(lat1, lon1, lat2, lon2, adjust=True, limit=45, wrap=False):
500 '''Approximate the distance between two points using the U{Equirectangular Approximation
501 / Projection<https://www.Movable-Type.co.UK/scripts/latlong.html#equirectangular>}.
503 This approximation is valid for short distance of several hundred Km or Miles, see
504 the B{C{limit}} keyword argument and L{LimitError}.
506 @arg lat1: Start latitude (C{degrees}).
507 @arg lon1: Start longitude (C{degrees}).
508 @arg lat2: End latitude (C{degrees}).
509 @arg lon2: End longitude (C{degrees}).
510 @kwarg adjust: Adjust the wrapped, unrolled longitudinal delta by the cosine of the mean
511 latitude (C{bool}).
512 @kwarg limit: Optional limit for lat- and longitudinal deltas (C{degrees}) or C{None}
513 or C{0} for unlimited.
514 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} and B{C{lon2}}
515 (C{bool}).
517 @return: A L{Distance4Tuple}C{(distance2, delta_lat, delta_lon, unroll_lon2)} with
518 C{distance2} in C{degrees squared}.
520 @raise LimitError: The lat- or longitudinal delta exceeds the B{C{-limit..limit}}
521 range and L{limiterrors<pygeodesy.limiterrors>} is C{True}.
523 @see: U{Local, flat earth approximation<https://www.EdWilliams.org/avform.htm#flat>},
524 functions L{equirectangular}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
525 L{cosineLaw}, L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar}, L{haversine},
526 L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2}, C{LatLon.distanceTo*}
527 and C{LatLon.equirectangularTo}.
528 '''
529 if wrap:
530 d_lon, lat2, ulon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap)
531 else:
532 d_lon, ulon2 = (lon2 - lon1), lon2
533 d_lat = lat2 - lat1
535 if limit and limit > 0 and limiterrors():
536 d = max(fabs(d_lat), fabs(d_lon))
537 if d > limit:
538 t = _SPACE_(_delta_, Fmt.PAREN_g(d), Fmt.exceeds_limit(limit))
539 s = unstr(equirectangular4, lat1, lon1, lat2, lon2,
540 limit=limit, wrap=wrap)
541 raise LimitError(s, txt=t)
543 if adjust: # scale delta lon
544 d_lon *= _scale_deg(lat1, lat2)
546 d2 = hypot2(d_lat, d_lon) # degrees squared!
547 return Distance4Tuple(d2, d_lat, d_lon, ulon2 - lon2)
550def euclidean(lat1, lon1, lat2, lon2, radius=R_M, adjust=True, wrap=False):
551 '''Approximate the C{Euclidean} distance between two (spherical) points.
553 @arg lat1: Start latitude (C{degrees}).
554 @arg lon1: Start longitude (C{degrees}).
555 @arg lat2: End latitude (C{degrees}).
556 @arg lon2: End longitude (C{degrees}).
557 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
558 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
559 L{a_f2Tuple}) to use.
560 @kwarg adjust: Adjust the longitudinal delta by the cosine of
561 the mean latitude (C{bool}).
562 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
563 and B{C{lon2}} (C{bool}).
565 @return: Distance (C{meter}, same units as B{C{radius}} or the
566 ellipsoid or datum axes).
568 @raise TypeError: Invalid B{C{radius}}.
570 @see: U{Distance between two (spherical) points
571 <https://www.EdWilliams.org/avform.htm#Dist>}, functions L{euclid},
572 L{euclidean_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
573 L{cosineLaw}, L{equirectangular}, L{flatLocal}/L{hubeny}, L{flatPolar},
574 L{haversine}, L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2},
575 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
576 '''
577 return _dS(euclidean_, radius, wrap, lat1, lon1, lat2, lon2, adjust=adjust)
580def euclidean_(phi2, phi1, lam21, adjust=True):
581 '''Approximate the I{angular} C{Euclidean} distance between two (spherical) points.
583 @arg phi2: End latitude (C{radians}).
584 @arg phi1: Start latitude (C{radians}).
585 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
586 @kwarg adjust: Adjust the longitudinal delta by the cosine
587 of the mean latitude (C{bool}).
589 @return: Angular distance (C{radians}).
591 @see: Functions L{euclid}, L{euclidean}, L{cosineAndoyerLambert_},
592 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
593 L{flatLocal_}/L{hubeny_}, L{flatPolar_}, L{haversine_},
594 L{thomas_} and L{vincentys_}.
595 '''
596 if adjust:
597 lam21 *= _scale_rad(phi2, phi1)
598 return euclid(phi2 - phi1, lam21)
601def excessAbc_(A, b, c):
602 '''Compute the I{spherical excess} C{E} of a (spherical) triangle from two sides
603 and the included (small) angle.
605 @arg A: An interior triangle angle (C{radians}).
606 @arg b: Frist adjacent triangle side (C{radians}).
607 @arg c: Second adjacent triangle side (C{radians}).
609 @return: Spherical excess (C{radians}).
611 @raise UnitError: Invalid B{C{A}}, B{C{b}} or B{C{c}}.
613 @see: Functions L{excessGirard_}, L{excessLHuilier_} and U{Spherical
614 trigonometry<https://WikiPedia.org/wiki/Spherical_trigonometry>}.
615 '''
616 A = Radians_(A=A)
617 b = Radians_(b=b) * _0_5
618 c = Radians_(c=c) * _0_5
620 sA, cA, sb, cb, sc, cc = sincos2_(A, b, c)
621 return atan2(sA * sb * sc, cb * cc + cA * sb * sc) * _2_0
624def excessCagnoli_(a, b, c):
625 '''Compute the I{spherical excess} C{E} of a (spherical) triangle using U{Cagnoli's
626 <https://Zenodo.org/record/35392>} (D.34) formula.
628 @arg a: First triangle side (C{radians}).
629 @arg b: Second triangle side (C{radians}).
630 @arg c: Third triangle side (C{radians}).
632 @return: Spherical excess (C{radians}).
634 @raise UnitError: Invalid B{C{a}}, B{C{b}} or B{C{c}}.
636 @see: Function L{excessLHuilier_} and U{Spherical trigonometry
637 <https://WikiPedia.org/wiki/Spherical_trigonometry>}.
638 '''
639 a = Radians_(a=a)
640 b = Radians_(b=b)
641 c = Radians_(c=c)
643 s = fsumf_(a, b, c) * _0_5
644 _s = sin
645 r = _s(s) * _s(s - a) * _s(s - b) * _s(s - c)
646 c = cos(a * _0_5) * cos(b * _0_5) * cos(c * _0_5)
647 r = asin(sqrt(r) * _0_5 / c) if c and r > 0 else _0_0
648 return Radians(Cagnoli=r * _2_0)
651def excessGirard_(A, B, C):
652 '''Compute the I{spherical excess} C{E} of a (spherical) triangle using U{Girard's
653 <https://MathWorld.Wolfram.com/GirardsSphericalExcessFormula.html>} formula.
655 @arg A: First interior triangle angle (C{radians}).
656 @arg B: Second interior triangle angle (C{radians}).
657 @arg C: Third interior triangle angle (C{radians}).
659 @return: Spherical excess (C{radians}).
661 @raise UnitError: Invalid B{C{A}}, B{C{B}} or B{C{C}}.
663 @see: Function L{excessLHuilier_} and U{Spherical trigonometry
664 <https://WikiPedia.org/wiki/Spherical_trigonometry>}.
665 '''
666 return Radians(Girard=fsumf_(Radians_(A=A),
667 Radians_(B=B),
668 Radians_(C=C), -PI))
671def excessLHuilier_(a, b, c):
672 '''Compute the I{spherical excess} C{E} of a (spherical) triangle using U{L'Huilier's
673 <https://MathWorld.Wolfram.com/LHuiliersTheorem.html>}'s Theorem.
675 @arg a: First triangle side (C{radians}).
676 @arg b: Second triangle side (C{radians}).
677 @arg c: Third triangle side (C{radians}).
679 @return: Spherical excess (C{radians}).
681 @raise UnitError: Invalid B{C{a}}, B{C{b}} or B{C{c}}.
683 @see: Function L{excessCagnoli_}, L{excessGirard_} and U{Spherical
684 trigonometry<https://WikiPedia.org/wiki/Spherical_trigonometry>}.
685 '''
686 a = Radians_(a=a)
687 b = Radians_(b=b)
688 c = Radians_(c=c)
690 s = fsumf_(a, b, c) * _0_5
691 _t = tan_2
692 r = _t(s) * _t(s - a) * _t(s - b) * _t(s - c)
693 r = atan(sqrt(r)) if r > 0 else _0_0
694 return Radians(LHuilier=r * _4_0)
697def excessKarney(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
698 '''Compute the surface area of a (spherical) quadrilateral bounded by a
699 segment of a great circle, two meridians and the equator using U{Karney's
700 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>}
701 method.
703 @arg lat1: Start latitude (C{degrees}).
704 @arg lon1: Start longitude (C{degrees}).
705 @arg lat2: End latitude (C{degrees}).
706 @arg lon2: End longitude (C{degrees}).
707 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
708 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
709 L{a_f2Tuple}) or C{None}.
710 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
711 B{C{lat2}} and B{C{lon2}} (C{bool}).
713 @return: Surface area, I{signed} (I{square} C{meter} or the same units as
714 B{C{radius}} I{squared}) or the I{spherical excess} (C{radians})
715 if C{B{radius}=0} or C{None}.
717 @raise TypeError: Invalid B{C{radius}}.
719 @raise UnitError: Invalid B{C{lat2}} or B{C{lat1}}.
721 @raise ValueError: Semi-circular longitudinal delta.
723 @see: Functions L{excessKarney_} and L{excessQuad}.
724 '''
725 return _eA(excessKarney_, radius, wrap, lat1, lon1, lat2, lon2)
728def excessKarney_(phi2, phi1, lam21):
729 '''Compute the I{spherical excess} C{E} of a (spherical) quadrilateral bounded by
730 a segment of a great circle, two meridians and the equator using U{Karney's
731 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>}
732 method.
734 @arg phi2: End latitude (C{radians}).
735 @arg phi1: Start latitude (C{radians}).
736 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
738 @return: Spherical excess, I{signed} (C{radians}).
740 @raise ValueError: Semi-circular longitudinal delta B{C{lam21}}.
742 @see: Function L{excessKarney} and U{Area of a spherical polygon
743 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>}.
744 '''
745 # from: Veness <https://www.Movable-Type.co.UK/scripts/latlong.html> Area
746 # method due to Karney: for each edge of the polygon,
747 #
748 # tan(Δλ / 2) · (tan(φ1 / 2) + tan(φ2 / 2))
749 # tan(E / 2) = -----------------------------------------
750 # 1 + tan(φ1 / 2) · tan(φ2 / 2)
751 #
752 # where E is the spherical excess of the trapezium obtained by extending
753 # the edge to the equator-circle vector for each edge (see also ***).
754 _t = tan_2
755 t2 = _t(phi2)
756 t1 = _t(phi1)
757 t = _t(lam21, lam21=None)
758 return Radians(Karney=atan2(t * (t1 + t2),
759 _1_0 + (t1 * t2)) * _2_0)
762# ***) Original post no longer available, following is a copy of the main part
763# <http://OSGeo-org.1560.x6.Nabble.com/Area-of-a-spherical-polygon-td3841625.html>
764#
765# The area of a polygon on a (unit) sphere is given by the spherical excess
766#
767# A = 2 * pi - sum(exterior angles)
768#
769# However this is badly conditioned if the polygon is small. In this case, use
770#
771# A = sum(S12{i, i+1}) over the edges of the polygon
772#
773# where S12 is the area of the quadrilateral bounded by an edge of the polygon,
774# two meridians and the equator, i.e. with vertices (phi1, lambda1), (phi2,
775# lambda2), (0, lambda1) and (0, lambda2). S12 is given by
776#
777# tan(S12 / 2) = tan(lambda21 / 2) * (tan(phi1 / 2) + tan(phi2 / 2)) /
778# (tan(phi1 / 2) * tan(phi2 / 2) + 1)
779#
780# = tan(lambda21 / 2) * tanh((Lamb(phi1) + Lamb(phi2)) / 2)
781#
782# where lambda21 = lambda2 - lambda1 and Lamb(x) is the Lambertian (or the
783# inverse Gudermannian) function
784#
785# Lambertian(x) = asinh(tan(x)) = atanh(sin(x)) = 2 * atanh(tan(x / 2))
786#
787# Notes: The formula for S12 is exact, except that...
788# - it is indeterminate if an edge is a semi-circle
789# - the formula for A applies only if the polygon does not include a pole
790# (if it does, then add +/- 2 * pi to the result)
791# - in the limit of small phi and lambda, S12 reduces to the trapezoidal
792# formula, S12 = (lambda2 - lambda1) * (phi1 + phi2) / 2
793# - I derived this result from the equation for the area of a spherical
794# triangle in terms of two edges and the included angle given by, e.g.
795# U{Todhunter, I. - Spherical Trigonometry (1871), Sec. 103, Eq. (2)
796# <http://Books.Google.com/books?id=3uBHAAAAIAAJ&pg=PA71>}
797# - I would be interested to know if this formula for S12 is already known
798# - Charles Karney
801def excessQuad(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
802 '''Compute the surface area of a (spherical) quadrilateral bounded by a segment
803 of a great circle, two meridians and the equator.
805 @arg lat1: Start latitude (C{degrees}).
806 @arg lon1: Start longitude (C{degrees}).
807 @arg lat2: End latitude (C{degrees}).
808 @arg lon2: End longitude (C{degrees}).
809 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
810 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
811 L{a_f2Tuple}) or C{None}.
812 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
813 B{C{lat2}} and B{C{lon2}} (C{bool}).
815 @return: Surface area, I{signed} (I{square} C{meter} or the same units as
816 B{C{radius}} I{squared}) or the I{spherical excess} (C{radians})
817 if C{B{radius}=0} or C{None}.
819 @raise TypeError: Invalid B{C{radius}}.
821 @raise UnitError: Invalid B{C{lat2}} or B{C{lat1}}.
823 @see: Function L{excessQuad_} and L{excessKarney}.
824 '''
825 return _eA(excessQuad_, radius, wrap, lat1, lon1, lat2, lon2)
828def excessQuad_(phi2, phi1, lam21):
829 '''Compute the I{spherical excess} C{E} of a (spherical) quadrilateral bounded
830 by a segment of a great circle, two meridians and the equator.
832 @arg phi2: End latitude (C{radians}).
833 @arg phi1: Start latitude (C{radians}).
834 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
836 @return: Spherical excess, I{signed} (C{radians}).
838 @see: Function L{excessQuad} and U{Spherical trigonometry
839 <https://WikiPedia.org/wiki/Spherical_trigonometry>}.
840 '''
841 s = sin((phi2 + phi1) * _0_5)
842 c = cos((phi2 - phi1) * _0_5)
843 return Radians(Quad=atan2(tan_2(lam21) * s, c) * _2_0)
846def flatLocal(lat1, lon1, lat2, lon2, datum=_WGS84, scaled=True, wrap=False):
847 '''Compute the distance between two (ellipsoidal) points using
848 the U{ellipsoidal Earth to plane projection<https://WikiPedia.org/
849 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>}
850 aka U{Hubeny<https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula.
852 @arg lat1: Start latitude (C{degrees}).
853 @arg lon1: Start longitude (C{degrees}).
854 @arg lat2: End latitude (C{degrees}).
855 @arg lon2: End longitude (C{degrees}).
856 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
857 L{Ellipsoid2} or L{a_f2Tuple}) to use.
858 @kwarg scaled: Scale prime_vertical by C{cos(B{phi})} (C{bool}),
859 see method L{pygeodesy.Ellipsoid.roc2_}.
860 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
861 B{C{lat2}} and B{C{lon2}} (C{bool}).
863 @return: Distance (C{meter}, same units as the B{C{datum}}'s
864 ellipsoid axes).
866 @raise TypeError: Invalid B{C{datum}}.
868 @note: The meridional and prime_vertical radii of curvature
869 are taken and scaled at the mean of both latitude.
871 @see: Functions L{flatLocal_} or L{hubeny_}, L{cosineLaw}, L{flatPolar},
872 L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
873 L{equirectangular}, L{euclidean}, L{haversine}, L{thomas},
874 L{vincentys}, method L{Ellipsoid.distance2} and U{local, flat
875 earth approximation<https://www.EdWilliams.org/avform.htm#flat>}.
876 '''
877 E = _ellipsoidal(datum, flatLocal)
878 return E._hubeny_2(*_d3(wrap, lat1, lon1, lat2, lon2),
879 scaled=scaled, squared=False) * E.a
881hubeny = flatLocal # PYCHOK for Karl Hubeny
884def flatLocal_(phi2, phi1, lam21, datum=_WGS84, scaled=True):
885 '''Compute the I{angular} distance between two (ellipsoidal) points using
886 the U{ellipsoidal Earth to plane projection<https://WikiPedia.org/
887 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>}
888 aka U{Hubeny<https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula.
890 @arg phi2: End latitude (C{radians}).
891 @arg phi1: Start latitude (C{radians}).
892 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
893 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
894 L{Ellipsoid2} or L{a_f2Tuple}) to use.
895 @kwarg scaled: Scale prime_vertical by C{cos(B{phi})} (C{bool}),
896 see method L{pygeodesy.Ellipsoid.roc2_}.
898 @return: Angular distance (C{radians}).
900 @raise TypeError: Invalid B{C{datum}}.
902 @note: The meridional and prime_vertical radii of curvature
903 are taken and scaled I{at the mean of both latitude}.
905 @see: Functions L{flatLocal} or L{hubeny}, L{cosineAndoyerLambert_},
906 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{flatPolar_},
907 L{euclidean_}, L{haversine_}, L{thomas_} and L{vincentys_} and
908 U{local, flat earth approximation
909 <https://www.EdWilliams.org/avform.htm#flat>}.
910 '''
911 E = _ellipsoidal(datum, flatLocal_)
912 return E._hubeny_2(phi2, phi1, lam21, scaled=scaled, squared=False)
914hubeny_ = flatLocal_ # PYCHOK for Karl Hubeny
917def flatPolar(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
918 '''Compute the distance between two (spherical) points using
919 the U{polar coordinate flat-Earth <https://WikiPedia.org/wiki/
920 Geographical_distance#Polar_coordinate_flat-Earth_formula>}
921 formula.
923 @arg lat1: Start latitude (C{degrees}).
924 @arg lon1: Start longitude (C{degrees}).
925 @arg lat2: End latitude (C{degrees}).
926 @arg lon2: End longitude (C{degrees}).
927 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
928 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
929 L{a_f2Tuple}) to use.
930 @kwarg wrap: If C{True}, wrap or I{normalize} and B{C{lat2}}
931 and B{C{lon2}} (C{bool}).
933 @return: Distance (C{meter}, same units as B{C{radius}} or the
934 ellipsoid or datum axes).
936 @raise TypeError: Invalid B{C{radius}}.
938 @see: Functions L{flatPolar_}, L{cosineAndoyerLambert},
939 L{cosineForsytheAndoyerLambert},L{cosineLaw},
940 L{flatLocal}/L{hubeny}, L{equirectangular},
941 L{euclidean}, L{haversine}, L{thomas} and
942 L{vincentys}.
943 '''
944 return _dS(flatPolar_, radius, wrap, lat1, lon1, lat2, lon2)
947def flatPolar_(phi2, phi1, lam21):
948 '''Compute the I{angular} distance between two (spherical) points
949 using the U{polar coordinate flat-Earth<https://WikiPedia.org/wiki/
950 Geographical_distance#Polar_coordinate_flat-Earth_formula>}
951 formula.
953 @arg phi2: End latitude (C{radians}).
954 @arg phi1: Start latitude (C{radians}).
955 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
957 @return: Angular distance (C{radians}).
959 @see: Functions L{flatPolar}, L{cosineAndoyerLambert_},
960 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
961 L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{haversine_},
962 L{thomas_} and L{vincentys_}.
963 '''
964 a = fabs(PI_2 - phi1) # co-latitude
965 b = fabs(PI_2 - phi2) # co-latitude
966 if a < b:
967 a, b = b, a
968 if a < EPS0:
969 a = _0_0
970 elif b > 0:
971 b = b / a # /= chokes PyChecker
972 c = b * cos(lam21) * _2_0
973 c = fsumf_(_1_0, b**2, -fabs(c))
974 a *= sqrt0(c)
975 return a
978def _hartzell(pov, los, earth, **kwds):
979 '''(INTERNAL) Helper for C{CartesianBase.hartzell} and C{LatLonBase.hartzell}.
980 '''
981 if earth is None:
982 earth = pov.datum
983 else:
984 earth = _spherical_datum(earth, name__=hartzell)
985 pov = pov.toDatum(earth)
986 h = pov.height
987 if h < 0: # EPS0
988 t = _SPACE_(Fmt.PARENSPACED(height=h), _inside_)
989 raise IntersectionError(pov=pov, earth=earth, txt=t)
990 return hartzell(pov, los=los, earth=earth, **kwds) if h > 0 else pov # EPS0
993def hartzell(pov, los=False, earth=_WGS84, **name_LatLon_and_kwds):
994 '''Compute the intersection of the earth's surface and a Line-Of-Sight from
995 a Point-Of-View in space.
997 @arg pov: Point-Of-View outside the earth (C{LatLon}, C{Cartesian},
998 L{Ecef9Tuple} or L{Vector3d}).
999 @kwarg los: Line-Of-Sight, I{direction} to earth (L{Los}, L{Vector3d}),
1000 C{True} for the I{normal, plumb} onto the surface or C{False}
1001 or C{None} to point to the center of the earth.
1002 @kwarg earth: The earth model (L{Datum}, L{Ellipsoid}, L{Ellipsoid2},
1003 L{a_f2Tuple} or a C{scalar} earth radius in C{meter}).
1004 @kwarg name_LatLon_and_kwds: Optional, overriding C{B{name}="hartzell"}
1005 (C{str}), class C{B{LatLon}=None} to return the intersection
1006 plus additional C{LatLon} keyword arguments, include the
1007 B{C{datum}} if different and to convert from B{C{earth}}.
1009 @return: The intersection (L{Vector3d}, B{C{pov}}'s C{cartesian type} or the
1010 given B{C{LatLon}} instance) with attribute C{height} set to the
1011 distance to the B{C{pov}}.
1013 @raise IntersectionError: Invalid B{C{pov}} or B{C{pov}} inside the earth or
1014 invalid B{C{los}} or B{C{los}} points outside or
1015 away from the earth.
1017 @raise TypeError: Invalid B{C{earth}}, C{ellipsoid} or C{datum}.
1019 @see: Class L{Los}, functions L{tyr3d} and L{hartzell4} and methods
1020 L{Ellipsoid.hartzell4} and any C{Cartesian.hartzell} and C{LatLon.hartzell}.
1021 '''
1022 n, LatLon_and_kwds = _name2__(name_LatLon_and_kwds, name__=hartzell)
1023 try:
1024 D = _spherical_datum(earth, name__=hartzell)
1025 r, h, i = _MODS.triaxials._hartzell3(pov, los, D.ellipsoid._triaxial)
1027 C = _MODS.cartesianBase.CartesianBase
1028 if LatLon_and_kwds:
1029 c = C(r, datum=D)
1030 r = c.toLatLon(**_xkwds(LatLon_and_kwds, height=h))
1031 elif isinstance(r, C):
1032 r.height = h
1033 if i:
1034 r._iteration = i
1035 except Exception as x:
1036 raise IntersectionError(pov=pov, los=los, earth=earth, cause=x,
1037 **LatLon_and_kwds)
1038 return _xnamed(r, n) if n else r
1041def haversine(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
1042 '''Compute the distance between two (spherical) points using the
1043 U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>} formula.
1045 @arg lat1: Start latitude (C{degrees}).
1046 @arg lon1: Start longitude (C{degrees}).
1047 @arg lat2: End latitude (C{degrees}).
1048 @arg lon2: End longitude (C{degrees}).
1049 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum}) or ellipsoid
1050 (L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}) to use.
1051 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} and
1052 B{C{lon2}} (C{bool}).
1054 @return: Distance (C{meter}, same units as B{C{radius}}).
1056 @raise TypeError: Invalid B{C{radius}}.
1058 @see: U{Distance between two (spherical) points
1059 <https://www.EdWilliams.org/avform.htm#Dist>}, functions
1060 L{cosineLaw}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
1061 L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar},
1062 L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2},
1063 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
1065 @note: See note at function L{vincentys_}.
1066 '''
1067 return _dS(haversine_, radius, wrap, lat1, lon1, lat2, lon2)
1070def haversine_(phi2, phi1, lam21):
1071 '''Compute the I{angular} distance between two (spherical) points using the
1072 U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>} formula.
1074 @arg phi2: End latitude (C{radians}).
1075 @arg phi1: Start latitude (C{radians}).
1076 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
1078 @return: Angular distance (C{radians}).
1080 @see: Functions L{haversine}, L{cosineAndoyerLambert_},
1081 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
1082 L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_},
1083 L{thomas_} and L{vincentys_}.
1085 @note: See note at function L{vincentys_}.
1086 '''
1087 def _hsin(rad):
1088 return sin(rad * _0_5)**2
1090 h = _hsin(phi2 - phi1) + cos(phi1) * cos(phi2) * _hsin(lam21) # haversine
1091 return atan2(sqrt0(h), sqrt0(_1_0 - h)) * _2_0 # == asin(sqrt(h)) * 2
1094def heightOf(angle, distance, radius=R_M):
1095 '''Determine the height above the (spherical) earth' surface after
1096 traveling along a straight line at a given tilt.
1098 @arg angle: Tilt angle above horizontal (C{degrees}).
1099 @arg distance: Distance along the line (C{meter} or same units as
1100 B{C{radius}}).
1101 @kwarg radius: Optional mean earth radius (C{meter}).
1103 @return: Height (C{meter}, same units as B{C{distance}} and B{C{radius}}).
1105 @raise ValueError: Invalid B{C{angle}}, B{C{distance}} or B{C{radius}}.
1107 @see: U{MultiDop geog_lib.GeogBeamHt<https://GitHub.com/NASA/MultiDop>}
1108 (U{Shapiro et al. 2009, JTECH
1109 <https://Journals.AMetSoc.org/doi/abs/10.1175/2009JTECHA1256.1>}
1110 and U{Potvin et al. 2012, JTECH
1111 <https://Journals.AMetSoc.org/doi/abs/10.1175/JTECH-D-11-00019.1>}).
1112 '''
1113 r = h = Radius(radius)
1114 d = fabs(Distance(distance))
1115 if d > h:
1116 d, h = h, d
1118 if d > EPS0: # and h > EPS0
1119 d = d / h # /= h chokes PyChecker
1120 s = sin(Phid(angle=angle, clip=_180_0))
1121 s = fsumf_(_1_0, s * d * _2_0, d**2)
1122 if s > 0:
1123 return h * sqrt(s) - r
1125 raise _ValueError(angle=angle, distance=distance, radius=radius)
1128def heightOrthometric(h_ll, N):
1129 '''Get the I{orthometric} height B{H}, the height above the geoid, earth surface.
1131 @arg h_ll: The height above the ellipsoid (C{meter}) or an I{ellipsoidal}
1132 location (C{LatLon} with a C{height} or C{h} attribute).
1133 @arg N: The I{geoid} height (C{meter}), the height of the geoid above the
1134 ellipsoid at the same B{C{h_ll}} location.
1136 @return: I{Orthometric} height C{B{H} = B{h} - B{N}} (C{meter}, same units
1137 as B{C{h}} and B{C{N}}).
1139 @see: U{Ellipsoid, Geoid, and Othometric Heights<https://www.NGS.NOAA.gov/
1140 GEOID/PRESENTATIONS/2007_02_24_CCPS/Roman_A_PLSC2007notes.pdf>}, page
1141 6 and module L{pygeodesy.geoids}.
1142 '''
1143 h = h_ll if _isHeight(h_ll) else _xattr(h_ll, height=_xattr(h_ll, h=0))
1144 return Height(H=Height(h=h) - Height(N=N))
1147def horizon(height, radius=R_M, refraction=False):
1148 '''Determine the distance to the horizon from a given altitude above the
1149 (spherical) earth.
1151 @arg height: Altitude (C{meter} or same units as B{C{radius}}).
1152 @kwarg radius: Optional mean earth radius (C{meter}).
1153 @kwarg refraction: Consider atmospheric refraction (C{bool}).
1155 @return: Distance (C{meter}, same units as B{C{height}} and B{C{radius}}).
1157 @raise ValueError: Invalid B{C{height}} or B{C{radius}}.
1159 @see: U{Distance to horizon<https://www.EdWilliams.org/avform.htm#Horizon>}.
1160 '''
1161 h, r = Height(height), Radius(radius)
1162 if min(h, r) < 0:
1163 raise _ValueError(height=height, radius=radius)
1165 d2 = ((r * 2.415750694528) if refraction else # 2.0 / 0.8279
1166 fsumf_(r, r, h)) * h
1167 return sqrt0(d2)
1170class _idllmn6(object): # see also .geodesicw._wargs, .latlonBase._toCartesian3, .vector2d._numpy
1171 '''(INTERNAL) Helper for C{intersection2} and C{intersections2}.
1172 '''
1173 @contextmanager # <https://www.Python.org/dev/peps/pep-0343/> Examples
1174 def __call__(self, datum, lat1, lon1, lat2, lon2, small, wrap, s, **kwds):
1175 try:
1176 if wrap:
1177 _, lat2, lon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap)
1178 kwds = _xkwds(kwds, wrap=wrap) # for _xError
1179 m = small if small is _100km else Meter_(small=small)
1180 n = _DUNDER_nameof(intersections2 if s else intersection2)
1181 if datum is None or euclidean(lat1, lon1, lat2, lon2) < m:
1182 d, m = None, _MODS.vector3d
1183 _i = m._intersects2 if s else m._intersect3d3
1184 elif _isRadius(datum) and datum < 0 and not s:
1185 d = _spherical_datum(-datum, name=n)
1186 m = _MODS.sphericalNvector
1187 _i = m.intersection
1188 else:
1189 d = _spherical_datum(datum, name=n)
1190 if d.isSpherical:
1191 m = _MODS.sphericalTrigonometry
1192 _i = m._intersects2 if s else m._intersect
1193 elif d.isEllipsoidal:
1194 try:
1195 if d.ellipsoid.geodesic:
1196 pass
1197 m = _MODS.ellipsoidalKarney
1198 except ImportError:
1199 m = _MODS.ellipsoidalExact
1200 _i = m._intersections2 if s else m._intersection3 # ellipsoidalBaseDI
1201 else:
1202 raise _TypeError(datum=datum)
1203 yield _i, d, lat2, lon2, m, n
1205 except (TypeError, ValueError) as x:
1206 raise _xError(x, lat1=lat1, lon1=lon1, datum=datum,
1207 lat2=lat2, lon2=lon2, small=small, **kwds)
1209_idllmn6 = _idllmn6() # PYCHOK singleton
1212def intersection2(lat1, lon1, bearing1,
1213 lat2, lon2, bearing2, datum=None, wrap=False, small=_100km): # was=True
1214 '''I{Conveniently} compute the intersection of two lines each defined
1215 by a (geodetic) point and a bearing from North, using either ...
1217 1) L{vector3d.intersection3d3} for B{C{small}} distances (below 100 Km
1218 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ...
1220 2) L{sphericalTrigonometry.intersection} for a spherical B{C{datum}}
1221 or a C{scalar B{datum}} representing the earth radius, conventionally
1222 in C{meter} or ...
1224 3) L{sphericalNvector.intersection} if B{C{datum}} is a I{negative}
1225 C{scalar}, (negative) earth radius, conventionally in C{meter} or ...
1227 4) L{ellipsoidalKarney.intersection3} for an ellipsoidal B{C{datum}}
1228 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>}
1229 is installed, otherwise ...
1231 5) L{ellipsoidalExact.intersection3}, provided B{C{datum}} is ellipsoidal.
1233 @arg lat1: Latitude of the first point (C{degrees}).
1234 @arg lon1: Longitude of the first point (C{degrees}).
1235 @arg bearing1: Bearing at the first point (compass C{degrees}).
1236 @arg lat2: Latitude of the second point (C{degrees}).
1237 @arg lon2: Longitude of the second point (C{degrees}).
1238 @arg bearing2: Bearing at the second point (compass C{degrees}).
1239 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid},
1240 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth
1241 radius (C{meter}, same units as B{C{radius1}} and
1242 B{C{radius2}}) or C{None}.
1243 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
1244 and B{C{lon2}} (C{bool}).
1245 @kwarg small: Upper limit for small distances (C{meter}).
1247 @return: A L{LatLon2Tuple}C{(lat, lon)} with the lat- and
1248 longitude of the intersection point.
1250 @raise IntersectionError: Ambiguous or infinite intersection
1251 or colinear, parallel or otherwise
1252 non-intersecting lines.
1254 @raise TypeError: Invalid B{C{datum}}.
1256 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{bearing1}},
1257 B{C{lat2}}, B{C{lon2}} or B{C{bearing2}}.
1259 @see: Method L{RhumbLine.intersection2}.
1261 @note: The returned intersections may be near-antipodal.
1262 '''
1263 b1 = Bearing(bearing1=bearing1)
1264 b2 = Bearing(bearing2=bearing2)
1265 with _idllmn6(datum, lat1, lon1, lat2, lon2,
1266 small, wrap, False, bearing1=b1, bearing2=b2) as t:
1267 _i, d, lat2, lon2, m, n = t
1268 if d is None:
1269 t, _, _ = _i(m.Vector3d(lon1, lat1, 0), b1,
1270 m.Vector3d(lon2, lat2, 0), b2, useZ=False)
1271 t = LatLon2Tuple(t.y, t.x, name=n)
1273 else:
1274 t = _i(m.LatLon(lat1, lon1, datum=d), b1,
1275 m.LatLon(lat2, lon2, datum=d), b2,
1276 LatLon=None, height=0, wrap=False)
1277 if isinstance(t, Intersection3Tuple): # ellipsoidal
1278 t, _, _ = t
1279 t = LatLon2Tuple(t.lat, t.lon, name=n)
1280 return t
1283def intersections2(lat1, lon1, radius1,
1284 lat2, lon2, radius2, datum=None, wrap=False, small=_100km): # was=True
1285 '''I{Conveniently} compute the intersections of two circles each defined
1286 by a (geodetic) center point and a radius, using either ...
1288 1) L{vector3d.intersections2} for B{C{small}} distances (below 100 Km
1289 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ...
1291 2) L{sphericalTrigonometry.intersections2} for a spherical B{C{datum}}
1292 or a C{scalar B{datum}} representing the earth radius, conventionally
1293 in C{meter} or ...
1295 3) L{ellipsoidalKarney.intersections2} for an ellipsoidal B{C{datum}}
1296 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>}
1297 is installed, otherwise ...
1299 4) L{ellipsoidalExact.intersections2}, provided B{C{datum}} is ellipsoidal.
1301 @arg lat1: Latitude of the first circle center (C{degrees}).
1302 @arg lon1: Longitude of the first circle center (C{degrees}).
1303 @arg radius1: Radius of the first circle (C{meter}, conventionally).
1304 @arg lat2: Latitude of the second circle center (C{degrees}).
1305 @arg lon2: Longitude of the second circle center (C{degrees}).
1306 @arg radius2: Radius of the second circle (C{meter}, same units as B{C{radius1}}).
1307 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid},
1308 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth
1309 radius (C{meter}, same units as B{C{radius1}} and
1310 B{C{radius2}}) or C{None}.
1311 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
1312 and B{C{lon2}} (C{bool}).
1313 @kwarg small: Upper limit for small distances (C{meter}).
1315 @return: 2-Tuple of the intersection points, each a
1316 L{LatLon2Tuple}C{(lat, lon)}. For abutting circles, the
1317 points are the same instance, aka the I{radical center}.
1319 @raise IntersectionError: Concentric, antipodal, invalid or
1320 non-intersecting circles or no
1321 convergence.
1323 @raise TypeError: Invalid B{C{datum}}.
1325 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{radius1}},
1326 B{C{lat2}}, B{C{lon2}} or B{C{radius2}}.
1327 '''
1328 r1 = Radius_(radius1=radius1)
1329 r2 = Radius_(radius2=radius2)
1330 with _idllmn6(datum, lat1, lon1, lat2, lon2,
1331 small, wrap, True, radius1=r1, radius2=r2) as t:
1332 _i, d, lat2, lon2, m, n = t
1333 if d is None:
1334 r1 = m2degrees(r1, radius=R_M, lat=lat1)
1335 r2 = m2degrees(r2, radius=R_M, lat=lat2)
1337 def _V2T(x, y, _, **unused): # _ == z unused
1338 return LatLon2Tuple(y, x, name=n)
1340 t = _i(m.Vector3d(lon1, lat1, 0), r1,
1341 m.Vector3d(lon2, lat2, 0), r2, sphere=False,
1342 Vector=_V2T)
1343 else:
1344 def _LL2T(lat, lon, **unused):
1345 return LatLon2Tuple(lat, lon, name=n)
1347 t = _i(m.LatLon(lat1, lon1, datum=d), r1,
1348 m.LatLon(lat2, lon2, datum=d), r2,
1349 LatLon=_LL2T, height=0, wrap=False)
1350 return t
1353def isantipode(lat1, lon1, lat2, lon2, eps=EPS):
1354 '''Check whether two points are I{antipodal}, on diametrically
1355 opposite sides of the earth.
1357 @arg lat1: Latitude of one point (C{degrees}).
1358 @arg lon1: Longitude of one point (C{degrees}).
1359 @arg lat2: Latitude of the other point (C{degrees}).
1360 @arg lon2: Longitude of the other point (C{degrees}).
1361 @kwarg eps: Tolerance for near-equality (C{degrees}).
1363 @return: C{True} if points are antipodal within the
1364 B{C{eps}} tolerance, C{False} otherwise.
1366 @see: Functions L{isantipode_} and L{antipode}.
1367 '''
1368 return (fabs(lat1 + lat2) <= eps and
1369 fabs(lon1 + lon2) <= eps) or _isequalTo(
1370 normal(lat1, lon1), antipode(lat2, lon2), eps)
1373def isantipode_(phi1, lam1, phi2, lam2, eps=EPS):
1374 '''Check whether two points are I{antipodal}, on diametrically
1375 opposite sides of the earth.
1377 @arg phi1: Latitude of one point (C{radians}).
1378 @arg lam1: Longitude of one point (C{radians}).
1379 @arg phi2: Latitude of the other point (C{radians}).
1380 @arg lam2: Longitude of the other point (C{radians}).
1381 @kwarg eps: Tolerance for near-equality (C{radians}).
1383 @return: C{True} if points are antipodal within the
1384 B{C{eps}} tolerance, C{False} otherwise.
1386 @see: Functions L{isantipode} and L{antipode_}.
1387 '''
1388 return (fabs(phi1 + phi2) <= eps and
1389 fabs(lam1 + lam2) <= eps) or _isequalTo_(
1390 normal_(phi1, lam1), antipode_(phi2, lam2), eps)
1393def _isequalTo(p1, p2, eps=EPS):
1394 '''Compare 2 point lat-/lons ignoring C{class}.
1395 '''
1396 return (fabs(p1.lat - p2.lat) <= eps and
1397 fabs(p1.lon - p2.lon) <= eps) if eps else (p1.latlon == p2.latlon)
1400def _isequalTo_(p1, p2, eps=EPS):
1401 '''(INTERNAL) Compare 2 point phi-/lams ignoring C{class}.
1402 '''
1403 return (fabs(p1.phi - p2.phi) <= eps and
1404 fabs(p1.lam - p2.lam) <= eps) if eps else (p1.philam == p2.philam)
1407def isnormal(lat, lon, eps=0):
1408 '''Check whether B{C{lat}} I{and} B{C{lon}} are within their
1409 respective I{normal} range in C{degrees}.
1411 @arg lat: Latitude (C{degrees}).
1412 @arg lon: Longitude (C{degrees}).
1413 @kwarg eps: Optional tolerance C{degrees}).
1415 @return: C{True} if C{(abs(B{lat}) + B{eps}) <= 90} and
1416 C{(abs(B{lon}) + B{eps}) <= 180}, C{False} othwerwise.
1418 @see: Functions L{isnormal_} and L{normal}.
1419 '''
1420 return (_90_0 - fabs(lat)) >= eps and _loneg(fabs(lon)) >= eps
1423def isnormal_(phi, lam, eps=0):
1424 '''Check whether B{C{phi}} I{and} B{C{lam}} are within their
1425 respective I{normal} range in C{radians}.
1427 @arg phi: Latitude (C{radians}).
1428 @arg lam: Longitude (C{radians}).
1429 @kwarg eps: Optional tolerance C{radians}).
1431 @return: C{True} if C{(abs(B{phi}) + B{eps}) <= PI/2} and
1432 C{(abs(B{lam}) + B{eps}) <= PI}, C{False} othwerwise.
1434 @see: Functions L{isnormal} and L{normal_}.
1435 '''
1436 return (PI_2 - fabs(phi)) >= eps and (PI - fabs(lam)) >= eps
1439def _normal2(a, b, n_2, n, n2):
1440 '''(INTERNAL) Helper for C{normal} and C{normal_}.
1441 '''
1442 if fabs(b) > n:
1443 b = remainder(b, n2)
1444 if fabs(a) > n_2:
1445 r = remainder(a, n)
1446 if r != a:
1447 a = -r
1448 b -= n if b > 0 else -n
1449 return float0_(a, b)
1452def normal(lat, lon, **name):
1453 '''Normalize a lat- I{and} longitude pair in C{degrees}.
1455 @arg lat: Latitude (C{degrees}).
1456 @arg lon: Longitude (C{degrees}).
1457 @kwarg name: Optional, overriding C{B{name}="normal"} (C{str}).
1459 @return: L{LatLon2Tuple}C{(lat, lon)} with C{abs(lat) <= 90}
1460 and C{abs(lon) <= 180}.
1462 @see: Functions L{normal_} and L{isnormal}.
1463 '''
1464 return LatLon2Tuple(*_normal2(lat, lon, _90_0, _180_0, _360_0),
1465 name=_name__(name, name__=normal))
1468def normal_(phi, lam, **name):
1469 '''Normalize a lat- I{and} longitude pair in C{radians}.
1471 @arg phi: Latitude (C{radians}).
1472 @arg lam: Longitude (C{radians}).
1473 @kwarg name: Optional, overriding C{B{name}="normal_"} (C{str}).
1475 @return: L{PhiLam2Tuple}C{(phi, lam)} with C{abs(phi) <= PI/2}
1476 and C{abs(lam) <= PI}.
1478 @see: Functions L{normal} and L{isnormal_}.
1479 '''
1480 return PhiLam2Tuple(*_normal2(phi, lam, PI_2, PI, PI2),
1481 name=_name__(name, name__=normal_))
1484def _opposes(d, m, n, n2):
1485 '''(INTERNAL) Helper for C{opposing} and C{opposing_}.
1486 '''
1487 d = d % n2 # -20 % 360 == 340, -1 % PI2 == PI2 - 1
1488 return False if d < m or d > (n2 - m) else (
1489 True if (n - m) < d < (n + m) else None)
1492def opposing(bearing1, bearing2, margin=_90_0):
1493 '''Compare the direction of two bearings given in C{degrees}.
1495 @arg bearing1: First bearing (compass C{degrees}).
1496 @arg bearing2: Second bearing (compass C{degrees}).
1497 @kwarg margin: Optional, interior angle bracket (C{degrees}).
1499 @return: C{True} if both bearings point in opposite, C{False} if
1500 in similar or C{None} if in I{perpendicular} directions.
1502 @see: Function L{opposing_}.
1503 '''
1504 m = Degrees_(margin=margin, low=EPS0, high=_90_0)
1505 return _opposes(bearing2 - bearing1, m, _180_0, _360_0)
1508def opposing_(radians1, radians2, margin=PI_2):
1509 '''Compare the direction of two bearings given in C{radians}.
1511 @arg radians1: First bearing (C{radians}).
1512 @arg radians2: Second bearing (C{radians}).
1513 @kwarg margin: Optional, interior angle bracket (C{radians}).
1515 @return: C{True} if both bearings point in opposite, C{False} if
1516 in similar or C{None} if in perpendicular directions.
1518 @see: Function L{opposing}.
1519 '''
1520 m = Radians_(margin=margin, low=EPS0, high=PI_2)
1521 return _opposes(radians2 - radians1, m, PI, PI2)
1524def _Propy(func, nargs, kwds):
1525 '''(INTERNAL) Helper for the C{frechet.[-]Frechet**} and
1526 C{hausdorff.[-]Hausdorff*} classes.
1527 '''
1528 try:
1529 _xcallable(distance=func)
1530 # assert _args_kwds_count2(func)[0] == nargs + int(ismethod(func))
1531 _ = func(*_0_0s(nargs), **kwds)
1532 except Exception as x:
1533 t = unstr(func, **kwds)
1534 raise _TypeError(t, cause=x)
1535 return func
1538def _radical2(d, r1, r2, **name): # in .ellipsoidalBaseDI, .sphericalTrigonometry, .vector3d
1539 # (INTERNAL) See C{radical2} below
1540 # assert d > EPS0
1541 r = fsumf_(_1_0, (r1 / d)**2, -(r2 / d)**2) * _0_5
1542 n = _name__(name, name__=radical2)
1543 return Radical2Tuple(max(_0_0, min(_1_0, r)), r * d, name=n)
1546def radical2(distance, radius1, radius2, **name):
1547 '''Compute the I{radical ratio} and I{radical line} of two
1548 U{intersecting circles<https://MathWorld.Wolfram.com/
1549 Circle-CircleIntersection.html>}.
1551 The I{radical line} is perpendicular to the axis thru the
1552 centers of the circles at C{(0, 0)} and C{(B{distance}, 0)}.
1554 @arg distance: Distance between the circle centers (C{scalar}).
1555 @arg radius1: Radius of the first circle (C{scalar}).
1556 @arg radius2: Radius of the second circle (C{scalar}).
1557 @kwarg name: Optional C{B{name}=NN} (C{str}).
1559 @return: A L{Radical2Tuple}C{(ratio, xline)} where C{0.0 <=
1560 ratio <= 1.0} and C{xline} is along the B{C{distance}}.
1562 @raise IntersectionError: The B{C{distance}} exceeds the sum
1563 of B{C{radius1}} and B{C{radius2}}.
1565 @raise UnitError: Invalid B{C{distance}}, B{C{radius1}} or
1566 B{C{radius2}}.
1568 @see: U{Circle-Circle Intersection
1569 <https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>}.
1570 '''
1571 d = Distance_(distance, low=_0_0)
1572 r1 = Radius_(radius1=radius1)
1573 r2 = Radius_(radius2=radius2)
1574 if d > (r1 + r2):
1575 raise IntersectionError(distance=d, radius1=r1, radius2=r2,
1576 txt=_too_(_distant_))
1577 return _radical2(d, r1, r2, **name) if d > EPS0 else \
1578 Radical2Tuple(_0_5, _0_0, **name)
1581class Radical2Tuple(_NamedTuple):
1582 '''2-Tuple C{(ratio, xline)} of the I{radical} C{ratio} and
1583 I{radical} C{xline}, both C{scalar} and C{0.0 <= ratio <= 1.0}
1584 '''
1585 _Names_ = (_ratio_, _xline_)
1586 _Units_ = ( Scalar, Scalar)
1589def _radistance(inst):
1590 '''(INTERNAL) Helper for the L{frechet._FrechetMeterRadians}
1591 and L{hausdorff._HausdorffMeterRedians} classes.
1592 '''
1593 wrap_, kwds_ = _xkwds_pop2(inst._kwds, wrap=False)
1594 func_ = inst._func_
1595 try: # calling lower-overhead C{func_}
1596 func_(0, _0_25, _0_5, **kwds_)
1597 wrap_ = _Wrap._philamop(wrap_)
1598 except TypeError:
1599 return inst.distance
1601 def _philam(p):
1602 try:
1603 return p.phi, p.lam
1604 except AttributeError: # no .phi or .lam
1605 return radians(p.lat), radians(p.lon)
1607 def _func_wrap(point1, point2):
1608 phi1, lam1 = wrap_(*_philam(point1))
1609 phi2, lam2 = wrap_(*_philam(point2))
1610 return func_(phi2, phi1, lam2 - lam1, **kwds_)
1612 inst._units = inst._units_
1613 return _func_wrap
1616def _scale_deg(lat1, lat2): # degrees
1617 # scale factor cos(mean of lats) for delta lon
1618 m = fabs(lat1 + lat2) * _0_5
1619 return cos(radians(m)) if m < _90_0 else _0_0
1622def _scale_rad(phi1, phi2): # radians, by .frechet, .hausdorff, .heights
1623 # scale factor cos(mean of phis) for delta lam
1624 m = fabs(phi1 + phi2) * _0_5
1625 return cos(m) if m < PI_2 else _0_0
1628def _sincosa6(phi2, phi1, lam21): # [4] in cosineLaw
1629 '''(INTERNAL) C{sin}es, C{cos}ines and C{acos}ine.
1630 '''
1631 s2, c2, s1, c1, _, c21 = sincos2_(phi2, phi1, lam21)
1632 return s2, c2, s1, c1, acos1(s1 * s2 + c1 * c2 * c21), c21
1635def thomas(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):
1636 '''Compute the distance between two (ellipsoidal) points using
1637 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>}
1638 formula.
1640 @arg lat1: Start latitude (C{degrees}).
1641 @arg lon1: Start longitude (C{degrees}).
1642 @arg lat2: End latitude (C{degrees}).
1643 @arg lon2: End longitude (C{degrees}).
1644 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
1645 L{Ellipsoid2} or L{a_f2Tuple}) to use.
1646 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
1647 B{C{lat2}} and B{C{lon2}} (C{bool}).
1649 @return: Distance (C{meter}, same units as the B{C{datum}}'s
1650 ellipsoid axes).
1652 @raise TypeError: Invalid B{C{datum}}.
1654 @see: Functions L{thomas_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
1655 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny},
1656 L{flatPolar}, L{haversine}, L{vincentys} and method L{Ellipsoid.distance2}.
1657 '''
1658 return _dE(thomas_, datum, wrap, lat1, lon1, lat2, lon2)
1661def thomas_(phi2, phi1, lam21, datum=_WGS84):
1662 '''Compute the I{angular} distance between two (ellipsoidal) points using
1663 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>}
1664 formula.
1666 @arg phi2: End latitude (C{radians}).
1667 @arg phi1: Start latitude (C{radians}).
1668 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
1669 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid},
1670 L{Ellipsoid2} or L{a_f2Tuple}).
1672 @return: Angular distance (C{radians}).
1674 @raise TypeError: Invalid B{C{datum}}.
1676 @see: Functions L{thomas}, L{cosineAndoyerLambert_},
1677 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
1678 L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_},
1679 L{haversine_} and L{vincentys_} and U{Geodesy-PHP
1680 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/
1681 Distance/ThomasFormula.php>}.
1682 '''
1683 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21)
1684 if r and isnon0(c1) and isnon0(c2):
1685 E = _ellipsoidal(datum, thomas_)
1686 if E.f:
1687 r1 = atan2(E.b_a * s1, c1)
1688 r2 = atan2(E.b_a * s2, c2)
1690 j = (r2 + r1) * _0_5
1691 k = (r2 - r1) * _0_5
1692 sj, cj, sk, ck, h, _ = sincos2_(j, k, lam21 * _0_5)
1694 h = fsumf_(sk**2, (ck * h)**2, -(sj * h)**2)
1695 u = _1_0 - h
1696 if isnon0(u) and isnon0(h):
1697 r = atan(sqrt0(h / u)) * 2 # == acos(1 - 2 * h)
1698 sr, cr = sincos2(r)
1699 if isnon0(sr):
1700 u = 2 * (sj * ck)**2 / u
1701 h = 2 * (sk * cj)**2 / h
1702 x = u + h
1703 y = u - h
1705 s = r / sr
1706 e = 4 * s**2
1707 d = 2 * cr
1708 a = e * d
1709 b = 2 * r
1710 c = s - (a - d) * _0_5
1711 f = E.f * _0_25
1713 t = fsumf_(a * x, -b * y, c * x**2, -d * y**2, e * x * y)
1714 r -= fsumf_(s * x, -y, -t * f * _0_25) * f * sr
1715 return r
1718def vincentys(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
1719 '''Compute the distance between two (spherical) points using
1720 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>}
1721 spherical formula.
1723 @arg lat1: Start latitude (C{degrees}).
1724 @arg lon1: Start longitude (C{degrees}).
1725 @arg lat2: End latitude (C{degrees}).
1726 @arg lon2: End longitude (C{degrees}).
1727 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
1728 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
1729 L{a_f2Tuple}) to use.
1730 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
1731 B{C{lat2}} and B{C{lon2}} (C{bool}).
1733 @return: Distance (C{meter}, same units as B{C{radius}}).
1735 @raise UnitError: Invalid B{C{radius}}.
1737 @see: Functions L{vincentys_}, L{cosineAndoyerLambert},
1738 L{cosineForsytheAndoyerLambert},L{cosineLaw}, L{equirectangular},
1739 L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar},
1740 L{haversine} and L{thomas} and methods L{Ellipsoid.distance2},
1741 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
1743 @note: See note at function L{vincentys_}.
1744 '''
1745 return _dS(vincentys_, radius, wrap, lat1, lon1, lat2, lon2)
1748def vincentys_(phi2, phi1, lam21):
1749 '''Compute the I{angular} distance between two (spherical) points using
1750 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>}
1751 spherical formula.
1753 @arg phi2: End latitude (C{radians}).
1754 @arg phi1: Start latitude (C{radians}).
1755 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
1757 @return: Angular distance (C{radians}).
1759 @see: Functions L{vincentys}, L{cosineAndoyerLambert_},
1760 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
1761 L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_},
1762 L{haversine_} and L{thomas_}.
1764 @note: Functions L{vincentys_}, L{haversine_} and L{cosineLaw_}
1765 produce equivalent results, but L{vincentys_} is suitable
1766 for antipodal points and slightly more expensive (M{3 cos,
1767 3 sin, 1 hypot, 1 atan2, 6 mul, 2 add}) than L{haversine_}
1768 (M{2 cos, 2 sin, 2 sqrt, 1 atan2, 5 mul, 1 add}) and
1769 L{cosineLaw_} (M{3 cos, 3 sin, 1 acos, 3 mul, 1 add}).
1770 '''
1771 s1, c1, s2, c2, s21, c21 = sincos2_(phi1, phi2, lam21)
1773 c = c2 * c21
1774 x = s1 * s2 + c1 * c
1775 y = c1 * s2 - s1 * c
1776 return atan2(hypot(c2 * s21, y), x)
1778# **) MIT License
1779#
1780# Copyright (C) 2016-2025 -- mrJean1 at Gmail -- All Rights Reserved.
1781#
1782# Permission is hereby granted, free of charge, to any person obtaining a
1783# copy of this software and associated documentation files (the "Software"),
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1787# Software is furnished to do so, subject to the following conditions:
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1789# The above copyright notice and this permission notice shall be included
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