Coverage for pygeodesy/formy.py: 98%
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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
9from pygeodesy.constants import EPS, EPS0, EPS1, PI, PI2, PI3, PI_2, R_M, \
10 _umod_PI2, float0, isnon0, remainder, _0_0, \
11 _0_125, _0_25, _0_5, _1_0, _2_0, _N_2_0, \
12 _4_0, _32_0, _90_0, _180_0, _360_0
13from pygeodesy.datums import Datum, Ellipsoid, _ellipsoidal_datum, \
14 _mean_radius, _spherical_datum, _WGS84
15# from pygeodesy.ellipsoids import Ellipsoid # from .datums
16from pygeodesy.errors import _AssertionError, IntersectionError, \
17 LimitError, limiterrors, _ValueError
18from pygeodesy.fmath import Fdot, euclid, fdot, hypot, hypot2, sqrt0
19from pygeodesy.fsums import fsum_, unstr
20from pygeodesy.interns import NN, _distant_, _inside_, _near_, _null_, \
21 _opposite_, _outside_, _too_
22from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS
23from pygeodesy.named import _NamedTuple, _xnamed
24from pygeodesy.namedTuples import Bearing2Tuple, Distance4Tuple, \
25 LatLon2Tuple, PhiLam2Tuple, Vector3Tuple
26# from pygeodesy.streprs import unstr # from .fsums
27from pygeodesy.units import Degrees_, Distance, Distance_, Height, Lam_, Lat, \
28 Lon, Phi_, Radians, Radians_, Radius, Radius_, \
29 Scalar, _100km
30from pygeodesy.utily import acos1, atan2b, atan2d, degrees2m, m2degrees, tan_2, \
31 sincos2, sincos2_, sincos2d_, unroll180, unrollPI
33from math import atan, atan2, cos, degrees, fabs, radians, sin, sqrt # pow
35__all__ = _ALL_LAZY.formy
36__version__ = '23.04.05'
38_ratio_ = 'ratio'
39_xline_ = 'xline'
42def _anti2(a, b, n_2, n, n2):
43 '''(INTERNAL) Helper for C{antipode} and C{antipode_}.
44 '''
45 r = remainder(a, n) if fabs(a) > n_2 else a
46 if r == a:
47 r = -r
48 b += n
49 if fabs(b) > n:
50 b = remainder(b, n2)
51 return float0(r, b)
54def antipode(lat, lon, name=NN):
55 '''Return the antipode, the point diametrically opposite
56 to a given point in C{degrees}.
58 @arg lat: Latitude (C{degrees}).
59 @arg lon: Longitude (C{degrees}).
60 @kwarg name: Optional name (C{str}).
62 @return: A L{LatLon2Tuple}C{(lat, lon)}.
64 @see: Functions L{antipode_} and L{normal} and U{Geosphere
65 <https://CRAN.R-Project.org/web/packages/geosphere/geosphere.pdf>}.
66 '''
67 return LatLon2Tuple(*_anti2(lat, lon, _90_0, _180_0, _360_0), name=name)
70def antipode_(phi, lam, name=NN):
71 '''Return the antipode, the point diametrically opposite
72 to a given point in C{radians}.
74 @arg phi: Latitude (C{radians}).
75 @arg lam: Longitude (C{radians}).
76 @kwarg name: Optional name (C{str}).
78 @return: A L{PhiLam2Tuple}C{(phi, lam)}.
80 @see: Functions L{antipode} and L{normal_} and U{Geosphere
81 <https://CRAN.R-Project.org/web/packages/geosphere/geosphere.pdf>}.
82 '''
83 return PhiLam2Tuple(*_anti2(phi, lam, PI_2, PI, PI2), name=name)
86def _area_or_(excess_, lat1, lat2, radius, d_lon, unused):
87 '''(INTERNAL) Helper for area and spherical excess.
88 '''
89 r = excess_(Phi_(lat2=lat2),
90 Phi_(lat1=lat1), radians(d_lon))
91 if radius:
92 r *= _mean_radius(radius, lat1, lat2)**2
93 return r
96def bearing(lat1, lon1, lat2, lon2, **options):
97 '''Compute the initial or final bearing (forward or reverse
98 azimuth) between a (spherical) start and end point.
100 @arg lat1: Start latitude (C{degrees}).
101 @arg lon1: Start longitude (C{degrees}).
102 @arg lat2: End latitude (C{degrees}).
103 @arg lon2: End longitude (C{degrees}).
104 @kwarg options: Optional keyword arguments for function
105 L{pygeodesy.bearing_}.
107 @return: Initial or final bearing (compass C{degrees360}) or
108 zero if start and end point coincide.
109 '''
110 r = bearing_(Phi_(lat1=lat1), Lam_(lon1=lon1),
111 Phi_(lat2=lat2), Lam_(lon2=lon2), **options)
112 return degrees(r)
115def bearing_(phi1, lam1, phi2, lam2, final=False, wrap=False):
116 '''Compute the initial or final bearing (forward or reverse azimuth)
117 between a (spherical) start and end point.
119 @arg phi1: Start latitude (C{radians}).
120 @arg lam1: Start longitude (C{radians}).
121 @arg phi2: End latitude (C{radians}).
122 @arg lam2: End longitude (C{radians}).
123 @kwarg final: Return final bearing if C{True}, initial otherwise (C{bool}).
124 @kwarg wrap: Wrap and L{pygeodesy.unrollPI} longitudes (C{bool}).
126 @return: Initial or final bearing (compass C{radiansPI2}) or zero if start
127 and end point coincide.
129 @see: U{Bearing<https://www.Movable-Type.co.UK/scripts/latlong.html>}, U{Course
130 between two points<https://www.EdWilliams.org/avform147.htm#Crs>} and
131 U{Bearing Between Two Points<https://web.Archive.org/web/20020630205931/
132 https://MathForum.org/library/drmath/view/55417.html>}.
133 '''
134 if final: # swap plus PI
135 phi1, lam1, phi2, lam2 = phi2, lam2, phi1, lam1
136 r = PI3
137 else:
138 r = PI2
140 db, _ = unrollPI(lam1, lam2, wrap=wrap)
141 sa1, ca1, sa2, ca2, sdb, cdb = sincos2_(phi1, phi2, db)
143 x = ca1 * sa2 - sa1 * ca2 * cdb
144 y = sdb * ca2
145 return _umod_PI2(atan2(y, x) + r) # .utily.wrapPI2
148def _bearingTo2(p1, p2, wrap=False): # for points.ispolar, sphericalTrigonometry.areaOf
149 '''(INTERNAL) Compute initial and final bearing.
150 '''
151 try: # for LatLon_ and ellipsoidal LatLon
152 return p1.bearingTo2(p2, wrap=wrap)
153 except AttributeError:
154 pass
155 # XXX spherical version, OK for ellipsoidal ispolar?
156 a1, b1 = p1.philam
157 a2, b2 = p2.philam
158 return Bearing2Tuple(degrees(bearing_(a1, b1, a2, b2, final=False, wrap=wrap)),
159 degrees(bearing_(a1, b1, a2, b2, final=True, wrap=wrap)),
160 name=_bearingTo2.__name__)
163def compassAngle(lat1, lon1, lat2, lon2, adjust=True, wrap=False):
164 '''Return the angle from North for the direction vector
165 M{(lon2 - lon1, lat2 - lat1)} between two points.
167 Suitable only for short, not near-polar vectors up to a few hundred
168 Km or Miles. Use function L{pygeodesy.bearing} for longer vectors.
170 @arg lat1: From latitude (C{degrees}).
171 @arg lon1: From longitude (C{degrees}).
172 @arg lat2: To latitude (C{degrees}).
173 @arg lon2: To longitude (C{degrees}).
174 @kwarg adjust: Adjust the longitudinal delta by the cosine of the
175 mean latitude (C{bool}).
176 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).
178 @return: Compass angle from North (C{degrees360}).
180 @note: Courtesy of Martin Schultz.
182 @see: U{Local, flat earth approximation
183 <https://www.EdWilliams.org/avform.htm#flat>}.
184 '''
185 d_lon, _ = unroll180(lon1, lon2, wrap=wrap)
186 if adjust: # scale delta lon
187 d_lon *= _scale_deg(lat1, lat2)
188 return atan2b(d_lon, lat2 - lat1)
191def cosineAndoyerLambert(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):
192 '''Compute the distance between two (ellipsoidal) points using the
193 U{Andoyer-Lambert correction<https://navlib.net/wp-content/uploads/
194 2013/10/admiralty-manual-of-navigation-vol-1-1964-english501c.pdf>} of the
195 U{Law of Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>}
196 fromula.
198 @arg lat1: Start latitude (C{degrees}).
199 @arg lon1: Start longitude (C{degrees}).
200 @arg lat2: End latitude (C{degrees}).
201 @arg lon2: End longitude (C{degrees}).
202 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid},
203 L{Ellipsoid2} or L{a_f2Tuple}).
204 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).
206 @return: Distance (C{meter}, same units as the B{C{datum}}'s
207 ellipsoid axes or C{radians} if B{C{datum}} is C{None}).
209 @raise TypeError: Invalid B{C{datum}}.
211 @see: Functions L{cosineAndoyerLambert_}, L{cosineForsytheAndoyerLambert},
212 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny},
213 L{flatPolar}, L{haversine}, L{thomas} and L{vincentys} and method
214 L{Ellipsoid.distance2}.
215 '''
216 return _distanceToE(cosineAndoyerLambert_, lat1, lat2, datum,
217 *unroll180(lon1, lon2, wrap=wrap))
220def cosineAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84):
221 '''Compute the I{angular} distance between two (ellipsoidal) points using the
222 U{Andoyer-Lambert correction<https://navlib.net/wp-content/uploads/2013/10/
223 admiralty-manual-of-navigation-vol-1-1964-english501c.pdf>} of the U{Law
224 of Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>}
225 fromula.
227 @arg phi2: End latitude (C{radians}).
228 @arg phi1: Start latitude (C{radians}).
229 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
230 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid},
231 L{Ellipsoid2} or L{a_f2Tuple}).
233 @return: Angular distance (C{radians}).
235 @raise TypeError: Invalid B{C{datum}}.
237 @see: Functions L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert_},
238 L{cosineLaw_}, L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
239 L{flatPolar_}, L{haversine_}, L{thomas_} and L{vincentys_} and U{Geodesy-PHP
240 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/
241 Distance/AndoyerLambert.php>}.
242 '''
243 s2, c2, s1, c1, r, c21 = _sincosa6(phi2, phi1, lam21)
244 if isnon0(c1) and isnon0(c2):
245 E = _ellipsoidal(datum, cosineAndoyerLambert_)
246 if E.f: # ellipsoidal
247 r2 = atan2(E.b_a * s2, c2)
248 r1 = atan2(E.b_a * s1, c1)
249 s2, c2, s1, c1 = sincos2_(r2, r1)
250 r = acos1(s1 * s2 + c1 * c2 * c21)
251 if r:
252 sr, _, sr_2, cr_2 = sincos2_(r, r * _0_5)
253 if isnon0(sr_2) and isnon0(cr_2):
254 s = (sr + r) * ((s1 - s2) / sr_2)**2
255 c = (sr - r) * ((s1 + s2) / cr_2)**2
256 r += (c - s) * E.f * _0_125
257 return r
260def cosineForsytheAndoyerLambert(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):
261 '''Compute the distance between two (ellipsoidal) points using the
262 U{Forsythe-Andoyer-Lambert correction<https://www2.UNB.Ca/gge/Pubs/TR77.pdf>} of
263 the U{Law of Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>}
264 formula.
266 @arg lat1: Start latitude (C{degrees}).
267 @arg lon1: Start longitude (C{degrees}).
268 @arg lat2: End latitude (C{degrees}).
269 @arg lon2: End longitude (C{degrees}).
270 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid},
271 L{Ellipsoid2} or L{a_f2Tuple}).
272 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).
274 @return: Distance (C{meter}, same units as the B{C{datum}}'s
275 ellipsoid axes or C{radians} if B{C{datum}} is C{None}).
277 @raise TypeError: Invalid B{C{datum}}.
279 @see: Functions L{cosineForsytheAndoyerLambert_}, L{cosineAndoyerLambert},
280 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny},
281 L{flatPolar}, L{haversine}, L{thomas} and L{vincentys} and method
282 L{Ellipsoid.distance2}.
283 '''
284 return _distanceToE(cosineForsytheAndoyerLambert_, lat1, lat2, datum,
285 *unroll180(lon1, lon2, wrap=wrap))
288def cosineForsytheAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84):
289 '''Compute the I{angular} distance between two (ellipsoidal) points using the
290 U{Forsythe-Andoyer-Lambert correction<https://www2.UNB.Ca/gge/Pubs/TR77.pdf>} of
291 the U{Law of Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>}
292 formula.
294 @arg phi2: End latitude (C{radians}).
295 @arg phi1: Start latitude (C{radians}).
296 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
297 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid},
298 L{Ellipsoid2} or L{a_f2Tuple}).
300 @return: Angular distance (C{radians}).
302 @raise TypeError: Invalid B{C{datum}}.
304 @see: Functions L{cosineForsytheAndoyerLambert}, L{cosineAndoyerLambert_},
305 L{cosineLaw_}, L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
306 L{flatPolar_}, L{haversine_}, L{thomas_} and L{vincentys_} and U{Geodesy-PHP
307 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/
308 Distance/ForsytheCorrection.php>}.
309 '''
310 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21)
311 if r and isnon0(c1) and isnon0(c2):
312 E = _ellipsoidal(datum, cosineForsytheAndoyerLambert_)
313 if E.f: # ellipsoidal
314 sr, cr, s2r, _ = sincos2_(r, r * _2_0)
315 if isnon0(sr) and fabs(cr) < EPS1:
316 s = (s1 + s2)**2 / (1 + cr)
317 t = (s1 - s2)**2 / (1 - cr)
318 x = s + t
319 y = s - t
321 s = 8 * r**2 / sr
322 a = 64 * r + _2_0 * s * cr # 16 * r**2 / tan(r)
323 d = 48 * sr + s # 8 * r**2 / tan(r)
324 b = -2 * d
325 e = 30 * s2r
326 c = fsum_(30 * r, e * _0_5, s * cr) # 8 * r**2 / tan(r)
328 t = fsum_( a * x, b * y, -c * x**2, d * x * y, e * y**2)
329 r += fsum_(-r * x, 3 * y * sr, t * E.f / _32_0) * E.f * _0_25
330 return r
333def cosineLaw(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
334 '''Compute the distance between two points using the
335 U{spherical Law of Cosines
336 <https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>}
337 formula.
339 @arg lat1: Start latitude (C{degrees}).
340 @arg lon1: Start longitude (C{degrees}).
341 @arg lat2: End latitude (C{degrees}).
342 @arg lon2: End longitude (C{degrees}).
343 @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter},
344 L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}).
345 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).
347 @return: Distance (C{meter}, same units as B{C{radius}} or the
348 ellipsoid or datum axes).
350 @raise TypeError: Invalid B{C{radius}}.
352 @see: Functions L{cosineLaw_}, L{cosineAndoyerLambert},
353 L{cosineForsytheAndoyerLambert}, L{equirectangular}, L{euclidean},
354 L{flatLocal}/L{hubeny}, L{flatPolar}, L{haversine}, L{thomas} and
355 L{vincentys} and method L{Ellipsoid.distance2}.
357 @note: See note at function L{vincentys_}.
358 '''
359 return _distanceToS(cosineLaw_, lat1, lat2, radius,
360 *unroll180(lon1, lon2, wrap=wrap))
363def cosineLaw_(phi2, phi1, lam21):
364 '''Compute the I{angular} distance between two points using the
365 U{spherical Law of Cosines
366 <https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>}
367 formula.
369 @arg phi2: End latitude (C{radians}).
370 @arg phi1: Start latitude (C{radians}).
371 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
373 @return: Angular distance (C{radians}).
375 @see: Functions L{cosineLaw}, L{cosineAndoyerLambert_},
376 L{cosineForsytheAndoyerLambert_}, L{equirectangular_},
377 L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_},
378 L{haversine_}, L{thomas_} and L{vincentys_}.
380 @note: See note at function L{vincentys_}.
381 '''
382 return _sincosa6(phi2, phi1, lam21)[4]
385def _distanceToE(func_, lat1, lat2, earth, d_lon, unused):
386 '''(INTERNAL) Helper for ellipsoidal distances.
387 '''
388 E = _ellipsoidal(earth, func_)
389 r = func_(Phi_(lat2=lat2),
390 Phi_(lat1=lat1), radians(d_lon), datum=E)
391 return r * E.a
394def _distanceToS(func_, lat1, lat2, earth, d_lon, unused, **adjust):
395 '''(INTERNAL) Helper for spherical distances.
396 '''
397 r = func_(Phi_(lat2=lat2),
398 Phi_(lat1=lat1), radians(d_lon), **adjust)
399 return r * _mean_radius(earth, lat1, lat2)
402def _ellipsoidal(earth, where):
403 '''(INTERNAL) Helper for distances.
404 '''
405 return earth if isinstance(earth, Ellipsoid) else (
406 earth if isinstance(earth, Datum) else
407 _ellipsoidal_datum(earth, name=where.__name__)).ellipsoid # PYCHOK indent
410def equirectangular(lat1, lon1, lat2, lon2, radius=R_M, **options):
411 '''Compute the distance between two points using
412 the U{Equirectangular Approximation / Projection
413 <https://www.Movable-Type.co.UK/scripts/latlong.html#equirectangular>}.
415 @arg lat1: Start latitude (C{degrees}).
416 @arg lon1: Start longitude (C{degrees}).
417 @arg lat2: End latitude (C{degrees}).
418 @arg lon2: End longitude (C{degrees}).
419 @kwarg radius: Mean earth radius, ellipsoid or datum
420 (C{meter}, L{Ellipsoid}, L{Ellipsoid2},
421 L{Datum} or L{a_f2Tuple}).
422 @kwarg options: Optional keyword arguments for function
423 L{equirectangular_}.
425 @return: Distance (C{meter}, same units as B{C{radius}} or
426 the ellipsoid or datum axes).
428 @raise TypeError: Invalid B{C{radius}}.
430 @see: Function L{equirectangular_} for more details, the
431 available B{C{options}}, errors, restrictions and other,
432 approximate or accurate distance functions.
433 '''
434 d = sqrt(equirectangular_(Lat(lat1=lat1), Lon(lon1=lon1),
435 Lat(lat2=lat2), Lon(lon2=lon2),
436 **options).distance2) # PYCHOK 4 vs 2-3
437 return degrees2m(d, radius=_mean_radius(radius, lat1, lat2))
440def equirectangular_(lat1, lon1, lat2, lon2,
441 adjust=True, limit=45, wrap=False):
442 '''Compute the distance between two points using
443 the U{Equirectangular Approximation / Projection
444 <https://www.Movable-Type.co.UK/scripts/latlong.html#equirectangular>}.
446 This approximation is valid for short distance of several
447 hundred Km or Miles, see the B{C{limit}} keyword argument and
448 the L{LimitError}.
450 @arg lat1: Start latitude (C{degrees}).
451 @arg lon1: Start longitude (C{degrees}).
452 @arg lat2: End latitude (C{degrees}).
453 @arg lon2: End longitude (C{degrees}).
454 @kwarg adjust: Adjust the wrapped, unrolled longitudinal delta
455 by the cosine of the mean latitude (C{bool}).
456 @kwarg limit: Optional limit for lat- and longitudinal deltas
457 (C{degrees}) or C{None} or C{0} for unlimited.
458 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).
460 @return: A L{Distance4Tuple}C{(distance2, delta_lat, delta_lon,
461 unroll_lon2)}.
463 @raise LimitError: If the lat- and/or longitudinal delta exceeds the
464 B{C{-limit..+limit}} range and L{pygeodesy.limiterrors}
465 set to C{True}.
467 @see: U{Local, flat earth approximation
468 <https://www.EdWilliams.org/avform.htm#flat>}, functions
469 L{equirectangular}, L{cosineAndoyerLambert},
470 L{cosineForsytheAndoyerLambert}, L{cosineLaw}, L{euclidean},
471 L{flatLocal}/L{hubeny}, L{flatPolar}, L{haversine}, L{thomas}
472 and L{vincentys} and methods L{Ellipsoid.distance2},
473 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
474 '''
475 d_lat = lat2 - lat1
476 d_lon, ulon2 = unroll180(lon1, lon2, wrap=wrap)
478 if limit and limit > 0 and limiterrors() and (fabs(d_lat) > limit or
479 fabs(d_lon) > limit):
480 t = unstr(equirectangular_, lat1, lon1, lat2, lon2, limit=limit)
481 raise LimitError('delta exceeds limit', txt=t)
483 if adjust: # scale delta lon
484 d_lon *= _scale_deg(lat1, lat2)
486 d2 = hypot2(d_lat, d_lon) # degrees squared!
487 return Distance4Tuple(d2, d_lat, d_lon, ulon2 - lon2)
490def euclidean(lat1, lon1, lat2, lon2, radius=R_M, adjust=True, wrap=False):
491 '''Approximate the C{Euclidean} distance between two (spherical) points.
493 @arg lat1: Start latitude (C{degrees}).
494 @arg lon1: Start longitude (C{degrees}).
495 @arg lat2: End latitude (C{degrees}).
496 @arg lon2: End longitude (C{degrees}).
497 @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter},
498 L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}).
499 @kwarg adjust: Adjust the longitudinal delta by the cosine of the
500 mean latitude (C{bool}).
501 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).
503 @return: Distance (C{meter}, same units as B{C{radius}} or the
504 ellipsoid or datum axes).
506 @raise TypeError: Invalid B{C{radius}}.
508 @see: U{Distance between two (spherical) points
509 <https://www.EdWilliams.org/avform.htm#Dist>}, functions L{euclid},
510 L{euclidean_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
511 L{cosineLaw}, L{equirectangular}, L{flatLocal}/L{hubeny}, L{flatPolar},
512 L{haversine}, L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2},
513 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
514 '''
515 return _distanceToS(euclidean_, lat1, lat2, radius,
516 *unroll180(lon1, lon2, wrap=wrap),
517 adjust=adjust)
520def euclidean_(phi2, phi1, lam21, adjust=True):
521 '''Approximate the I{angular} C{Euclidean} distance between two
522 (spherical) points.
524 @arg phi2: End latitude (C{radians}).
525 @arg phi1: Start latitude (C{radians}).
526 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
527 @kwarg adjust: Adjust the longitudinal delta by the cosine
528 of the mean latitude (C{bool}).
530 @return: Angular distance (C{radians}).
532 @see: Functions L{euclid}, L{euclidean}, L{cosineAndoyerLambert_},
533 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{equirectangular_},
534 L{flatLocal_}/L{hubeny_}, L{flatPolar_}, L{haversine_}, L{thomas_}
535 and L{vincentys_}.
536 '''
537 if adjust:
538 lam21 *= _scale_rad(phi2, phi1)
539 return euclid(phi2 - phi1, lam21)
542def excessAbc_(A, b, c):
543 '''Compute the I{spherical excess} C{E} of a (spherical) triangle
544 from two sides and the included angle.
546 @arg A: An interior triangle angle (C{radians}).
547 @arg b: Frist adjacent triangle side (C{radians}).
548 @arg c: Second adjacent triangle side (C{radians}).
550 @return: Spherical excess (C{radians}).
552 @raise UnitError: Invalid B{C{A}}, B{C{b}} or B{C{c}}.
554 @see: Functions L{excessGirard_}, L{excessLHuilier_} and U{Spherical
555 trigonometry<https://WikiPedia.org/wiki/Spherical_trigonometry>}.
556 '''
557 sA, cA, sb, cb, sc, cc = sincos2_(Radians_(A=A), Radians_(b=b) * _0_5,
558 Radians_(c=c) * _0_5)
559 return atan2(sA * sb * sc, cb * cc + cA * sb * sc) * _2_0
562def excessGirard_(A, B, C):
563 '''Compute the I{spherical excess} C{E} of a (spherical) triangle using
564 U{Girard's<https://MathWorld.Wolfram.com/GirardsSphericalExcessFormula.html>}
565 formula.
567 @arg A: First interior triangle angle (C{radians}).
568 @arg B: Second interior triangle angle (C{radians}).
569 @arg C: Third interior triangle angle (C{radians}).
571 @return: Spherical excess (C{radians}).
573 @raise UnitError: Invalid B{C{A}}, B{C{B}} or B{C{C}}.
575 @see: Function L{excessLHuilier_} and U{Spherical trigonometry
576 <https://WikiPedia.org/wiki/Spherical_trigonometry>}.
577 '''
578 return Radians(Girard=fsum_(Radians_(A=A),
579 Radians_(B=B),
580 Radians_(C=C), -PI))
583def excessLHuilier_(a, b, c):
584 '''Compute the I{spherical excess} C{E} of a (spherical) triangle using
585 U{L'Huilier's<https://MathWorld.Wolfram.com/LHuiliersTheorem.html>}
586 Theorem.
588 @arg a: First triangle side (C{radians}).
589 @arg b: Second triangle side (C{radians}).
590 @arg c: Third triangle side (C{radians}).
592 @return: Spherical excess (C{radians}).
594 @raise UnitError: Invalid B{C{a}}, B{C{b}} or B{C{c}}.
596 @see: Function L{excessGirard_} and U{Spherical trigonometry
597 <https://WikiPedia.org/wiki/Spherical_trigonometry>}.
598 '''
599 a = Radians_(a=a)
600 b = Radians_(b=b)
601 c = Radians_(c=c)
603 s = fsum_(a, b, c) * _0_5
604 r = tan_2(s) * tan_2(s - a) * tan_2(s - b) * tan_2(s - c)
605 r = atan(sqrt(r)) if r > 0 else _0_0
606 return Radians(LHuilier=r * _4_0)
609def excessKarney(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
610 '''Compute the surface area of a (spherical) quadrilateral bounded by a
611 segment of a great circle, two meridians and the equator using U{Karney's
612 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>}
613 method.
615 @arg lat1: Start latitude (C{degrees}).
616 @arg lon1: Start longitude (C{degrees}).
617 @arg lat2: End latitude (C{degrees}).
618 @arg lon2: End longitude (C{degrees}).
619 @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter}, L{Ellipsoid},
620 L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}) or C{None}.
621 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).
623 @return: Surface area, I{signed} (I{square} C{meter} or the same units as
624 B{C{radius}} I{squared}) or the I{spherical excess} (C{radians})
625 if C{B{radius}=0} or C{None}.
627 @raise TypeError: Invalid B{C{radius}}.
629 @raise UnitError: Invalid B{C{lat2}} or B{C{lat1}}.
631 @raise ValueError: Semi-circular longitudinal delta.
633 @see: Functions L{excessKarney_} and L{excessQuad}.
634 '''
635 return _area_or_(excessKarney_, lat1, lat2, radius,
636 *unroll180(lon1, lon2, wrap=wrap))
639def excessKarney_(phi2, phi1, lam21):
640 '''Compute the I{spherical excess} C{E} of a (spherical) quadrilateral bounded
641 by a segment of a great circle, two meridians and the equator using U{Karney's
642 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>}
643 method.
645 @arg phi2: End latitude (C{radians}).
646 @arg phi1: Start latitude (C{radians}).
647 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
649 @return: Spherical excess, I{signed} (C{radians}).
651 @raise ValueError: Semi-circular longitudinal delta B{C{lam21}}.
653 @see: Function L{excessKarney} and U{Area of a spherical polygon
654 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>}.
655 '''
656 # from: Veness <https://www.Movable-Type.co.UK/scripts/latlong.html> Area
657 # method due to Karney: for each edge of the polygon,
658 #
659 # tan(Δλ / 2) · (tan(φ1 / 2) + tan(φ2 / 2))
660 # tan(E / 2) = -----------------------------------------
661 # 1 + tan(φ1 / 2) · tan(φ2 / 2)
662 #
663 # where E is the spherical excess of the trapezium obtained by extending
664 # the edge to the equator-circle vector for each edge (see also ***).
665 t2 = tan_2(phi2)
666 t1 = tan_2(phi1)
667 t = tan_2(lam21, lam21=None)
668 return Radians(Karney=atan2(t * (t1 + t2),
669 _1_0 + (t1 * t2)) * _2_0)
672# ***) Original post no longer available, following is a copy of the main part
673# <http://OSGeo-org.1560.x6.Nabble.com/Area-of-a-spherical-polygon-td3841625.html>
674#
675# The area of a polygon on a (unit) sphere is given by the spherical excess
676#
677# A = 2 * pi - sum(exterior angles)
678#
679# However this is badly conditioned if the polygon is small. In this case, use
680#
681# A = sum(S12{i, i+1}) over the edges of the polygon
682#
683# where S12 is the area of the quadrilateral bounded by an edge of the polygon,
684# two meridians and the equator, i.e. with vertices (phi1, lambda1), (phi2,
685# lambda2), (0, lambda1) and (0, lambda2). S12 is given by
686#
687# tan(S12 / 2) = tan(lambda21 / 2) * (tan(phi1 / 2) + tan(phi2 / 2)) /
688# (tan(phi1 / 2) * tan(phi2 / 2) + 1)
689#
690# = tan(lambda21 / 2) * tanh((Lambertian(phi1) +
691# Lambertian(phi2)) / 2)
692#
693# where lambda21 = lambda2 - lambda1 and lamb(x) is the Lambertian (or
694# inverse Gudermannian) function
695#
696# Lambertian(x) = asinh(tan(x)) = atanh(sin(x)) = 2 * atanh(tan(x / 2))
697#
698# Notes: The formula for S12 is exact, except that...
699# - it is indeterminate if an edge is a semi-circle
700# - the formula for A applies only if the polygon does not include a pole
701# (if it does, then add +/- 2 * pi to the result)
702# - in the limit of small phi and lambda, S12 reduces to the trapezoidal
703# formula, S12 = (lambda2 - lambda1) * (phi1 + phi2) / 2
704# - I derived this result from the equation for the area of a spherical
705# triangle in terms of two edges and the included angle given by, e.g.
706# U{Todhunter, I. - Spherical Trigonometry (1871), Sec. 103, Eq. (2)
707# <http://Books.Google.com/books?id=3uBHAAAAIAAJ&pg=PA71>}
708# - I would be interested to know if this formula for S12 is already known
709# - Charles Karney
712def excessQuad(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
713 '''Compute the surface area of a (spherical) quadrilateral bounded by a segment
714 of a great circle, two meridians and the equator.
716 @arg lat1: Start latitude (C{degrees}).
717 @arg lon1: Start longitude (C{degrees}).
718 @arg lat2: End latitude (C{degrees}).
719 @arg lon2: End longitude (C{degrees}).
720 @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter},
721 L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}) or C{None}.
722 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).
724 @return: Surface area, I{signed} (I{square} C{meter} or the same units as
725 B{C{radius}} I{squared}) or the I{spherical excess} (C{radians})
726 if C{B{radius}=0} or C{None}.
728 @raise TypeError: Invalid B{C{radius}}.
730 @raise UnitError: Invalid B{C{lat2}} or B{C{lat1}}.
732 @see: Function L{excessQuad_} and L{excessKarney}.
733 '''
734 return _area_or_(excessQuad_, lat1, lat2, radius,
735 *unroll180(lon1, lon2, wrap=wrap))
738def excessQuad_(phi2, phi1, lam21):
739 '''Compute the I{spherical excess} C{E} of a (spherical) quadrilateral bounded
740 by a segment of a great circle, two meridians and the equator.
742 @arg phi2: End latitude (C{radians}).
743 @arg phi1: Start latitude (C{radians}).
744 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
746 @return: Spherical excess, I{signed} (C{radians}).
748 @see: Function L{excessQuad}, U{Spherical trigonometry
749 <https://WikiPedia.org/wiki/Spherical_trigonometry>}.
750 '''
751 s = sin((phi2 + phi1) * _0_5)
752 c = cos((phi2 - phi1) * _0_5)
753 return Radians(Quad=atan2(tan_2(lam21) * s, c) * _2_0)
756def flatLocal(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):
757 '''Compute the distance between two (ellipsoidal) points using
758 the U{ellipsoidal Earth to plane projection<https://WikiPedia.org/
759 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>}
760 aka U{Hubeny<https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula.
762 @arg lat1: Start latitude (C{degrees}).
763 @arg lon1: Start longitude (C{degrees}).
764 @arg lat2: End latitude (C{degrees}).
765 @arg lon2: End longitude (C{degrees}).
766 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid},
767 L{Ellipsoid2} or L{a_f2Tuple}).
768 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).
770 @return: Distance (C{meter}, same units as the B{C{datum}}'s
771 ellipsoid axes).
773 @raise TypeError: Invalid B{C{datum}}.
775 @note: The meridional and prime_vertical radii of curvature
776 are taken and scaled at the mean of both latitude.
778 @see: Functions L{flatLocal_} or L{hubeny_}, L{cosineLaw}, L{flatPolar},
779 L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
780 L{equirectangular}, L{euclidean}, L{haversine}, L{thomas},
781 L{vincentys}, method L{Ellipsoid.distance2} and U{local, flat
782 earth approximation<https://www.EdWilliams.org/avform.htm#flat>}.
783 '''
784 d, _ = unroll180(lon1, lon2, wrap=wrap)
785 return flatLocal_(Phi_(lat2=lat2),
786 Phi_(lat1=lat1), radians(d), datum=datum)
788hubeny = flatLocal # PYCHOK for Karl Hubeny
791def flatLocal_(phi2, phi1, lam21, datum=_WGS84):
792 '''Compute the I{angular} distance between two (ellipsoidal) points using
793 the U{ellipsoidal Earth to plane projection<https://WikiPedia.org/
794 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>}
795 aka U{Hubeny<https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula.
797 @arg phi2: End latitude (C{radians}).
798 @arg phi1: Start latitude (C{radians}).
799 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
800 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid},
801 L{Ellipsoid2} or L{a_f2Tuple}).
803 @return: Angular distance (C{radians}).
805 @raise TypeError: Invalid B{C{datum}}.
807 @note: The meridional and prime_vertical radii of curvature
808 are taken and scaled I{at the mean of both latitude}.
810 @see: Functions L{flatLocal} or L{hubeny}, L{cosineAndoyerLambert_},
811 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{flatPolar_},
812 L{equirectangular_}, L{euclidean_}, L{haversine_}, L{thomas_}
813 and L{vincentys_} and U{local, flat earth approximation
814 <https://www.EdWilliams.org/avform.htm#flat>}.
815 '''
816 E = _ellipsoidal(datum, flatLocal_)
817 m, n = E.roc2_((phi2 + phi1) * _0_5, scaled=True)
818 return hypot(m * (phi2 - phi1), n * lam21)
820hubeny_ = flatLocal_ # PYCHOK for Karl Hubeny
823def flatPolar(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
824 '''Compute the distance between two (spherical) points using
825 the U{polar coordinate flat-Earth <https://WikiPedia.org/wiki/
826 Geographical_distance#Polar_coordinate_flat-Earth_formula>}
827 formula.
829 @arg lat1: Start latitude (C{degrees}).
830 @arg lon1: Start longitude (C{degrees}).
831 @arg lat2: End latitude (C{degrees}).
832 @arg lon2: End longitude (C{degrees}).
833 @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter},
834 L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}).
835 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).
837 @return: Distance (C{meter}, same units as B{C{radius}} or the
838 ellipsoid or datum axes).
840 @raise TypeError: Invalid B{C{radius}}.
842 @see: Functions L{flatPolar_}, L{cosineAndoyerLambert},
843 L{cosineForsytheAndoyerLambert},L{cosineLaw},
844 L{flatLocal}/L{hubeny}, L{equirectangular},
845 L{euclidean}, L{haversine}, L{thomas} and
846 L{vincentys}.
847 '''
848 return _distanceToS(flatPolar_, lat1, lat2, radius,
849 *unroll180(lon1, lon2, wrap=wrap))
852def flatPolar_(phi2, phi1, lam21):
853 '''Compute the I{angular} distance between two (spherical) points
854 using the U{polar coordinate flat-Earth<https://WikiPedia.org/wiki/
855 Geographical_distance#Polar_coordinate_flat-Earth_formula>}
856 formula.
858 @arg phi2: End latitude (C{radians}).
859 @arg phi1: Start latitude (C{radians}).
860 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
862 @return: Angular distance (C{radians}).
864 @see: Functions L{flatPolar}, L{cosineAndoyerLambert_},
865 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
866 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
867 L{haversine_}, L{thomas_} and L{vincentys_}.
868 '''
869 a = fabs(PI_2 - phi1) # co-latitude
870 b = fabs(PI_2 - phi2) # co-latitude
871 if a < b:
872 a, b = b, a
873 if a < EPS0:
874 a = _0_0
875 elif b > 0:
876 b = b / a # /= chokes PyChecker
877 c = b * cos(lam21) * _2_0
878 c = fsum_(_1_0, b**2, -fabs(c))
879 a *= sqrt0(c)
880 return a
883def hartzell(pov, los=None, earth=_WGS84, name=NN, **LatLon_and_kwds):
884 '''Compute the intersection of the earth's surface and a Line-Of-Sight
885 from a Point-Of-View in space.
887 @arg pov: Point-Of-View outside the earth (C{Cartesian}, L{Ecef9Tuple}
888 or L{Vector3d}).
889 @kwarg los: Line-Of-Sight, I{direction} to earth (L{Vector3d}) or
890 C{None} to point to the earth' center.
891 @kwarg earth: The earth model (L{Datum}, L{Ellipsoid}, L{Ellipsoid2},
892 L{a_f2Tuple} or C{scalar} radius in C{meter}).
893 @kwarg name: Optional name (C{str}).
894 @kwarg LatLon_and_kwds: Optional C{LatLon} class for the intersection
895 point plus C{LatLon} keyword arguments, include
896 B{C{datum}} if different from B{C{earth}}.
898 @return: The earth intersection (L{Vector3d}, C{Cartesian type} of
899 B{C{pov}} or B{C{LatLon}}).
901 @raise IntersectionError: Null B{C{pov}} or B{C{los}} vector, B{C{pov}}
902 is inside the earth or B{C{los}} points outside
903 the earth or points in an opposite direction.
905 @raise TypeError: Invalid B{C{pov}}, B{C{los}} or B{C{earth}}.
907 @see: Function L{pygeodesy.hartzell4}, L{pygeodesy.tyr3d} for B{C{los}} and
908 U{I{Satellite Line-of-Sight Intersection with Earth}<https://StephenHartzell.
909 Medium.com/satellite-line-of-sight-intersection-with-earth-d786b4a6a9b6>}.
910 '''
911 def _Error(txt):
912 return IntersectionError(pov=pov, los=los, earth=earth, txt=txt)
914 D = earth if isinstance(earth, Datum) else \
915 _spherical_datum(earth, name=hartzell.__name__)
916 E = D.ellipsoid
918 if E.b > E.a: # PYCHOK no cover
919 try:
920 t = _MODS.triaxials
921 r, _ = t._hartzell3d2(pov, los, t.Triaxial_(E.a, E.a, E.b))
922 except Exception as x:
923 raise _Error(str(x))
924 else:
925 a2 = b2 = E.a2 # earth' x, y, ...
926 c2 = E.b2 # ... z semi-axis squared
927 q2 = E.b2_a2 # == c2 / a2
928 bc = E.a * E.b # == b * c
930 V3 = _MODS.vector3d._otherV3d
931 p3 = V3(pov=pov)
932 u3 = V3(los=los) if los else p3.negate()
933 u3 = u3.unit() # unit vector, opposing signs
935 x2, y2, z2 = p3.x2y2z2 # p3.times_(p3).xyz
936 ux, vy, wz = u3.times_(p3).xyz
937 u2, v2, w2 = u3.x2y2z2 # u3.times_(u3).xyz
939 t = c2, c2, b2
940 m = fdot(t, u2, v2, w2) # a2 factored out
941 if m < EPS0: # zero or near-null LOS vector
942 raise _Error(_near_(_null_))
944 # a2 and b2 factored out, b2 == a2 and b2 / a2 == 1
945 r = fsum_(b2 * w2, c2 * v2, -v2 * z2, vy * wz * 2,
946 c2 * u2, -u2 * z2, -w2 * x2, ux * wz * 2,
947 -w2 * y2, -u2 * y2 * q2, -v2 * x2 * q2, ux * vy * 2 * q2, floats=True)
948 if r > 0:
949 r = sqrt(r) * bc # == a * a * b * c / a2
950 elif r < 0: # LOS pointing away from or missing the earth
951 raise _Error(_opposite_ if max(ux, vy, wz) > 0 else _outside_)
953 d = Fdot(t, ux, vy, wz).fadd_(r).fover(m) # -r for antipode, a2 factored out
954 if d > 0: # POV inside or LOS missing, outside the earth
955 s = fsum_(_1_0, x2 / a2, y2 / b2, z2 / c2, _N_2_0, floats=True) # like _sideOf
956 raise _Error(_outside_ if s > 0 else _inside_)
957 elif fsum_(x2, y2, z2) < d**2: # d past earth center
958 raise _Error(_too_(_distant_))
960 r = p3.minus(u3.times(d))
961# h = p3.minus(r).length # distance to ellipsoid
963 r = _xnamed(r, name or hartzell.__name__)
964 if LatLon_and_kwds:
965 c = _MODS.cartesianBase.CartesianBase(r, datum=D, name=r.name)
966 r = c.toLatLon(**LatLon_and_kwds)
967 return r
970def haversine(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
971 '''Compute the distance between two (spherical) points using the
972 U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>}
973 formula.
975 @arg lat1: Start latitude (C{degrees}).
976 @arg lon1: Start longitude (C{degrees}).
977 @arg lat2: End latitude (C{degrees}).
978 @arg lon2: End longitude (C{degrees}).
979 @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter},
980 L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}).
981 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).
983 @return: Distance (C{meter}, same units as B{C{radius}}).
985 @raise TypeError: Invalid B{C{radius}}.
987 @see: U{Distance between two (spherical) points
988 <https://www.EdWilliams.org/avform.htm#Dist>}, functions
989 L{cosineLaw}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
990 L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar},
991 L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2},
992 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
994 @note: See note at function L{vincentys_}.
995 '''
996 return _distanceToS(haversine_, lat1, lat2, radius,
997 *unroll180(lon1, lon2, wrap=wrap))
1000def haversine_(phi2, phi1, lam21):
1001 '''Compute the I{angular} distance between two (spherical) points
1002 using the U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>}
1003 formula.
1005 @arg phi2: End latitude (C{radians}).
1006 @arg phi1: Start latitude (C{radians}).
1007 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
1009 @return: Angular distance (C{radians}).
1011 @see: Functions L{haversine}, L{cosineAndoyerLambert_},
1012 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
1013 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
1014 L{flatPolar_}, L{thomas_} and L{vincentys_}.
1016 @note: See note at function L{vincentys_}.
1017 '''
1018 def _hsin(rad):
1019 return sin(rad * _0_5)**2
1021 h = _hsin(phi2 - phi1) + cos(phi1) * cos(phi2) * _hsin(lam21) # haversine
1022 return atan2(sqrt0(h), sqrt0(_1_0 - h)) * _2_0 # == asin(sqrt(h)) * 2
1025def heightOf(angle, distance, radius=R_M):
1026 '''Determine the height above the (spherical) earth' surface after
1027 traveling along a straight line at a given tilt.
1029 @arg angle: Tilt angle above horizontal (C{degrees}).
1030 @arg distance: Distance along the line (C{meter} or same units as
1031 B{C{radius}}).
1032 @kwarg radius: Optional mean earth radius (C{meter}).
1034 @return: Height (C{meter}, same units as B{C{distance}} and B{C{radius}}).
1036 @raise ValueError: Invalid B{C{angle}}, B{C{distance}} or B{C{radius}}.
1038 @see: U{MultiDop geog_lib.GeogBeamHt<https://GitHub.com/NASA/MultiDop>}
1039 (U{Shapiro et al. 2009, JTECH
1040 <https://Journals.AMetSoc.org/doi/abs/10.1175/2009JTECHA1256.1>}
1041 and U{Potvin et al. 2012, JTECH
1042 <https://Journals.AMetSoc.org/doi/abs/10.1175/JTECH-D-11-00019.1>}).
1043 '''
1044 r = h = Radius(radius)
1045 d = fabs(Distance(distance))
1046 if d > h:
1047 d, h = h, d
1049 if d > EPS0: # and h > EPS0
1050 d = d / h # /= h chokes PyChecker
1051 s = sin(Phi_(angle=angle, clip=_180_0))
1052 s = fsum_(_1_0, _2_0 * s * d, d**2)
1053 if s > 0:
1054 return h * sqrt(s) - r
1056 raise _ValueError(angle=angle, distance=distance, radius=radius)
1059def horizon(height, radius=R_M, refraction=False):
1060 '''Determine the distance to the horizon from a given altitude
1061 above the (spherical) earth.
1063 @arg height: Altitude (C{meter} or same units as B{C{radius}}).
1064 @kwarg radius: Optional mean earth radius (C{meter}).
1065 @kwarg refraction: Consider atmospheric refraction (C{bool}).
1067 @return: Distance (C{meter}, same units as B{C{height}} and B{C{radius}}).
1069 @raise ValueError: Invalid B{C{height}} or B{C{radius}}.
1071 @see: U{Distance to horizon<https://www.EdWilliams.org/avform.htm#Horizon>}.
1072 '''
1073 h, r = Height(height), Radius(radius)
1074 if min(h, r) < 0:
1075 raise _ValueError(height=height, radius=radius)
1077 if refraction:
1078 d2 = 2.415750694528 * h * r # 2.0 / 0.8279
1079 else:
1080 d2 = h * fsum_(r, r, h)
1081 return sqrt0(d2)
1084def intersections2(lat1, lon1, radius1,
1085 lat2, lon2, radius2, datum=None, wrap=True):
1086 '''Conveniently compute the intersections of two circles each defined
1087 by a (geodetic) center point and a radius, using either ...
1089 1) L{vector3d.intersections2} for small distances (below 100 KM or
1090 about 0.9 degrees) or if no B{C{datum}} is specified, or ...
1092 2) L{sphericalTrigonometry.intersections2} for a spherical B{C{datum}}
1093 or if B{C{datum}} is a C{scalar} representing the earth radius,
1094 conventionally in C{meter} or ...
1096 3) L{ellipsoidalKarney.intersections2} for an ellipsoidal B{C{datum}}
1097 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>}
1098 is installed, otherwise ...
1100 4) L{ellipsoidalExact.intersections2}, provided B{C{datum}} is ellipsoidal.
1102 @arg lat1: Latitude of the first circle center (C{degrees}).
1103 @arg lon1: Longitude of the first circle center (C{degrees}).
1104 @arg radius1: Radius of the first circle (C{meter}, conventionally).
1105 @arg lat2: Latitude of the second circle center (C{degrees}).
1106 @arg lon2: Longitude of the second circle center (C{degrees}).
1107 @arg radius2: Radius of the second circle (C{meter}, same units as B{C{radius1}}).
1108 @kwarg datum: Optional ellipsoidal or spherical datum (L{Datum},
1109 L{Ellipsoid}, L{Ellipsoid2}, L{a_f2Tuple} or
1110 C{scalar} earth radius in C{meter}, same units as
1111 B{C{radius1}} and B{C{radius2}}) or C{None}.
1112 @kwarg wrap: Wrap and unroll longitudes (C{bool}).
1114 @return: 2-Tuple of the intersection points, each a
1115 L{LatLon2Tuple}C{(lat, lon)}. For abutting circles,
1116 the points are the same instance, aka I{radical center}.
1118 @raise IntersectionError: Concentric, antipodal, invalid or
1119 non-intersecting circles or no
1120 convergence.
1122 @raise TypeError: Invalid B{C{datum}}.
1124 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{radius1}},
1125 B{C{lat2}}, B{C{lon2}} or B{C{radius2}}.
1126 '''
1127 if datum is None or euclidean(lat1, lon1, lat2, lon2, radius=R_M,
1128 adjust=True, wrap=wrap) < _100km:
1129 def _V2T(x, y, _, **unused): # _ == z unused
1130 return LatLon2Tuple(y, x, name=intersections2.__name__)
1132 r1 = m2degrees(Radius_(radius1=radius1), radius=R_M, lat=lat1)
1133 r2 = m2degrees(Radius_(radius2=radius2), radius=R_M, lat=lat2)
1135 _, lon2 = unroll180(lon1, lon2, wrap=wrap)
1136 m = _MODS.vector3d
1137 t = m.intersections2(m.Vector3d(lon1, lat1, 0), r1,
1138 m.Vector3d(lon2, lat2, 0), r2, sphere=False,
1139 Vector=_V2T)
1140 else:
1141 def _LL2T(lat, lon, **unused):
1142 return LatLon2Tuple(lat, lon, name=intersections2.__name__)
1144 d = _spherical_datum(datum, name=intersections2.__name__)
1145 if d.isSpherical:
1146 m = _MODS.sphericalTrigonometry
1147 elif d.isEllipsoidal:
1148 try:
1149 if d.ellipsoid.geodesic:
1150 pass
1151 m = _MODS.ellipsoidalKarney
1152 except ImportError:
1153 m = _MODS.ellipsoidalExact
1154 else:
1155 raise _AssertionError(datum=d)
1157 t = m.intersections2(m.LatLon(lat1, lon1, datum=d), radius1,
1158 m.LatLon(lat2, lon2, datum=d), radius2,
1159 LatLon=_LL2T, height=0, wrap=wrap)
1160 return t
1163def isantipode(lat1, lon1, lat2, lon2, eps=EPS):
1164 '''Check whether two points are antipodal, on diametrically
1165 opposite sides of the earth.
1167 @arg lat1: Latitude of one point (C{degrees}).
1168 @arg lon1: Longitude of one point (C{degrees}).
1169 @arg lat2: Latitude of the other point (C{degrees}).
1170 @arg lon2: Longitude of the other point (C{degrees}).
1171 @kwarg eps: Tolerance for near-equality (C{degrees}).
1173 @return: C{True} if points are antipodal within the
1174 B{C{eps}} tolerance, C{False} otherwise.
1176 @see: Functions L{isantipode_} and L{antipode}.
1177 '''
1178 return True if (fabs(lat1 + lat2) <= eps and
1179 fabs(lon1 + lon2) <= eps) else \
1180 _MODS.latlonBase._isequalTo(antipode(lat1, lon1),
1181 normal(lat2, lon2), eps=eps)
1184def isantipode_(phi1, lam1, phi2, lam2, eps=EPS):
1185 '''Check whether two points are antipodal, on diametrically
1186 opposite sides of the earth.
1188 @arg phi1: Latitude of one point (C{radians}).
1189 @arg lam1: Longitude of one point (C{radians}).
1190 @arg phi2: Latitude of the other point (C{radians}).
1191 @arg lam2: Longitude of the other point (C{radians}).
1192 @kwarg eps: Tolerance for near-equality (C{radians}).
1194 @return: C{True} if points are antipodal within the
1195 B{C{eps}} tolerance, C{False} otherwise.
1197 @see: Functions L{isantipode} and L{antipode_}.
1198 '''
1199 return True if (fabs(phi1 + phi2) <= eps and
1200 fabs(lam1 + lam2) <= eps) else \
1201 _MODS.latlonBase._isequalTo_(antipode_(phi1, lam1),
1202 normal_(phi2, lam2), eps=eps)
1205def isnormal(lat, lon, eps=0):
1206 '''Check whether B{C{lat}} I{and} B{C{lon}} are within the I{normal}
1207 range in C{degrees}.
1209 @arg lat: Latitude (C{degrees}).
1210 @arg lon: Longitude (C{degrees}).
1211 @kwarg eps: Optional tolerance below C{90} and C{180 degrees}.
1213 @return: C{True} if C{(abs(B{lat}) + B{eps}) <= 90} and
1214 C{(abs(B{lon}) + B{eps}) <= 180}, C{False} othwerwise.
1216 @see: Functions L{isnormal_} and L{normal}.
1217 '''
1218 return (_90_0 - fabs(lat)) >= eps and (_180_0 - fabs(lon)) >= eps
1221def isnormal_(phi, lam, eps=0):
1222 '''Check whether B{C{phi}} I{and} B{C{lam}} are within the I{normal}
1223 range in C{radians}.
1225 @arg phi: Latitude (C{radians}).
1226 @arg lam: Longitude (C{radians}).
1227 @kwarg eps: Optional tolerance below C{PI/2} and C{PI radians}.
1229 @return: C{True} if C{(abs(B{phi}) + B{eps}) <= PI/2} and
1230 C{(abs(B{lam}) + B{eps}) <= PI}, C{False} othwerwise.
1232 @see: Functions L{isnormal} and L{normal_}.
1233 '''
1234 return (PI_2 - fabs(phi)) >= eps and (PI - fabs(lam)) >= eps
1237def latlon2n_xyz(lat, lon, name=NN):
1238 '''Convert lat-, longitude to C{n-vector} (I{normal} to the
1239 earth's surface) X, Y and Z components.
1241 @arg lat: Latitude (C{degrees}).
1242 @arg lon: Longitude (C{degrees}).
1243 @kwarg name: Optional name (C{str}).
1245 @return: A L{Vector3Tuple}C{(x, y, z)}.
1247 @see: Function L{philam2n_xyz}.
1249 @note: These are C{n-vector} x, y and z components,
1250 I{NOT} geocentric ECEF x, y and z coordinates!
1251 '''
1252 return _2n_xyz(name, *sincos2d_(lat, lon))
1255def _normal2(a, b, n_2, n, n2):
1256 '''(INTERNAL) Helper for C{normal} and C{normal_}.
1257 '''
1258 if fabs(b) > n:
1259 b = remainder(b, n2)
1260 r = remainder(a, n) if fabs(a) > n_2 else a
1261 if r != a:
1262 r = -r
1263 b -= n if b > 0 else -n
1264 return float0(r, b)
1267def normal(lat, lon, name=NN):
1268 '''Normalize a lat- I{and} longitude pair in C{degrees}.
1270 @arg lat: Latitude (C{degrees}).
1271 @arg lon: Longitude (C{degrees}).
1272 @kwarg name: Optional name (C{str}).
1274 @return: L{LatLon2Tuple}C{(lat, lon)} with C{abs(lat) <= 90}
1275 and C{abs(lon) <= 180}.
1277 @see: Functions L{normal_} and L{isnormal}.
1278 '''
1279 return LatLon2Tuple(*_normal2(lat, lon, _90_0, _180_0, _360_0), name=name)
1282def normal_(phi, lam, name=NN):
1283 '''Normalize a lat- I{and} longitude pair in C{radians}.
1285 @arg phi: Latitude (C{radians}).
1286 @arg lam: Longitude (C{radians}).
1287 @kwarg name: Optional name (C{str}).
1289 @return: L{PhiLam2Tuple}C{(phi, lam)} with C{abs(phi) <= PI/2}
1290 and C{abs(lam) <= PI}.
1292 @see: Functions L{normal} and L{isnormal_}.
1293 '''
1294 return PhiLam2Tuple(*_normal2(phi, lam, PI_2, PI, PI2), name=name)
1297def _2n_xyz(name, sa, ca, sb, cb):
1298 '''(INTERNAL) Helper for C{latlon2n_xyz} and C{philam2n_xyz}.
1299 '''
1300 # Kenneth Gade eqn 3, but using right-handed
1301 # vector x -> 0°E,0°N, y -> 90°E,0°N, z -> 90°N
1302 return Vector3Tuple(ca * cb, ca * sb, sa, name=name)
1305def n_xyz2latlon(x, y, z, name=NN):
1306 '''Convert C{n-vector} components to lat- and longitude in C{degrees}.
1308 @arg x: X component (C{scalar}).
1309 @arg y: Y component (C{scalar}).
1310 @arg z: Z component (C{scalar}).
1311 @kwarg name: Optional name (C{str}).
1313 @return: A L{LatLon2Tuple}C{(lat, lon)}.
1315 @see: Function L{n_xyz2philam}.
1316 '''
1317 return LatLon2Tuple(atan2d(z, hypot(x, y)), atan2d(y, x), name=name)
1320def n_xyz2philam(x, y, z, name=NN):
1321 '''Convert C{n-vector} components to lat- and longitude in C{radians}.
1323 @arg x: X component (C{scalar}).
1324 @arg y: Y component (C{scalar}).
1325 @arg z: Z component (C{scalar}).
1326 @kwarg name: Optional name (C{str}).
1328 @return: A L{PhiLam2Tuple}C{(phi, lam)}.
1330 @see: Function L{n_xyz2latlon}.
1331 '''
1332 return PhiLam2Tuple(atan2(z, hypot(x, y)), atan2(y, x), name=name)
1335def _opposes(d, m, n, n2):
1336 '''(INETNAL) Helper for C{opposing} and C{opposing_}.
1337 '''
1338 d = d % n2 # -20 % 360 == 340, -1 % PI2 == PI2 - 1
1339 return False if d < m or d > (n2 - m) else (
1340 True if (n - m) < d < (n + m) else None)
1343def opposing(bearing1, bearing2, margin=_90_0):
1344 '''Compare the direction of two bearings given in C{degrees}.
1346 @arg bearing1: First bearing (compass C{degrees}).
1347 @arg bearing2: Second bearing (compass C{degrees}).
1348 @kwarg margin: Optional, interior angle bracket (C{degrees}).
1350 @return: C{True} if both bearings point in opposite, C{False} if
1351 in similar or C{None} if in perpendicular directions.
1353 @see: Function L{opposing_}.
1354 '''
1355 m = Degrees_(margin=margin, low=EPS0, high=_90_0)
1356 return _opposes(bearing2 - bearing1, m,_180_0, _360_0)
1359def opposing_(radians1, radians2, margin=PI_2):
1360 '''Compare the direction of two bearings given in C{radians}.
1362 @arg radians1: First bearing (C{radians}).
1363 @arg radians2: Second bearing (C{radians}).
1364 @kwarg margin: Optional, interior angle bracket (C{radians}).
1366 @return: C{True} if both bearings point in opposite, C{False} if
1367 in similar or C{None} if in perpendicular directions.
1369 @see: Function L{opposing}.
1370 '''
1371 m = Radians_(margin=margin, low=EPS0, high=PI_2)
1372 return _opposes(radians2 - radians1, m, PI, PI2)
1375def philam2n_xyz(phi, lam, name=NN):
1376 '''Convert lat-, longitude to C{n-vector} (I{normal} to the
1377 earth's surface) X, Y and Z components.
1379 @arg phi: Latitude (C{radians}).
1380 @arg lam: Longitude (C{radians}).
1381 @kwarg name: Optional name (C{str}).
1383 @return: A L{Vector3Tuple}C{(x, y, z)}.
1385 @see: Function L{latlon2n_xyz}.
1387 @note: These are C{n-vector} x, y and z components,
1388 I{NOT} geocentric ECEF x, y and z coordinates!
1389 '''
1390 return _2n_xyz(name, *sincos2_(phi, lam))
1393def _radical2(d, r1, r2): # in .ellipsoidalBaseDI, .sphericalTrigonometry, .vector3d
1394 # (INTERNAL) See C{radical2} below
1395 # assert d > EPS0
1396 r = fsum_(_1_0, (r1 / d)**2, -(r2 / d)**2) * _0_5
1397 return Radical2Tuple(max(_0_0, min(_1_0, r)), r * d)
1400def radical2(distance, radius1, radius2):
1401 '''Compute the I{radical ratio} and I{radical line} of two
1402 U{intersecting circles<https://MathWorld.Wolfram.com/
1403 Circle-CircleIntersection.html>}.
1405 The I{radical line} is perpendicular to the axis thru the
1406 centers of the circles at C{(0, 0)} and C{(B{distance}, 0)}.
1408 @arg distance: Distance between the circle centers (C{scalar}).
1409 @arg radius1: Radius of the first circle (C{scalar}).
1410 @arg radius2: Radius of the second circle (C{scalar}).
1412 @return: A L{Radical2Tuple}C{(ratio, xline)} where C{0.0 <=
1413 ratio <= 1.0} and C{xline} is along the B{C{distance}}.
1415 @raise IntersectionError: The B{C{distance}} exceeds the sum
1416 of B{C{radius1}} and B{C{radius2}}.
1418 @raise UnitError: Invalid B{C{distance}}, B{C{radius1}} or
1419 B{C{radius2}}.
1421 @see: U{Circle-Circle Intersection
1422 <https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>}.
1423 '''
1424 d = Distance_(distance, low=_0_0)
1425 r1 = Radius_(radius1=radius1)
1426 r2 = Radius_(radius2=radius2)
1427 if d > (r1 + r2):
1428 raise IntersectionError(distance=d, radius1=r1, radius2=r2,
1429 txt=_too_(_distant_))
1430 return _radical2(d, r1, r2) if d > EPS0 else \
1431 Radical2Tuple(_0_5, _0_0)
1434class Radical2Tuple(_NamedTuple):
1435 '''2-Tuple C{(ratio, xline)} of the I{radical} C{ratio} and
1436 I{radical} C{xline}, both C{scalar} and C{0.0 <= ratio <= 1.0}
1437 '''
1438 _Names_ = (_ratio_, _xline_)
1439 _Units_ = ( Scalar, Scalar)
1442def _scale_deg(lat1, lat2): # degrees
1443 # scale factor cos(mean of lats) for delta lon
1444 m = fabs(lat1 + lat2) * _0_5
1445 return cos(radians(m)) if m < 90 else _0_0
1448def _scale_rad(phi1, phi2): # radians, by .frechet, .hausdorff, .heights
1449 # scale factor cos(mean of phis) for delta lam
1450 m = fabs(phi1 + phi2) * _0_5
1451 return cos(m) if m < PI_2 else _0_0
1454def _sincosa6(phi2, phi1, lam21):
1455 '''(INTERNAL) C{sin}es, C{cos}ines and C{acos}ine.
1456 '''
1457 s2, c2, s1, c1, _, c21 = sincos2_(phi2, phi1, lam21)
1458 return s2, c2, s1, c1, acos1(s1 * s2 + c1 * c2 * c21), c21
1461def thomas(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):
1462 '''Compute the distance between two (ellipsoidal) points using
1463 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>}
1464 formula.
1466 @arg lat1: Start latitude (C{degrees}).
1467 @arg lon1: Start longitude (C{degrees}).
1468 @arg lat2: End latitude (C{degrees}).
1469 @arg lon2: End longitude (C{degrees}).
1470 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid},
1471 L{Ellipsoid2} or L{a_f2Tuple}).
1472 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).
1474 @return: Distance (C{meter}, same units as the B{C{datum}}'s
1475 ellipsoid axes).
1477 @raise TypeError: Invalid B{C{datum}}.
1479 @see: Functions L{thomas_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
1480 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny},
1481 L{flatPolar}, L{haversine}, L{vincentys} and method L{Ellipsoid.distance2}.
1482 '''
1483 return _distanceToE(thomas_, lat1, lat2, datum,
1484 *unroll180(lon1, lon2, wrap=wrap))
1487def thomas_(phi2, phi1, lam21, datum=_WGS84):
1488 '''Compute the I{angular} distance between two (ellipsoidal) points using
1489 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>}
1490 formula.
1492 @arg phi2: End latitude (C{radians}).
1493 @arg phi1: Start latitude (C{radians}).
1494 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
1495 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid},
1496 L{Ellipsoid2} or L{a_f2Tuple}).
1498 @return: Angular distance (C{radians}).
1500 @raise TypeError: Invalid B{C{datum}}.
1502 @see: Functions L{thomas}, L{cosineAndoyerLambert_},
1503 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
1504 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
1505 L{flatPolar_}, L{haversine_} and L{vincentys_} and U{Geodesy-PHP
1506 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/
1507 Distance/ThomasFormula.php>}.
1508 '''
1509 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21)
1510 if r and isnon0(c1) and isnon0(c2):
1511 E = _ellipsoidal(datum, thomas_)
1512 if E.f:
1513 r1 = atan2(E.b_a * s1, c1)
1514 r2 = atan2(E.b_a * s2, c2)
1516 j = (r2 + r1) * _0_5
1517 k = (r2 - r1) * _0_5
1518 sj, cj, sk, ck, h, _ = sincos2_(j, k, lam21 * _0_5)
1520 h = fsum_(sk**2, (ck * h)**2, -(sj * h)**2)
1521 u = _1_0 - h
1522 if isnon0(u) and isnon0(h):
1523 r = atan(sqrt0(h / u)) * _2_0 # == acos(1 - 2 * h)
1524 sr, cr = sincos2(r)
1525 if isnon0(sr):
1526 u = 2 * (sj * ck)**2 / u
1527 h = 2 * (sk * cj)**2 / h
1528 x = u + h
1529 y = u - h
1531 s = r / sr
1532 e = 4 * s**2
1533 d = 2 * cr
1534 a = e * d
1535 b = 2 * r
1536 c = s - (a - d) * _0_5
1537 f = E.f * _0_25
1539 t = fsum_(a * x, -b * y, c * x**2, -d * y**2, e * x * y)
1540 r -= fsum_(s * x, -y, -t * f * _0_25) * f * sr
1541 return r
1544def vincentys(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
1545 '''Compute the distance between two (spherical) points using
1546 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>}
1547 spherical formula.
1549 @arg lat1: Start latitude (C{degrees}).
1550 @arg lon1: Start longitude (C{degrees}).
1551 @arg lat2: End latitude (C{degrees}).
1552 @arg lon2: End longitude (C{degrees}).
1553 @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter},
1554 L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}).
1555 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).
1557 @return: Distance (C{meter}, same units as B{C{radius}}).
1559 @raise UnitError: Invalid B{C{radius}}.
1561 @see: Functions L{vincentys_}, L{cosineAndoyerLambert},
1562 L{cosineForsytheAndoyerLambert},L{cosineLaw}, L{equirectangular},
1563 L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar},
1564 L{haversine} and L{thomas} and methods L{Ellipsoid.distance2},
1565 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
1567 @note: See note at function L{vincentys_}.
1568 '''
1569 return _distanceToS(vincentys_, lat1, lat2, radius,
1570 *unroll180(lon1, lon2, wrap=wrap))
1573def vincentys_(phi2, phi1, lam21):
1574 '''Compute the I{angular} distance between two (spherical) points using
1575 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>}
1576 spherical formula.
1578 @arg phi2: End latitude (C{radians}).
1579 @arg phi1: Start latitude (C{radians}).
1580 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
1582 @return: Angular distance (C{radians}).
1584 @see: Functions L{vincentys}, L{cosineAndoyerLambert_},
1585 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
1586 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
1587 L{flatPolar_}, L{haversine_} and L{thomas_}.
1589 @note: Functions L{vincentys_}, L{haversine_} and L{cosineLaw_}
1590 produce equivalent results, but L{vincentys_} is suitable
1591 for antipodal points and slightly more expensive (M{3 cos,
1592 3 sin, 1 hypot, 1 atan2, 6 mul, 2 add}) than L{haversine_}
1593 (M{2 cos, 2 sin, 2 sqrt, 1 atan2, 5 mul, 1 add}) and
1594 L{cosineLaw_} (M{3 cos, 3 sin, 1 acos, 3 mul, 1 add}).
1595 '''
1596 s1, c1, s2, c2, s21, c21 = sincos2_(phi1, phi2, lam21)
1598 c = c2 * c21
1599 x = s1 * s2 + c1 * c
1600 y = c1 * s2 - s1 * c
1601 return atan2(hypot(c2 * s21, y), x)
1603# **) MIT License
1604#
1605# Copyright (C) 2016-2023 -- mrJean1 at Gmail -- All Rights Reserved.
1606#
1607# Permission is hereby granted, free of charge, to any person obtaining a
1608# copy of this software and associated documentation files (the "Software"),
1609# to deal in the Software without restriction, including without limitation
1610# the rights to use, copy, modify, merge, publish, distribute, sublicense,
1611# and/or sell copies of the Software, and to permit persons to whom the
1612# Software is furnished to do so, subject to the following conditions:
1613#
1614# The above copyright notice and this permission notice shall be included
1615# in all copies or substantial portions of the Software.
1616#
1617# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
1618# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
1619# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
1620# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
1621# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
1622# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
1623# OTHER DEALINGS IN THE SOFTWARE.