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
9# from pygeodesy.basics import isscalar # from .fsums
10from pygeodesy.constants import EPS, EPS0, EPS1, PI, PI2, PI3, PI_2, R_M, \
11 _umod_PI2, float0, isnon0, remainder, _0_0, \
12 _0_125, _0_25, _0_5, _1_0, _2_0, _N_2_0, \
13 _4_0, _32_0, _90_0, _180_0, _360_0
14from pygeodesy.datums import Datum, Ellipsoid, _ellipsoidal_datum, \
15 _mean_radius, _spherical_datum, _WGS84
16# from pygeodesy.ellipsoids import Ellipsoid # from .datums
17from pygeodesy.errors import _AssertionError, IntersectionError, LimitError, \
18 limiterrors, _ValueError, _xError
19from pygeodesy.fmath import Fdot, euclid, fdot, hypot, hypot2, sqrt0
20from pygeodesy.fsums import fsum_, isscalar, unstr
21from pygeodesy.interns import NN, _distant_, _inside_, _near_, _null_, \
22 _opposite_, _outside_, _too_
23from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS
24from pygeodesy.named import _NamedTuple, _xnamed
25from pygeodesy.namedTuples import Bearing2Tuple, Distance4Tuple, \
26 Intersection3Tuple, LatLon2Tuple, \
27 PhiLam2Tuple, Vector3Tuple
28# from pygeodesy.streprs import unstr # from .fsums
29from pygeodesy.units import Bearing, Degrees_, Distance, Distance_, Height, \
30 Lam_, Lat, Lon, Phi_, Radians, Radians_, Radius, \
31 Radius_, Scalar, _100km
32from pygeodesy.utily import acos1, atan2b, atan2d, degrees2m, m2degrees, tan_2, \
33 sincos2, sincos2_, sincos2d_, unroll180, unrollPI
35from math import atan, atan2, cos, degrees, fabs, radians, sin, sqrt # pow
37__all__ = _ALL_LAZY.formy
38__version__ = '23.04.11'
40_ratio_ = 'ratio'
41_xline_ = 'xline'
44def _anti2(a, b, n_2, n, n2):
45 '''(INTERNAL) Helper for C{antipode} and C{antipode_}.
46 '''
47 r = remainder(a, n) if fabs(a) > n_2 else a
48 if r == a:
49 r = -r
50 b += n
51 if fabs(b) > n:
52 b = remainder(b, n2)
53 return float0(r, b)
56def antipode(lat, lon, name=NN):
57 '''Return the antipode, the point diametrically opposite
58 to a given point in C{degrees}.
60 @arg lat: Latitude (C{degrees}).
61 @arg lon: Longitude (C{degrees}).
62 @kwarg name: Optional name (C{str}).
64 @return: A L{LatLon2Tuple}C{(lat, lon)}.
66 @see: Functions L{antipode_} and L{normal} and U{Geosphere
67 <https://CRAN.R-Project.org/web/packages/geosphere/geosphere.pdf>}.
68 '''
69 return LatLon2Tuple(*_anti2(lat, lon, _90_0, _180_0, _360_0), name=name)
72def antipode_(phi, lam, name=NN):
73 '''Return the antipode, the point diametrically opposite
74 to a given point in C{radians}.
76 @arg phi: Latitude (C{radians}).
77 @arg lam: Longitude (C{radians}).
78 @kwarg name: Optional name (C{str}).
80 @return: A L{PhiLam2Tuple}C{(phi, lam)}.
82 @see: Functions L{antipode} and L{normal_} and U{Geosphere
83 <https://CRAN.R-Project.org/web/packages/geosphere/geosphere.pdf>}.
84 '''
85 return PhiLam2Tuple(*_anti2(phi, lam, PI_2, PI, PI2), name=name)
88def _area_or_(excess_, lat1, lat2, radius, d_lon, unused):
89 '''(INTERNAL) Helper for area and spherical excess.
90 '''
91 r = excess_(Phi_(lat2=lat2),
92 Phi_(lat1=lat1), radians(d_lon))
93 if radius:
94 r *= _mean_radius(radius, lat1, lat2)**2
95 return r
98def bearing(lat1, lon1, lat2, lon2, **options):
99 '''Compute the initial or final bearing (forward or reverse
100 azimuth) between a (spherical) start and end point.
102 @arg lat1: Start latitude (C{degrees}).
103 @arg lon1: Start longitude (C{degrees}).
104 @arg lat2: End latitude (C{degrees}).
105 @arg lon2: End longitude (C{degrees}).
106 @kwarg options: Optional keyword arguments for function
107 L{pygeodesy.bearing_}.
109 @return: Initial or final bearing (compass C{degrees360}) or
110 zero if start and end point coincide.
111 '''
112 r = bearing_(Phi_(lat1=lat1), Lam_(lon1=lon1),
113 Phi_(lat2=lat2), Lam_(lon2=lon2), **options)
114 return degrees(r)
117def bearing_(phi1, lam1, phi2, lam2, final=False, wrap=False):
118 '''Compute the initial or final bearing (forward or reverse azimuth)
119 between a (spherical) start and end point.
121 @arg phi1: Start latitude (C{radians}).
122 @arg lam1: Start longitude (C{radians}).
123 @arg phi2: End latitude (C{radians}).
124 @arg lam2: End longitude (C{radians}).
125 @kwarg final: Return final bearing if C{True}, initial otherwise (C{bool}).
126 @kwarg wrap: Wrap and L{pygeodesy.unrollPI} longitudes (C{bool}).
128 @return: Initial or final bearing (compass C{radiansPI2}) or zero if start
129 and end point coincide.
131 @see: U{Bearing<https://www.Movable-Type.co.UK/scripts/latlong.html>}, U{Course
132 between two points<https://www.EdWilliams.org/avform147.htm#Crs>} and
133 U{Bearing Between Two Points<https://web.Archive.org/web/20020630205931/
134 https://MathForum.org/library/drmath/view/55417.html>}.
135 '''
136 if final: # swap plus PI
137 phi1, lam1, phi2, lam2 = phi2, lam2, phi1, lam1
138 r = PI3
139 else:
140 r = PI2
142 db, _ = unrollPI(lam1, lam2, wrap=wrap)
143 sa1, ca1, sa2, ca2, sdb, cdb = sincos2_(phi1, phi2, db)
145 x = ca1 * sa2 - sa1 * ca2 * cdb
146 y = sdb * ca2
147 return _umod_PI2(atan2(y, x) + r) # .utily.wrapPI2
150def _bearingTo2(p1, p2, wrap=False): # for points.ispolar, sphericalTrigonometry.areaOf
151 '''(INTERNAL) Compute initial and final bearing.
152 '''
153 try: # for LatLon_ and ellipsoidal LatLon
154 return p1.bearingTo2(p2, wrap=wrap)
155 except AttributeError:
156 pass
157 # XXX spherical version, OK for ellipsoidal ispolar?
158 a1, b1 = p1.philam
159 a2, b2 = p2.philam
160 return Bearing2Tuple(degrees(bearing_(a1, b1, a2, b2, final=False, wrap=wrap)),
161 degrees(bearing_(a1, b1, a2, b2, final=True, wrap=wrap)),
162 name=_bearingTo2.__name__)
165def compassAngle(lat1, lon1, lat2, lon2, adjust=True, wrap=False):
166 '''Return the angle from North for the direction vector
167 M{(lon2 - lon1, lat2 - lat1)} between two points.
169 Suitable only for short, not near-polar vectors up to a few hundred
170 Km or Miles. Use function L{pygeodesy.bearing} for longer vectors.
172 @arg lat1: From latitude (C{degrees}).
173 @arg lon1: From longitude (C{degrees}).
174 @arg lat2: To latitude (C{degrees}).
175 @arg lon2: To longitude (C{degrees}).
176 @kwarg adjust: Adjust the longitudinal delta by the cosine of the
177 mean latitude (C{bool}).
178 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).
180 @return: Compass angle from North (C{degrees360}).
182 @note: Courtesy of Martin Schultz.
184 @see: U{Local, flat earth approximation
185 <https://www.EdWilliams.org/avform.htm#flat>}.
186 '''
187 d_lon, _ = unroll180(lon1, lon2, wrap=wrap)
188 if adjust: # scale delta lon
189 d_lon *= _scale_deg(lat1, lat2)
190 return atan2b(d_lon, lat2 - lat1)
193def cosineAndoyerLambert(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):
194 '''Compute the distance between two (ellipsoidal) points using the
195 U{Andoyer-Lambert correction<https://navlib.net/wp-content/uploads/
196 2013/10/admiralty-manual-of-navigation-vol-1-1964-english501c.pdf>} of the
197 U{Law of Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>}
198 fromula.
200 @arg lat1: Start latitude (C{degrees}).
201 @arg lon1: Start longitude (C{degrees}).
202 @arg lat2: End latitude (C{degrees}).
203 @arg lon2: End longitude (C{degrees}).
204 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid},
205 L{Ellipsoid2} or L{a_f2Tuple}).
206 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).
208 @return: Distance (C{meter}, same units as the B{C{datum}}'s
209 ellipsoid axes or C{radians} if B{C{datum}} is C{None}).
211 @raise TypeError: Invalid B{C{datum}}.
213 @see: Functions L{cosineAndoyerLambert_}, L{cosineForsytheAndoyerLambert},
214 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny},
215 L{flatPolar}, L{haversine}, L{thomas} and L{vincentys} and method
216 L{Ellipsoid.distance2}.
217 '''
218 return _distanceToE(cosineAndoyerLambert_, lat1, lat2, datum,
219 *unroll180(lon1, lon2, wrap=wrap))
222def cosineAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84):
223 '''Compute the I{angular} distance between two (ellipsoidal) points using the
224 U{Andoyer-Lambert correction<https://navlib.net/wp-content/uploads/2013/10/
225 admiralty-manual-of-navigation-vol-1-1964-english501c.pdf>} of the U{Law
226 of Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>}
227 fromula.
229 @arg phi2: End latitude (C{radians}).
230 @arg phi1: Start latitude (C{radians}).
231 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
232 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid},
233 L{Ellipsoid2} or L{a_f2Tuple}).
235 @return: Angular distance (C{radians}).
237 @raise TypeError: Invalid B{C{datum}}.
239 @see: Functions L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert_},
240 L{cosineLaw_}, L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
241 L{flatPolar_}, L{haversine_}, L{thomas_} and L{vincentys_} and U{Geodesy-PHP
242 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/
243 Distance/AndoyerLambert.php>}.
244 '''
245 s2, c2, s1, c1, r, c21 = _sincosa6(phi2, phi1, lam21)
246 if isnon0(c1) and isnon0(c2):
247 E = _ellipsoidal(datum, cosineAndoyerLambert_)
248 if E.f: # ellipsoidal
249 r2 = atan2(E.b_a * s2, c2)
250 r1 = atan2(E.b_a * s1, c1)
251 s2, c2, s1, c1 = sincos2_(r2, r1)
252 r = acos1(s1 * s2 + c1 * c2 * c21)
253 if r:
254 sr, _, sr_2, cr_2 = sincos2_(r, r * _0_5)
255 if isnon0(sr_2) and isnon0(cr_2):
256 s = (sr + r) * ((s1 - s2) / sr_2)**2
257 c = (sr - r) * ((s1 + s2) / cr_2)**2
258 r += (c - s) * E.f * _0_125
259 return r
262def cosineForsytheAndoyerLambert(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):
263 '''Compute the distance between two (ellipsoidal) points using the
264 U{Forsythe-Andoyer-Lambert correction<https://www2.UNB.Ca/gge/Pubs/TR77.pdf>} of
265 the U{Law of Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>}
266 formula.
268 @arg lat1: Start latitude (C{degrees}).
269 @arg lon1: Start longitude (C{degrees}).
270 @arg lat2: End latitude (C{degrees}).
271 @arg lon2: End longitude (C{degrees}).
272 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid},
273 L{Ellipsoid2} or L{a_f2Tuple}).
274 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).
276 @return: Distance (C{meter}, same units as the B{C{datum}}'s
277 ellipsoid axes or C{radians} if B{C{datum}} is C{None}).
279 @raise TypeError: Invalid B{C{datum}}.
281 @see: Functions L{cosineForsytheAndoyerLambert_}, L{cosineAndoyerLambert},
282 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny},
283 L{flatPolar}, L{haversine}, L{thomas} and L{vincentys} and method
284 L{Ellipsoid.distance2}.
285 '''
286 return _distanceToE(cosineForsytheAndoyerLambert_, lat1, lat2, datum,
287 *unroll180(lon1, lon2, wrap=wrap))
290def cosineForsytheAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84):
291 '''Compute the I{angular} distance between two (ellipsoidal) points using the
292 U{Forsythe-Andoyer-Lambert correction<https://www2.UNB.Ca/gge/Pubs/TR77.pdf>} of
293 the U{Law of Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>}
294 formula.
296 @arg phi2: End latitude (C{radians}).
297 @arg phi1: Start latitude (C{radians}).
298 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
299 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid},
300 L{Ellipsoid2} or L{a_f2Tuple}).
302 @return: Angular distance (C{radians}).
304 @raise TypeError: Invalid B{C{datum}}.
306 @see: Functions L{cosineForsytheAndoyerLambert}, L{cosineAndoyerLambert_},
307 L{cosineLaw_}, L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
308 L{flatPolar_}, L{haversine_}, L{thomas_} and L{vincentys_} and U{Geodesy-PHP
309 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/
310 Distance/ForsytheCorrection.php>}.
311 '''
312 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21)
313 if r and isnon0(c1) and isnon0(c2):
314 E = _ellipsoidal(datum, cosineForsytheAndoyerLambert_)
315 if E.f: # ellipsoidal
316 sr, cr, s2r, _ = sincos2_(r, r * _2_0)
317 if isnon0(sr) and fabs(cr) < EPS1:
318 s = (s1 + s2)**2 / (1 + cr)
319 t = (s1 - s2)**2 / (1 - cr)
320 x = s + t
321 y = s - t
323 s = 8 * r**2 / sr
324 a = 64 * r + _2_0 * s * cr # 16 * r**2 / tan(r)
325 d = 48 * sr + s # 8 * r**2 / tan(r)
326 b = -2 * d
327 e = 30 * s2r
328 c = fsum_(30 * r, e * _0_5, s * cr) # 8 * r**2 / tan(r)
330 t = fsum_( a * x, b * y, -c * x**2, d * x * y, e * y**2)
331 r += fsum_(-r * x, 3 * y * sr, t * E.f / _32_0) * E.f * _0_25
332 return r
335def cosineLaw(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
336 '''Compute the distance between two points using the
337 U{spherical Law of Cosines
338 <https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>}
339 formula.
341 @arg lat1: Start latitude (C{degrees}).
342 @arg lon1: Start longitude (C{degrees}).
343 @arg lat2: End latitude (C{degrees}).
344 @arg lon2: End longitude (C{degrees}).
345 @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter},
346 L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}).
347 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).
349 @return: Distance (C{meter}, same units as B{C{radius}} or the
350 ellipsoid or datum axes).
352 @raise TypeError: Invalid B{C{radius}}.
354 @see: Functions L{cosineLaw_}, L{cosineAndoyerLambert},
355 L{cosineForsytheAndoyerLambert}, L{equirectangular}, L{euclidean},
356 L{flatLocal}/L{hubeny}, L{flatPolar}, L{haversine}, L{thomas} and
357 L{vincentys} and method L{Ellipsoid.distance2}.
359 @note: See note at function L{vincentys_}.
360 '''
361 return _distanceToS(cosineLaw_, lat1, lat2, radius,
362 *unroll180(lon1, lon2, wrap=wrap))
365def cosineLaw_(phi2, phi1, lam21):
366 '''Compute the I{angular} distance between two points using the
367 U{spherical Law of Cosines
368 <https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>}
369 formula.
371 @arg phi2: End latitude (C{radians}).
372 @arg phi1: Start latitude (C{radians}).
373 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
375 @return: Angular distance (C{radians}).
377 @see: Functions L{cosineLaw}, L{cosineAndoyerLambert_},
378 L{cosineForsytheAndoyerLambert_}, L{equirectangular_},
379 L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_},
380 L{haversine_}, L{thomas_} and L{vincentys_}.
382 @note: See note at function L{vincentys_}.
383 '''
384 return _sincosa6(phi2, phi1, lam21)[4]
387def _distanceToE(func_, lat1, lat2, earth, d_lon, unused):
388 '''(INTERNAL) Helper for ellipsoidal distances.
389 '''
390 E = _ellipsoidal(earth, func_)
391 r = func_(Phi_(lat2=lat2),
392 Phi_(lat1=lat1), radians(d_lon), datum=E)
393 return r * E.a
396def _distanceToS(func_, lat1, lat2, earth, d_lon, unused, **adjust):
397 '''(INTERNAL) Helper for spherical distances.
398 '''
399 r = func_(Phi_(lat2=lat2),
400 Phi_(lat1=lat1), radians(d_lon), **adjust)
401 return r * _mean_radius(earth, lat1, lat2)
404def _ellipsoidal(earth, where):
405 '''(INTERNAL) Helper for distances.
406 '''
407 return earth if isinstance(earth, Ellipsoid) else (
408 earth if isinstance(earth, Datum) else
409 _ellipsoidal_datum(earth, name=where.__name__)).ellipsoid # PYCHOK indent
412def equirectangular(lat1, lon1, lat2, lon2, radius=R_M, **options):
413 '''Compute the distance between two points using
414 the U{Equirectangular Approximation / Projection
415 <https://www.Movable-Type.co.UK/scripts/latlong.html#equirectangular>}.
417 @arg lat1: Start latitude (C{degrees}).
418 @arg lon1: Start longitude (C{degrees}).
419 @arg lat2: End latitude (C{degrees}).
420 @arg lon2: End longitude (C{degrees}).
421 @kwarg radius: Mean earth radius, ellipsoid or datum
422 (C{meter}, L{Ellipsoid}, L{Ellipsoid2},
423 L{Datum} or L{a_f2Tuple}).
424 @kwarg options: Optional keyword arguments for function
425 L{equirectangular_}.
427 @return: Distance (C{meter}, same units as B{C{radius}} or
428 the ellipsoid or datum axes).
430 @raise TypeError: Invalid B{C{radius}}.
432 @see: Function L{equirectangular_} for more details, the
433 available B{C{options}}, errors, restrictions and other,
434 approximate or accurate distance functions.
435 '''
436 d = sqrt(equirectangular_(Lat(lat1=lat1), Lon(lon1=lon1),
437 Lat(lat2=lat2), Lon(lon2=lon2),
438 **options).distance2) # PYCHOK 4 vs 2-3
439 return degrees2m(d, radius=_mean_radius(radius, lat1, lat2))
442def equirectangular_(lat1, lon1, lat2, lon2,
443 adjust=True, limit=45, wrap=False):
444 '''Compute the distance between two points using
445 the U{Equirectangular Approximation / Projection
446 <https://www.Movable-Type.co.UK/scripts/latlong.html#equirectangular>}.
448 This approximation is valid for short distance of several
449 hundred Km or Miles, see the B{C{limit}} keyword argument and
450 the L{LimitError}.
452 @arg lat1: Start latitude (C{degrees}).
453 @arg lon1: Start longitude (C{degrees}).
454 @arg lat2: End latitude (C{degrees}).
455 @arg lon2: End longitude (C{degrees}).
456 @kwarg adjust: Adjust the wrapped, unrolled longitudinal delta
457 by the cosine of the mean latitude (C{bool}).
458 @kwarg limit: Optional limit for lat- and longitudinal deltas
459 (C{degrees}) or C{None} or C{0} for unlimited.
460 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).
462 @return: A L{Distance4Tuple}C{(distance2, delta_lat, delta_lon,
463 unroll_lon2)}.
465 @raise LimitError: If the lat- and/or longitudinal delta exceeds the
466 B{C{-limit..+limit}} range and L{pygeodesy.limiterrors}
467 set to C{True}.
469 @see: U{Local, flat earth approximation
470 <https://www.EdWilliams.org/avform.htm#flat>}, functions
471 L{equirectangular}, L{cosineAndoyerLambert},
472 L{cosineForsytheAndoyerLambert}, L{cosineLaw}, L{euclidean},
473 L{flatLocal}/L{hubeny}, L{flatPolar}, L{haversine}, L{thomas}
474 and L{vincentys} and methods L{Ellipsoid.distance2},
475 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
476 '''
477 d_lat = lat2 - lat1
478 d_lon, ulon2 = unroll180(lon1, lon2, wrap=wrap)
480 if limit and limit > 0 and limiterrors() and (fabs(d_lat) > limit or
481 fabs(d_lon) > limit):
482 t = unstr(equirectangular_, lat1, lon1, lat2, lon2, limit=limit)
483 raise LimitError('delta exceeds limit', txt=t)
485 if adjust: # scale delta lon
486 d_lon *= _scale_deg(lat1, lat2)
488 d2 = hypot2(d_lat, d_lon) # degrees squared!
489 return Distance4Tuple(d2, d_lat, d_lon, ulon2 - lon2)
492def euclidean(lat1, lon1, lat2, lon2, radius=R_M, adjust=True, wrap=False):
493 '''Approximate the C{Euclidean} distance between two (spherical) points.
495 @arg lat1: Start latitude (C{degrees}).
496 @arg lon1: Start longitude (C{degrees}).
497 @arg lat2: End latitude (C{degrees}).
498 @arg lon2: End longitude (C{degrees}).
499 @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter},
500 L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}).
501 @kwarg adjust: Adjust the longitudinal delta by the cosine of the
502 mean latitude (C{bool}).
503 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).
505 @return: Distance (C{meter}, same units as B{C{radius}} or the
506 ellipsoid or datum axes).
508 @raise TypeError: Invalid B{C{radius}}.
510 @see: U{Distance between two (spherical) points
511 <https://www.EdWilliams.org/avform.htm#Dist>}, functions L{euclid},
512 L{euclidean_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
513 L{cosineLaw}, L{equirectangular}, L{flatLocal}/L{hubeny}, L{flatPolar},
514 L{haversine}, L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2},
515 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
516 '''
517 return _distanceToS(euclidean_, lat1, lat2, radius,
518 *unroll180(lon1, lon2, wrap=wrap),
519 adjust=adjust)
522def euclidean_(phi2, phi1, lam21, adjust=True):
523 '''Approximate the I{angular} C{Euclidean} distance between two
524 (spherical) points.
526 @arg phi2: End latitude (C{radians}).
527 @arg phi1: Start latitude (C{radians}).
528 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
529 @kwarg adjust: Adjust the longitudinal delta by the cosine
530 of the mean latitude (C{bool}).
532 @return: Angular distance (C{radians}).
534 @see: Functions L{euclid}, L{euclidean}, L{cosineAndoyerLambert_},
535 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{equirectangular_},
536 L{flatLocal_}/L{hubeny_}, L{flatPolar_}, L{haversine_}, L{thomas_}
537 and L{vincentys_}.
538 '''
539 if adjust:
540 lam21 *= _scale_rad(phi2, phi1)
541 return euclid(phi2 - phi1, lam21)
544def excessAbc_(A, b, c):
545 '''Compute the I{spherical excess} C{E} of a (spherical) triangle
546 from two sides and the included angle.
548 @arg A: An interior triangle angle (C{radians}).
549 @arg b: Frist adjacent triangle side (C{radians}).
550 @arg c: Second adjacent triangle side (C{radians}).
552 @return: Spherical excess (C{radians}).
554 @raise UnitError: Invalid B{C{A}}, B{C{b}} or B{C{c}}.
556 @see: Functions L{excessGirard_}, L{excessLHuilier_} and U{Spherical
557 trigonometry<https://WikiPedia.org/wiki/Spherical_trigonometry>}.
558 '''
559 sA, cA, sb, cb, sc, cc = sincos2_(Radians_(A=A), Radians_(b=b) * _0_5,
560 Radians_(c=c) * _0_5)
561 return atan2(sA * sb * sc, cb * cc + cA * sb * sc) * _2_0
564def excessGirard_(A, B, C):
565 '''Compute the I{spherical excess} C{E} of a (spherical) triangle using
566 U{Girard's<https://MathWorld.Wolfram.com/GirardsSphericalExcessFormula.html>}
567 formula.
569 @arg A: First interior triangle angle (C{radians}).
570 @arg B: Second interior triangle angle (C{radians}).
571 @arg C: Third interior triangle angle (C{radians}).
573 @return: Spherical excess (C{radians}).
575 @raise UnitError: Invalid B{C{A}}, B{C{B}} or B{C{C}}.
577 @see: Function L{excessLHuilier_} and U{Spherical trigonometry
578 <https://WikiPedia.org/wiki/Spherical_trigonometry>}.
579 '''
580 return Radians(Girard=fsum_(Radians_(A=A),
581 Radians_(B=B),
582 Radians_(C=C), -PI))
585def excessLHuilier_(a, b, c):
586 '''Compute the I{spherical excess} C{E} of a (spherical) triangle using
587 U{L'Huilier's<https://MathWorld.Wolfram.com/LHuiliersTheorem.html>}
588 Theorem.
590 @arg a: First triangle side (C{radians}).
591 @arg b: Second triangle side (C{radians}).
592 @arg c: Third triangle side (C{radians}).
594 @return: Spherical excess (C{radians}).
596 @raise UnitError: Invalid B{C{a}}, B{C{b}} or B{C{c}}.
598 @see: Function L{excessGirard_} and U{Spherical trigonometry
599 <https://WikiPedia.org/wiki/Spherical_trigonometry>}.
600 '''
601 a = Radians_(a=a)
602 b = Radians_(b=b)
603 c = Radians_(c=c)
605 s = fsum_(a, b, c) * _0_5
606 r = tan_2(s) * tan_2(s - a) * tan_2(s - b) * tan_2(s - c)
607 r = atan(sqrt(r)) if r > 0 else _0_0
608 return Radians(LHuilier=r * _4_0)
611def excessKarney(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
612 '''Compute the surface area of a (spherical) quadrilateral bounded by a
613 segment of a great circle, two meridians and the equator using U{Karney's
614 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>}
615 method.
617 @arg lat1: Start latitude (C{degrees}).
618 @arg lon1: Start longitude (C{degrees}).
619 @arg lat2: End latitude (C{degrees}).
620 @arg lon2: End longitude (C{degrees}).
621 @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter}, L{Ellipsoid},
622 L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}) or C{None}.
623 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).
625 @return: Surface area, I{signed} (I{square} C{meter} or the same units as
626 B{C{radius}} I{squared}) or the I{spherical excess} (C{radians})
627 if C{B{radius}=0} or C{None}.
629 @raise TypeError: Invalid B{C{radius}}.
631 @raise UnitError: Invalid B{C{lat2}} or B{C{lat1}}.
633 @raise ValueError: Semi-circular longitudinal delta.
635 @see: Functions L{excessKarney_} and L{excessQuad}.
636 '''
637 return _area_or_(excessKarney_, lat1, lat2, radius,
638 *unroll180(lon1, lon2, wrap=wrap))
641def excessKarney_(phi2, phi1, lam21):
642 '''Compute the I{spherical excess} C{E} of a (spherical) quadrilateral bounded
643 by a segment of a great circle, two meridians and the equator using U{Karney's
644 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>}
645 method.
647 @arg phi2: End latitude (C{radians}).
648 @arg phi1: Start latitude (C{radians}).
649 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
651 @return: Spherical excess, I{signed} (C{radians}).
653 @raise ValueError: Semi-circular longitudinal delta B{C{lam21}}.
655 @see: Function L{excessKarney} and U{Area of a spherical polygon
656 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>}.
657 '''
658 # from: Veness <https://www.Movable-Type.co.UK/scripts/latlong.html> Area
659 # method due to Karney: for each edge of the polygon,
660 #
661 # tan(Δλ / 2) · (tan(φ1 / 2) + tan(φ2 / 2))
662 # tan(E / 2) = -----------------------------------------
663 # 1 + tan(φ1 / 2) · tan(φ2 / 2)
664 #
665 # where E is the spherical excess of the trapezium obtained by extending
666 # the edge to the equator-circle vector for each edge (see also ***).
667 t2 = tan_2(phi2)
668 t1 = tan_2(phi1)
669 t = tan_2(lam21, lam21=None)
670 return Radians(Karney=atan2(t * (t1 + t2),
671 _1_0 + (t1 * t2)) * _2_0)
674# ***) Original post no longer available, following is a copy of the main part
675# <http://OSGeo-org.1560.x6.Nabble.com/Area-of-a-spherical-polygon-td3841625.html>
676#
677# The area of a polygon on a (unit) sphere is given by the spherical excess
678#
679# A = 2 * pi - sum(exterior angles)
680#
681# However this is badly conditioned if the polygon is small. In this case, use
682#
683# A = sum(S12{i, i+1}) over the edges of the polygon
684#
685# where S12 is the area of the quadrilateral bounded by an edge of the polygon,
686# two meridians and the equator, i.e. with vertices (phi1, lambda1), (phi2,
687# lambda2), (0, lambda1) and (0, lambda2). S12 is given by
688#
689# tan(S12 / 2) = tan(lambda21 / 2) * (tan(phi1 / 2) + tan(phi2 / 2)) /
690# (tan(phi1 / 2) * tan(phi2 / 2) + 1)
691#
692# = tan(lambda21 / 2) * tanh((Lambertian(phi1) +
693# Lambertian(phi2)) / 2)
694#
695# where lambda21 = lambda2 - lambda1 and lamb(x) is the Lambertian (or
696# inverse Gudermannian) function
697#
698# Lambertian(x) = asinh(tan(x)) = atanh(sin(x)) = 2 * atanh(tan(x / 2))
699#
700# Notes: The formula for S12 is exact, except that...
701# - it is indeterminate if an edge is a semi-circle
702# - the formula for A applies only if the polygon does not include a pole
703# (if it does, then add +/- 2 * pi to the result)
704# - in the limit of small phi and lambda, S12 reduces to the trapezoidal
705# formula, S12 = (lambda2 - lambda1) * (phi1 + phi2) / 2
706# - I derived this result from the equation for the area of a spherical
707# triangle in terms of two edges and the included angle given by, e.g.
708# U{Todhunter, I. - Spherical Trigonometry (1871), Sec. 103, Eq. (2)
709# <http://Books.Google.com/books?id=3uBHAAAAIAAJ&pg=PA71>}
710# - I would be interested to know if this formula for S12 is already known
711# - Charles Karney
714def excessQuad(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
715 '''Compute the surface area of a (spherical) quadrilateral bounded by a segment
716 of a great circle, two meridians and the equator.
718 @arg lat1: Start latitude (C{degrees}).
719 @arg lon1: Start longitude (C{degrees}).
720 @arg lat2: End latitude (C{degrees}).
721 @arg lon2: End longitude (C{degrees}).
722 @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter},
723 L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}) or C{None}.
724 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).
726 @return: Surface area, I{signed} (I{square} C{meter} or the same units as
727 B{C{radius}} I{squared}) or the I{spherical excess} (C{radians})
728 if C{B{radius}=0} or C{None}.
730 @raise TypeError: Invalid B{C{radius}}.
732 @raise UnitError: Invalid B{C{lat2}} or B{C{lat1}}.
734 @see: Function L{excessQuad_} and L{excessKarney}.
735 '''
736 return _area_or_(excessQuad_, lat1, lat2, radius,
737 *unroll180(lon1, lon2, wrap=wrap))
740def excessQuad_(phi2, phi1, lam21):
741 '''Compute the I{spherical excess} C{E} of a (spherical) quadrilateral bounded
742 by a segment of a great circle, two meridians and the equator.
744 @arg phi2: End latitude (C{radians}).
745 @arg phi1: Start latitude (C{radians}).
746 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
748 @return: Spherical excess, I{signed} (C{radians}).
750 @see: Function L{excessQuad}, U{Spherical trigonometry
751 <https://WikiPedia.org/wiki/Spherical_trigonometry>}.
752 '''
753 s = sin((phi2 + phi1) * _0_5)
754 c = cos((phi2 - phi1) * _0_5)
755 return Radians(Quad=atan2(tan_2(lam21) * s, c) * _2_0)
758def flatLocal(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):
759 '''Compute the distance between two (ellipsoidal) points using
760 the U{ellipsoidal Earth to plane projection<https://WikiPedia.org/
761 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>}
762 aka U{Hubeny<https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula.
764 @arg lat1: Start latitude (C{degrees}).
765 @arg lon1: Start longitude (C{degrees}).
766 @arg lat2: End latitude (C{degrees}).
767 @arg lon2: End longitude (C{degrees}).
768 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid},
769 L{Ellipsoid2} or L{a_f2Tuple}).
770 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).
772 @return: Distance (C{meter}, same units as the B{C{datum}}'s
773 ellipsoid axes).
775 @raise TypeError: Invalid B{C{datum}}.
777 @note: The meridional and prime_vertical radii of curvature
778 are taken and scaled at the mean of both latitude.
780 @see: Functions L{flatLocal_} or L{hubeny_}, L{cosineLaw}, L{flatPolar},
781 L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
782 L{equirectangular}, L{euclidean}, L{haversine}, L{thomas},
783 L{vincentys}, method L{Ellipsoid.distance2} and U{local, flat
784 earth approximation<https://www.EdWilliams.org/avform.htm#flat>}.
785 '''
786 d, _ = unroll180(lon1, lon2, wrap=wrap)
787 return flatLocal_(Phi_(lat2=lat2),
788 Phi_(lat1=lat1), radians(d), datum=datum)
790hubeny = flatLocal # PYCHOK for Karl Hubeny
793def flatLocal_(phi2, phi1, lam21, datum=_WGS84):
794 '''Compute the I{angular} distance between two (ellipsoidal) points using
795 the U{ellipsoidal Earth to plane projection<https://WikiPedia.org/
796 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>}
797 aka U{Hubeny<https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula.
799 @arg phi2: End latitude (C{radians}).
800 @arg phi1: Start latitude (C{radians}).
801 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
802 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid},
803 L{Ellipsoid2} or L{a_f2Tuple}).
805 @return: Angular distance (C{radians}).
807 @raise TypeError: Invalid B{C{datum}}.
809 @note: The meridional and prime_vertical radii of curvature
810 are taken and scaled I{at the mean of both latitude}.
812 @see: Functions L{flatLocal} or L{hubeny}, L{cosineAndoyerLambert_},
813 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{flatPolar_},
814 L{equirectangular_}, L{euclidean_}, L{haversine_}, L{thomas_}
815 and L{vincentys_} and U{local, flat earth approximation
816 <https://www.EdWilliams.org/avform.htm#flat>}.
817 '''
818 E = _ellipsoidal(datum, flatLocal_)
819 m, n = E.roc2_((phi2 + phi1) * _0_5, scaled=True)
820 return hypot(m * (phi2 - phi1), n * lam21)
822hubeny_ = flatLocal_ # PYCHOK for Karl Hubeny
825def flatPolar(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
826 '''Compute the distance between two (spherical) points using
827 the U{polar coordinate flat-Earth <https://WikiPedia.org/wiki/
828 Geographical_distance#Polar_coordinate_flat-Earth_formula>}
829 formula.
831 @arg lat1: Start latitude (C{degrees}).
832 @arg lon1: Start longitude (C{degrees}).
833 @arg lat2: End latitude (C{degrees}).
834 @arg lon2: End longitude (C{degrees}).
835 @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter},
836 L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}).
837 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).
839 @return: Distance (C{meter}, same units as B{C{radius}} or the
840 ellipsoid or datum axes).
842 @raise TypeError: Invalid B{C{radius}}.
844 @see: Functions L{flatPolar_}, L{cosineAndoyerLambert},
845 L{cosineForsytheAndoyerLambert},L{cosineLaw},
846 L{flatLocal}/L{hubeny}, L{equirectangular},
847 L{euclidean}, L{haversine}, L{thomas} and
848 L{vincentys}.
849 '''
850 return _distanceToS(flatPolar_, lat1, lat2, radius,
851 *unroll180(lon1, lon2, wrap=wrap))
854def flatPolar_(phi2, phi1, lam21):
855 '''Compute the I{angular} distance between two (spherical) points
856 using the U{polar coordinate flat-Earth<https://WikiPedia.org/wiki/
857 Geographical_distance#Polar_coordinate_flat-Earth_formula>}
858 formula.
860 @arg phi2: End latitude (C{radians}).
861 @arg phi1: Start latitude (C{radians}).
862 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
864 @return: Angular distance (C{radians}).
866 @see: Functions L{flatPolar}, L{cosineAndoyerLambert_},
867 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
868 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
869 L{haversine_}, L{thomas_} and L{vincentys_}.
870 '''
871 a = fabs(PI_2 - phi1) # co-latitude
872 b = fabs(PI_2 - phi2) # co-latitude
873 if a < b:
874 a, b = b, a
875 if a < EPS0:
876 a = _0_0
877 elif b > 0:
878 b = b / a # /= chokes PyChecker
879 c = b * cos(lam21) * _2_0
880 c = fsum_(_1_0, b**2, -fabs(c))
881 a *= sqrt0(c)
882 return a
885def hartzell(pov, los=None, earth=_WGS84, name=NN, **LatLon_and_kwds):
886 '''Compute the intersection of the earth's surface and a Line-Of-Sight
887 from a Point-Of-View in space.
889 @arg pov: Point-Of-View outside the earth (C{Cartesian}, L{Ecef9Tuple}
890 or L{Vector3d}).
891 @kwarg los: Line-Of-Sight, I{direction} to earth (L{Vector3d}) or
892 C{None} to point to the earth' center.
893 @kwarg earth: The earth model (L{Datum}, L{Ellipsoid}, L{Ellipsoid2},
894 L{a_f2Tuple} or C{scalar} radius in C{meter}).
895 @kwarg name: Optional name (C{str}).
896 @kwarg LatLon_and_kwds: Optional C{LatLon} class for the intersection
897 point plus C{LatLon} keyword arguments, include
898 B{C{datum}} if different from B{C{earth}}.
900 @return: The earth intersection (L{Vector3d}, C{Cartesian type} of
901 B{C{pov}} or B{C{LatLon}}).
903 @raise IntersectionError: Null B{C{pov}} or B{C{los}} vector, B{C{pov}}
904 is inside the earth or B{C{los}} points outside
905 the earth or points in an opposite direction.
907 @raise TypeError: Invalid B{C{pov}}, B{C{los}} or B{C{earth}}.
909 @see: Function L{pygeodesy.hartzell4}, L{pygeodesy.tyr3d} for B{C{los}} and
910 U{I{Satellite Line-of-Sight Intersection with Earth}<https://StephenHartzell.
911 Medium.com/satellite-line-of-sight-intersection-with-earth-d786b4a6a9b6>}.
912 '''
913 def _Error(txt):
914 return IntersectionError(pov=pov, los=los, earth=earth, txt=txt)
916 D = earth if isinstance(earth, Datum) else \
917 _spherical_datum(earth, name=hartzell.__name__)
918 E = D.ellipsoid
920 if E.b > E.a: # PYCHOK no cover
921 try:
922 t = _MODS.triaxials
923 r, _ = t._hartzell3d2(pov, los, t.Triaxial_(E.a, E.a, E.b))
924 except Exception as x:
925 raise _Error(str(x))
926 else:
927 a2 = b2 = E.a2 # earth' x, y, ...
928 c2 = E.b2 # ... z semi-axis squared
929 q2 = E.b2_a2 # == c2 / a2
930 bc = E.a * E.b # == b * c
932 V3 = _MODS.vector3d._otherV3d
933 p3 = V3(pov=pov)
934 u3 = V3(los=los) if los else p3.negate()
935 u3 = u3.unit() # unit vector, opposing signs
937 x2, y2, z2 = p3.x2y2z2 # p3.times_(p3).xyz
938 ux, vy, wz = u3.times_(p3).xyz
939 u2, v2, w2 = u3.x2y2z2 # u3.times_(u3).xyz
941 t = c2, c2, b2
942 m = fdot(t, u2, v2, w2) # a2 factored out
943 if m < EPS0: # zero or near-null LOS vector
944 raise _Error(_near_(_null_))
946 # a2 and b2 factored out, b2 == a2 and b2 / a2 == 1
947 r = fsum_(b2 * w2, c2 * v2, -v2 * z2, vy * wz * 2,
948 c2 * u2, -u2 * z2, -w2 * x2, ux * wz * 2,
949 -w2 * y2, -u2 * y2 * q2, -v2 * x2 * q2, ux * vy * 2 * q2, floats=True)
950 if r > 0:
951 r = sqrt(r) * bc # == a * a * b * c / a2
952 elif r < 0: # LOS pointing away from or missing the earth
953 raise _Error(_opposite_ if max(ux, vy, wz) > 0 else _outside_)
955 d = Fdot(t, ux, vy, wz).fadd_(r).fover(m) # -r for antipode, a2 factored out
956 if d > 0: # POV inside or LOS missing, outside the earth
957 s = fsum_(_1_0, x2 / a2, y2 / b2, z2 / c2, _N_2_0, floats=True) # like _sideOf
958 raise _Error(_outside_ if s > 0 else _inside_)
959 elif fsum_(x2, y2, z2) < d**2: # d past earth center
960 raise _Error(_too_(_distant_))
962 r = p3.minus(u3.times(d))
963# h = p3.minus(r).length # distance to ellipsoid
965 r = _xnamed(r, name or hartzell.__name__)
966 if LatLon_and_kwds:
967 c = _MODS.cartesianBase.CartesianBase(r, datum=D, name=r.name)
968 r = c.toLatLon(**LatLon_and_kwds)
969 return r
972def haversine(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
973 '''Compute the distance between two (spherical) points using the
974 U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>}
975 formula.
977 @arg lat1: Start latitude (C{degrees}).
978 @arg lon1: Start longitude (C{degrees}).
979 @arg lat2: End latitude (C{degrees}).
980 @arg lon2: End longitude (C{degrees}).
981 @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter},
982 L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}).
983 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).
985 @return: Distance (C{meter}, same units as B{C{radius}}).
987 @raise TypeError: Invalid B{C{radius}}.
989 @see: U{Distance between two (spherical) points
990 <https://www.EdWilliams.org/avform.htm#Dist>}, functions
991 L{cosineLaw}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
992 L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar},
993 L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2},
994 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
996 @note: See note at function L{vincentys_}.
997 '''
998 return _distanceToS(haversine_, lat1, lat2, radius,
999 *unroll180(lon1, lon2, wrap=wrap))
1002def haversine_(phi2, phi1, lam21):
1003 '''Compute the I{angular} distance between two (spherical) points
1004 using the U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>}
1005 formula.
1007 @arg phi2: End latitude (C{radians}).
1008 @arg phi1: Start latitude (C{radians}).
1009 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
1011 @return: Angular distance (C{radians}).
1013 @see: Functions L{haversine}, L{cosineAndoyerLambert_},
1014 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
1015 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
1016 L{flatPolar_}, L{thomas_} and L{vincentys_}.
1018 @note: See note at function L{vincentys_}.
1019 '''
1020 def _hsin(rad):
1021 return sin(rad * _0_5)**2
1023 h = _hsin(phi2 - phi1) + cos(phi1) * cos(phi2) * _hsin(lam21) # haversine
1024 return atan2(sqrt0(h), sqrt0(_1_0 - h)) * _2_0 # == asin(sqrt(h)) * 2
1027def heightOf(angle, distance, radius=R_M):
1028 '''Determine the height above the (spherical) earth' surface after
1029 traveling along a straight line at a given tilt.
1031 @arg angle: Tilt angle above horizontal (C{degrees}).
1032 @arg distance: Distance along the line (C{meter} or same units as
1033 B{C{radius}}).
1034 @kwarg radius: Optional mean earth radius (C{meter}).
1036 @return: Height (C{meter}, same units as B{C{distance}} and B{C{radius}}).
1038 @raise ValueError: Invalid B{C{angle}}, B{C{distance}} or B{C{radius}}.
1040 @see: U{MultiDop geog_lib.GeogBeamHt<https://GitHub.com/NASA/MultiDop>}
1041 (U{Shapiro et al. 2009, JTECH
1042 <https://Journals.AMetSoc.org/doi/abs/10.1175/2009JTECHA1256.1>}
1043 and U{Potvin et al. 2012, JTECH
1044 <https://Journals.AMetSoc.org/doi/abs/10.1175/JTECH-D-11-00019.1>}).
1045 '''
1046 r = h = Radius(radius)
1047 d = fabs(Distance(distance))
1048 if d > h:
1049 d, h = h, d
1051 if d > EPS0: # and h > EPS0
1052 d = d / h # /= h chokes PyChecker
1053 s = sin(Phi_(angle=angle, clip=_180_0))
1054 s = fsum_(_1_0, _2_0 * s * d, d**2)
1055 if s > 0:
1056 return h * sqrt(s) - r
1058 raise _ValueError(angle=angle, distance=distance, radius=radius)
1061def horizon(height, radius=R_M, refraction=False):
1062 '''Determine the distance to the horizon from a given altitude
1063 above the (spherical) earth.
1065 @arg height: Altitude (C{meter} or same units as B{C{radius}}).
1066 @kwarg radius: Optional mean earth radius (C{meter}).
1067 @kwarg refraction: Consider atmospheric refraction (C{bool}).
1069 @return: Distance (C{meter}, same units as B{C{height}} and B{C{radius}}).
1071 @raise ValueError: Invalid B{C{height}} or B{C{radius}}.
1073 @see: U{Distance to horizon<https://www.EdWilliams.org/avform.htm#Horizon>}.
1074 '''
1075 h, r = Height(height), Radius(radius)
1076 if min(h, r) < 0:
1077 raise _ValueError(height=height, radius=radius)
1079 if refraction:
1080 d2 = 2.415750694528 * h * r # 2.0 / 0.8279
1081 else:
1082 d2 = h * fsum_(r, r, h)
1083 return sqrt0(d2)
1086def _idlmn5(datum, lat1, lon1, lat2, lon2, wrap, s):
1087 '''(INTERNAL) Helper for C{intersection2} and C{intersections2}.
1088 '''
1089 n = (intersections2 if s else intersection2).__name__
1090 if datum is None or euclidean(lat1, lon1, lat2, lon2, radius=R_M,
1091 adjust=True, wrap=wrap) < _100km:
1092 d, m = None, _MODS.vector3d
1093 _i = m._intersects2 if s else m._intersect3d3
1094 _, lon2 = unroll180(lon1, lon2, wrap=wrap)
1095 elif isscalar(datum) and datum < 0 and not s:
1096 d = _spherical_datum(-datum, name=n)
1097 m = _MODS.sphericalNvector
1098 _i = m.intersection
1099 else:
1100 d = _spherical_datum(datum, name=n)
1101 if d.isSpherical:
1102 m = _MODS.sphericalTrigonometry
1103 _i = m._intersects2 if s else m._intersect
1104 elif d.isEllipsoidal:
1105 try:
1106 if d.ellipsoid.geodesic:
1107 pass
1108 m = _MODS.ellipsoidalKarney
1109 except ImportError:
1110 m = _MODS.ellipsoidalExact
1111 _i = m._intersections2 if s else m._intersection3 # ellispoidalBaseDi
1112 else:
1113 raise _AssertionError(datum=datum)
1114 return _i, d, lon2, m, n
1117def intersection2(lat1, lon1, bearing1,
1118 lat2, lon2, bearing2, datum=None, wrap=True):
1119 '''I{Conveniently} compute the intersection of two lines each defined
1120 by a (geodetic) point and a bearing from North, using either ...
1122 1) L{vector3d.intersection3d3} for small distances (below 100 KM or
1123 about 0.9 degrees) or if no B{C{datum}} is specified, or ...
1125 2) L{sphericalTrigonometry.intersection} for a spherical B{C{datum}}
1126 or if B{C{datum}} is a C{scalar} representing the earth
1127 radius, conventionally in C{meter} or ...
1129 3) L{sphericalNvector.intersection} if B{C{datum}} is a I{negative}
1130 C{scalar}, (negative) earth radius, conventionally in C{meter} or ...
1132 4) L{ellipsoidalKarney.intersection3} for an ellipsoidal B{C{datum}}
1133 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>}
1134 is installed, otherwise ...
1136 5) L{ellipsoidalExact.intersection3}, provided B{C{datum}} is ellipsoidal.
1138 @arg lat1: Latitude of the first point (C{degrees}).
1139 @arg lon1: Longitude of the first point (C{degrees}).
1140 @arg bearing1: Bearing at the first point (compass C{degrees}).
1141 @arg lat2: Latitude of the second point (C{degrees}).
1142 @arg lon2: Longitude of the second point (C{degrees}).
1143 @arg bearing2: Bearing at the second point (compass C{degrees}).
1144 @kwarg datum: Optional ellipsoidal or spherical datum (L{Datum},
1145 L{Ellipsoid}, L{Ellipsoid2}, L{a_f2Tuple} or
1146 C{scalar} earth radius in C{meter}) or C{None}.
1147 @kwarg wrap: Wrap and unroll longitudes (C{bool}).
1149 @return: A L{LatLon2Tuple}C{(lat, lon)} with the lat- and
1150 longitude of the intersection point.
1152 @raise IntersectionError: Ambiguous or infinite intersection
1153 or colinear, parallel or otherwise
1154 non-intersecting lines.
1156 @raise TypeError: Invalid B{C{datum}}.
1158 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{bearing1}},
1159 B{C{lat2}}, B{C{lon2}} or B{C{bearing2}}.
1161 @see: Method L{RhumbLine.intersection2}.
1163 @note: The returned intersections may be antipodal or near-antipodal.
1164 '''
1165 b1, b2 = Bearing(bearing1=bearing1), Bearing(bearing2=bearing2)
1166 try:
1167 _i, d, l2, m, n = _idlmn5(datum, lat1, lon1, lat2, lon2, wrap, False)
1168 if d is None:
1169 t, _, _ = _i(m.Vector3d(lon1, lat1, 0), b1,
1170 m.Vector3d(l2, lat2, 0), b2, useZ=False)
1171 t = LatLon2Tuple(t.y, t.x, name=n)
1173 else:
1174 t = _i(m.LatLon(lat1, lon1, datum=d), b1,
1175 m.LatLon(lat2, lon2, datum=d), b2, height=0, wrap=wrap)
1176 if isinstance(t, Intersection3Tuple): # ellipsoidal
1177 t, _, _ = t
1178 t = LatLon2Tuple(t.lat, t.lon, name=n)
1180 except (TypeError, ValueError) as x:
1181 raise _xError(x, lat1=lat1, lon1=lon1, bearing1=bearing1,
1182 lat2=lat2, lon2=lon2, bearing2=bearing2,
1183 datum=datum, wrap=wrap)
1184 return t
1187def intersections2(lat1, lon1, radius1,
1188 lat2, lon2, radius2, datum=None, wrap=True):
1189 '''I{Conveniently} compute the intersections of two circles each defined
1190 by a (geodetic) center point and a radius, using either ...
1192 1) L{vector3d.intersections2} for small distances (below 100 KM or
1193 about 0.9 degrees) or if no B{C{datum}} is specified, or ...
1195 2) L{sphericalTrigonometry.intersections2} for a spherical B{C{datum}}
1196 or if B{C{datum}} is a C{scalar} representing the earth radius,
1197 conventionally in C{meter} or ...
1199 3) L{ellipsoidalKarney.intersections2} for an ellipsoidal B{C{datum}}
1200 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>}
1201 is installed, otherwise ...
1203 4) L{ellipsoidalExact.intersections2}, provided B{C{datum}} is ellipsoidal.
1205 @arg lat1: Latitude of the first circle center (C{degrees}).
1206 @arg lon1: Longitude of the first circle center (C{degrees}).
1207 @arg radius1: Radius of the first circle (C{meter}, conventionally).
1208 @arg lat2: Latitude of the second circle center (C{degrees}).
1209 @arg lon2: Longitude of the second circle center (C{degrees}).
1210 @arg radius2: Radius of the second circle (C{meter}, same units as B{C{radius1}}).
1211 @kwarg datum: Optional ellipsoidal or spherical datum (L{Datum},
1212 L{Ellipsoid}, L{Ellipsoid2}, L{a_f2Tuple} or
1213 C{scalar} earth radius in C{meter}, same units as
1214 B{C{radius1}} and B{C{radius2}}) or C{None}.
1215 @kwarg wrap: Wrap and unroll longitudes (C{bool}).
1217 @return: 2-Tuple of the intersection points, each a
1218 L{LatLon2Tuple}C{(lat, lon)}. For abutting circles, the
1219 points are the same instance, aka the I{radical center}.
1221 @raise IntersectionError: Concentric, antipodal, invalid or
1222 non-intersecting circles or no
1223 convergence.
1225 @raise TypeError: Invalid B{C{datum}}.
1227 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{radius1}},
1228 B{C{lat2}}, B{C{lon2}} or B{C{radius2}}.
1229 '''
1230 r1, r2 = Radius_(radius1=radius1), Radius_(radius2=radius2)
1231 try:
1232 _i, d, l2, m, n = _idlmn5(datum, lat1, lon1, lat2, lon2, wrap, True)
1233 if d is None:
1234 r1 = m2degrees(r1, radius=R_M, lat=lat1)
1235 r2 = m2degrees(r2, radius=R_M, lat=lat2)
1237 def _V2T(x, y, _, **unused): # _ == z unused
1238 return LatLon2Tuple(y, x, name=n)
1240 t = _i(m.Vector3d(lon1, lat1, 0), r1,
1241 m.Vector3d(l2, lat2, 0), r2, sphere=False,
1242 Vector=_V2T)
1243 else:
1245 def _LL2T(lat, lon, **unused):
1246 return LatLon2Tuple(lat, lon, name=n)
1248 t = _i(m.LatLon(lat1, lon1, datum=d), r1,
1249 m.LatLon(lat2, lon2, datum=d), r2,
1250 LatLon=_LL2T, height=0, wrap=wrap)
1252 except (TypeError, ValueError) as x:
1253 raise _xError(x, lat1=lat1, lon1=lon1, radius1=radius1,
1254 lat2=lat2, lon2=lon2, radius2=radius2,
1255 datum=datum, wrap=wrap)
1256 return t
1259def isantipode(lat1, lon1, lat2, lon2, eps=EPS):
1260 '''Check whether two points are antipodal, on diametrically
1261 opposite sides of the earth.
1263 @arg lat1: Latitude of one point (C{degrees}).
1264 @arg lon1: Longitude of one point (C{degrees}).
1265 @arg lat2: Latitude of the other point (C{degrees}).
1266 @arg lon2: Longitude of the other point (C{degrees}).
1267 @kwarg eps: Tolerance for near-equality (C{degrees}).
1269 @return: C{True} if points are antipodal within the
1270 B{C{eps}} tolerance, C{False} otherwise.
1272 @see: Functions L{isantipode_} and L{antipode}.
1273 '''
1274 return True if (fabs(lat1 + lat2) <= eps and
1275 fabs(lon1 + lon2) <= eps) else \
1276 _MODS.latlonBase._isequalTo(antipode(lat1, lon1),
1277 normal(lat2, lon2), eps=eps)
1280def isantipode_(phi1, lam1, phi2, lam2, eps=EPS):
1281 '''Check whether two points are antipodal, on diametrically
1282 opposite sides of the earth.
1284 @arg phi1: Latitude of one point (C{radians}).
1285 @arg lam1: Longitude of one point (C{radians}).
1286 @arg phi2: Latitude of the other point (C{radians}).
1287 @arg lam2: Longitude of the other point (C{radians}).
1288 @kwarg eps: Tolerance for near-equality (C{radians}).
1290 @return: C{True} if points are antipodal within the
1291 B{C{eps}} tolerance, C{False} otherwise.
1293 @see: Functions L{isantipode} and L{antipode_}.
1294 '''
1295 return True if (fabs(phi1 + phi2) <= eps and
1296 fabs(lam1 + lam2) <= eps) else \
1297 _MODS.latlonBase._isequalTo_(antipode_(phi1, lam1),
1298 normal_(phi2, lam2), eps=eps)
1301def isnormal(lat, lon, eps=0):
1302 '''Check whether B{C{lat}} I{and} B{C{lon}} are within the I{normal}
1303 range in C{degrees}.
1305 @arg lat: Latitude (C{degrees}).
1306 @arg lon: Longitude (C{degrees}).
1307 @kwarg eps: Optional tolerance below C{90} and C{180 degrees}.
1309 @return: C{True} if C{(abs(B{lat}) + B{eps}) <= 90} and
1310 C{(abs(B{lon}) + B{eps}) <= 180}, C{False} othwerwise.
1312 @see: Functions L{isnormal_} and L{normal}.
1313 '''
1314 return (_90_0 - fabs(lat)) >= eps and (_180_0 - fabs(lon)) >= eps
1317def isnormal_(phi, lam, eps=0):
1318 '''Check whether B{C{phi}} I{and} B{C{lam}} are within the I{normal}
1319 range in C{radians}.
1321 @arg phi: Latitude (C{radians}).
1322 @arg lam: Longitude (C{radians}).
1323 @kwarg eps: Optional tolerance below C{PI/2} and C{PI radians}.
1325 @return: C{True} if C{(abs(B{phi}) + B{eps}) <= PI/2} and
1326 C{(abs(B{lam}) + B{eps}) <= PI}, C{False} othwerwise.
1328 @see: Functions L{isnormal} and L{normal_}.
1329 '''
1330 return (PI_2 - fabs(phi)) >= eps and (PI - fabs(lam)) >= eps
1333def latlon2n_xyz(lat, lon, name=NN):
1334 '''Convert lat-, longitude to C{n-vector} (I{normal} to the
1335 earth's surface) X, Y and Z components.
1337 @arg lat: Latitude (C{degrees}).
1338 @arg lon: Longitude (C{degrees}).
1339 @kwarg name: Optional name (C{str}).
1341 @return: A L{Vector3Tuple}C{(x, y, z)}.
1343 @see: Function L{philam2n_xyz}.
1345 @note: These are C{n-vector} x, y and z components,
1346 I{NOT} geocentric ECEF x, y and z coordinates!
1347 '''
1348 return _2n_xyz(name, *sincos2d_(lat, lon))
1351def _normal2(a, b, n_2, n, n2):
1352 '''(INTERNAL) Helper for C{normal} and C{normal_}.
1353 '''
1354 if fabs(b) > n:
1355 b = remainder(b, n2)
1356 r = remainder(a, n) if fabs(a) > n_2 else a
1357 if r != a:
1358 r = -r
1359 b -= n if b > 0 else -n
1360 return float0(r, b)
1363def normal(lat, lon, name=NN):
1364 '''Normalize a lat- I{and} longitude pair in C{degrees}.
1366 @arg lat: Latitude (C{degrees}).
1367 @arg lon: Longitude (C{degrees}).
1368 @kwarg name: Optional name (C{str}).
1370 @return: L{LatLon2Tuple}C{(lat, lon)} with C{abs(lat) <= 90}
1371 and C{abs(lon) <= 180}.
1373 @see: Functions L{normal_} and L{isnormal}.
1374 '''
1375 return LatLon2Tuple(*_normal2(lat, lon, _90_0, _180_0, _360_0), name=name)
1378def normal_(phi, lam, name=NN):
1379 '''Normalize a lat- I{and} longitude pair in C{radians}.
1381 @arg phi: Latitude (C{radians}).
1382 @arg lam: Longitude (C{radians}).
1383 @kwarg name: Optional name (C{str}).
1385 @return: L{PhiLam2Tuple}C{(phi, lam)} with C{abs(phi) <= PI/2}
1386 and C{abs(lam) <= PI}.
1388 @see: Functions L{normal} and L{isnormal_}.
1389 '''
1390 return PhiLam2Tuple(*_normal2(phi, lam, PI_2, PI, PI2), name=name)
1393def _2n_xyz(name, sa, ca, sb, cb):
1394 '''(INTERNAL) Helper for C{latlon2n_xyz} and C{philam2n_xyz}.
1395 '''
1396 # Kenneth Gade eqn 3, but using right-handed
1397 # vector x -> 0°E,0°N, y -> 90°E,0°N, z -> 90°N
1398 return Vector3Tuple(ca * cb, ca * sb, sa, name=name)
1401def n_xyz2latlon(x, y, z, name=NN):
1402 '''Convert C{n-vector} components to lat- and longitude in C{degrees}.
1404 @arg x: X component (C{scalar}).
1405 @arg y: Y component (C{scalar}).
1406 @arg z: Z component (C{scalar}).
1407 @kwarg name: Optional name (C{str}).
1409 @return: A L{LatLon2Tuple}C{(lat, lon)}.
1411 @see: Function L{n_xyz2philam}.
1412 '''
1413 return LatLon2Tuple(atan2d(z, hypot(x, y)), atan2d(y, x), name=name)
1416def n_xyz2philam(x, y, z, name=NN):
1417 '''Convert C{n-vector} components to lat- and longitude in C{radians}.
1419 @arg x: X component (C{scalar}).
1420 @arg y: Y component (C{scalar}).
1421 @arg z: Z component (C{scalar}).
1422 @kwarg name: Optional name (C{str}).
1424 @return: A L{PhiLam2Tuple}C{(phi, lam)}.
1426 @see: Function L{n_xyz2latlon}.
1427 '''
1428 return PhiLam2Tuple(atan2(z, hypot(x, y)), atan2(y, x), name=name)
1431def _opposes(d, m, n, n2):
1432 '''(INETNAL) Helper for C{opposing} and C{opposing_}.
1433 '''
1434 d = d % n2 # -20 % 360 == 340, -1 % PI2 == PI2 - 1
1435 return False if d < m or d > (n2 - m) else (
1436 True if (n - m) < d < (n + m) else None)
1439def opposing(bearing1, bearing2, margin=_90_0):
1440 '''Compare the direction of two bearings given in C{degrees}.
1442 @arg bearing1: First bearing (compass C{degrees}).
1443 @arg bearing2: Second bearing (compass C{degrees}).
1444 @kwarg margin: Optional, interior angle bracket (C{degrees}).
1446 @return: C{True} if both bearings point in opposite, C{False} if
1447 in similar or C{None} if in perpendicular directions.
1449 @see: Function L{opposing_}.
1450 '''
1451 m = Degrees_(margin=margin, low=EPS0, high=_90_0)
1452 return _opposes(bearing2 - bearing1, m,_180_0, _360_0)
1455def opposing_(radians1, radians2, margin=PI_2):
1456 '''Compare the direction of two bearings given in C{radians}.
1458 @arg radians1: First bearing (C{radians}).
1459 @arg radians2: Second bearing (C{radians}).
1460 @kwarg margin: Optional, interior angle bracket (C{radians}).
1462 @return: C{True} if both bearings point in opposite, C{False} if
1463 in similar or C{None} if in perpendicular directions.
1465 @see: Function L{opposing}.
1466 '''
1467 m = Radians_(margin=margin, low=EPS0, high=PI_2)
1468 return _opposes(radians2 - radians1, m, PI, PI2)
1471def philam2n_xyz(phi, lam, name=NN):
1472 '''Convert lat-, longitude to C{n-vector} (I{normal} to the
1473 earth's surface) X, Y and Z components.
1475 @arg phi: Latitude (C{radians}).
1476 @arg lam: Longitude (C{radians}).
1477 @kwarg name: Optional name (C{str}).
1479 @return: A L{Vector3Tuple}C{(x, y, z)}.
1481 @see: Function L{latlon2n_xyz}.
1483 @note: These are C{n-vector} x, y and z components,
1484 I{NOT} geocentric ECEF x, y and z coordinates!
1485 '''
1486 return _2n_xyz(name, *sincos2_(phi, lam))
1489def _radical2(d, r1, r2): # in .ellipsoidalBaseDI, .sphericalTrigonometry, .vector3d
1490 # (INTERNAL) See C{radical2} below
1491 # assert d > EPS0
1492 r = fsum_(_1_0, (r1 / d)**2, -(r2 / d)**2) * _0_5
1493 return Radical2Tuple(max(_0_0, min(_1_0, r)), r * d)
1496def radical2(distance, radius1, radius2):
1497 '''Compute the I{radical ratio} and I{radical line} of two
1498 U{intersecting circles<https://MathWorld.Wolfram.com/
1499 Circle-CircleIntersection.html>}.
1501 The I{radical line} is perpendicular to the axis thru the
1502 centers of the circles at C{(0, 0)} and C{(B{distance}, 0)}.
1504 @arg distance: Distance between the circle centers (C{scalar}).
1505 @arg radius1: Radius of the first circle (C{scalar}).
1506 @arg radius2: Radius of the second circle (C{scalar}).
1508 @return: A L{Radical2Tuple}C{(ratio, xline)} where C{0.0 <=
1509 ratio <= 1.0} and C{xline} is along the B{C{distance}}.
1511 @raise IntersectionError: The B{C{distance}} exceeds the sum
1512 of B{C{radius1}} and B{C{radius2}}.
1514 @raise UnitError: Invalid B{C{distance}}, B{C{radius1}} or
1515 B{C{radius2}}.
1517 @see: U{Circle-Circle Intersection
1518 <https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>}.
1519 '''
1520 d = Distance_(distance, low=_0_0)
1521 r1 = Radius_(radius1=radius1)
1522 r2 = Radius_(radius2=radius2)
1523 if d > (r1 + r2):
1524 raise IntersectionError(distance=d, radius1=r1, radius2=r2,
1525 txt=_too_(_distant_))
1526 return _radical2(d, r1, r2) if d > EPS0 else \
1527 Radical2Tuple(_0_5, _0_0)
1530class Radical2Tuple(_NamedTuple):
1531 '''2-Tuple C{(ratio, xline)} of the I{radical} C{ratio} and
1532 I{radical} C{xline}, both C{scalar} and C{0.0 <= ratio <= 1.0}
1533 '''
1534 _Names_ = (_ratio_, _xline_)
1535 _Units_ = ( Scalar, Scalar)
1538def _scale_deg(lat1, lat2): # degrees
1539 # scale factor cos(mean of lats) for delta lon
1540 m = fabs(lat1 + lat2) * _0_5
1541 return cos(radians(m)) if m < 90 else _0_0
1544def _scale_rad(phi1, phi2): # radians, by .frechet, .hausdorff, .heights
1545 # scale factor cos(mean of phis) for delta lam
1546 m = fabs(phi1 + phi2) * _0_5
1547 return cos(m) if m < PI_2 else _0_0
1550def _sincosa6(phi2, phi1, lam21):
1551 '''(INTERNAL) C{sin}es, C{cos}ines and C{acos}ine.
1552 '''
1553 s2, c2, s1, c1, _, c21 = sincos2_(phi2, phi1, lam21)
1554 return s2, c2, s1, c1, acos1(s1 * s2 + c1 * c2 * c21), c21
1557def thomas(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):
1558 '''Compute the distance between two (ellipsoidal) points using
1559 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>}
1560 formula.
1562 @arg lat1: Start latitude (C{degrees}).
1563 @arg lon1: Start longitude (C{degrees}).
1564 @arg lat2: End latitude (C{degrees}).
1565 @arg lon2: End longitude (C{degrees}).
1566 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid},
1567 L{Ellipsoid2} or L{a_f2Tuple}).
1568 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).
1570 @return: Distance (C{meter}, same units as the B{C{datum}}'s
1571 ellipsoid axes).
1573 @raise TypeError: Invalid B{C{datum}}.
1575 @see: Functions L{thomas_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
1576 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny},
1577 L{flatPolar}, L{haversine}, L{vincentys} and method L{Ellipsoid.distance2}.
1578 '''
1579 return _distanceToE(thomas_, lat1, lat2, datum,
1580 *unroll180(lon1, lon2, wrap=wrap))
1583def thomas_(phi2, phi1, lam21, datum=_WGS84):
1584 '''Compute the I{angular} distance between two (ellipsoidal) points using
1585 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>}
1586 formula.
1588 @arg phi2: End latitude (C{radians}).
1589 @arg phi1: Start latitude (C{radians}).
1590 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
1591 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid},
1592 L{Ellipsoid2} or L{a_f2Tuple}).
1594 @return: Angular distance (C{radians}).
1596 @raise TypeError: Invalid B{C{datum}}.
1598 @see: Functions L{thomas}, L{cosineAndoyerLambert_},
1599 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
1600 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
1601 L{flatPolar_}, L{haversine_} and L{vincentys_} and U{Geodesy-PHP
1602 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/
1603 Distance/ThomasFormula.php>}.
1604 '''
1605 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21)
1606 if r and isnon0(c1) and isnon0(c2):
1607 E = _ellipsoidal(datum, thomas_)
1608 if E.f:
1609 r1 = atan2(E.b_a * s1, c1)
1610 r2 = atan2(E.b_a * s2, c2)
1612 j = (r2 + r1) * _0_5
1613 k = (r2 - r1) * _0_5
1614 sj, cj, sk, ck, h, _ = sincos2_(j, k, lam21 * _0_5)
1616 h = fsum_(sk**2, (ck * h)**2, -(sj * h)**2)
1617 u = _1_0 - h
1618 if isnon0(u) and isnon0(h):
1619 r = atan(sqrt0(h / u)) * _2_0 # == acos(1 - 2 * h)
1620 sr, cr = sincos2(r)
1621 if isnon0(sr):
1622 u = 2 * (sj * ck)**2 / u
1623 h = 2 * (sk * cj)**2 / h
1624 x = u + h
1625 y = u - h
1627 s = r / sr
1628 e = 4 * s**2
1629 d = 2 * cr
1630 a = e * d
1631 b = 2 * r
1632 c = s - (a - d) * _0_5
1633 f = E.f * _0_25
1635 t = fsum_(a * x, -b * y, c * x**2, -d * y**2, e * x * y)
1636 r -= fsum_(s * x, -y, -t * f * _0_25) * f * sr
1637 return r
1640def vincentys(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
1641 '''Compute the distance between two (spherical) points using
1642 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>}
1643 spherical formula.
1645 @arg lat1: Start latitude (C{degrees}).
1646 @arg lon1: Start longitude (C{degrees}).
1647 @arg lat2: End latitude (C{degrees}).
1648 @arg lon2: End longitude (C{degrees}).
1649 @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter},
1650 L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}).
1651 @kwarg wrap: Wrap and L{pygeodesy.unroll180} longitudes (C{bool}).
1653 @return: Distance (C{meter}, same units as B{C{radius}}).
1655 @raise UnitError: Invalid B{C{radius}}.
1657 @see: Functions L{vincentys_}, L{cosineAndoyerLambert},
1658 L{cosineForsytheAndoyerLambert},L{cosineLaw}, L{equirectangular},
1659 L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar},
1660 L{haversine} and L{thomas} and methods L{Ellipsoid.distance2},
1661 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
1663 @note: See note at function L{vincentys_}.
1664 '''
1665 return _distanceToS(vincentys_, lat1, lat2, radius,
1666 *unroll180(lon1, lon2, wrap=wrap))
1669def vincentys_(phi2, phi1, lam21):
1670 '''Compute the I{angular} distance between two (spherical) points using
1671 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>}
1672 spherical formula.
1674 @arg phi2: End latitude (C{radians}).
1675 @arg phi1: Start latitude (C{radians}).
1676 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
1678 @return: Angular distance (C{radians}).
1680 @see: Functions L{vincentys}, L{cosineAndoyerLambert_},
1681 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
1682 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
1683 L{flatPolar_}, L{haversine_} and L{thomas_}.
1685 @note: Functions L{vincentys_}, L{haversine_} and L{cosineLaw_}
1686 produce equivalent results, but L{vincentys_} is suitable
1687 for antipodal points and slightly more expensive (M{3 cos,
1688 3 sin, 1 hypot, 1 atan2, 6 mul, 2 add}) than L{haversine_}
1689 (M{2 cos, 2 sin, 2 sqrt, 1 atan2, 5 mul, 1 add}) and
1690 L{cosineLaw_} (M{3 cos, 3 sin, 1 acos, 3 mul, 1 add}).
1691 '''
1692 s1, c1, s2, c2, s21, c21 = sincos2_(phi1, phi2, lam21)
1694 c = c2 * c21
1695 x = s1 * s2 + c1 * c
1696 y = c1 * s2 - s1 * c
1697 return atan2(hypot(c2 * s21, y), x)
1699# **) MIT License
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