Coverage for pygeodesy/formy.py: 99%
409 statements
« prev ^ index » next coverage.py v7.2.2, created at 2023-05-20 11:54 -0400
« prev ^ index » next coverage.py v7.2.2, created at 2023-05-20 11:54 -0400
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, \
12 _0_0, _0_125, _0_25, _0_5, _1_0, _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 IntersectionError, LimitError, limiterrors, \
18 _TypeError, _ValueError, \
19 _xError, _xkwds, _xkwds_pop
20from pygeodesy.fmath import euclid, hypot, hypot2, sqrt0
21from pygeodesy.fsums import fsumf_, isscalar
22from pygeodesy.interns import NN, _distant_, _SPACE_, _too_
23from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS
24from pygeodesy.named import _NamedTuple, _xnamed, Fmt, unstr
25from pygeodesy.namedTuples import Bearing2Tuple, Distance4Tuple, \
26 Intersection3Tuple, LatLon2Tuple, \
27 PhiLam2Tuple, Vector3Tuple
28# from pygeodesy.streprs import Fmt, unstr # from .named
29from pygeodesy.units import Bearing, Degrees_, Distance, Distance_, Height, \
30 Lam_, Lat, Lon, Meter_, Phi_, Radians, Radians_, \
31 Radius, Radius_, Scalar, _100km
32from pygeodesy.utily import acos1, atan2b, atan2d, degrees2m, m2degrees, \
33 tan_2, sincos2, sincos2_, sincos2d_, _Wrap
35from contextlib import contextmanager
36from math import atan, atan2, cos, degrees, fabs, radians, sin, sqrt # pow
38__all__ = _ALL_LAZY.formy
39__version__ = '23.05.15'
41_delta_ = 'delta'
42_D2_R2 = (PI / _180_0)**2 # degrees- to radians-squared
43_EWGS84 = _WGS84.ellipsoid
44_ratio_ = 'ratio'
45_xline_ = 'xline'
48def _anti2(a, b, n_2, n, n2):
49 '''(INTERNAL) Helper for C{antipode} and C{antipode_}.
50 '''
51 r = remainder(a, n) if fabs(a) > n_2 else a
52 if r == a:
53 r = -r
54 b += n
55 if fabs(b) > n:
56 b = remainder(b, n2)
57 return float0_(r, b)
60def antipode(lat, lon, name=NN):
61 '''Return the antipode, the point diametrically opposite
62 to a given point in C{degrees}.
64 @arg lat: Latitude (C{degrees}).
65 @arg lon: Longitude (C{degrees}).
66 @kwarg name: Optional name (C{str}).
68 @return: A L{LatLon2Tuple}C{(lat, lon)}.
70 @see: Functions L{antipode_} and L{normal} and U{Geosphere
71 <https://CRAN.R-Project.org/web/packages/geosphere/geosphere.pdf>}.
72 '''
73 return LatLon2Tuple(*_anti2(lat, lon, _90_0, _180_0, _360_0), name=name)
76def antipode_(phi, lam, name=NN):
77 '''Return the antipode, the point diametrically opposite
78 to a given point in C{radians}.
80 @arg phi: Latitude (C{radians}).
81 @arg lam: Longitude (C{radians}).
82 @kwarg name: Optional name (C{str}).
84 @return: A L{PhiLam2Tuple}C{(phi, lam)}.
86 @see: Functions L{antipode} and L{normal_} and U{Geosphere
87 <https://CRAN.R-Project.org/web/packages/geosphere/geosphere.pdf>}.
88 '''
89 return PhiLam2Tuple(*_anti2(phi, lam, PI_2, PI, PI2), name=name)
92def bearing(lat1, lon1, lat2, lon2, **final_wrap):
93 '''Compute the initial or final bearing (forward or reverse
94 azimuth) between a (spherical) start and end point.
96 @arg lat1: Start latitude (C{degrees}).
97 @arg lon1: Start longitude (C{degrees}).
98 @arg lat2: End latitude (C{degrees}).
99 @arg lon2: End longitude (C{degrees}).
100 @kwarg final_wrap: Optional keyword arguments for function
101 L{pygeodesy.bearing_}.
103 @return: Initial or final bearing (compass C{degrees360}) or
104 zero if start and end point coincide.
105 '''
106 r = bearing_(Phi_(lat1=lat1), Lam_(lon1=lon1),
107 Phi_(lat2=lat2), Lam_(lon2=lon2), **final_wrap)
108 return degrees(r)
111def bearing_(phi1, lam1, phi2, lam2, final=False, wrap=False):
112 '''Compute the initial or final bearing (forward or reverse azimuth)
113 between a (spherical) start and end point.
115 @arg phi1: Start latitude (C{radians}).
116 @arg lam1: Start longitude (C{radians}).
117 @arg phi2: End latitude (C{radians}).
118 @arg lam2: End longitude (C{radians}).
119 @kwarg final: Return final bearing if C{True}, initial otherwise (C{bool}).
120 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{phi2}} and
121 B{C{lam2}} (C{bool}).
123 @return: Initial or final bearing (compass C{radiansPI2}) or zero if start
124 and end point coincide.
126 @see: U{Bearing<https://www.Movable-Type.co.UK/scripts/latlong.html>}, U{Course
127 between two points<https://www.EdWilliams.org/avform147.htm#Crs>} and
128 U{Bearing Between Two Points<https://web.Archive.org/web/20020630205931/
129 https://MathForum.org/library/drmath/view/55417.html>}.
130 '''
131 db, phi2, lam2 = _Wrap.philam3(lam1, phi2, lam2, wrap)
132 if final: # swap plus PI
133 phi1, lam1, phi2, lam2, db = phi2, lam2, phi1, lam1, -db
134 r = PI3
135 else:
136 r = PI2
137 sa1, ca1, sa2, ca2, sdb, cdb = sincos2_(phi1, phi2, db)
139 x = ca1 * sa2 - sa1 * ca2 * cdb
140 y = sdb * ca2
141 return _umod_PI2(atan2(y, x) + r) # .utily.wrapPI2
144def _bearingTo2(p1, p2, wrap=False): # for points.ispolar, sphericalTrigonometry.areaOf
145 '''(INTERNAL) Compute initial and final bearing.
146 '''
147 try: # for LatLon_ and ellipsoidal LatLon
148 return p1.bearingTo2(p2, wrap=wrap)
149 except AttributeError:
150 pass
151 # XXX spherical version, OK for ellipsoidal ispolar?
152 a1, b1 = p1.philam
153 a2, b2 = p2.philam
154 return Bearing2Tuple(degrees(bearing_(a1, b1, a2, b2, final=False, wrap=wrap)),
155 degrees(bearing_(a1, b1, a2, b2, final=True, wrap=wrap)),
156 name=_bearingTo2.__name__)
159def compassAngle(lat1, lon1, lat2, lon2, adjust=True, wrap=False):
160 '''Return the angle from North for the direction vector M{(lon2 - lon1,
161 lat2 - lat1)} between two points.
163 Suitable only for short, not near-polar vectors up to a few hundred
164 Km or Miles. Use function L{pygeodesy.bearing} for longer vectors.
166 @arg lat1: From latitude (C{degrees}).
167 @arg lon1: From longitude (C{degrees}).
168 @arg lat2: To latitude (C{degrees}).
169 @arg lon2: To longitude (C{degrees}).
170 @kwarg adjust: Adjust the longitudinal delta by the cosine of the
171 mean latitude (C{bool}).
172 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
173 and B{C{lon2}} (C{bool}).
175 @return: Compass angle from North (C{degrees360}).
177 @note: Courtesy of Martin Schultz.
179 @see: U{Local, flat earth approximation
180 <https://www.EdWilliams.org/avform.htm#flat>}.
181 '''
182 d_lon, lat2, lon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap)
183 if adjust: # scale delta lon
184 d_lon *= _scale_deg(lat1, lat2)
185 return atan2b(d_lon, lat2 - lat1)
188def cosineAndoyerLambert(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):
189 '''Compute the distance between two (ellipsoidal) points using the
190 U{Andoyer-Lambert correction<https://NavLib.net/wp-content/uploads/2013/10/
191 admiralty-manual-of-navigation-vol-1-1964-english501c.pdf>} of the U{Law of
192 Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula.
194 @arg lat1: Start latitude (C{degrees}).
195 @arg lon1: Start longitude (C{degrees}).
196 @arg lat2: End latitude (C{degrees}).
197 @arg lon2: End longitude (C{degrees}).
198 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
199 L{Ellipsoid2} or L{a_f2Tuple}) to use.
200 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
201 B{C{lat2}} and B{C{lon2}} (C{bool}).
203 @return: Distance (C{meter}, same units as the B{C{datum}}'s
204 ellipsoid axes or C{radians} if B{C{datum}} is C{None}).
206 @raise TypeError: Invalid B{C{datum}}.
208 @see: Functions L{cosineAndoyerLambert_}, L{cosineForsytheAndoyerLambert},
209 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny},
210 L{flatPolar}, L{haversine}, L{thomas} and L{vincentys} and method
211 L{Ellipsoid.distance2}.
212 '''
213 return _dE(cosineAndoyerLambert_, datum, wrap, lat1, lon1, lat2, lon2)
216def cosineAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84):
217 '''Compute the I{angular} distance between two (ellipsoidal) points using the
218 U{Andoyer-Lambert correction<https://NavLib.net/wp-content/uploads/2013/10/
219 admiralty-manual-of-navigation-vol-1-1964-english501c.pdf>} of the U{Law of
220 Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula.
222 @arg phi2: End latitude (C{radians}).
223 @arg phi1: Start latitude (C{radians}).
224 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
225 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
226 L{Ellipsoid2} or L{a_f2Tuple}) to use.
228 @return: Angular distance (C{radians}).
230 @raise TypeError: Invalid B{C{datum}}.
232 @see: Functions L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert_},
233 L{cosineLaw_}, L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
234 L{flatPolar_}, L{haversine_}, L{thomas_} and L{vincentys_} and U{Geodesy-PHP
235 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/Distance/
236 AndoyerLambert.php>}.
237 '''
238 s2, c2, s1, c1, r, c21 = _sincosa6(phi2, phi1, lam21)
239 if isnon0(c1) and isnon0(c2):
240 E = _ellipsoidal(datum, cosineAndoyerLambert_)
241 if E.f: # ellipsoidal
242 r2 = atan2(E.b_a * s2, c2)
243 r1 = atan2(E.b_a * s1, c1)
244 s2, c2, s1, c1 = sincos2_(r2, r1)
245 r = acos1(s1 * s2 + c1 * c2 * c21)
246 if r:
247 sr, _, sr_2, cr_2 = sincos2_(r, r * _0_5)
248 if isnon0(sr_2) and isnon0(cr_2):
249 s = (sr + r) * ((s1 - s2) / sr_2)**2
250 c = (sr - r) * ((s1 + s2) / cr_2)**2
251 r += (c - s) * E.f * _0_125
252 return r
255def cosineForsytheAndoyerLambert(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):
256 '''Compute the distance between two (ellipsoidal) points using the
257 U{Forsythe-Andoyer-Lambert correction<https://www2.UNB.Ca/gge/Pubs/TR77.pdf>} of
258 the U{Law of Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>}
259 formula.
261 @arg lat1: Start latitude (C{degrees}).
262 @arg lon1: Start longitude (C{degrees}).
263 @arg lat2: End latitude (C{degrees}).
264 @arg lon2: End longitude (C{degrees}).
265 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
266 L{Ellipsoid2} or L{a_f2Tuple}) to use.
267 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
268 B{C{lat2}} and B{C{lon2}} (C{bool}).
270 @return: Distance (C{meter}, same units as the B{C{datum}}'s
271 ellipsoid axes or C{radians} if B{C{datum}} is C{None}).
273 @raise TypeError: Invalid B{C{datum}}.
275 @see: Functions L{cosineForsytheAndoyerLambert_}, L{cosineAndoyerLambert},
276 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny},
277 L{flatPolar}, L{haversine}, L{thomas} and L{vincentys} and method
278 L{Ellipsoid.distance2}.
279 '''
280 return _dE(cosineForsytheAndoyerLambert_, datum, wrap, lat1, lon1, lat2, lon2)
283def cosineForsytheAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84):
284 '''Compute the I{angular} distance between two (ellipsoidal) points using the
285 U{Forsythe-Andoyer-Lambert correction<https://www2.UNB.Ca/gge/Pubs/TR77.pdf>} of
286 the U{Law of Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>}
287 formula.
289 @arg phi2: End latitude (C{radians}).
290 @arg phi1: Start latitude (C{radians}).
291 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
292 @kwarg datum: Datum (L{Datum}) or ellipsoid to use (L{Ellipsoid},
293 L{Ellipsoid2} or L{a_f2Tuple}).
295 @return: Angular distance (C{radians}).
297 @raise TypeError: Invalid B{C{datum}}.
299 @see: Functions L{cosineForsytheAndoyerLambert}, L{cosineAndoyerLambert_},
300 L{cosineLaw_}, L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
301 L{flatPolar_}, L{haversine_}, L{thomas_} and L{vincentys_} and U{Geodesy-PHP
302 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/
303 Distance/ForsytheCorrection.php>}.
304 '''
305 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21)
306 if r and isnon0(c1) and isnon0(c2):
307 E = _ellipsoidal(datum, cosineForsytheAndoyerLambert_)
308 if E.f: # ellipsoidal
309 sr, cr, s2r, _ = sincos2_(r, r * 2)
310 if isnon0(sr) and fabs(cr) < EPS1:
311 s = (s1 + s2)**2 / (1 + cr)
312 t = (s1 - s2)**2 / (1 - cr)
313 x = s + t
314 y = s - t
316 s = 8 * r**2 / sr
317 a = 64 * r + s * cr * 2 # 16 * r**2 / tan(r)
318 d = 48 * sr + s # 8 * r**2 / tan(r)
319 b = -2 * d
320 e = 30 * s2r
321 c = fsumf_(30 * r, e * _0_5, s * cr) # 8 * r**2 / tan(r)
323 t = fsumf_( a * x, b * y, -c * x**2, d * x * y, e * y**2)
324 r += fsumf_(-r * x, 3 * y * sr, t * E.f / _32_0) * E.f * _0_25
325 return r
328def cosineLaw(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
329 '''Compute the distance between two points using the U{spherical Law of
330 Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>}
331 formula.
333 @arg lat1: Start latitude (C{degrees}).
334 @arg lon1: Start longitude (C{degrees}).
335 @arg lat2: End latitude (C{degrees}).
336 @arg lon2: End longitude (C{degrees}).
337 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
338 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
339 L{a_f2Tuple}) to use.
340 @kwarg wrap: If C{True}, wrap or I{normalize} and B{C{lat2}}
341 and B{C{lon2}} (C{bool}).
343 @return: Distance (C{meter}, same units as B{C{radius}} or the
344 ellipsoid or datum axes).
346 @raise TypeError: Invalid B{C{radius}}.
348 @see: Functions L{cosineLaw_}, L{cosineAndoyerLambert},
349 L{cosineForsytheAndoyerLambert}, L{equirectangular}, L{euclidean},
350 L{flatLocal}/L{hubeny}, L{flatPolar}, L{haversine}, L{thomas} and
351 L{vincentys} and method L{Ellipsoid.distance2}.
353 @note: See note at function L{vincentys_}.
354 '''
355 return _dS(cosineLaw_, radius, wrap, lat1, lon1, lat2, lon2)
358def cosineLaw_(phi2, phi1, lam21):
359 '''Compute the I{angular} distance between two points using the U{spherical
360 Law of Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>}
361 formula.
363 @arg phi2: End latitude (C{radians}).
364 @arg phi1: Start latitude (C{radians}).
365 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
367 @return: Angular distance (C{radians}).
369 @see: Functions L{cosineLaw}, L{cosineAndoyerLambert_},
370 L{cosineForsytheAndoyerLambert_}, L{equirectangular_},
371 L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_},
372 L{haversine_}, L{thomas_} and L{vincentys_}.
374 @note: See note at function L{vincentys_}.
375 '''
376 return _sincosa6(phi2, phi1, lam21)[4]
379def _d3(wrap, lat1, lon1, lat2, lon2):
380 '''(INTERNAL) Helper for _dE, _dS and _eA.
381 '''
382 if wrap:
383 d_lon, lat2, _ = _Wrap.latlon3(lon1, lat2, lon2, wrap)
384 return radians(lat2), Phi_(lat1=lat1), radians(d_lon)
385 else: # for backward compaibility
386 return Phi_(lat2=lat2), Phi_(lat1=lat1), Phi_(d_lon=lon2 - lon1)
389def _dE(func_, earth, *wrap_lls):
390 '''(INTERNAL) Helper for ellipsoidal distances.
391 '''
392 E = _ellipsoidal(earth, func_)
393 r = func_(*_d3(*wrap_lls), datum=E)
394 return r * E.a
397def _dS(func_, radius, *wrap_lls, **adjust):
398 '''(INTERNAL) Helper for spherical distances.
399 '''
400 r = func_(*_d3(*wrap_lls), **adjust)
401 if radius is not R_M:
402 _, lat1, _, lat2, _ = wrap_lls
403 radius = _mean_radius(radius, lat1, lat2)
404 return r * radius
407def _eA(excess_, radius, *wrap_lls):
408 '''(INTERNAL) Helper for spherical excess or area.
409 '''
410 r = excess_(*_d3(*wrap_lls))
411 if radius:
412 _, lat1, _, lat2, _ = wrap_lls
413 r *= _mean_radius(radius, lat1, lat2)**2
414 return r
417def _ellipsoidal(earth, where):
418 '''(INTERNAL) Helper for distances.
419 '''
420 return _EWGS84 if earth in (_WGS84, _EWGS84) else (
421 earth if isinstance(earth, Ellipsoid) else
422 (earth if isinstance(earth, Datum) else # PYCHOK indent
423 _ellipsoidal_datum(earth, name=where.__name__)).ellipsoid)
426def equirectangular(lat1, lon1, lat2, lon2, radius=R_M, **adjust_limit_wrap):
427 '''Compute the distance between two points using
428 the U{Equirectangular Approximation / Projection
429 <https://www.Movable-Type.co.UK/scripts/latlong.html#equirectangular>}.
431 @arg lat1: Start latitude (C{degrees}).
432 @arg lon1: Start longitude (C{degrees}).
433 @arg lat2: End latitude (C{degrees}).
434 @arg lon2: End longitude (C{degrees}).
435 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
436 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
437 L{a_f2Tuple}).
438 @kwarg adjust_limit_wrap: Optional keyword arguments for
439 function L{equirectangular_}.
441 @return: Distance (C{meter}, same units as B{C{radius}} or
442 the ellipsoid or datum axes).
444 @raise TypeError: Invalid B{C{radius}}.
446 @see: Function L{equirectangular_} for more details, the
447 available B{C{options}}, errors, restrictions and other,
448 approximate or accurate distance functions.
449 '''
450 d = sqrt(equirectangular_(Lat(lat1=lat1), Lon(lon1=lon1),
451 Lat(lat2=lat2), Lon(lon2=lon2),
452 **adjust_limit_wrap).distance2) # PYCHOK 4 vs 2-3
453 return degrees2m(d, radius=_mean_radius(radius, lat1, lat2))
456def _equirectangular(lat1, lon1, lat2, lon2, **adjust_limit_wrap):
457 '''(INTERNAL) Helper for the L{frechet._FrecherMeterRadians}
458 and L{hausdorff._HausdorffMeterRedians} classes.
459 '''
460 return equirectangular_(lat1, lon1, lat2, lon2, **adjust_limit_wrap).distance2 * _D2_R2
463def equirectangular_(lat1, lon1, lat2, lon2, adjust=True, limit=45, wrap=False):
464 '''Compute the distance between two points using the U{Equirectangular
465 Approximation / Projection
466 <https://www.Movable-Type.co.UK/scripts/latlong.html#equirectangular>}.
468 This approximation is valid for short distance of several hundred Km
469 or Miles, see the B{C{limit}} keyword argument and L{LimitError}.
471 @arg lat1: Start latitude (C{degrees}).
472 @arg lon1: Start longitude (C{degrees}).
473 @arg lat2: End latitude (C{degrees}).
474 @arg lon2: End longitude (C{degrees}).
475 @kwarg adjust: Adjust the wrapped, unrolled longitudinal delta
476 by the cosine of the mean latitude (C{bool}).
477 @kwarg limit: Optional limit for lat- and longitudinal deltas
478 (C{degrees}) or C{None} or C{0} for unlimited.
479 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
480 and B{C{lon2}} (C{bool}).
482 @return: A L{Distance4Tuple}C{(distance2, delta_lat, delta_lon,
483 unroll_lon2)} in C{degrees squared}.
485 @raise LimitError: If the lat- and/or longitudinal delta exceeds the
486 B{C{-limit..limit}} range and L{pygeodesy.limiterrors}
487 set to C{True}.
489 @see: U{Local, flat earth approximation
490 <https://www.EdWilliams.org/avform.htm#flat>}, functions
491 L{equirectangular}, L{cosineAndoyerLambert},
492 L{cosineForsytheAndoyerLambert}, L{cosineLaw}, L{euclidean},
493 L{flatLocal}/L{hubeny}, L{flatPolar}, L{haversine}, L{thomas}
494 and L{vincentys} and methods L{Ellipsoid.distance2},
495 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
496 '''
497 d_lon, lat2, ulon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap)
498 d_lat = lat2 - lat1
500 if limit and limit > 0 and limiterrors():
501 d = max(fabs(d_lat), fabs(d_lon))
502 if d > limit:
503 t = _SPACE_(_delta_, Fmt.PAREN_g(d), Fmt.exceeds_limit(limit))
504 s = unstr(equirectangular_, lat1, lon1, lat2, lon2,
505 limit=limit, wrap=wrap)
506 raise LimitError(s, txt=t)
508 if adjust: # scale delta lon
509 d_lon *= _scale_deg(lat1, lat2)
511 d2 = hypot2(d_lat, d_lon) # degrees squared!
512 return Distance4Tuple(d2, d_lat, d_lon, ulon2 - lon2)
515def euclidean(lat1, lon1, lat2, lon2, radius=R_M, adjust=True, wrap=False):
516 '''Approximate the C{Euclidean} distance between two (spherical) points.
518 @arg lat1: Start latitude (C{degrees}).
519 @arg lon1: Start longitude (C{degrees}).
520 @arg lat2: End latitude (C{degrees}).
521 @arg lon2: End longitude (C{degrees}).
522 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
523 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
524 L{a_f2Tuple}) to use.
525 @kwarg adjust: Adjust the longitudinal delta by the cosine of
526 the mean latitude (C{bool}).
527 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
528 and B{C{lon2}} (C{bool}).
530 @return: Distance (C{meter}, same units as B{C{radius}} or the
531 ellipsoid or datum axes).
533 @raise TypeError: Invalid B{C{radius}}.
535 @see: U{Distance between two (spherical) points
536 <https://www.EdWilliams.org/avform.htm#Dist>}, functions L{euclid},
537 L{euclidean_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
538 L{cosineLaw}, L{equirectangular}, L{flatLocal}/L{hubeny}, L{flatPolar},
539 L{haversine}, L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2},
540 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
541 '''
542 return _dS(euclidean_, radius, wrap, lat1, lon1, lat2, lon2, adjust=adjust)
545def euclidean_(phi2, phi1, lam21, adjust=True):
546 '''Approximate the I{angular} C{Euclidean} distance between two
547 (spherical) points.
549 @arg phi2: End latitude (C{radians}).
550 @arg phi1: Start latitude (C{radians}).
551 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
552 @kwarg adjust: Adjust the longitudinal delta by the cosine
553 of the mean latitude (C{bool}).
555 @return: Angular distance (C{radians}).
557 @see: Functions L{euclid}, L{euclidean}, L{cosineAndoyerLambert_},
558 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{equirectangular_},
559 L{flatLocal_}/L{hubeny_}, L{flatPolar_}, L{haversine_}, L{thomas_}
560 and L{vincentys_}.
561 '''
562 if adjust:
563 lam21 *= _scale_rad(phi2, phi1)
564 return euclid(phi2 - phi1, lam21)
567def excessAbc_(A, b, c):
568 '''Compute the I{spherical excess} C{E} of a (spherical) triangle
569 from two sides and the included angle.
571 @arg A: An interior triangle angle (C{radians}).
572 @arg b: Frist adjacent triangle side (C{radians}).
573 @arg c: Second adjacent triangle side (C{radians}).
575 @return: Spherical excess (C{radians}).
577 @raise UnitError: Invalid B{C{A}}, B{C{b}} or B{C{c}}.
579 @see: Functions L{excessGirard_}, L{excessLHuilier_} and U{Spherical
580 trigonometry<https://WikiPedia.org/wiki/Spherical_trigonometry>}.
581 '''
582 sA, cA, sb, cb, sc, cc = sincos2_(Radians_(A=A), Radians_(b=b) * _0_5,
583 Radians_(c=c) * _0_5)
584 return atan2(sA * sb * sc, cb * cc + cA * sb * sc) * _2_0
587def excessGirard_(A, B, C):
588 '''Compute the I{spherical excess} C{E} of a (spherical) triangle using
589 U{Girard's<https://MathWorld.Wolfram.com/GirardsSphericalExcessFormula.html>}
590 formula.
592 @arg A: First interior triangle angle (C{radians}).
593 @arg B: Second interior triangle angle (C{radians}).
594 @arg C: Third interior triangle angle (C{radians}).
596 @return: Spherical excess (C{radians}).
598 @raise UnitError: Invalid B{C{A}}, B{C{B}} or B{C{C}}.
600 @see: Function L{excessLHuilier_} and U{Spherical trigonometry
601 <https://WikiPedia.org/wiki/Spherical_trigonometry>}.
602 '''
603 return Radians(Girard=fsumf_(Radians_(A=A),
604 Radians_(B=B),
605 Radians_(C=C), -PI))
608def excessLHuilier_(a, b, c):
609 '''Compute the I{spherical excess} C{E} of a (spherical) triangle using
610 U{L'Huilier's<https://MathWorld.Wolfram.com/LHuiliersTheorem.html>}
611 Theorem.
613 @arg a: First triangle side (C{radians}).
614 @arg b: Second triangle side (C{radians}).
615 @arg c: Third triangle side (C{radians}).
617 @return: Spherical excess (C{radians}).
619 @raise UnitError: Invalid B{C{a}}, B{C{b}} or B{C{c}}.
621 @see: Function L{excessGirard_} and U{Spherical trigonometry
622 <https://WikiPedia.org/wiki/Spherical_trigonometry>}.
623 '''
624 a = Radians_(a=a)
625 b = Radians_(b=b)
626 c = Radians_(c=c)
628 s = fsumf_(a, b, c) * _0_5
629 r = tan_2(s) * tan_2(s - a) * tan_2(s - b) * tan_2(s - c)
630 r = atan(sqrt(r)) if r > 0 else _0_0
631 return Radians(LHuilier=r * _4_0)
634def excessKarney(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
635 '''Compute the surface area of a (spherical) quadrilateral bounded by a
636 segment of a great circle, two meridians and the equator using U{Karney's
637 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>}
638 method.
640 @arg lat1: Start latitude (C{degrees}).
641 @arg lon1: Start longitude (C{degrees}).
642 @arg lat2: End latitude (C{degrees}).
643 @arg lon2: End longitude (C{degrees}).
644 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
645 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
646 L{a_f2Tuple}) or C{None}.
647 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
648 B{C{lat2}} and B{C{lon2}} (C{bool}).
650 @return: Surface area, I{signed} (I{square} C{meter} or the same units as
651 B{C{radius}} I{squared}) or the I{spherical excess} (C{radians})
652 if C{B{radius}=0} or C{None}.
654 @raise TypeError: Invalid B{C{radius}}.
656 @raise UnitError: Invalid B{C{lat2}} or B{C{lat1}}.
658 @raise ValueError: Semi-circular longitudinal delta.
660 @see: Functions L{excessKarney_} and L{excessQuad}.
661 '''
662 return _eA(excessKarney_, radius, wrap, lat1, lon1, lat2, lon2)
665def excessKarney_(phi2, phi1, lam21):
666 '''Compute the I{spherical excess} C{E} of a (spherical) quadrilateral bounded
667 by a segment of a great circle, two meridians and the equator using U{Karney's
668 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>}
669 method.
671 @arg phi2: End latitude (C{radians}).
672 @arg phi1: Start latitude (C{radians}).
673 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
675 @return: Spherical excess, I{signed} (C{radians}).
677 @raise ValueError: Semi-circular longitudinal delta B{C{lam21}}.
679 @see: Function L{excessKarney} and U{Area of a spherical polygon
680 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>}.
681 '''
682 # from: Veness <https://www.Movable-Type.co.UK/scripts/latlong.html> Area
683 # method due to Karney: for each edge of the polygon,
684 #
685 # tan(Δλ / 2) · (tan(φ1 / 2) + tan(φ2 / 2))
686 # tan(E / 2) = -----------------------------------------
687 # 1 + tan(φ1 / 2) · tan(φ2 / 2)
688 #
689 # where E is the spherical excess of the trapezium obtained by extending
690 # the edge to the equator-circle vector for each edge (see also ***).
691 t2 = tan_2(phi2)
692 t1 = tan_2(phi1)
693 t = tan_2(lam21, lam21=None)
694 return Radians(Karney=atan2(t * (t1 + t2),
695 _1_0 + (t1 * t2)) * _2_0)
698# ***) Original post no longer available, following is a copy of the main part
699# <http://OSGeo-org.1560.x6.Nabble.com/Area-of-a-spherical-polygon-td3841625.html>
700#
701# The area of a polygon on a (unit) sphere is given by the spherical excess
702#
703# A = 2 * pi - sum(exterior angles)
704#
705# However this is badly conditioned if the polygon is small. In this case, use
706#
707# A = sum(S12{i, i+1}) over the edges of the polygon
708#
709# where S12 is the area of the quadrilateral bounded by an edge of the polygon,
710# two meridians and the equator, i.e. with vertices (phi1, lambda1), (phi2,
711# lambda2), (0, lambda1) and (0, lambda2). S12 is given by
712#
713# tan(S12 / 2) = tan(lambda21 / 2) * (tan(phi1 / 2) + tan(phi2 / 2)) /
714# (tan(phi1 / 2) * tan(phi2 / 2) + 1)
715#
716# = tan(lambda21 / 2) * tanh((Lambertian(phi1) +
717# Lambertian(phi2)) / 2)
718#
719# where lambda21 = lambda2 - lambda1 and lamb(x) is the Lambertian (or
720# inverse Gudermannian) function
721#
722# Lambertian(x) = asinh(tan(x)) = atanh(sin(x)) = 2 * atanh(tan(x / 2))
723#
724# Notes: The formula for S12 is exact, except that...
725# - it is indeterminate if an edge is a semi-circle
726# - the formula for A applies only if the polygon does not include a pole
727# (if it does, then add +/- 2 * pi to the result)
728# - in the limit of small phi and lambda, S12 reduces to the trapezoidal
729# formula, S12 = (lambda2 - lambda1) * (phi1 + phi2) / 2
730# - I derived this result from the equation for the area of a spherical
731# triangle in terms of two edges and the included angle given by, e.g.
732# U{Todhunter, I. - Spherical Trigonometry (1871), Sec. 103, Eq. (2)
733# <http://Books.Google.com/books?id=3uBHAAAAIAAJ&pg=PA71>}
734# - I would be interested to know if this formula for S12 is already known
735# - Charles Karney
738def excessQuad(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
739 '''Compute the surface area of a (spherical) quadrilateral bounded by a segment
740 of a great circle, two meridians and the equator.
742 @arg lat1: Start latitude (C{degrees}).
743 @arg lon1: Start longitude (C{degrees}).
744 @arg lat2: End latitude (C{degrees}).
745 @arg lon2: End longitude (C{degrees}).
746 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
747 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
748 L{a_f2Tuple}) or C{None}.
749 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
750 B{C{lat2}} and B{C{lon2}} (C{bool}).
752 @return: Surface area, I{signed} (I{square} C{meter} or the same units as
753 B{C{radius}} I{squared}) or the I{spherical excess} (C{radians})
754 if C{B{radius}=0} or C{None}.
756 @raise TypeError: Invalid B{C{radius}}.
758 @raise UnitError: Invalid B{C{lat2}} or B{C{lat1}}.
760 @see: Function L{excessQuad_} and L{excessKarney}.
761 '''
762 return _eA(excessQuad_, radius, wrap, lat1, lon1, lat2, lon2)
765def excessQuad_(phi2, phi1, lam21):
766 '''Compute the I{spherical excess} C{E} of a (spherical) quadrilateral bounded
767 by a segment of a great circle, two meridians and the equator.
769 @arg phi2: End latitude (C{radians}).
770 @arg phi1: Start latitude (C{radians}).
771 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
773 @return: Spherical excess, I{signed} (C{radians}).
775 @see: Function L{excessQuad}, U{Spherical trigonometry
776 <https://WikiPedia.org/wiki/Spherical_trigonometry>}.
777 '''
778 s = sin((phi2 + phi1) * _0_5)
779 c = cos((phi2 - phi1) * _0_5)
780 return Radians(Quad=atan2(tan_2(lam21) * s, c) * _2_0)
783def flatLocal(lat1, lon1, lat2, lon2, datum=_WGS84, scaled=True, wrap=False):
784 '''Compute the distance between two (ellipsoidal) points using
785 the U{ellipsoidal Earth to plane projection<https://WikiPedia.org/
786 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>}
787 aka U{Hubeny<https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula.
789 @arg lat1: Start latitude (C{degrees}).
790 @arg lon1: Start longitude (C{degrees}).
791 @arg lat2: End latitude (C{degrees}).
792 @arg lon2: End longitude (C{degrees}).
793 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
794 L{Ellipsoid2} or L{a_f2Tuple}) to use.
795 @kwarg scaled: Scale prime_vertical by C{cos(B{phi})} (C{bool}),
796 see method L{pygeodesy.Ellipsoid.roc2_}.
797 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
798 B{C{lat2}} and B{C{lon2}} (C{bool}).
800 @return: Distance (C{meter}, same units as the B{C{datum}}'s
801 ellipsoid axes).
803 @raise TypeError: Invalid B{C{datum}}.
805 @note: The meridional and prime_vertical radii of curvature
806 are taken and scaled at the mean of both latitude.
808 @see: Functions L{flatLocal_} or L{hubeny_}, L{cosineLaw}, L{flatPolar},
809 L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
810 L{equirectangular}, L{euclidean}, L{haversine}, L{thomas},
811 L{vincentys}, method L{Ellipsoid.distance2} and U{local, flat
812 earth approximation<https://www.EdWilliams.org/avform.htm#flat>}.
813 '''
814 E = _ellipsoidal(datum, flatLocal)
815 return E._hubeny_2(*_d3(wrap, lat1, lon1, lat2, lon2),
816 scaled=scaled, squared=False) * E.a
818hubeny = flatLocal # PYCHOK for Karl Hubeny
821def flatLocal_(phi2, phi1, lam21, datum=_WGS84, scaled=True):
822 '''Compute the I{angular} distance between two (ellipsoidal) points using
823 the U{ellipsoidal Earth to plane projection<https://WikiPedia.org/
824 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>}
825 aka U{Hubeny<https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula.
827 @arg phi2: End latitude (C{radians}).
828 @arg phi1: Start latitude (C{radians}).
829 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
830 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
831 L{Ellipsoid2} or L{a_f2Tuple}) to use.
832 @kwarg scaled: Scale prime_vertical by C{cos(B{phi})} (C{bool}),
833 see method L{pygeodesy.Ellipsoid.roc2_}.
835 @return: Angular distance (C{radians}).
837 @raise TypeError: Invalid B{C{datum}}.
839 @note: The meridional and prime_vertical radii of curvature
840 are taken and scaled I{at the mean of both latitude}.
842 @see: Functions L{flatLocal} or L{hubeny}, L{cosineAndoyerLambert_},
843 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{flatPolar_},
844 L{equirectangular_}, L{euclidean_}, L{haversine_}, L{thomas_}
845 and L{vincentys_} and U{local, flat earth approximation
846 <https://www.EdWilliams.org/avform.htm#flat>}.
847 '''
848 E = _ellipsoidal(datum, flatLocal_)
849 return E._hubeny_2(phi2, phi1, lam21, scaled=scaled, squared=False)
851hubeny_ = flatLocal_ # PYCHOK for Karl Hubeny
854def flatPolar(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
855 '''Compute the distance between two (spherical) points using
856 the U{polar coordinate flat-Earth <https://WikiPedia.org/wiki/
857 Geographical_distance#Polar_coordinate_flat-Earth_formula>}
858 formula.
860 @arg lat1: Start latitude (C{degrees}).
861 @arg lon1: Start longitude (C{degrees}).
862 @arg lat2: End latitude (C{degrees}).
863 @arg lon2: End longitude (C{degrees}).
864 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
865 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
866 L{a_f2Tuple}) to use.
867 @kwarg wrap: If C{True}, wrap or I{normalize} and B{C{lat2}}
868 and B{C{lon2}} (C{bool}).
870 @return: Distance (C{meter}, same units as B{C{radius}} or the
871 ellipsoid or datum axes).
873 @raise TypeError: Invalid B{C{radius}}.
875 @see: Functions L{flatPolar_}, L{cosineAndoyerLambert},
876 L{cosineForsytheAndoyerLambert},L{cosineLaw},
877 L{flatLocal}/L{hubeny}, L{equirectangular},
878 L{euclidean}, L{haversine}, L{thomas} and
879 L{vincentys}.
880 '''
881 return _dS(flatPolar_, radius, wrap, lat1, lon1, lat2, lon2)
884def flatPolar_(phi2, phi1, lam21):
885 '''Compute the I{angular} distance between two (spherical) points
886 using the U{polar coordinate flat-Earth<https://WikiPedia.org/wiki/
887 Geographical_distance#Polar_coordinate_flat-Earth_formula>}
888 formula.
890 @arg phi2: End latitude (C{radians}).
891 @arg phi1: Start latitude (C{radians}).
892 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
894 @return: Angular distance (C{radians}).
896 @see: Functions L{flatPolar}, L{cosineAndoyerLambert_},
897 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
898 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
899 L{haversine_}, L{thomas_} and L{vincentys_}.
900 '''
901 a = fabs(PI_2 - phi1) # co-latitude
902 b = fabs(PI_2 - phi2) # co-latitude
903 if a < b:
904 a, b = b, a
905 if a < EPS0:
906 a = _0_0
907 elif b > 0:
908 b = b / a # /= chokes PyChecker
909 c = b * cos(lam21) * _2_0
910 c = fsumf_(_1_0, b**2, -fabs(c))
911 a *= sqrt0(c)
912 return a
915def hartzell(pov, los=None, earth=_WGS84, name=NN, **LatLon_and_kwds):
916 '''Compute the intersection of the earth's surface and a Line-Of-Sight
917 from a Point-Of-View in space.
919 @arg pov: Point-Of-View outside the earth (C{Cartesian}, L{Ecef9Tuple}
920 or L{Vector3d}).
921 @kwarg los: Line-Of-Sight, I{direction} to earth (L{Vector3d}) or
922 C{None} to point to the earth' center.
923 @kwarg earth: The earth model (L{Datum}, L{Ellipsoid}, L{Ellipsoid2},
924 L{a_f2Tuple} or C{scalar} radius in C{meter}).
925 @kwarg name: Optional name (C{str}).
926 @kwarg LatLon_and_kwds: Optional C{LatLon} class for the intersection
927 point plus C{LatLon} keyword arguments, include
928 B{C{datum}} if different from B{C{earth}}.
930 @return: The earth intersection (L{Vector3d}, C{Cartesian type} of
931 B{C{pov}} or B{C{LatLon}}).
933 @raise IntersectionError: Null B{C{pov}} or B{C{los}} vector, B{C{pov}}
934 is inside the earth or B{C{los}} points outside
935 the earth or points in an opposite direction.
937 @raise TypeError: Invalid B{C{pov}}, B{C{los}} or B{C{earth}}.
939 @see: Function L{pygeodesy.hartzell4}, L{pygeodesy.tyr3d} for B{C{los}},
940 method L{Ellipsoid.hartzell4} and U{I{Satellite Line-of-Sight
941 Intersection with Earth}<https://StephenHartzell.Medium.com/
942 satellite-line-of-sight-intersection-with-earth-d786b4a6a9b6>}.
943 '''
944 D = earth if isinstance(earth, Datum) else \
945 _spherical_datum(earth, name=hartzell.__name__)
946 try:
947 r, _ = _MODS.triaxials._hartzell3d2(pov, los, D.ellipsoid._triaxial)
948 except Exception as x:
949 raise IntersectionError(pov=pov, los=los, earth=earth, cause=x)
951# else:
952# E = D.ellipsoid
953# # Triaxial(a, b, c) == (E.a, E.a, E.b)
954#
955# def _Error(txt):
956# return IntersectionError(pov=pov, los=los, earth=earth, txt=txt)
957#
958# a2 = b2 = E.a2 # earth' x, y, ...
959# c2 = E.b2 # ... z semi-axis squared
960# q2 = E.b2_a2 # == c2 / a2
961# bc = E.a * E.b # == b * c
962#
963# V3 = _MODS.vector3d._otherV3d
964# p3 = V3(pov=pov)
965# u3 = V3(los=los) if los else p3.negate()
966# u3 = u3.unit() # unit vector, opposing signs
967#
968# x2, y2, z2 = p3.x2y2z2 # p3.times_(p3).xyz
969# ux, vy, wz = u3.times_(p3).xyz
970# u2, v2, w2 = u3.x2y2z2 # u3.times_(u3).xyz
971#
972# t = c2, c2, b2
973# m = fdot(t, u2, v2, w2) # a2 factored out
974# if m < EPS0: # zero or near-null LOS vector
975# raise _Error(_near_(_null_))
976#
977# # a2 and b2 factored out, b2 == a2 and b2 / a2 == 1
978# r = fsumf_(b2 * w2, c2 * v2, -v2 * z2, vy * wz * 2,
979# c2 * u2, -u2 * z2, -w2 * x2, ux * wz * 2,
980# -w2 * y2, -u2 * y2 * q2, -v2 * x2 * q2, ux * vy * 2 * q2)
981# if r > 0:
982# r = sqrt(r) * bc # == a * a * b * c / a2
983# elif r < 0: # LOS pointing away from or missing the earth
984# raise _Error(_opposite_ if max(ux, vy, wz) > 0 else _outside_)
985#
986# d = Fdot(t, ux, vy, wz).fadd_(r).fover(m) # -r for antipode, a2 factored out
987# if d > 0: # POV inside or LOS missing, outside the earth
988# s = fsumf_(_1_0, x2 / a2, y2 / b2, z2 / c2, _N_2_0) # like _sideOf
989# raise _Error(_outside_ if s > 0 else _inside_)
990# elif fsumf_(x2, y2, z2) < d**2: # d past earth center
991# raise _Error(_too_(_distant_))
992#
993# r = p3.minus(u3.times(d))
994# # h = p3.minus(r).length # distance to ellipsoid
996 r = _xnamed(r, name or hartzell.__name__)
997 if LatLon_and_kwds:
998 c = _MODS.cartesianBase.CartesianBase(r, datum=D, name=r.name)
999 r = c.toLatLon(**LatLon_and_kwds)
1000 return r
1003def haversine(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
1004 '''Compute the distance between two (spherical) points using the
1005 U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>}
1006 formula.
1008 @arg lat1: Start latitude (C{degrees}).
1009 @arg lon1: Start longitude (C{degrees}).
1010 @arg lat2: End latitude (C{degrees}).
1011 @arg lon2: End longitude (C{degrees}).
1012 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
1013 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
1014 L{a_f2Tuple}) to use.
1015 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
1016 B{C{lat2}} and B{C{lon2}} (C{bool}).
1018 @return: Distance (C{meter}, same units as B{C{radius}}).
1020 @raise TypeError: Invalid B{C{radius}}.
1022 @see: U{Distance between two (spherical) points
1023 <https://www.EdWilliams.org/avform.htm#Dist>}, functions
1024 L{cosineLaw}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
1025 L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar},
1026 L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2},
1027 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
1029 @note: See note at function L{vincentys_}.
1030 '''
1031 return _dS(haversine_, radius, wrap, lat1, lon1, lat2, lon2)
1034def haversine_(phi2, phi1, lam21):
1035 '''Compute the I{angular} distance between two (spherical) points
1036 using the U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>}
1037 formula.
1039 @arg phi2: End latitude (C{radians}).
1040 @arg phi1: Start latitude (C{radians}).
1041 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
1043 @return: Angular distance (C{radians}).
1045 @see: Functions L{haversine}, L{cosineAndoyerLambert_},
1046 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
1047 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
1048 L{flatPolar_}, L{thomas_} and L{vincentys_}.
1050 @note: See note at function L{vincentys_}.
1051 '''
1052 def _hsin(rad):
1053 return sin(rad * _0_5)**2
1055 h = _hsin(phi2 - phi1) + cos(phi1) * cos(phi2) * _hsin(lam21) # haversine
1056 return atan2(sqrt0(h), sqrt0(_1_0 - h)) * _2_0 # == asin(sqrt(h)) * 2
1059def heightOf(angle, distance, radius=R_M):
1060 '''Determine the height above the (spherical) earth' surface after
1061 traveling along a straight line at a given tilt.
1063 @arg angle: Tilt angle above horizontal (C{degrees}).
1064 @arg distance: Distance along the line (C{meter} or same units as
1065 B{C{radius}}).
1066 @kwarg radius: Optional mean earth radius (C{meter}).
1068 @return: Height (C{meter}, same units as B{C{distance}} and B{C{radius}}).
1070 @raise ValueError: Invalid B{C{angle}}, B{C{distance}} or B{C{radius}}.
1072 @see: U{MultiDop geog_lib.GeogBeamHt<https://GitHub.com/NASA/MultiDop>}
1073 (U{Shapiro et al. 2009, JTECH
1074 <https://Journals.AMetSoc.org/doi/abs/10.1175/2009JTECHA1256.1>}
1075 and U{Potvin et al. 2012, JTECH
1076 <https://Journals.AMetSoc.org/doi/abs/10.1175/JTECH-D-11-00019.1>}).
1077 '''
1078 r = h = Radius(radius)
1079 d = fabs(Distance(distance))
1080 if d > h:
1081 d, h = h, d
1083 if d > EPS0: # and h > EPS0
1084 d = d / h # /= h chokes PyChecker
1085 s = sin(Phi_(angle=angle, clip=_180_0))
1086 s = fsumf_(_1_0, _2_0 * s * d, d**2)
1087 if s > 0:
1088 return h * sqrt(s) - r
1090 raise _ValueError(angle=angle, distance=distance, radius=radius)
1093def horizon(height, radius=R_M, refraction=False):
1094 '''Determine the distance to the horizon from a given altitude
1095 above the (spherical) earth.
1097 @arg height: Altitude (C{meter} or same units as B{C{radius}}).
1098 @kwarg radius: Optional mean earth radius (C{meter}).
1099 @kwarg refraction: Consider atmospheric refraction (C{bool}).
1101 @return: Distance (C{meter}, same units as B{C{height}} and B{C{radius}}).
1103 @raise ValueError: Invalid B{C{height}} or B{C{radius}}.
1105 @see: U{Distance to horizon<https://www.EdWilliams.org/avform.htm#Horizon>}.
1106 '''
1107 h, r = Height(height), Radius(radius)
1108 if min(h, r) < 0:
1109 raise _ValueError(height=height, radius=radius)
1111 if refraction:
1112 d2 = 2.415750694528 * h * r # 2.0 / 0.8279
1113 else:
1114 d2 = h * fsumf_(r, r, h)
1115 return sqrt0(d2)
1118class _idllmn6(object): # see also .geodesicw._wargs, .vector2d._numpy
1119 '''(INTERNAL) Helper for C{intersection2} and C{intersections2}.
1120 '''
1121 @contextmanager # <https://www.python.org/dev/peps/pep-0343/> Examples
1122 def __call__(self, datum, lat1, lon1, lat2, lon2, small, wrap, s, **kwds):
1123 try:
1124 if wrap:
1125 _, lat2, lon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap)
1126 kwds = _xkwds(kwds, wrap=wrap) # for _xError
1127 m = small if small is _100km else Meter_(small=small)
1128 n = (intersections2 if s else intersection2).__name__
1129 if datum is None or euclidean(lat1, lon1, lat2, lon2) < m:
1130 d, m = None, _MODS.vector3d
1131 _i = m._intersects2 if s else m._intersect3d3
1132 elif isscalar(datum) and datum < 0 and not s:
1133 d = _spherical_datum(-datum, name=n)
1134 m = _MODS.sphericalNvector
1135 _i = m.intersection
1136 else:
1137 d = _spherical_datum(datum, name=n)
1138 if d.isSpherical:
1139 m = _MODS.sphericalTrigonometry
1140 _i = m._intersects2 if s else m._intersect
1141 elif d.isEllipsoidal:
1142 try:
1143 if d.ellipsoid.geodesic:
1144 pass
1145 m = _MODS.ellipsoidalKarney
1146 except ImportError:
1147 m = _MODS.ellipsoidalExact
1148 _i = m._intersections2 if s else m._intersection3 # ellispoidalBaseDI
1149 else:
1150 raise _TypeError(datum=datum)
1151 yield _i, d, lat2, lon2, m, n
1153 except (TypeError, ValueError) as x:
1154 raise _xError(x, lat1=lat1, lon1=lon1, datum=datum,
1155 lat2=lat2, lon2=lon2, small=small, **kwds)
1157_idllmn6 = _idllmn6() # PYCHOK singleton
1160def intersection2(lat1, lon1, bearing1,
1161 lat2, lon2, bearing2, datum=None, wrap=False, small=_100km): # was=True
1162 '''I{Conveniently} compute the intersection of two lines each defined
1163 by a (geodetic) point and a bearing from North, using either ...
1165 1) L{vector3d.intersection3d3} for B{C{small}} distances (below 100 Km
1166 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ...
1168 2) L{sphericalTrigonometry.intersection} for a spherical B{C{datum}}
1169 or a C{scalar B{datum}} representing the earth radius, conventionally
1170 in C{meter} or ...
1172 3) L{sphericalNvector.intersection} if B{C{datum}} is a I{negative}
1173 C{scalar}, (negative) earth radius, conventionally in C{meter} or ...
1175 4) L{ellipsoidalKarney.intersection3} for an ellipsoidal B{C{datum}}
1176 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>}
1177 is installed, otherwise ...
1179 5) L{ellipsoidalExact.intersection3}, provided B{C{datum}} is ellipsoidal.
1181 @arg lat1: Latitude of the first point (C{degrees}).
1182 @arg lon1: Longitude of the first point (C{degrees}).
1183 @arg bearing1: Bearing at the first point (compass C{degrees}).
1184 @arg lat2: Latitude of the second point (C{degrees}).
1185 @arg lon2: Longitude of the second point (C{degrees}).
1186 @arg bearing2: Bearing at the second point (compass C{degrees}).
1187 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid},
1188 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth
1189 radius (C{meter}, same units as B{C{radius1}} and
1190 B{C{radius2}}) or C{None}.
1191 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
1192 and B{C{lon2}} (C{bool}).
1193 @kwarg small: Upper limit for small distances (C{meter}).
1195 @return: A L{LatLon2Tuple}C{(lat, lon)} with the lat- and
1196 longitude of the intersection point.
1198 @raise IntersectionError: Ambiguous or infinite intersection
1199 or colinear, parallel or otherwise
1200 non-intersecting lines.
1202 @raise TypeError: Invalid B{C{datum}}.
1204 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{bearing1}},
1205 B{C{lat2}}, B{C{lon2}} or B{C{bearing2}}.
1207 @see: Method L{RhumbLine.intersection2}.
1209 @note: The returned intersections may be near-antipodal.
1210 '''
1211 b1 = Bearing(bearing1=bearing1)
1212 b2 = Bearing(bearing2=bearing2)
1213 with _idllmn6(datum, lat1, lon1, lat2, lon2,
1214 small, wrap, False, bearing1=b1, bearing2=b2) as t:
1215 _i, d, lat2, lon2, m, n = t
1216 if d is None:
1217 t, _, _ = _i(m.Vector3d(lon1, lat1, 0), b1,
1218 m.Vector3d(lon2, lat2, 0), b2, useZ=False)
1219 t = LatLon2Tuple(t.y, t.x, name=n)
1221 else:
1222 t = _i(m.LatLon(lat1, lon1, datum=d), b1,
1223 m.LatLon(lat2, lon2, datum=d), b2, height=0, wrap=False)
1224 if isinstance(t, Intersection3Tuple): # ellipsoidal
1225 t, _, _ = t
1226 t = LatLon2Tuple(t.lat, t.lon, name=n)
1227 return t
1230def intersections2(lat1, lon1, radius1,
1231 lat2, lon2, radius2, datum=None, wrap=False, small=_100km): # was=True
1232 '''I{Conveniently} compute the intersections of two circles each defined
1233 by a (geodetic) center point and a radius, using either ...
1235 1) L{vector3d.intersections2} for B{C{small}} distances (below 100 Km
1236 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ...
1238 2) L{sphericalTrigonometry.intersections2} for a spherical B{C{datum}}
1239 or a C{scalar B{datum}} representing the earth radius, conventionally
1240 in C{meter} or ...
1242 3) L{ellipsoidalKarney.intersections2} for an ellipsoidal B{C{datum}}
1243 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>}
1244 is installed, otherwise ...
1246 4) L{ellipsoidalExact.intersections2}, provided B{C{datum}} is ellipsoidal.
1248 @arg lat1: Latitude of the first circle center (C{degrees}).
1249 @arg lon1: Longitude of the first circle center (C{degrees}).
1250 @arg radius1: Radius of the first circle (C{meter}, conventionally).
1251 @arg lat2: Latitude of the second circle center (C{degrees}).
1252 @arg lon2: Longitude of the second circle center (C{degrees}).
1253 @arg radius2: Radius of the second circle (C{meter}, same units as B{C{radius1}}).
1254 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid},
1255 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth
1256 radius (C{meter}, same units as B{C{radius1}} and
1257 B{C{radius2}}) or C{None}.
1258 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
1259 and B{C{lon2}} (C{bool}).
1260 @kwarg small: Upper limit for small distances (C{meter}).
1262 @return: 2-Tuple of the intersection points, each a
1263 L{LatLon2Tuple}C{(lat, lon)}. For abutting circles, the
1264 points are the same instance, aka the I{radical center}.
1266 @raise IntersectionError: Concentric, antipodal, invalid or
1267 non-intersecting circles or no
1268 convergence.
1270 @raise TypeError: Invalid B{C{datum}}.
1272 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{radius1}},
1273 B{C{lat2}}, B{C{lon2}} or B{C{radius2}}.
1274 '''
1275 r1 = Radius_(radius1=radius1)
1276 r2 = Radius_(radius2=radius2)
1277 with _idllmn6(datum, lat1, lon1, lat2, lon2,
1278 small, wrap, True, radius1=r1, radius2=r2) as t:
1279 _i, d, lat2, lon2, m, n = t
1280 if d is None:
1281 r1 = m2degrees(r1, radius=R_M, lat=lat1)
1282 r2 = m2degrees(r2, radius=R_M, lat=lat2)
1284 def _V2T(x, y, _, **unused): # _ == z unused
1285 return LatLon2Tuple(y, x, name=n)
1287 t = _i(m.Vector3d(lon1, lat1, 0), r1,
1288 m.Vector3d(lon2, lat2, 0), r2, sphere=False,
1289 Vector=_V2T)
1290 else:
1291 def _LL2T(lat, lon, **unused):
1292 return LatLon2Tuple(lat, lon, name=n)
1294 t = _i(m.LatLon(lat1, lon1, datum=d), r1,
1295 m.LatLon(lat2, lon2, datum=d), r2,
1296 LatLon=_LL2T, height=0, wrap=False)
1297 return t
1300def isantipode(lat1, lon1, lat2, lon2, eps=EPS):
1301 '''Check whether two points are I{antipodal}, on diametrically
1302 opposite sides of the earth.
1304 @arg lat1: Latitude of one point (C{degrees}).
1305 @arg lon1: Longitude of one point (C{degrees}).
1306 @arg lat2: Latitude of the other point (C{degrees}).
1307 @arg lon2: Longitude of the other point (C{degrees}).
1308 @kwarg eps: Tolerance for near-equality (C{degrees}).
1310 @return: C{True} if points are antipodal within the
1311 B{C{eps}} tolerance, C{False} otherwise.
1313 @see: Functions L{isantipode_} and L{antipode}.
1314 '''
1315 return (fabs(lat1 + lat2) <= eps and
1316 fabs(lon1 + lon2) <= eps) or _isequalTo(
1317 normal(lat1, lon1), antipode(lat2, lon2), eps)
1320def isantipode_(phi1, lam1, phi2, lam2, eps=EPS):
1321 '''Check whether two points are I{antipodal}, on diametrically
1322 opposite sides of the earth.
1324 @arg phi1: Latitude of one point (C{radians}).
1325 @arg lam1: Longitude of one point (C{radians}).
1326 @arg phi2: Latitude of the other point (C{radians}).
1327 @arg lam2: Longitude of the other point (C{radians}).
1328 @kwarg eps: Tolerance for near-equality (C{radians}).
1330 @return: C{True} if points are antipodal within the
1331 B{C{eps}} tolerance, C{False} otherwise.
1333 @see: Functions L{isantipode} and L{antipode_}.
1334 '''
1335 return (fabs(phi1 + phi2) <= eps and
1336 fabs(lam1 + lam2) <= eps) or _isequalTo_(
1337 normal_(phi1, lam1), antipode_(phi2, lam2), eps)
1340def _isequalTo(p1, p2, eps=EPS):
1341 '''Compare 2 point lat-/lons ignoring C{class}.
1342 '''
1343 return (fabs(p1.lat - p2.lat) <= eps and
1344 fabs(p1.lon - p2.lon) <= eps) if eps else (p1.latlon == p2.latlon)
1347def _isequalTo_(p1, p2, eps=EPS):
1348 '''(INTERNAL) Compare 2 point phi-/lams ignoring C{class}.
1349 '''
1350 return (fabs(p1.phi - p2.phi) <= eps and
1351 fabs(p1.lam - p2.lam) <= eps) if eps else (p1.philam == p2.philam)
1354def isnormal(lat, lon, eps=0):
1355 '''Check whether B{C{lat}} I{and} B{C{lon}} are within their
1356 respective I{normal} range in C{degrees}.
1358 @arg lat: Latitude (C{degrees}).
1359 @arg lon: Longitude (C{degrees}).
1360 @kwarg eps: Optional tolerance C{degrees}).
1362 @return: C{True} if C{(abs(B{lat}) + B{eps}) <= 90} and
1363 C{(abs(B{lon}) + B{eps}) <= 180}, C{False} othwerwise.
1365 @see: Functions L{isnormal_} and L{normal}.
1366 '''
1367 return (_90_0 - fabs(lat)) >= eps and (_180_0 - fabs(lon)) >= eps
1370def isnormal_(phi, lam, eps=0):
1371 '''Check whether B{C{phi}} I{and} B{C{lam}} are within their
1372 respective I{normal} range in C{radians}.
1374 @arg phi: Latitude (C{radians}).
1375 @arg lam: Longitude (C{radians}).
1376 @kwarg eps: Optional tolerance C{radians}).
1378 @return: C{True} if C{(abs(B{phi}) + B{eps}) <= PI/2} and
1379 C{(abs(B{lam}) + B{eps}) <= PI}, C{False} othwerwise.
1381 @see: Functions L{isnormal} and L{normal_}.
1382 '''
1383 return (PI_2 - fabs(phi)) >= eps and (PI - fabs(lam)) >= eps
1386def latlon2n_xyz(lat, lon, name=NN):
1387 '''Convert lat-, longitude to C{n-vector} (I{normal} to the
1388 earth's surface) X, Y and Z components.
1390 @arg lat: Latitude (C{degrees}).
1391 @arg lon: Longitude (C{degrees}).
1392 @kwarg name: Optional name (C{str}).
1394 @return: A L{Vector3Tuple}C{(x, y, z)}.
1396 @see: Function L{philam2n_xyz}.
1398 @note: These are C{n-vector} x, y and z components,
1399 I{NOT} geocentric ECEF x, y and z coordinates!
1400 '''
1401 return _2n_xyz(name, *sincos2d_(lat, lon))
1404def _normal2(a, b, n_2, n, n2):
1405 '''(INTERNAL) Helper for C{normal} and C{normal_}.
1406 '''
1407 if fabs(b) > n:
1408 b = remainder(b, n2)
1409 if fabs(a) > n_2:
1410 r = remainder(a, n)
1411 if r != a:
1412 a = -r
1413 b -= n if b > 0 else -n
1414 return float0_(a, b)
1417def normal(lat, lon, name=NN):
1418 '''Normalize a lat- I{and} longitude pair in C{degrees}.
1420 @arg lat: Latitude (C{degrees}).
1421 @arg lon: Longitude (C{degrees}).
1422 @kwarg name: Optional name (C{str}).
1424 @return: L{LatLon2Tuple}C{(lat, lon)} with C{abs(lat) <= 90}
1425 and C{abs(lon) <= 180}.
1427 @see: Functions L{normal_} and L{isnormal}.
1428 '''
1429 return LatLon2Tuple(*_normal2(lat, lon, _90_0, _180_0, _360_0),
1430 name=name or normal.__name__)
1433def normal_(phi, lam, name=NN):
1434 '''Normalize a lat- I{and} longitude pair in C{radians}.
1436 @arg phi: Latitude (C{radians}).
1437 @arg lam: Longitude (C{radians}).
1438 @kwarg name: Optional name (C{str}).
1440 @return: L{PhiLam2Tuple}C{(phi, lam)} with C{abs(phi) <= PI/2}
1441 and C{abs(lam) <= PI}.
1443 @see: Functions L{normal} and L{isnormal_}.
1444 '''
1445 return PhiLam2Tuple(*_normal2(phi, lam, PI_2, PI, PI2),
1446 name=name or normal_.__name__)
1449def _2n_xyz(name, sa, ca, sb, cb):
1450 '''(INTERNAL) Helper for C{latlon2n_xyz} and C{philam2n_xyz}.
1451 '''
1452 # Kenneth Gade eqn 3, but using right-handed
1453 # vector x -> 0°E,0°N, y -> 90°E,0°N, z -> 90°N
1454 return Vector3Tuple(ca * cb, ca * sb, sa, name=name)
1457def n_xyz2latlon(x, y, z, name=NN):
1458 '''Convert C{n-vector} components to lat- and longitude in C{degrees}.
1460 @arg x: X component (C{scalar}).
1461 @arg y: Y component (C{scalar}).
1462 @arg z: Z component (C{scalar}).
1463 @kwarg name: Optional name (C{str}).
1465 @return: A L{LatLon2Tuple}C{(lat, lon)}.
1467 @see: Function L{n_xyz2philam}.
1468 '''
1469 return LatLon2Tuple(atan2d(z, hypot(x, y)), atan2d(y, x), name=name)
1472def n_xyz2philam(x, y, z, name=NN):
1473 '''Convert C{n-vector} components to lat- and longitude in C{radians}.
1475 @arg x: X component (C{scalar}).
1476 @arg y: Y component (C{scalar}).
1477 @arg z: Z component (C{scalar}).
1478 @kwarg name: Optional name (C{str}).
1480 @return: A L{PhiLam2Tuple}C{(phi, lam)}.
1482 @see: Function L{n_xyz2latlon}.
1483 '''
1484 return PhiLam2Tuple(atan2(z, hypot(x, y)), atan2(y, x), name=name)
1487def _opposes(d, m, n, n2):
1488 '''(INETNAL) Helper for C{opposing} and C{opposing_}.
1489 '''
1490 d = d % n2 # -20 % 360 == 340, -1 % PI2 == PI2 - 1
1491 return False if d < m or d > (n2 - m) else (
1492 True if (n - m) < d < (n + m) else None)
1495def opposing(bearing1, bearing2, margin=_90_0):
1496 '''Compare the direction of two bearings given in C{degrees}.
1498 @arg bearing1: First bearing (compass C{degrees}).
1499 @arg bearing2: Second bearing (compass C{degrees}).
1500 @kwarg margin: Optional, interior angle bracket (C{degrees}).
1502 @return: C{True} if both bearings point in opposite, C{False} if
1503 in similar or C{None} if in perpendicular directions.
1505 @see: Function L{opposing_}.
1506 '''
1507 m = Degrees_(margin=margin, low=EPS0, high=_90_0)
1508 return _opposes(bearing2 - bearing1, m,_180_0, _360_0)
1511def opposing_(radians1, radians2, margin=PI_2):
1512 '''Compare the direction of two bearings given in C{radians}.
1514 @arg radians1: First bearing (C{radians}).
1515 @arg radians2: Second bearing (C{radians}).
1516 @kwarg margin: Optional, interior angle bracket (C{radians}).
1518 @return: C{True} if both bearings point in opposite, C{False} if
1519 in similar or C{None} if in perpendicular directions.
1521 @see: Function L{opposing}.
1522 '''
1523 m = Radians_(margin=margin, low=EPS0, high=PI_2)
1524 return _opposes(radians2 - radians1, m, PI, PI2)
1527def philam2n_xyz(phi, lam, name=NN):
1528 '''Convert lat-, longitude to C{n-vector} (I{normal} to the
1529 earth's surface) X, Y and Z components.
1531 @arg phi: Latitude (C{radians}).
1532 @arg lam: Longitude (C{radians}).
1533 @kwarg name: Optional name (C{str}).
1535 @return: A L{Vector3Tuple}C{(x, y, z)}.
1537 @see: Function L{latlon2n_xyz}.
1539 @note: These are C{n-vector} x, y and z components,
1540 I{NOT} geocentric ECEF x, y and z coordinates!
1541 '''
1542 return _2n_xyz(name, *sincos2_(phi, lam))
1545def _radical2(d, r1, r2): # in .ellipsoidalBaseDI, .sphericalTrigonometry, .vector3d
1546 # (INTERNAL) See C{radical2} below
1547 # assert d > EPS0
1548 r = fsumf_(_1_0, (r1 / d)**2, -(r2 / d)**2) * _0_5
1549 return Radical2Tuple(max(_0_0, min(_1_0, r)), r * d)
1552def radical2(distance, radius1, radius2):
1553 '''Compute the I{radical ratio} and I{radical line} of two
1554 U{intersecting circles<https://MathWorld.Wolfram.com/
1555 Circle-CircleIntersection.html>}.
1557 The I{radical line} is perpendicular to the axis thru the
1558 centers of the circles at C{(0, 0)} and C{(B{distance}, 0)}.
1560 @arg distance: Distance between the circle centers (C{scalar}).
1561 @arg radius1: Radius of the first circle (C{scalar}).
1562 @arg radius2: Radius of the second circle (C{scalar}).
1564 @return: A L{Radical2Tuple}C{(ratio, xline)} where C{0.0 <=
1565 ratio <= 1.0} and C{xline} is along the B{C{distance}}.
1567 @raise IntersectionError: The B{C{distance}} exceeds the sum
1568 of B{C{radius1}} and B{C{radius2}}.
1570 @raise UnitError: Invalid B{C{distance}}, B{C{radius1}} or
1571 B{C{radius2}}.
1573 @see: U{Circle-Circle Intersection
1574 <https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>}.
1575 '''
1576 d = Distance_(distance, low=_0_0)
1577 r1 = Radius_(radius1=radius1)
1578 r2 = Radius_(radius2=radius2)
1579 if d > (r1 + r2):
1580 raise IntersectionError(distance=d, radius1=r1, radius2=r2,
1581 txt=_too_(_distant_))
1582 return _radical2(d, r1, r2) if d > EPS0 else \
1583 Radical2Tuple(_0_5, _0_0)
1586class Radical2Tuple(_NamedTuple):
1587 '''2-Tuple C{(ratio, xline)} of the I{radical} C{ratio} and
1588 I{radical} C{xline}, both C{scalar} and C{0.0 <= ratio <= 1.0}
1589 '''
1590 _Names_ = (_ratio_, _xline_)
1591 _Units_ = ( Scalar, Scalar)
1594def _radistance(inst):
1595 '''(INTERNAL) Helper for the L{frechet._FrecherMeterRadians}
1596 and L{hausdorff._HausdorffMeterRedians} classes.
1597 '''
1598 kwds_ = _xkwds(inst._kwds, wrap=False)
1599 wrap_ = _xkwds_pop(kwds_, wrap=False)
1600 func_ = inst._func_
1601 try: # calling lower-overhead C{func_}
1602 func_(0, _0_25, _0_5, **kwds_)
1603 wrap_ = _Wrap._philamop(wrap_)
1604 except TypeError:
1605 return inst.distance
1607 _rad = radians
1609 def _philam(p):
1610 return _rad(p.lat), _rad(p.lon)
1612 def _radistance(point1, point2):
1613 try:
1614 phi1, lam1 = wrap_(point1.phi, point1.lam)
1615 phi2, lam2 = wrap_(point2.phi, point2.lam)
1616 except AttributeError: # no 'phi or .lam
1617 phi1, lam1 = wrap_(*_philam(point1))
1618 phi2, lam2 = wrap_(*_philam(point2))
1619 return func_(phi2, phi1, lam2 - lam1, **kwds_)
1621 inst._units = inst._units_
1622 return _radistance
1625def _scale_deg(lat1, lat2): # degrees
1626 # scale factor cos(mean of lats) for delta lon
1627 m = fabs(lat1 + lat2) * _0_5
1628 return cos(radians(m)) if m < 90 else _0_0
1631def _scale_rad(phi1, phi2): # radians, by .frechet, .hausdorff, .heights
1632 # scale factor cos(mean of phis) for delta lam
1633 m = fabs(phi1 + phi2) * _0_5
1634 return cos(m) if m < PI_2 else _0_0
1637def _sincosa6(phi2, phi1, lam21): # [4] in cosineLaw
1638 '''(INTERNAL) C{sin}es, C{cos}ines and C{acos}ine.
1639 '''
1640 s2, c2, s1, c1, _, c21 = sincos2_(phi2, phi1, lam21)
1641 return s2, c2, s1, c1, acos1(s1 * s2 + c1 * c2 * c21), c21
1644def thomas(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):
1645 '''Compute the distance between two (ellipsoidal) points using
1646 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>}
1647 formula.
1649 @arg lat1: Start latitude (C{degrees}).
1650 @arg lon1: Start longitude (C{degrees}).
1651 @arg lat2: End latitude (C{degrees}).
1652 @arg lon2: End longitude (C{degrees}).
1653 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
1654 L{Ellipsoid2} or L{a_f2Tuple}) to use.
1655 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
1656 B{C{lat2}} and B{C{lon2}} (C{bool}).
1658 @return: Distance (C{meter}, same units as the B{C{datum}}'s
1659 ellipsoid axes).
1661 @raise TypeError: Invalid B{C{datum}}.
1663 @see: Functions L{thomas_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
1664 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny},
1665 L{flatPolar}, L{haversine}, L{vincentys} and method L{Ellipsoid.distance2}.
1666 '''
1667 return _dE(thomas_, datum, wrap, lat1, lon1, lat2, lon2)
1670def thomas_(phi2, phi1, lam21, datum=_WGS84):
1671 '''Compute the I{angular} distance between two (ellipsoidal) points using
1672 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>}
1673 formula.
1675 @arg phi2: End latitude (C{radians}).
1676 @arg phi1: Start latitude (C{radians}).
1677 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
1678 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid},
1679 L{Ellipsoid2} or L{a_f2Tuple}).
1681 @return: Angular distance (C{radians}).
1683 @raise TypeError: Invalid B{C{datum}}.
1685 @see: Functions L{thomas}, L{cosineAndoyerLambert_},
1686 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
1687 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
1688 L{flatPolar_}, L{haversine_} and L{vincentys_} and U{Geodesy-PHP
1689 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/
1690 Distance/ThomasFormula.php>}.
1691 '''
1692 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21)
1693 if r and isnon0(c1) and isnon0(c2):
1694 E = _ellipsoidal(datum, thomas_)
1695 if E.f:
1696 r1 = atan2(E.b_a * s1, c1)
1697 r2 = atan2(E.b_a * s2, c2)
1699 j = (r2 + r1) * _0_5
1700 k = (r2 - r1) * _0_5
1701 sj, cj, sk, ck, h, _ = sincos2_(j, k, lam21 * _0_5)
1703 h = fsumf_(sk**2, (ck * h)**2, -(sj * h)**2)
1704 u = _1_0 - h
1705 if isnon0(u) and isnon0(h):
1706 r = atan(sqrt0(h / u)) * 2 # == acos(1 - 2 * h)
1707 sr, cr = sincos2(r)
1708 if isnon0(sr):
1709 u = 2 * (sj * ck)**2 / u
1710 h = 2 * (sk * cj)**2 / h
1711 x = u + h
1712 y = u - h
1714 s = r / sr
1715 e = 4 * s**2
1716 d = 2 * cr
1717 a = e * d
1718 b = 2 * r
1719 c = s - (a - d) * _0_5
1720 f = E.f * _0_25
1722 t = fsumf_(a * x, -b * y, c * x**2, -d * y**2, e * x * y)
1723 r -= fsumf_(s * x, -y, -t * f * _0_25) * f * sr
1724 return r
1727def vincentys(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
1728 '''Compute the distance between two (spherical) points using
1729 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>}
1730 spherical formula.
1732 @arg lat1: Start latitude (C{degrees}).
1733 @arg lon1: Start longitude (C{degrees}).
1734 @arg lat2: End latitude (C{degrees}).
1735 @arg lon2: End longitude (C{degrees}).
1736 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
1737 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
1738 L{a_f2Tuple}) to use.
1739 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
1740 B{C{lat2}} and B{C{lon2}} (C{bool}).
1742 @return: Distance (C{meter}, same units as B{C{radius}}).
1744 @raise UnitError: Invalid B{C{radius}}.
1746 @see: Functions L{vincentys_}, L{cosineAndoyerLambert},
1747 L{cosineForsytheAndoyerLambert},L{cosineLaw}, L{equirectangular},
1748 L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar},
1749 L{haversine} and L{thomas} and methods L{Ellipsoid.distance2},
1750 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
1752 @note: See note at function L{vincentys_}.
1753 '''
1754 return _dS(vincentys_, radius, wrap, lat1, lon1, lat2, lon2)
1757def vincentys_(phi2, phi1, lam21):
1758 '''Compute the I{angular} distance between two (spherical) points using
1759 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>}
1760 spherical formula.
1762 @arg phi2: End latitude (C{radians}).
1763 @arg phi1: Start latitude (C{radians}).
1764 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
1766 @return: Angular distance (C{radians}).
1768 @see: Functions L{vincentys}, L{cosineAndoyerLambert_},
1769 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
1770 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
1771 L{flatPolar_}, L{haversine_} and L{thomas_}.
1773 @note: Functions L{vincentys_}, L{haversine_} and L{cosineLaw_}
1774 produce equivalent results, but L{vincentys_} is suitable
1775 for antipodal points and slightly more expensive (M{3 cos,
1776 3 sin, 1 hypot, 1 atan2, 6 mul, 2 add}) than L{haversine_}
1777 (M{2 cos, 2 sin, 2 sqrt, 1 atan2, 5 mul, 1 add}) and
1778 L{cosineLaw_} (M{3 cos, 3 sin, 1 acos, 3 mul, 1 add}).
1779 '''
1780 s1, c1, s2, c2, s21, c21 = sincos2_(phi1, phi2, lam21)
1782 c = c2 * c21
1783 x = s1 * s2 + c1 * c
1784 y = c1 * s2 - s1 * c
1785 return atan2(hypot(c2 * s21, y), x)
1787# **) MIT License
1788#
1789# Copyright (C) 2016-2023 -- mrJean1 at Gmail -- All Rights Reserved.
1790#
1791# Permission is hereby granted, free of charge, to any person obtaining a
1792# copy of this software and associated documentation files (the "Software"),
1793# to deal in the Software without restriction, including without limitation
1794# the rights to use, copy, modify, merge, publish, distribute, sublicense,
1795# and/or sell copies of the Software, and to permit persons to whom the
1796# Software is furnished to do so, subject to the following conditions:
1797#
1798# The above copyright notice and this permission notice shall be included
1799# in all copies or substantial portions of the Software.
1800#
1801# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
1802# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
1803# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
1804# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
1805# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
1806# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
1807# OTHER DEALINGS IN THE SOFTWARE.