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.cartesianBase import CartesianBase # _MODS
10from pygeodesy.constants import EPS, EPS0, EPS1, PI, PI2, PI3, PI_2, R_M, \
11 _0_0s, float0_, isnon0, remainder, _umod_PI2, \
12 _0_0, _0_125, _0_25, _0_5, _1_0, _2_0, _4_0, \
13 _32_0, _90_0, _180_0, _360_0
14from pygeodesy.datums import Datum, Ellipsoid, _ellipsoidal_datum, \
15 _mean_radius, _spherical_datum, _WGS84, _EWGS84
16# from pygeodesy.ellipsoids import Ellipsoid, _EWGS84 # from .datums
17from pygeodesy.errors import IntersectionError, LimitError, limiterrors, \
18 _TypeError, _ValueError, _xattr, _xError, \
19 _xcallable,_xkwds, _xkwds_pop2
20from pygeodesy.fmath import euclid, hypot, hypot2, sqrt0
21from pygeodesy.fsums import fsumf_, Fmt, unstr
22# from pygeodesy.internals import _dunder_nameof # from .named
23from pygeodesy.interns import _delta_, _distant_, _inside_, _SPACE_, _too_
24from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS
25from pygeodesy.named import _name__, _name2__, _NamedTuple, _xnamed, \
26 _dunder_nameof
27from pygeodesy.namedTuples import Bearing2Tuple, Distance4Tuple, LatLon2Tuple, \
28 Intersection3Tuple, PhiLam2Tuple, Vector3Tuple
29# from pygeodesy.streprs import Fmt, unstr # from .fsums
30# from pygeodesy.triaxials import _hartzell3 # _MODS
31from pygeodesy.units import _isHeight, _isRadius, Bearing, Degrees_, Distance, \
32 Distance_, Height, Lam_, Lat, Lon, Meter_, Phi_, \
33 Radians, Radians_, Radius, Radius_, Scalar, _100km
34from pygeodesy.utily import acos1, atan2b, atan2d, degrees2m, _loneg, m2degrees, \
35 tan_2, sincos2, sincos2_, sincos2d_, _Wrap
36# from pygeodesy.vector3d import _otherV3d # _MODS
37# from pygeodesy.vector3dBase import _xyz_y_z3 # _MODS
38# from pygeodesy import ellipsoidalExact, ellipsoidalKarney, vector3d, \
39# sphericalNvector, sphericalTrigonometry # _MODS
41from contextlib import contextmanager
42from math import asin, atan, atan2, cos, degrees, fabs, radians, sin, sqrt # pow
44__all__ = _ALL_LAZY.formy
45__version__ = '24.05.28'
47_RADIANS2 = (PI / _180_0)**2 # degrees- to radians-squared
48_ratio_ = 'ratio'
49_xline_ = 'xline'
52def _anti2(a, b, n_2, n, n2):
53 '''(INTERNAL) Helper for C{antipode} and C{antipode_}.
54 '''
55 r = remainder(a, n) if fabs(a) > n_2 else a
56 if r == a:
57 r = -r
58 b += n
59 if fabs(b) > n:
60 b = remainder(b, n2)
61 return float0_(r, b)
64def antipode(lat, lon, **name):
65 '''Return the antipode, the point diametrically opposite
66 to a given point in C{degrees}.
68 @arg lat: Latitude (C{degrees}).
69 @arg lon: Longitude (C{degrees}).
70 @kwarg name: Optional C{B{name}=NN} (C{str}).
72 @return: A L{LatLon2Tuple}C{(lat, lon)}.
74 @see: Functions L{antipode_} and L{normal} and U{Geosphere
75 <https://CRAN.R-Project.org/web/packages/geosphere/geosphere.pdf>}.
76 '''
77 return LatLon2Tuple(*_anti2(lat, lon, _90_0, _180_0, _360_0), **name)
80def antipode_(phi, lam, **name):
81 '''Return the antipode, the point diametrically opposite
82 to a given point in C{radians}.
84 @arg phi: Latitude (C{radians}).
85 @arg lam: Longitude (C{radians}).
86 @kwarg name: Optional C{B{name}=NN} (C{str}).
88 @return: A L{PhiLam2Tuple}C{(phi, lam)}.
90 @see: Functions L{antipode} and L{normal_} and U{Geosphere
91 <https://CRAN.R-Project.org/web/packages/geosphere/geosphere.pdf>}.
92 '''
93 return PhiLam2Tuple(*_anti2(phi, lam, PI_2, PI, PI2), **name)
96def bearing(lat1, lon1, lat2, lon2, **final_wrap):
97 '''Compute the initial or final bearing (forward or reverse
98 azimuth) between a (spherical) start and end point.
100 @arg lat1: Start latitude (C{degrees}).
101 @arg lon1: Start longitude (C{degrees}).
102 @arg lat2: End latitude (C{degrees}).
103 @arg lon2: End longitude (C{degrees}).
104 @kwarg final_wrap: Optional keyword arguments for function
105 L{pygeodesy.bearing_}.
107 @return: Initial or final bearing (compass C{degrees360}) or
108 zero if start and end point coincide.
109 '''
110 r = bearing_(Phi_(lat1=lat1), Lam_(lon1=lon1),
111 Phi_(lat2=lat2), Lam_(lon2=lon2), **final_wrap)
112 return degrees(r)
115def bearing_(phi1, lam1, phi2, lam2, final=False, wrap=False):
116 '''Compute the initial or final bearing (forward or reverse azimuth)
117 between a (spherical) start and end point.
119 @arg phi1: Start latitude (C{radians}).
120 @arg lam1: Start longitude (C{radians}).
121 @arg phi2: End latitude (C{radians}).
122 @arg lam2: End longitude (C{radians}).
123 @kwarg final: If C{True}, return the final, otherwise the initial
124 bearing (C{bool}).
125 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{phi2}}
126 and B{C{lam2}} (C{bool}).
128 @return: Initial or final bearing (compass C{radiansPI2}) or zero if
129 start and end point coincide.
131 @see: U{Bearing<https://www.Movable-Type.co.UK/scripts/latlong.html>},
132 U{Course between two points<https://www.EdWilliams.org/avform147.htm#Crs>}
133 and U{Bearing Between Two Points<https://web.Archive.org/web/20020630205931/
134 https://MathForum.org/library/drmath/view/55417.html>}.
135 '''
136 db, phi2, lam2 = _Wrap.philam3(lam1, phi2, lam2, wrap)
137 if final: # swap plus PI
138 phi1, lam1, phi2, lam2, db = phi2, lam2, phi1, lam1, -db
139 r = PI3
140 else:
141 r = PI2
142 sa1, ca1, sa2, ca2, sdb, cdb = sincos2_(phi1, phi2, db)
144 x = ca1 * sa2 - sa1 * ca2 * cdb
145 y = sdb * ca2
146 return _umod_PI2(atan2(y, x) + r) # .utily.wrapPI2
149def _bearingTo2(p1, p2, wrap=False): # for points.ispolar, sphericalTrigonometry.areaOf
150 '''(INTERNAL) Compute initial and final bearing.
151 '''
152 try: # for LatLon_ and ellipsoidal LatLon
153 return p1.bearingTo2(p2, wrap=wrap)
154 except AttributeError:
155 pass
156 # XXX spherical version, OK for ellipsoidal ispolar?
157 t = p1.philam + p2.philam
158 return Bearing2Tuple(degrees(bearing_(*t, final=False, wrap=wrap)),
159 degrees(bearing_(*t, final=True, wrap=wrap)),
160 name__=_bearingTo2)
163def compassAngle(lat1, lon1, lat2, lon2, adjust=True, wrap=False):
164 '''Return the angle from North for the direction vector M{(lon2 - lon1,
165 lat2 - lat1)} between two points.
167 Suitable only for short, not near-polar vectors up to a few hundred
168 Km or Miles. Use function L{pygeodesy.bearing} for longer vectors.
170 @arg lat1: From latitude (C{degrees}).
171 @arg lon1: From longitude (C{degrees}).
172 @arg lat2: To latitude (C{degrees}).
173 @arg lon2: To longitude (C{degrees}).
174 @kwarg adjust: Adjust the longitudinal delta by the cosine of the
175 mean latitude (C{bool}).
176 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
177 and B{C{lon2}} (C{bool}).
179 @return: Compass angle from North (C{degrees360}).
181 @note: Courtesy of Martin Schultz.
183 @see: U{Local, flat earth approximation
184 <https://www.EdWilliams.org/avform.htm#flat>}.
185 '''
186 d_lon, lat2, lon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap)
187 if adjust: # scale delta lon
188 d_lon *= _scale_deg(lat1, lat2)
189 return atan2b(d_lon, lat2 - lat1)
192def cosineAndoyerLambert(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):
193 '''Compute the distance between two (ellipsoidal) points using the U{Andoyer-Lambert
194 <https://books.google.com/books?id=x2UiAQAAIAAJ>} correction of the U{Law of
195 Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula.
197 @arg lat1: Start latitude (C{degrees}).
198 @arg lon1: Start longitude (C{degrees}).
199 @arg lat2: End latitude (C{degrees}).
200 @arg lon2: End longitude (C{degrees}).
201 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
202 L{Ellipsoid2} or L{a_f2Tuple}) to use.
203 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
204 B{C{lat2}} and B{C{lon2}} (C{bool}).
206 @return: Distance (C{meter}, same units as the B{C{datum}}'s
207 ellipsoid axes or C{radians} if B{C{datum}} is C{None}).
209 @raise TypeError: Invalid B{C{datum}}.
211 @see: Functions L{cosineAndoyerLambert_}, L{cosineForsytheAndoyerLambert},
212 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny},
213 L{flatPolar}, L{haversine}, L{thomas} and L{vincentys} and method
214 L{Ellipsoid.distance2}.
215 '''
216 return _dE(cosineAndoyerLambert_, datum, wrap, lat1, lon1, lat2, lon2)
219def cosineAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84):
220 '''Compute the I{angular} distance between two (ellipsoidal) points using the U{Andoyer-Lambert
221 <https://books.google.com/books?id=x2UiAQAAIAAJ>} correction of the U{Law of
222 Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula.
224 @arg phi2: End latitude (C{radians}).
225 @arg phi1: Start latitude (C{radians}).
226 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
227 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
228 L{Ellipsoid2} or L{a_f2Tuple}) to use.
230 @return: Angular distance (C{radians}).
232 @raise TypeError: Invalid B{C{datum}}.
234 @see: Functions L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert_},
235 L{cosineLaw_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_},
236 L{haversine_}, L{thomas_} and L{vincentys_} and U{Geodesy-PHP
237 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/Distance/
238 AndoyerLambert.php>}.
239 '''
240 s2, c2, s1, c1, r, c21 = _sincosa6(phi2, phi1, lam21)
241 if isnon0(c1) and isnon0(c2):
242 E = _ellipsoidal(datum, cosineAndoyerLambert_)
243 if E.f: # ellipsoidal
244 r2 = atan2(E.b_a * s2, c2)
245 r1 = atan2(E.b_a * s1, c1)
246 s2, c2, s1, c1 = sincos2_(r2, r1)
247 r = acos1(s1 * s2 + c1 * c2 * c21)
248 if r:
249 sr, _, sr_2, cr_2 = sincos2_(r, r * _0_5)
250 if isnon0(sr_2) and isnon0(cr_2):
251 s = (sr + r) * ((s1 - s2) / sr_2)**2
252 c = (sr - r) * ((s1 + s2) / cr_2)**2
253 r += (c - s) * E.f * _0_125
254 return r
257def cosineForsytheAndoyerLambert(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):
258 '''Compute the distance between two (ellipsoidal) points using the U{Forsythe-Andoyer-Lambert
259 <https://www2.UNB.Ca/gge/Pubs/TR77.pdf>} correction of the U{Law of Cosines
260 <https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula.
262 @arg lat1: Start latitude (C{degrees}).
263 @arg lon1: Start longitude (C{degrees}).
264 @arg lat2: End latitude (C{degrees}).
265 @arg lon2: End longitude (C{degrees}).
266 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
267 L{Ellipsoid2} or L{a_f2Tuple}) to use.
268 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
269 B{C{lat2}} and B{C{lon2}} (C{bool}).
271 @return: Distance (C{meter}, same units as the B{C{datum}}'s
272 ellipsoid axes or C{radians} if B{C{datum}} is C{None}).
274 @raise TypeError: Invalid B{C{datum}}.
276 @see: Functions L{cosineForsytheAndoyerLambert_}, L{cosineAndoyerLambert},
277 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny},
278 L{flatPolar}, L{haversine}, L{thomas} and L{vincentys} and method
279 L{Ellipsoid.distance2}.
280 '''
281 return _dE(cosineForsytheAndoyerLambert_, datum, wrap, lat1, lon1, lat2, lon2)
284def cosineForsytheAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84):
285 '''Compute the I{angular} distance between two (ellipsoidal) points using the
286 U{Forsythe-Andoyer-Lambert<https://www2.UNB.Ca/gge/Pubs/TR77.pdf>} correction of
287 the U{Law of Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>}
288 formula.
290 @arg phi2: End latitude (C{radians}).
291 @arg phi1: Start latitude (C{radians}).
292 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
293 @kwarg datum: Datum (L{Datum}) or ellipsoid to use (L{Ellipsoid},
294 L{Ellipsoid2} or L{a_f2Tuple}).
296 @return: Angular distance (C{radians}).
298 @raise TypeError: Invalid B{C{datum}}.
300 @see: Functions L{cosineForsytheAndoyerLambert}, L{cosineAndoyerLambert_},
301 L{cosineLaw_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_},
302 L{haversine_}, L{thomas_} and L{vincentys_} and U{Geodesy-PHP
303 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/
304 Distance/ForsytheCorrection.php>}.
305 '''
306 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21)
307 if r and isnon0(c1) and isnon0(c2):
308 E = _ellipsoidal(datum, cosineForsytheAndoyerLambert_)
309 if E.f: # ellipsoidal
310 sr, cr, s2r, _ = sincos2_(r, r * 2)
311 if isnon0(sr) and fabs(cr) < EPS1:
312 s = (s1 + s2)**2 / (1 + cr)
313 t = (s1 - s2)**2 / (1 - cr)
314 x = s + t
315 y = s - t
317 s = 8 * r**2 / sr
318 a = 64 * r + s * cr * 2 # 16 * r**2 / tan(r)
319 d = 48 * sr + s # 8 * r**2 / tan(r)
320 b = -2 * d
321 e = 30 * s2r
322 c = fsumf_(30 * r, e * _0_5, s * cr) # 8 * r**2 / tan(r)
323 t = fsumf_( a * x, e * y**2, b * y, -c * x**2, d * x * y)
325 r += fsumf_(-r * x, 3 * y * sr, t * E.f / _32_0) * E.f * _0_25
326 return r
329def cosineLaw(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
330 '''Compute the distance between two points using the U{spherical Law of Cosines
331 <https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} 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 Law of
360 Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula.
362 @arg phi2: End latitude (C{radians}).
363 @arg phi1: Start latitude (C{radians}).
364 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
366 @return: Angular distance (C{radians}).
368 @see: Functions L{cosineLaw}, L{cosineAndoyerLambert_},
369 L{cosineForsytheAndoyerLambert_}, L{euclidean_},
370 L{flatLocal_}/L{hubeny_}, L{flatPolar_}, L{haversine_},
371 L{thomas_} and L{vincentys_}.
373 @note: See note at function L{vincentys_}.
374 '''
375 return _sincosa6(phi2, phi1, lam21)[4]
378def _d3(wrap, lat1, lon1, lat2, lon2):
379 '''(INTERNAL) Helper for _dE, _dS and _eA.
380 '''
381 if wrap:
382 d_lon, lat2, _ = _Wrap.latlon3(lon1, lat2, lon2, wrap)
383 return radians(lat2), Phi_(lat1=lat1), radians(d_lon)
384 else: # for backward compaibility
385 return Phi_(lat2=lat2), Phi_(lat1=lat1), Phi_(d_lon=lon2 - lon1)
388def _dE(func_, earth, *wrap_lls):
389 '''(INTERNAL) Helper for ellipsoidal distances.
390 '''
391 E = _ellipsoidal(earth, func_)
392 r = func_(*_d3(*wrap_lls), datum=E)
393 return r * E.a
396def _dS(func_, radius, *wrap_lls, **adjust):
397 '''(INTERNAL) Helper for spherical distances.
398 '''
399 r = func_(*_d3(*wrap_lls), **adjust)
400 if radius is not R_M:
401 _, lat1, _, lat2, _ = wrap_lls
402 radius = _mean_radius(radius, lat1, lat2)
403 return r * radius
406def _eA(excess_, radius, *wrap_lls):
407 '''(INTERNAL) Helper for spherical excess or area.
408 '''
409 r = excess_(*_d3(*wrap_lls))
410 if radius:
411 _, lat1, _, lat2, _ = wrap_lls
412 r *= _mean_radius(radius, lat1, lat2)**2
413 return r
416def _ellipsoidal(earth, where):
417 '''(INTERNAL) Helper for distances.
418 '''
419 return _EWGS84 if earth in (_WGS84, _EWGS84) else (
420 earth if isinstance(earth, Ellipsoid) else
421 (earth if isinstance(earth, Datum) else # PYCHOK indent
422 _ellipsoidal_datum(earth, name__=where)).ellipsoid)
425def equirectangular(lat1, lon1, lat2, lon2, radius=R_M, **adjust_limit_wrap):
426 '''Compute the distance between two points using the U{Equirectangular Approximation
427 / Projection<https://www.Movable-Type.co.UK/scripts/latlong.html#equirectangular>}.
429 @arg lat1: Start latitude (C{degrees}).
430 @arg lon1: Start longitude (C{degrees}).
431 @arg lat2: End latitude (C{degrees}).
432 @arg lon2: End longitude (C{degrees}).
433 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum}) or ellipsoid
434 (L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}).
435 @kwarg adjust_limit_wrap: Optional keyword arguments for function L{equirectangular4}.
437 @return: Distance (C{meter}, same units as B{C{radius}} or the ellipsoid or datum axes).
439 @raise TypeError: Invalid B{C{radius}}.
441 @see: Function L{equirectangular4} for more details, the available B{C{options}},
442 errors, restrictions and other, approximate or accurate distance functions.
443 '''
444 d = sqrt(equirectangular4(Lat(lat1=lat1), Lon(lon1=lon1),
445 Lat(lat2=lat2), Lon(lon2=lon2),
446 **adjust_limit_wrap).distance2) # PYCHOK 4 vs 2-3
447 return degrees2m(d, radius=_mean_radius(radius, lat1, lat2))
450def _equirectangular(lat1, lon1, lat2, lon2, **adjust_limit_wrap):
451 '''(INTERNAL) Helper for the L{frechet._FrechetMeterRadians}
452 and L{hausdorff._HausdorffMeterRedians} classes.
453 '''
454 return equirectangular4(lat1, lon1, lat2, lon2, **adjust_limit_wrap).distance2 * _RADIANS2
457def equirectangular4(lat1, lon1, lat2, lon2, adjust=True, limit=45, wrap=False):
458 '''Compute the distance between two points using the U{Equirectangular Approximation
459 / Projection<https://www.Movable-Type.co.UK/scripts/latlong.html#equirectangular>}.
461 This approximation is valid for short distance of several hundred Km
462 or Miles, see the B{C{limit}} keyword argument and L{LimitError}.
464 @arg lat1: Start latitude (C{degrees}).
465 @arg lon1: Start longitude (C{degrees}).
466 @arg lat2: End latitude (C{degrees}).
467 @arg lon2: End longitude (C{degrees}).
468 @kwarg adjust: Adjust the wrapped, unrolled longitudinal delta by the cosine
469 of the mean latitude (C{bool}).
470 @kwarg limit: Optional limit for lat- and longitudinal deltas (C{degrees}) or
471 C{None} or C{0} for unlimited.
472 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} and
473 B{C{lon2}} (C{bool}).
475 @return: A L{Distance4Tuple}C{(distance2, delta_lat, delta_lon, unroll_lon2)}
476 in C{degrees squared}.
478 @raise LimitError: If the lat- and/or longitudinal delta exceeds the
479 B{C{-limit..limit}} range and L{pygeodesy.limiterrors}
480 set to C{True}.
482 @see: U{Local, flat earth approximation
483 <https://www.EdWilliams.org/avform.htm#flat>}, functions
484 L{equirectangular}, L{cosineAndoyerLambert},
485 L{cosineForsytheAndoyerLambert}, L{cosineLaw}, L{euclidean},
486 L{flatLocal}/L{hubeny}, L{flatPolar}, L{haversine}, L{thomas}
487 and L{vincentys} and methods L{Ellipsoid.distance2},
488 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
489 '''
490 d_lon, lat2, ulon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap)
491 d_lat = lat2 - lat1
493 if limit and limit > 0 and limiterrors():
494 d = max(fabs(d_lat), fabs(d_lon))
495 if d > limit:
496 t = _SPACE_(_delta_, Fmt.PAREN_g(d), Fmt.exceeds_limit(limit))
497 s = unstr(equirectangular4, lat1, lon1, lat2, lon2,
498 limit=limit, wrap=wrap)
499 raise LimitError(s, txt=t)
501 if adjust: # scale delta lon
502 d_lon *= _scale_deg(lat1, lat2)
504 d2 = hypot2(d_lat, d_lon) # degrees squared!
505 return Distance4Tuple(d2, d_lat, d_lon, ulon2 - lon2)
508def euclidean(lat1, lon1, lat2, lon2, radius=R_M, adjust=True, wrap=False):
509 '''Approximate the C{Euclidean} distance between two (spherical) points.
511 @arg lat1: Start latitude (C{degrees}).
512 @arg lon1: Start longitude (C{degrees}).
513 @arg lat2: End latitude (C{degrees}).
514 @arg lon2: End longitude (C{degrees}).
515 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
516 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
517 L{a_f2Tuple}) to use.
518 @kwarg adjust: Adjust the longitudinal delta by the cosine of
519 the mean latitude (C{bool}).
520 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
521 and B{C{lon2}} (C{bool}).
523 @return: Distance (C{meter}, same units as B{C{radius}} or the
524 ellipsoid or datum axes).
526 @raise TypeError: Invalid B{C{radius}}.
528 @see: U{Distance between two (spherical) points
529 <https://www.EdWilliams.org/avform.htm#Dist>}, functions L{euclid},
530 L{euclidean_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
531 L{cosineLaw}, L{equirectangular}, L{flatLocal}/L{hubeny}, L{flatPolar},
532 L{haversine}, L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2},
533 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
534 '''
535 return _dS(euclidean_, radius, wrap, lat1, lon1, lat2, lon2, adjust=adjust)
538def euclidean_(phi2, phi1, lam21, adjust=True):
539 '''Approximate the I{angular} C{Euclidean} distance between two (spherical) points.
541 @arg phi2: End latitude (C{radians}).
542 @arg phi1: Start latitude (C{radians}).
543 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
544 @kwarg adjust: Adjust the longitudinal delta by the cosine
545 of the mean latitude (C{bool}).
547 @return: Angular distance (C{radians}).
549 @see: Functions L{euclid}, L{euclidean}, L{cosineAndoyerLambert_},
550 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
551 L{flatLocal_}/L{hubeny_}, L{flatPolar_}, L{haversine_},
552 L{thomas_} and L{vincentys_}.
553 '''
554 if adjust:
555 lam21 *= _scale_rad(phi2, phi1)
556 return euclid(phi2 - phi1, lam21)
559def excessAbc_(A, b, c):
560 '''Compute the I{spherical excess} C{E} of a (spherical) triangle from two sides
561 and the included (small) angle.
563 @arg A: An interior triangle angle (C{radians}).
564 @arg b: Frist adjacent triangle side (C{radians}).
565 @arg c: Second adjacent triangle side (C{radians}).
567 @return: Spherical excess (C{radians}).
569 @raise UnitError: Invalid B{C{A}}, B{C{b}} or B{C{c}}.
571 @see: Functions L{excessGirard_}, L{excessLHuilier_} and U{Spherical
572 trigonometry<https://WikiPedia.org/wiki/Spherical_trigonometry>}.
573 '''
574 A = Radians_(A=A)
575 b = Radians_(b=b) * _0_5
576 c = Radians_(c=c) * _0_5
578 sA, cA, sb, cb, sc, cc = sincos2_(A, b, c)
579 return atan2(sA * sb * sc, cb * cc + cA * sb * sc) * _2_0
582def excessCagnoli_(a, b, c):
583 '''Compute the I{spherical excess} C{E} of a (spherical) triangle using U{Cagnoli's
584 <https://Zenodo.org/record/35392>} (D.34) formula.
586 @arg a: First triangle side (C{radians}).
587 @arg b: Second triangle side (C{radians}).
588 @arg c: Third triangle side (C{radians}).
590 @return: Spherical excess (C{radians}).
592 @raise UnitError: Invalid B{C{a}}, B{C{b}} or B{C{c}}.
594 @see: Function L{excessLHuilier_} and U{Spherical trigonometry
595 <https://WikiPedia.org/wiki/Spherical_trigonometry>}.
596 '''
597 a = Radians_(a=a)
598 b = Radians_(b=b)
599 c = Radians_(c=c)
601 s = fsumf_(a, b, c) * _0_5
602 _s = sin
603 r = _s(s) * _s(s - a) * _s(s - b) * _s(s - c)
604 c = cos(a * _0_5) * cos(b * _0_5) * cos(c * _0_5)
605 r = asin(sqrt(r) * _0_5 / c) if c and r > 0 else _0_0
606 return Radians(Cagnoli=r * _2_0)
609def excessGirard_(A, B, C):
610 '''Compute the I{spherical excess} C{E} of a (spherical) triangle using U{Girard's
611 <https://MathWorld.Wolfram.com/GirardsSphericalExcessFormula.html>} formula.
613 @arg A: First interior triangle angle (C{radians}).
614 @arg B: Second interior triangle angle (C{radians}).
615 @arg C: Third interior triangle angle (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{excessLHuilier_} and U{Spherical trigonometry
622 <https://WikiPedia.org/wiki/Spherical_trigonometry>}.
623 '''
624 return Radians(Girard=fsumf_(Radians_(A=A),
625 Radians_(B=B),
626 Radians_(C=C), -PI))
629def excessLHuilier_(a, b, c):
630 '''Compute the I{spherical excess} C{E} of a (spherical) triangle using U{L'Huilier's
631 <https://MathWorld.Wolfram.com/LHuiliersTheorem.html>}'s Theorem.
633 @arg a: First triangle side (C{radians}).
634 @arg b: Second triangle side (C{radians}).
635 @arg c: Third triangle side (C{radians}).
637 @return: Spherical excess (C{radians}).
639 @raise UnitError: Invalid B{C{a}}, B{C{b}} or B{C{c}}.
641 @see: Function L{excessCagnoli_}, L{excessGirard_} and U{Spherical
642 trigonometry<https://WikiPedia.org/wiki/Spherical_trigonometry>}.
643 '''
644 a = Radians_(a=a)
645 b = Radians_(b=b)
646 c = Radians_(c=c)
648 s = fsumf_(a, b, c) * _0_5
649 _t = tan_2
650 r = _t(s) * _t(s - a) * _t(s - b) * _t(s - c)
651 r = atan(sqrt(r)) if r > 0 else _0_0
652 return Radians(LHuilier=r * _4_0)
655def excessKarney(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
656 '''Compute the surface area of a (spherical) quadrilateral bounded by a
657 segment of a great circle, two meridians and the equator using U{Karney's
658 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>}
659 method.
661 @arg lat1: Start latitude (C{degrees}).
662 @arg lon1: Start longitude (C{degrees}).
663 @arg lat2: End latitude (C{degrees}).
664 @arg lon2: End longitude (C{degrees}).
665 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
666 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
667 L{a_f2Tuple}) or C{None}.
668 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
669 B{C{lat2}} and B{C{lon2}} (C{bool}).
671 @return: Surface area, I{signed} (I{square} C{meter} or the same units as
672 B{C{radius}} I{squared}) or the I{spherical excess} (C{radians})
673 if C{B{radius}=0} or C{None}.
675 @raise TypeError: Invalid B{C{radius}}.
677 @raise UnitError: Invalid B{C{lat2}} or B{C{lat1}}.
679 @raise ValueError: Semi-circular longitudinal delta.
681 @see: Functions L{excessKarney_} and L{excessQuad}.
682 '''
683 return _eA(excessKarney_, radius, wrap, lat1, lon1, lat2, lon2)
686def excessKarney_(phi2, phi1, lam21):
687 '''Compute the I{spherical excess} C{E} of a (spherical) quadrilateral bounded by
688 a segment of a great circle, two meridians and the equator using U{Karney's
689 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>}
690 method.
692 @arg phi2: End latitude (C{radians}).
693 @arg phi1: Start latitude (C{radians}).
694 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
696 @return: Spherical excess, I{signed} (C{radians}).
698 @raise ValueError: Semi-circular longitudinal delta B{C{lam21}}.
700 @see: Function L{excessKarney} and U{Area of a spherical polygon
701 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>}.
702 '''
703 # from: Veness <https://www.Movable-Type.co.UK/scripts/latlong.html> Area
704 # method due to Karney: for each edge of the polygon,
705 #
706 # tan(Δλ / 2) · (tan(φ1 / 2) + tan(φ2 / 2))
707 # tan(E / 2) = -----------------------------------------
708 # 1 + tan(φ1 / 2) · tan(φ2 / 2)
709 #
710 # where E is the spherical excess of the trapezium obtained by extending
711 # the edge to the equator-circle vector for each edge (see also ***).
712 _t = tan_2
713 t2 = _t(phi2)
714 t1 = _t(phi1)
715 t = _t(lam21, lam21=None)
716 return Radians(Karney=atan2(t * (t1 + t2),
717 _1_0 + (t1 * t2)) * _2_0)
720# ***) Original post no longer available, following is a copy of the main part
721# <http://OSGeo-org.1560.x6.Nabble.com/Area-of-a-spherical-polygon-td3841625.html>
722#
723# The area of a polygon on a (unit) sphere is given by the spherical excess
724#
725# A = 2 * pi - sum(exterior angles)
726#
727# However this is badly conditioned if the polygon is small. In this case, use
728#
729# A = sum(S12{i, i+1}) over the edges of the polygon
730#
731# where S12 is the area of the quadrilateral bounded by an edge of the polygon,
732# two meridians and the equator, i.e. with vertices (phi1, lambda1), (phi2,
733# lambda2), (0, lambda1) and (0, lambda2). S12 is given by
734#
735# tan(S12 / 2) = tan(lambda21 / 2) * (tan(phi1 / 2) + tan(phi2 / 2)) /
736# (tan(phi1 / 2) * tan(phi2 / 2) + 1)
737#
738# = tan(lambda21 / 2) * tanh((Lamb(phi1) + Lamb(phi2)) / 2)
739#
740# where lambda21 = lambda2 - lambda1 and Lamb(x) is the Lambertian (or the
741# inverse Gudermannian) function
742#
743# Lambertian(x) = asinh(tan(x)) = atanh(sin(x)) = 2 * atanh(tan(x / 2))
744#
745# Notes: The formula for S12 is exact, except that...
746# - it is indeterminate if an edge is a semi-circle
747# - the formula for A applies only if the polygon does not include a pole
748# (if it does, then add +/- 2 * pi to the result)
749# - in the limit of small phi and lambda, S12 reduces to the trapezoidal
750# formula, S12 = (lambda2 - lambda1) * (phi1 + phi2) / 2
751# - I derived this result from the equation for the area of a spherical
752# triangle in terms of two edges and the included angle given by, e.g.
753# U{Todhunter, I. - Spherical Trigonometry (1871), Sec. 103, Eq. (2)
754# <http://Books.Google.com/books?id=3uBHAAAAIAAJ&pg=PA71>}
755# - I would be interested to know if this formula for S12 is already known
756# - Charles Karney
759def excessQuad(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
760 '''Compute the surface area of a (spherical) quadrilateral bounded by a segment
761 of a great circle, two meridians and the equator.
763 @arg lat1: Start latitude (C{degrees}).
764 @arg lon1: Start longitude (C{degrees}).
765 @arg lat2: End latitude (C{degrees}).
766 @arg lon2: End longitude (C{degrees}).
767 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
768 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
769 L{a_f2Tuple}) or C{None}.
770 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
771 B{C{lat2}} and B{C{lon2}} (C{bool}).
773 @return: Surface area, I{signed} (I{square} C{meter} or the same units as
774 B{C{radius}} I{squared}) or the I{spherical excess} (C{radians})
775 if C{B{radius}=0} or C{None}.
777 @raise TypeError: Invalid B{C{radius}}.
779 @raise UnitError: Invalid B{C{lat2}} or B{C{lat1}}.
781 @see: Function L{excessQuad_} and L{excessKarney}.
782 '''
783 return _eA(excessQuad_, radius, wrap, lat1, lon1, lat2, lon2)
786def excessQuad_(phi2, phi1, lam21):
787 '''Compute the I{spherical excess} C{E} of a (spherical) quadrilateral bounded
788 by a segment of a great circle, two meridians and the equator.
790 @arg phi2: End latitude (C{radians}).
791 @arg phi1: Start latitude (C{radians}).
792 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
794 @return: Spherical excess, I{signed} (C{radians}).
796 @see: Function L{excessQuad} and U{Spherical trigonometry
797 <https://WikiPedia.org/wiki/Spherical_trigonometry>}.
798 '''
799 s = sin((phi2 + phi1) * _0_5)
800 c = cos((phi2 - phi1) * _0_5)
801 return Radians(Quad=atan2(tan_2(lam21) * s, c) * _2_0)
804def flatLocal(lat1, lon1, lat2, lon2, datum=_WGS84, scaled=True, wrap=False):
805 '''Compute the distance between two (ellipsoidal) points using
806 the U{ellipsoidal Earth to plane projection<https://WikiPedia.org/
807 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>}
808 aka U{Hubeny<https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula.
810 @arg lat1: Start latitude (C{degrees}).
811 @arg lon1: Start longitude (C{degrees}).
812 @arg lat2: End latitude (C{degrees}).
813 @arg lon2: End longitude (C{degrees}).
814 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
815 L{Ellipsoid2} or L{a_f2Tuple}) to use.
816 @kwarg scaled: Scale prime_vertical by C{cos(B{phi})} (C{bool}),
817 see method L{pygeodesy.Ellipsoid.roc2_}.
818 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
819 B{C{lat2}} and B{C{lon2}} (C{bool}).
821 @return: Distance (C{meter}, same units as the B{C{datum}}'s
822 ellipsoid axes).
824 @raise TypeError: Invalid B{C{datum}}.
826 @note: The meridional and prime_vertical radii of curvature
827 are taken and scaled at the mean of both latitude.
829 @see: Functions L{flatLocal_} or L{hubeny_}, L{cosineLaw}, L{flatPolar},
830 L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
831 L{equirectangular}, L{euclidean}, L{haversine}, L{thomas},
832 L{vincentys}, method L{Ellipsoid.distance2} and U{local, flat
833 earth approximation<https://www.EdWilliams.org/avform.htm#flat>}.
834 '''
835 E = _ellipsoidal(datum, flatLocal)
836 return E._hubeny_2(*_d3(wrap, lat1, lon1, lat2, lon2),
837 scaled=scaled, squared=False) * E.a
839hubeny = flatLocal # PYCHOK for Karl Hubeny
842def flatLocal_(phi2, phi1, lam21, datum=_WGS84, scaled=True):
843 '''Compute the I{angular} distance between two (ellipsoidal) points using
844 the U{ellipsoidal Earth to plane projection<https://WikiPedia.org/
845 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>}
846 aka U{Hubeny<https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula.
848 @arg phi2: End latitude (C{radians}).
849 @arg phi1: Start latitude (C{radians}).
850 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
851 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
852 L{Ellipsoid2} or L{a_f2Tuple}) to use.
853 @kwarg scaled: Scale prime_vertical by C{cos(B{phi})} (C{bool}),
854 see method L{pygeodesy.Ellipsoid.roc2_}.
856 @return: Angular distance (C{radians}).
858 @raise TypeError: Invalid B{C{datum}}.
860 @note: The meridional and prime_vertical radii of curvature
861 are taken and scaled I{at the mean of both latitude}.
863 @see: Functions L{flatLocal} or L{hubeny}, L{cosineAndoyerLambert_},
864 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{flatPolar_},
865 L{euclidean_}, L{haversine_}, L{thomas_} and L{vincentys_} and
866 U{local, flat earth approximation
867 <https://www.EdWilliams.org/avform.htm#flat>}.
868 '''
869 E = _ellipsoidal(datum, flatLocal_)
870 return E._hubeny_2(phi2, phi1, lam21, scaled=scaled, squared=False)
872hubeny_ = flatLocal_ # PYCHOK for Karl Hubeny
875def flatPolar(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
876 '''Compute the distance between two (spherical) points using
877 the U{polar coordinate flat-Earth <https://WikiPedia.org/wiki/
878 Geographical_distance#Polar_coordinate_flat-Earth_formula>}
879 formula.
881 @arg lat1: Start latitude (C{degrees}).
882 @arg lon1: Start longitude (C{degrees}).
883 @arg lat2: End latitude (C{degrees}).
884 @arg lon2: End longitude (C{degrees}).
885 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
886 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
887 L{a_f2Tuple}) to use.
888 @kwarg wrap: If C{True}, wrap or I{normalize} and B{C{lat2}}
889 and B{C{lon2}} (C{bool}).
891 @return: Distance (C{meter}, same units as B{C{radius}} or the
892 ellipsoid or datum axes).
894 @raise TypeError: Invalid B{C{radius}}.
896 @see: Functions L{flatPolar_}, L{cosineAndoyerLambert},
897 L{cosineForsytheAndoyerLambert},L{cosineLaw},
898 L{flatLocal}/L{hubeny}, L{equirectangular},
899 L{euclidean}, L{haversine}, L{thomas} and
900 L{vincentys}.
901 '''
902 return _dS(flatPolar_, radius, wrap, lat1, lon1, lat2, lon2)
905def flatPolar_(phi2, phi1, lam21):
906 '''Compute the I{angular} distance between two (spherical) points
907 using the U{polar coordinate flat-Earth<https://WikiPedia.org/wiki/
908 Geographical_distance#Polar_coordinate_flat-Earth_formula>}
909 formula.
911 @arg phi2: End latitude (C{radians}).
912 @arg phi1: Start latitude (C{radians}).
913 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
915 @return: Angular distance (C{radians}).
917 @see: Functions L{flatPolar}, L{cosineAndoyerLambert_},
918 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
919 L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{haversine_},
920 L{thomas_} and L{vincentys_}.
921 '''
922 a = fabs(PI_2 - phi1) # co-latitude
923 b = fabs(PI_2 - phi2) # co-latitude
924 if a < b:
925 a, b = b, a
926 if a < EPS0:
927 a = _0_0
928 elif b > 0:
929 b = b / a # /= chokes PyChecker
930 c = b * cos(lam21) * _2_0
931 c = fsumf_(_1_0, b**2, -fabs(c))
932 a *= sqrt0(c)
933 return a
936def _hartzell(pov, los, earth, **kwds):
937 '''(INTERNAL) Helper for C{CartesianBase.hartzell} and C{LatLonBase.hartzell}.
938 '''
939 if earth is None:
940 earth = pov.datum
941 else:
942 earth = _spherical_datum(earth, name__=hartzell)
943 pov = pov.toDatum(earth)
944 h = pov.height
945 if h < 0: # EPS0
946 t = _SPACE_(Fmt.PARENSPACED(height=h), _inside_)
947 raise IntersectionError(pov=pov, earth=earth, txt=t)
948 return hartzell(pov, los=los, earth=earth, **kwds) if h > 0 else pov # EPS0
951def hartzell(pov, los=False, earth=_WGS84, **name_LatLon_and_kwds):
952 '''Compute the intersection of the earth's surface and a Line-Of-Sight from
953 a Point-Of-View in space.
955 @arg pov: Point-Of-View outside the earth (C{LatLon}, C{Cartesian},
956 L{Ecef9Tuple} or L{Vector3d}).
957 @kwarg los: Line-Of-Sight, I{direction} to earth (L{Los}, L{Vector3d}),
958 C{True} for the I{normal, plumb} onto the surface or C{False}
959 or C{None} to point to the center of the earth.
960 @kwarg earth: The earth model (L{Datum}, L{Ellipsoid}, L{Ellipsoid2},
961 L{a_f2Tuple} or a C{scalar} earth radius in C{meter}).
962 @kwarg name_LatLon_and_kwds: Optional, overriding C{B{name}="hartzell"}
963 (C{str}), class C{B{LatLon}=None} to return the intersection
964 plus additional C{LatLon} keyword arguments, include the
965 B{C{datum}} if different and to convert from B{C{earth}}.
967 @return: The intersection (L{Vector3d}, B{C{pov}}'s C{cartesian type} or the
968 given B{C{LatLon}} instance) with attribute C{heigth} set to the
969 distance to the B{C{pov}}.
971 @raise IntersectionError: Invalid B{C{pov}} or B{C{pov}} inside the earth or
972 invalid B{C{los}} or B{C{los}} points outside or
973 away from the earth.
975 @raise TypeError: Invalid B{C{earth}}, C{ellipsoid} or C{datum}.
977 @see: Class L{Los}, functions L{tyr3d} and L{hartzell4} and methods
978 L{Ellipsoid.hartzell4} and any C{Cartesian.hartzell} and C{LatLon.hartzell}.
979 '''
980 n, LatLon_and_kwds = _name2__(name_LatLon_and_kwds, name__=hartzell)
981 try:
982 D = _spherical_datum(earth, name__=hartzell)
983 r, h, i = _MODS.triaxials._hartzell3(pov, los, D.ellipsoid._triaxial)
985 C = _MODS.cartesianBase.CartesianBase
986 if LatLon_and_kwds:
987 c = C(r, datum=D)
988 r = c.toLatLon(**_xkwds(LatLon_and_kwds, height=h))
989 elif isinstance(r, C):
990 r.height = h
991 if i:
992 r._iteration = i
993 except Exception as x:
994 raise IntersectionError(pov=pov, los=los, earth=earth, cause=x,
995 **LatLon_and_kwds)
996 return _xnamed(r, n) if n else r
999def haversine(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
1000 '''Compute the distance between two (spherical) points using the
1001 U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>}
1002 formula.
1004 @arg lat1: Start latitude (C{degrees}).
1005 @arg lon1: Start longitude (C{degrees}).
1006 @arg lat2: End latitude (C{degrees}).
1007 @arg lon2: End longitude (C{degrees}).
1008 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
1009 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
1010 L{a_f2Tuple}) to use.
1011 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
1012 B{C{lat2}} and B{C{lon2}} (C{bool}).
1014 @return: Distance (C{meter}, same units as B{C{radius}}).
1016 @raise TypeError: Invalid B{C{radius}}.
1018 @see: U{Distance between two (spherical) points
1019 <https://www.EdWilliams.org/avform.htm#Dist>}, functions
1020 L{cosineLaw}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
1021 L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar},
1022 L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2},
1023 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
1025 @note: See note at function L{vincentys_}.
1026 '''
1027 return _dS(haversine_, radius, wrap, lat1, lon1, lat2, lon2)
1030def haversine_(phi2, phi1, lam21):
1031 '''Compute the I{angular} distance between two (spherical) points
1032 using the U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>}
1033 formula.
1035 @arg phi2: End latitude (C{radians}).
1036 @arg phi1: Start latitude (C{radians}).
1037 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
1039 @return: Angular distance (C{radians}).
1041 @see: Functions L{haversine}, L{cosineAndoyerLambert_},
1042 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
1043 L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_},
1044 L{thomas_} and L{vincentys_}.
1046 @note: See note at function L{vincentys_}.
1047 '''
1048 def _hsin(rad):
1049 return sin(rad * _0_5)**2
1051 h = _hsin(phi2 - phi1) + cos(phi1) * cos(phi2) * _hsin(lam21) # haversine
1052 return atan2(sqrt0(h), sqrt0(_1_0 - h)) * _2_0 # == asin(sqrt(h)) * 2
1055def heightOf(angle, distance, radius=R_M):
1056 '''Determine the height above the (spherical) earth' surface after
1057 traveling along a straight line at a given tilt.
1059 @arg angle: Tilt angle above horizontal (C{degrees}).
1060 @arg distance: Distance along the line (C{meter} or same units as
1061 B{C{radius}}).
1062 @kwarg radius: Optional mean earth radius (C{meter}).
1064 @return: Height (C{meter}, same units as B{C{distance}} and B{C{radius}}).
1066 @raise ValueError: Invalid B{C{angle}}, B{C{distance}} or B{C{radius}}.
1068 @see: U{MultiDop geog_lib.GeogBeamHt<https://GitHub.com/NASA/MultiDop>}
1069 (U{Shapiro et al. 2009, JTECH
1070 <https://Journals.AMetSoc.org/doi/abs/10.1175/2009JTECHA1256.1>}
1071 and U{Potvin et al. 2012, JTECH
1072 <https://Journals.AMetSoc.org/doi/abs/10.1175/JTECH-D-11-00019.1>}).
1073 '''
1074 r = h = Radius(radius)
1075 d = fabs(Distance(distance))
1076 if d > h:
1077 d, h = h, d
1079 if d > EPS0: # and h > EPS0
1080 d = d / h # /= h chokes PyChecker
1081 s = sin(Phi_(angle=angle, clip=_180_0))
1082 s = fsumf_(_1_0, s * d * _2_0, d**2)
1083 if s > 0:
1084 return h * sqrt(s) - r
1086 raise _ValueError(angle=angle, distance=distance, radius=radius)
1089def heightOrthometric(h_ll, N):
1090 '''Get the I{orthometric} height B{H}, the height above the geoid, earth surface.
1092 @arg h_ll: The height above the ellipsoid (C{meter}) or an I{ellipsoidal}
1093 location (C{LatLon} with a C{height} or C{h} attribute).
1094 @arg N: The I{geoid} height (C{meter}), the height of the geoid above the
1095 ellipsoid at the same B{C{h_ll}} location.
1097 @return: I{Orthometric} height C{B{H} = B{h} - B{N}} (C{meter}, same units
1098 as B{C{h}} and B{C{N}}).
1100 @see: U{Ellipsoid, Geoid, and Othometric Heights<https://www.NGS.NOAA.gov/
1101 GEOID/PRESENTATIONS/2007_02_24_CCPS/Roman_A_PLSC2007notes.pdf>}, page
1102 6 and module L{pygeodesy.geoids}.
1103 '''
1104 h = h_ll if _isHeight(h_ll) else _xattr(h_ll, height=_xattr(h_ll, h=0))
1105 return Height(H=Height(h=h) - Height(N=N))
1108def horizon(height, radius=R_M, refraction=False):
1109 '''Determine the distance to the horizon from a given altitude above the
1110 (spherical) earth.
1112 @arg height: Altitude (C{meter} or same units as B{C{radius}}).
1113 @kwarg radius: Optional mean earth radius (C{meter}).
1114 @kwarg refraction: Consider atmospheric refraction (C{bool}).
1116 @return: Distance (C{meter}, same units as B{C{height}} and B{C{radius}}).
1118 @raise ValueError: Invalid B{C{height}} or B{C{radius}}.
1120 @see: U{Distance to horizon<https://www.EdWilliams.org/avform.htm#Horizon>}.
1121 '''
1122 h, r = Height(height), Radius(radius)
1123 if min(h, r) < 0:
1124 raise _ValueError(height=height, radius=radius)
1126 d2 = ((r * 2.415750694528) if refraction else # 2.0 / 0.8279
1127 fsumf_(r, r, h)) * h
1128 return sqrt0(d2)
1131class _idllmn6(object): # see also .geodesicw._wargs, .latlonBase._toCartesian3, .vector2d._numpy
1132 '''(INTERNAL) Helper for C{intersection2} and C{intersections2}.
1133 '''
1134 @contextmanager # <https://www.Python.org/dev/peps/pep-0343/> Examples
1135 def __call__(self, datum, lat1, lon1, lat2, lon2, small, wrap, s, **kwds):
1136 try:
1137 if wrap:
1138 _, lat2, lon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap)
1139 kwds = _xkwds(kwds, wrap=wrap) # for _xError
1140 m = small if small is _100km else Meter_(small=small)
1141 n = _dunder_nameof(intersections2 if s else intersection2)
1142 if datum is None or euclidean(lat1, lon1, lat2, lon2) < m:
1143 d, m = None, _MODS.vector3d
1144 _i = m._intersects2 if s else m._intersect3d3
1145 elif _isRadius(datum) and datum < 0 and not s:
1146 d = _spherical_datum(-datum, name=n)
1147 m = _MODS.sphericalNvector
1148 _i = m.intersection
1149 else:
1150 d = _spherical_datum(datum, name=n)
1151 if d.isSpherical:
1152 m = _MODS.sphericalTrigonometry
1153 _i = m._intersects2 if s else m._intersect
1154 elif d.isEllipsoidal:
1155 try:
1156 if d.ellipsoid.geodesic:
1157 pass
1158 m = _MODS.ellipsoidalKarney
1159 except ImportError:
1160 m = _MODS.ellipsoidalExact
1161 _i = m._intersections2 if s else m._intersection3 # ellispoidalBaseDI
1162 else:
1163 raise _TypeError(datum=datum)
1164 yield _i, d, lat2, lon2, m, n
1166 except (TypeError, ValueError) as x:
1167 raise _xError(x, lat1=lat1, lon1=lon1, datum=datum,
1168 lat2=lat2, lon2=lon2, small=small, **kwds)
1170_idllmn6 = _idllmn6() # PYCHOK singleton
1173def intersection2(lat1, lon1, bearing1,
1174 lat2, lon2, bearing2, datum=None, wrap=False, small=_100km): # was=True
1175 '''I{Conveniently} compute the intersection of two lines each defined
1176 by a (geodetic) point and a bearing from North, using either ...
1178 1) L{vector3d.intersection3d3} for B{C{small}} distances (below 100 Km
1179 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ...
1181 2) L{sphericalTrigonometry.intersection} for a spherical B{C{datum}}
1182 or a C{scalar B{datum}} representing the earth radius, conventionally
1183 in C{meter} or ...
1185 3) L{sphericalNvector.intersection} if B{C{datum}} is a I{negative}
1186 C{scalar}, (negative) earth radius, conventionally in C{meter} or ...
1188 4) L{ellipsoidalKarney.intersection3} for an ellipsoidal B{C{datum}}
1189 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>}
1190 is installed, otherwise ...
1192 5) L{ellipsoidalExact.intersection3}, provided B{C{datum}} is ellipsoidal.
1194 @arg lat1: Latitude of the first point (C{degrees}).
1195 @arg lon1: Longitude of the first point (C{degrees}).
1196 @arg bearing1: Bearing at the first point (compass C{degrees}).
1197 @arg lat2: Latitude of the second point (C{degrees}).
1198 @arg lon2: Longitude of the second point (C{degrees}).
1199 @arg bearing2: Bearing at the second point (compass C{degrees}).
1200 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid},
1201 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth
1202 radius (C{meter}, same units as B{C{radius1}} and
1203 B{C{radius2}}) or C{None}.
1204 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
1205 and B{C{lon2}} (C{bool}).
1206 @kwarg small: Upper limit for small distances (C{meter}).
1208 @return: A L{LatLon2Tuple}C{(lat, lon)} with the lat- and
1209 longitude of the intersection point.
1211 @raise IntersectionError: Ambiguous or infinite intersection
1212 or colinear, parallel or otherwise
1213 non-intersecting lines.
1215 @raise TypeError: Invalid B{C{datum}}.
1217 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{bearing1}},
1218 B{C{lat2}}, B{C{lon2}} or B{C{bearing2}}.
1220 @see: Method L{RhumbLine.intersection2}.
1222 @note: The returned intersections may be near-antipodal.
1223 '''
1224 b1 = Bearing(bearing1=bearing1)
1225 b2 = Bearing(bearing2=bearing2)
1226 with _idllmn6(datum, lat1, lon1, lat2, lon2,
1227 small, wrap, False, bearing1=b1, bearing2=b2) as t:
1228 _i, d, lat2, lon2, m, n = t
1229 if d is None:
1230 t, _, _ = _i(m.Vector3d(lon1, lat1, 0), b1,
1231 m.Vector3d(lon2, lat2, 0), b2, useZ=False)
1232 t = LatLon2Tuple(t.y, t.x, name=n)
1234 else:
1235 t = _i(m.LatLon(lat1, lon1, datum=d), b1,
1236 m.LatLon(lat2, lon2, datum=d), b2, height=0, wrap=False)
1237 if isinstance(t, Intersection3Tuple): # ellipsoidal
1238 t, _, _ = t
1239 t = LatLon2Tuple(t.lat, t.lon, name=n)
1240 return t
1243def intersections2(lat1, lon1, radius1,
1244 lat2, lon2, radius2, datum=None, wrap=False, small=_100km): # was=True
1245 '''I{Conveniently} compute the intersections of two circles each defined
1246 by a (geodetic) center point and a radius, using either ...
1248 1) L{vector3d.intersections2} for B{C{small}} distances (below 100 Km
1249 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ...
1251 2) L{sphericalTrigonometry.intersections2} for a spherical B{C{datum}}
1252 or a C{scalar B{datum}} representing the earth radius, conventionally
1253 in C{meter} or ...
1255 3) L{ellipsoidalKarney.intersections2} for an ellipsoidal B{C{datum}}
1256 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>}
1257 is installed, otherwise ...
1259 4) L{ellipsoidalExact.intersections2}, provided B{C{datum}} is ellipsoidal.
1261 @arg lat1: Latitude of the first circle center (C{degrees}).
1262 @arg lon1: Longitude of the first circle center (C{degrees}).
1263 @arg radius1: Radius of the first circle (C{meter}, conventionally).
1264 @arg lat2: Latitude of the second circle center (C{degrees}).
1265 @arg lon2: Longitude of the second circle center (C{degrees}).
1266 @arg radius2: Radius of the second circle (C{meter}, same units as B{C{radius1}}).
1267 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid},
1268 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth
1269 radius (C{meter}, same units as B{C{radius1}} and
1270 B{C{radius2}}) or C{None}.
1271 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
1272 and B{C{lon2}} (C{bool}).
1273 @kwarg small: Upper limit for small distances (C{meter}).
1275 @return: 2-Tuple of the intersection points, each a
1276 L{LatLon2Tuple}C{(lat, lon)}. For abutting circles, the
1277 points are the same instance, aka the I{radical center}.
1279 @raise IntersectionError: Concentric, antipodal, invalid or
1280 non-intersecting circles or no
1281 convergence.
1283 @raise TypeError: Invalid B{C{datum}}.
1285 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{radius1}},
1286 B{C{lat2}}, B{C{lon2}} or B{C{radius2}}.
1287 '''
1288 r1 = Radius_(radius1=radius1)
1289 r2 = Radius_(radius2=radius2)
1290 with _idllmn6(datum, lat1, lon1, lat2, lon2,
1291 small, wrap, True, radius1=r1, radius2=r2) as t:
1292 _i, d, lat2, lon2, m, n = t
1293 if d is None:
1294 r1 = m2degrees(r1, radius=R_M, lat=lat1)
1295 r2 = m2degrees(r2, radius=R_M, lat=lat2)
1297 def _V2T(x, y, _, **unused): # _ == z unused
1298 return LatLon2Tuple(y, x, name=n)
1300 t = _i(m.Vector3d(lon1, lat1, 0), r1,
1301 m.Vector3d(lon2, lat2, 0), r2, sphere=False,
1302 Vector=_V2T)
1303 else:
1304 def _LL2T(lat, lon, **unused):
1305 return LatLon2Tuple(lat, lon, name=n)
1307 t = _i(m.LatLon(lat1, lon1, datum=d), r1,
1308 m.LatLon(lat2, lon2, datum=d), r2,
1309 LatLon=_LL2T, height=0, wrap=False)
1310 return t
1313def isantipode(lat1, lon1, lat2, lon2, eps=EPS):
1314 '''Check whether two points are I{antipodal}, on diametrically
1315 opposite sides of the earth.
1317 @arg lat1: Latitude of one point (C{degrees}).
1318 @arg lon1: Longitude of one point (C{degrees}).
1319 @arg lat2: Latitude of the other point (C{degrees}).
1320 @arg lon2: Longitude of the other point (C{degrees}).
1321 @kwarg eps: Tolerance for near-equality (C{degrees}).
1323 @return: C{True} if points are antipodal within the
1324 B{C{eps}} tolerance, C{False} otherwise.
1326 @see: Functions L{isantipode_} and L{antipode}.
1327 '''
1328 return (fabs(lat1 + lat2) <= eps and
1329 fabs(lon1 + lon2) <= eps) or _isequalTo(
1330 normal(lat1, lon1), antipode(lat2, lon2), eps)
1333def isantipode_(phi1, lam1, phi2, lam2, eps=EPS):
1334 '''Check whether two points are I{antipodal}, on diametrically
1335 opposite sides of the earth.
1337 @arg phi1: Latitude of one point (C{radians}).
1338 @arg lam1: Longitude of one point (C{radians}).
1339 @arg phi2: Latitude of the other point (C{radians}).
1340 @arg lam2: Longitude of the other point (C{radians}).
1341 @kwarg eps: Tolerance for near-equality (C{radians}).
1343 @return: C{True} if points are antipodal within the
1344 B{C{eps}} tolerance, C{False} otherwise.
1346 @see: Functions L{isantipode} and L{antipode_}.
1347 '''
1348 return (fabs(phi1 + phi2) <= eps and
1349 fabs(lam1 + lam2) <= eps) or _isequalTo_(
1350 normal_(phi1, lam1), antipode_(phi2, lam2), eps)
1353def _isequalTo(p1, p2, eps=EPS):
1354 '''Compare 2 point lat-/lons ignoring C{class}.
1355 '''
1356 return (fabs(p1.lat - p2.lat) <= eps and
1357 fabs(p1.lon - p2.lon) <= eps) if eps else (p1.latlon == p2.latlon)
1360def _isequalTo_(p1, p2, eps=EPS):
1361 '''(INTERNAL) Compare 2 point phi-/lams ignoring C{class}.
1362 '''
1363 return (fabs(p1.phi - p2.phi) <= eps and
1364 fabs(p1.lam - p2.lam) <= eps) if eps else (p1.philam == p2.philam)
1367def isnormal(lat, lon, eps=0):
1368 '''Check whether B{C{lat}} I{and} B{C{lon}} are within their
1369 respective I{normal} range in C{degrees}.
1371 @arg lat: Latitude (C{degrees}).
1372 @arg lon: Longitude (C{degrees}).
1373 @kwarg eps: Optional tolerance C{degrees}).
1375 @return: C{True} if C{(abs(B{lat}) + B{eps}) <= 90} and
1376 C{(abs(B{lon}) + B{eps}) <= 180}, C{False} othwerwise.
1378 @see: Functions L{isnormal_} and L{normal}.
1379 '''
1380 return (_90_0 - fabs(lat)) >= eps and _loneg(fabs(lon)) >= eps
1383def isnormal_(phi, lam, eps=0):
1384 '''Check whether B{C{phi}} I{and} B{C{lam}} are within their
1385 respective I{normal} range in C{radians}.
1387 @arg phi: Latitude (C{radians}).
1388 @arg lam: Longitude (C{radians}).
1389 @kwarg eps: Optional tolerance C{radians}).
1391 @return: C{True} if C{(abs(B{phi}) + B{eps}) <= PI/2} and
1392 C{(abs(B{lam}) + B{eps}) <= PI}, C{False} othwerwise.
1394 @see: Functions L{isnormal} and L{normal_}.
1395 '''
1396 return (PI_2 - fabs(phi)) >= eps and (PI - fabs(lam)) >= eps
1399def latlon2n_xyz(lat, lon, **name):
1400 '''Convert lat-, longitude to C{n-vector} (I{normal} to the
1401 earth's surface) X, Y and Z components.
1403 @arg lat: Latitude (C{degrees}).
1404 @arg lon: Longitude (C{degrees}).
1405 @kwarg name: Optional C{B{name}=NN} (C{str}).
1407 @return: A L{Vector3Tuple}C{(x, y, z)}.
1409 @see: Function L{philam2n_xyz}.
1411 @note: These are C{n-vector} x, y and z components,
1412 I{NOT} geocentric ECEF x, y and z coordinates!
1413 '''
1414 return _2n_xyz(name, *sincos2d_(lat, lon))
1417def _normal2(a, b, n_2, n, n2):
1418 '''(INTERNAL) Helper for C{normal} and C{normal_}.
1419 '''
1420 if fabs(b) > n:
1421 b = remainder(b, n2)
1422 if fabs(a) > n_2:
1423 r = remainder(a, n)
1424 if r != a:
1425 a = -r
1426 b -= n if b > 0 else -n
1427 return float0_(a, b)
1430def normal(lat, lon, **name):
1431 '''Normalize a lat- I{and} longitude pair in C{degrees}.
1433 @arg lat: Latitude (C{degrees}).
1434 @arg lon: Longitude (C{degrees}).
1435 @kwarg name: Optional, overriding C{B{name}="normal"} (C{str}).
1437 @return: L{LatLon2Tuple}C{(lat, lon)} with C{abs(lat) <= 90}
1438 and C{abs(lon) <= 180}.
1440 @see: Functions L{normal_} and L{isnormal}.
1441 '''
1442 return LatLon2Tuple(*_normal2(lat, lon, _90_0, _180_0, _360_0),
1443 name=_name__(name, name__=normal))
1446def normal_(phi, lam, **name):
1447 '''Normalize a lat- I{and} longitude pair in C{radians}.
1449 @arg phi: Latitude (C{radians}).
1450 @arg lam: Longitude (C{radians}).
1451 @kwarg name: Optional, overriding C{B{name}="normal_"} (C{str}).
1453 @return: L{PhiLam2Tuple}C{(phi, lam)} with C{abs(phi) <= PI/2}
1454 and C{abs(lam) <= PI}.
1456 @see: Functions L{normal} and L{isnormal_}.
1457 '''
1458 return PhiLam2Tuple(*_normal2(phi, lam, PI_2, PI, PI2),
1459 name=_name__(name, name__=normal_))
1462def _2n_xyz(name, sa, ca, sb, cb): # name always **name
1463 '''(INTERNAL) Helper for C{latlon2n_xyz} and C{philam2n_xyz}.
1464 '''
1465 # Kenneth Gade eqn 3, but using right-handed
1466 # vector x -> 0°E,0°N, y -> 90°E,0°N, z -> 90°N
1467 return Vector3Tuple(ca * cb, ca * sb, sa, **name)
1470def n_xyz2latlon(x, y, z, **name):
1471 '''Convert C{n-vector} components to lat- and longitude in C{degrees}.
1473 @arg x: X component (C{scalar}).
1474 @arg y: Y component (C{scalar}).
1475 @arg z: Z component (C{scalar}).
1476 @kwarg name: Optional C{B{name}=NN} (C{str}).
1478 @return: A L{LatLon2Tuple}C{(lat, lon)}.
1480 @see: Function L{n_xyz2philam}.
1481 '''
1482 return LatLon2Tuple(atan2d(z, hypot(x, y)), atan2d(y, x), **name)
1485def n_xyz2philam(x, y, z, **name):
1486 '''Convert C{n-vector} components to lat- and longitude in C{radians}.
1488 @arg x: X component (C{scalar}).
1489 @arg y: Y component (C{scalar}).
1490 @arg z: Z component (C{scalar}).
1491 @kwarg name: Optional C{B{name}=NN} (C{str}).
1493 @return: A L{PhiLam2Tuple}C{(phi, lam)}.
1495 @see: Function L{n_xyz2latlon}.
1496 '''
1497 return PhiLam2Tuple(atan2(z, hypot(x, y)), atan2(y, x), **name)
1500def _opposes(d, m, n, n2):
1501 '''(INTERNAL) Helper for C{opposing} and C{opposing_}.
1502 '''
1503 d = d % n2 # -20 % 360 == 340, -1 % PI2 == PI2 - 1
1504 return False if d < m or d > (n2 - m) else (
1505 True if (n - m) < d < (n + m) else None)
1508def opposing(bearing1, bearing2, margin=_90_0):
1509 '''Compare the direction of two bearings given in C{degrees}.
1511 @arg bearing1: First bearing (compass C{degrees}).
1512 @arg bearing2: Second bearing (compass C{degrees}).
1513 @kwarg margin: Optional, interior angle bracket (C{degrees}).
1515 @return: C{True} if both bearings point in opposite, C{False} if
1516 in similar or C{None} if in I{perpendicular} directions.
1518 @see: Function L{opposing_}.
1519 '''
1520 m = Degrees_(margin=margin, low=EPS0, high=_90_0)
1521 return _opposes(bearing2 - bearing1, m, _180_0, _360_0)
1524def opposing_(radians1, radians2, margin=PI_2):
1525 '''Compare the direction of two bearings given in C{radians}.
1527 @arg radians1: First bearing (C{radians}).
1528 @arg radians2: Second bearing (C{radians}).
1529 @kwarg margin: Optional, interior angle bracket (C{radians}).
1531 @return: C{True} if both bearings point in opposite, C{False} if
1532 in similar or C{None} if in perpendicular directions.
1534 @see: Function L{opposing}.
1535 '''
1536 m = Radians_(margin=margin, low=EPS0, high=PI_2)
1537 return _opposes(radians2 - radians1, m, PI, PI2)
1540def philam2n_xyz(phi, lam, **name):
1541 '''Convert lat-, longitude to C{n-vector} (I{normal} to the
1542 earth's surface) X, Y and Z components.
1544 @arg phi: Latitude (C{radians}).
1545 @arg lam: Longitude (C{radians}).
1546 @kwarg name: Optional name (C{str}).
1548 @return: A L{Vector3Tuple}C{(x, y, z)}.
1550 @see: Function L{latlon2n_xyz}.
1552 @note: These are C{n-vector} x, y and z components,
1553 I{NOT} geocentric ECEF x, y and z coordinates!
1554 '''
1555 return _2n_xyz(name, *sincos2_(phi, lam))
1558def _Propy(func, nargs, kwds):
1559 '''(INTERNAL) Helper for the C{frechet.[-]Frechet**} and
1560 C{hausdorff.[-]Hausdorff*} classes.
1561 '''
1562 try:
1563 _xcallable(distance=func)
1564 # assert _args_kwds_count2(func)[0] == nargs + int(ismethod(func))
1565 _ = func(*_0_0s(nargs), **kwds)
1566 except Exception as x:
1567 t = unstr(func, **kwds)
1568 raise _TypeError(t, cause=x)
1569 return func
1572def _radical2(d, r1, r2, **name): # in .ellipsoidalBaseDI, .sphericalTrigonometry, .vector3d
1573 # (INTERNAL) See C{radical2} below
1574 # assert d > EPS0
1575 r = fsumf_(_1_0, (r1 / d)**2, -(r2 / d)**2) * _0_5
1576 n = _name__(name, name__=radical2)
1577 return Radical2Tuple(max(_0_0, min(_1_0, r)), r * d, name=n)
1580def radical2(distance, radius1, radius2, **name):
1581 '''Compute the I{radical ratio} and I{radical line} of two
1582 U{intersecting circles<https://MathWorld.Wolfram.com/
1583 Circle-CircleIntersection.html>}.
1585 The I{radical line} is perpendicular to the axis thru the
1586 centers of the circles at C{(0, 0)} and C{(B{distance}, 0)}.
1588 @arg distance: Distance between the circle centers (C{scalar}).
1589 @arg radius1: Radius of the first circle (C{scalar}).
1590 @arg radius2: Radius of the second circle (C{scalar}).
1591 @kwarg name: Optional C{B{name}=NN} (C{str}).
1593 @return: A L{Radical2Tuple}C{(ratio, xline)} where C{0.0 <=
1594 ratio <= 1.0} and C{xline} is along the B{C{distance}}.
1596 @raise IntersectionError: The B{C{distance}} exceeds the sum
1597 of B{C{radius1}} and B{C{radius2}}.
1599 @raise UnitError: Invalid B{C{distance}}, B{C{radius1}} or
1600 B{C{radius2}}.
1602 @see: U{Circle-Circle Intersection
1603 <https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>}.
1604 '''
1605 d = Distance_(distance, low=_0_0)
1606 r1 = Radius_(radius1=radius1)
1607 r2 = Radius_(radius2=radius2)
1608 if d > (r1 + r2):
1609 raise IntersectionError(distance=d, radius1=r1, radius2=r2,
1610 txt=_too_(_distant_))
1611 return _radical2(d, r1, r2, **name) if d > EPS0 else \
1612 Radical2Tuple(_0_5, _0_0, **name)
1615class Radical2Tuple(_NamedTuple):
1616 '''2-Tuple C{(ratio, xline)} of the I{radical} C{ratio} and
1617 I{radical} C{xline}, both C{scalar} and C{0.0 <= ratio <= 1.0}
1618 '''
1619 _Names_ = (_ratio_, _xline_)
1620 _Units_ = ( Scalar, Scalar)
1623def _radistance(inst):
1624 '''(INTERNAL) Helper for the L{frechet._FrechetMeterRadians}
1625 and L{hausdorff._HausdorffMeterRedians} classes.
1626 '''
1627 wrap_, kwds_ = _xkwds_pop2(inst._kwds, wrap=False)
1628 func_ = inst._func_
1629 try: # calling lower-overhead C{func_}
1630 func_(0, _0_25, _0_5, **kwds_)
1631 wrap_ = _Wrap._philamop(wrap_)
1632 except TypeError:
1633 return inst.distance
1635 def _philam(p):
1636 try:
1637 return p.phi, p.lam
1638 except AttributeError: # no .phi or .lam
1639 return radians(p.lat), radians(p.lon)
1641 def _func_wrap(point1, point2):
1642 phi1, lam1 = wrap_(*_philam(point1))
1643 phi2, lam2 = wrap_(*_philam(point2))
1644 return func_(phi2, phi1, lam2 - lam1, **kwds_)
1646 inst._units = inst._units_
1647 return _func_wrap
1650def _scale_deg(lat1, lat2): # degrees
1651 # scale factor cos(mean of lats) for delta lon
1652 m = fabs(lat1 + lat2) * _0_5
1653 return cos(radians(m)) if m < 90 else _0_0
1656def _scale_rad(phi1, phi2): # radians, by .frechet, .hausdorff, .heights
1657 # scale factor cos(mean of phis) for delta lam
1658 m = fabs(phi1 + phi2) * _0_5
1659 return cos(m) if m < PI_2 else _0_0
1662def _sincosa6(phi2, phi1, lam21): # [4] in cosineLaw
1663 '''(INTERNAL) C{sin}es, C{cos}ines and C{acos}ine.
1664 '''
1665 s2, c2, s1, c1, _, c21 = sincos2_(phi2, phi1, lam21)
1666 return s2, c2, s1, c1, acos1(s1 * s2 + c1 * c2 * c21), c21
1669def thomas(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):
1670 '''Compute the distance between two (ellipsoidal) points using
1671 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>}
1672 formula.
1674 @arg lat1: Start latitude (C{degrees}).
1675 @arg lon1: Start longitude (C{degrees}).
1676 @arg lat2: End latitude (C{degrees}).
1677 @arg lon2: End longitude (C{degrees}).
1678 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
1679 L{Ellipsoid2} or L{a_f2Tuple}) to use.
1680 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
1681 B{C{lat2}} and B{C{lon2}} (C{bool}).
1683 @return: Distance (C{meter}, same units as the B{C{datum}}'s
1684 ellipsoid axes).
1686 @raise TypeError: Invalid B{C{datum}}.
1688 @see: Functions L{thomas_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
1689 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny},
1690 L{flatPolar}, L{haversine}, L{vincentys} and method L{Ellipsoid.distance2}.
1691 '''
1692 return _dE(thomas_, datum, wrap, lat1, lon1, lat2, lon2)
1695def thomas_(phi2, phi1, lam21, datum=_WGS84):
1696 '''Compute the I{angular} distance between two (ellipsoidal) points using
1697 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>}
1698 formula.
1700 @arg phi2: End latitude (C{radians}).
1701 @arg phi1: Start latitude (C{radians}).
1702 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
1703 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid},
1704 L{Ellipsoid2} or L{a_f2Tuple}).
1706 @return: Angular distance (C{radians}).
1708 @raise TypeError: Invalid B{C{datum}}.
1710 @see: Functions L{thomas}, L{cosineAndoyerLambert_},
1711 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
1712 L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_},
1713 L{haversine_} and L{vincentys_} and U{Geodesy-PHP
1714 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/
1715 Distance/ThomasFormula.php>}.
1716 '''
1717 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21)
1718 if r and isnon0(c1) and isnon0(c2):
1719 E = _ellipsoidal(datum, thomas_)
1720 if E.f:
1721 r1 = atan2(E.b_a * s1, c1)
1722 r2 = atan2(E.b_a * s2, c2)
1724 j = (r2 + r1) * _0_5
1725 k = (r2 - r1) * _0_5
1726 sj, cj, sk, ck, h, _ = sincos2_(j, k, lam21 * _0_5)
1728 h = fsumf_(sk**2, (ck * h)**2, -(sj * h)**2)
1729 u = _1_0 - h
1730 if isnon0(u) and isnon0(h):
1731 r = atan(sqrt0(h / u)) * 2 # == acos(1 - 2 * h)
1732 sr, cr = sincos2(r)
1733 if isnon0(sr):
1734 u = 2 * (sj * ck)**2 / u
1735 h = 2 * (sk * cj)**2 / h
1736 x = u + h
1737 y = u - h
1739 s = r / sr
1740 e = 4 * s**2
1741 d = 2 * cr
1742 a = e * d
1743 b = 2 * r
1744 c = s - (a - d) * _0_5
1745 f = E.f * _0_25
1747 t = fsumf_(a * x, -b * y, c * x**2, -d * y**2, e * x * y)
1748 r -= fsumf_(s * x, -y, -t * f * _0_25) * f * sr
1749 return r
1752def vincentys(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
1753 '''Compute the distance between two (spherical) points using
1754 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>}
1755 spherical formula.
1757 @arg lat1: Start latitude (C{degrees}).
1758 @arg lon1: Start longitude (C{degrees}).
1759 @arg lat2: End latitude (C{degrees}).
1760 @arg lon2: End longitude (C{degrees}).
1761 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
1762 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
1763 L{a_f2Tuple}) to use.
1764 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
1765 B{C{lat2}} and B{C{lon2}} (C{bool}).
1767 @return: Distance (C{meter}, same units as B{C{radius}}).
1769 @raise UnitError: Invalid B{C{radius}}.
1771 @see: Functions L{vincentys_}, L{cosineAndoyerLambert},
1772 L{cosineForsytheAndoyerLambert},L{cosineLaw}, L{equirectangular},
1773 L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar},
1774 L{haversine} and L{thomas} and methods L{Ellipsoid.distance2},
1775 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
1777 @note: See note at function L{vincentys_}.
1778 '''
1779 return _dS(vincentys_, radius, wrap, lat1, lon1, lat2, lon2)
1782def vincentys_(phi2, phi1, lam21):
1783 '''Compute the I{angular} distance between two (spherical) points using
1784 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>}
1785 spherical formula.
1787 @arg phi2: End latitude (C{radians}).
1788 @arg phi1: Start latitude (C{radians}).
1789 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
1791 @return: Angular distance (C{radians}).
1793 @see: Functions L{vincentys}, L{cosineAndoyerLambert_},
1794 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
1795 L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_},
1796 L{haversine_} and L{thomas_}.
1798 @note: Functions L{vincentys_}, L{haversine_} and L{cosineLaw_}
1799 produce equivalent results, but L{vincentys_} is suitable
1800 for antipodal points and slightly more expensive (M{3 cos,
1801 3 sin, 1 hypot, 1 atan2, 6 mul, 2 add}) than L{haversine_}
1802 (M{2 cos, 2 sin, 2 sqrt, 1 atan2, 5 mul, 1 add}) and
1803 L{cosineLaw_} (M{3 cos, 3 sin, 1 acos, 3 mul, 1 add}).
1804 '''
1805 s1, c1, s2, c2, s21, c21 = sincos2_(phi1, phi2, lam21)
1807 c = c2 * c21
1808 x = s1 * s2 + c1 * c
1809 y = c1 * s2 - s1 * c
1810 return atan2(hypot(c2 * s21, y), x)
1812# **) MIT License
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1814# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved.
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