Coverage for pygeodesy/formy.py: 98%
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2# -*- coding: utf-8 -*-
4u'''Formulary of basic geodesy functions and approximations.
5'''
6# make sure int/int division yields float quotient, see .basics
7from __future__ import division as _; del _ # PYCHOK semicolon
9# from pygeodesy.basics import W_args_kwds_count2
10# from pygeodesy.cartesianBase import CartesianBase # _MODS
11from pygeodesy.constants import EPS, EPS0, EPS1, PI, PI2, PI3, PI_2, R_M, \
12 _0_0s, float0_, isnon0, remainder, _umod_PI2, \
13 _0_0, _0_125, _0_25, _0_5, _1_0, _2_0, _4_0, \
14 _32_0, _90_0, _180_0, _360_0
15from pygeodesy.datums import Datum, Ellipsoid, _ellipsoidal_datum, \
16 _mean_radius, _spherical_datum, _WGS84, _EWGS84
17# from pygeodesy.ellipsoids import Ellipsoid, _EWGS84 # from .datums
18from pygeodesy.errors import IntersectionError, LimitError, limiterrors, \
19 _TypeError, _ValueError, _xattr, _xError, \
20 _xcallable,_xkwds, _xkwds_pop2
21from pygeodesy.fmath import euclid, hypot, hypot2, sqrt0
22from pygeodesy.fsums import fsumf_, Fmt, unstr
23# from pygeodesy.internals import _DUNDER_nameof # from .named
24from pygeodesy.interns import _delta_, _distant_, _inside_, _SPACE_, _too_
25from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS
26from pygeodesy.named import _name__, _name2__, _NamedTuple, _xnamed, \
27 _DUNDER_nameof
28from pygeodesy.namedTuples import Bearing2Tuple, Distance4Tuple, LatLon2Tuple, \
29 Intersection3Tuple, PhiLam2Tuple, Vector3Tuple
30# from pygeodesy.streprs import Fmt, unstr # from .fsums
31# from pygeodesy.triaxials import _hartzell3 # _MODS
32from pygeodesy.units import _isHeight, _isRadius, Bearing, Degrees_, Distance, \
33 Distance_, Height, Lamd, Lat, Lon, Meter_, Phid, \
34 Radians, Radians_, Radius, Radius_, Scalar, _100km
35from pygeodesy.utily import acos1, atan2b, atan2d, degrees2m, _loneg, m2degrees, \
36 tan_2, sincos2, sincos2_, sincos2d_, _Wrap
37# from pygeodesy.vector3d import _otherV3d # _MODS
38# from pygeodesy.vector3dBase import _xyz_y_z3 # _MODS
39# from pygeodesy import ellipsoidalExact, ellipsoidalKarney, vector3d, \
40# sphericalNvector, sphericalTrigonometry # _MODS
42from contextlib import contextmanager
43from math import asin, atan, atan2, cos, degrees, fabs, radians, sin, sqrt # pow
45__all__ = _ALL_LAZY.formy
46__version__ = '24.10.14'
48_RADIANS2 = (PI / _180_0)**2 # degrees- to radians-squared
49_ratio_ = 'ratio'
50_xline_ = 'xline'
53def _anti2(a, b, n_2, n, n2):
54 '''(INTERNAL) Helper for C{antipode} and C{antipode_}.
55 '''
56 r = remainder(a, n) if fabs(a) > n_2 else a
57 if r == a:
58 r = -r
59 b += n
60 if fabs(b) > n:
61 b = remainder(b, n2)
62 return float0_(r, b)
65def antipode(lat, lon, **name):
66 '''Return the antipode, the point diametrically opposite to a given
67 point in C{degrees}.
69 @arg lat: Latitude (C{degrees}).
70 @arg lon: Longitude (C{degrees}).
71 @kwarg name: Optional C{B{name}=NN} (C{str}).
73 @return: A L{LatLon2Tuple}C{(lat, lon)}.
75 @see: Functions L{antipode_} and L{normal} and U{Geosphere
76 <https://CRAN.R-Project.org/web/packages/geosphere/geosphere.pdf>}.
77 '''
78 return LatLon2Tuple(*_anti2(lat, lon, _90_0, _180_0, _360_0), **name)
81def antipode_(phi, lam, **name):
82 '''Return the antipode, the point diametrically opposite to a given
83 point in C{radians}.
85 @arg phi: Latitude (C{radians}).
86 @arg lam: Longitude (C{radians}).
87 @kwarg name: Optional C{B{name}=NN} (C{str}).
89 @return: A L{PhiLam2Tuple}C{(phi, lam)}.
91 @see: Functions L{antipode} and L{normal_} and U{Geosphere
92 <https://CRAN.R-Project.org/web/packages/geosphere/geosphere.pdf>}.
93 '''
94 return PhiLam2Tuple(*_anti2(phi, lam, PI_2, PI, PI2), **name)
97def bearing(lat1, lon1, lat2, lon2, **final_wrap):
98 '''Compute the initial or final bearing (forward or reverse azimuth)
99 between two (spherical) points.
101 @arg lat1: Start latitude (C{degrees}).
102 @arg lon1: Start longitude (C{degrees}).
103 @arg lat2: End latitude (C{degrees}).
104 @arg lon2: End longitude (C{degrees}).
105 @kwarg final_wrap: Optional keyword arguments for function
106 L{pygeodesy.bearing_}.
108 @return: Initial or final bearing (compass C{degrees360}) or zero if
109 both points coincide.
110 '''
111 r = bearing_(Phid(lat1=lat1), Lamd(lon1=lon1),
112 Phid(lat2=lat2), Lamd(lon2=lon2), **final_wrap)
113 return degrees(r)
116def bearing_(phi1, lam1, phi2, lam2, final=False, wrap=False):
117 '''Compute the initial or final bearing (forward or reverse azimuth) between
118 two (spherical) points.
120 @arg phi1: Start latitude (C{radians}).
121 @arg lam1: Start longitude (C{radians}).
122 @arg phi2: End latitude (C{radians}).
123 @arg lam2: End longitude (C{radians}).
124 @kwarg final: If C{True}, return the final, otherwise the initial bearing
125 (C{bool}).
126 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{phi2}} and
127 B{C{lam2}} (C{bool}).
129 @return: Initial or final bearing (compass C{radiansPI2}) or zero if both
130 are coincident.
132 @see: U{Bearing<https://www.Movable-Type.co.UK/scripts/latlong.html>}, U{Course
133 between two points<https://www.EdWilliams.org/avform147.htm#Crs>} and
134 U{Bearing Between Two Points<https://web.Archive.org/web/20020630205931/
135 https://MathForum.org/library/drmath/view/55417.html>}.
136 '''
137 db, phi2, lam2 = _Wrap.philam3(lam1, phi2, lam2, wrap)
138 if final: # swap plus PI
139 phi1, lam1, phi2, lam2, db = phi2, lam2, phi1, lam1, -db
140 r = PI3
141 else:
142 r = PI2
143 sa1, ca1, sa2, ca2, sdb, cdb = sincos2_(phi1, phi2, db)
145 x = ca1 * sa2 - sa1 * ca2 * cdb
146 y = sdb * ca2
147 return _umod_PI2(atan2(y, x) + r) # .utily.wrapPI2
150def _bearingTo2(p1, p2, wrap=False): # for points.ispolar, sphericalTrigonometry.areaOf
151 '''(INTERNAL) Compute initial and final bearing.
152 '''
153 try: # for LatLon_ and ellipsoidal LatLon
154 return p1.bearingTo2(p2, wrap=wrap)
155 except AttributeError:
156 pass
157 # XXX spherical version, OK for ellipsoidal ispolar?
158 t = p1.philam + p2.philam
159 return Bearing2Tuple(degrees(bearing_(*t, final=False, wrap=wrap)),
160 degrees(bearing_(*t, final=True, wrap=wrap)),
161 name__=_bearingTo2)
164def compassAngle(lat1, lon1, lat2, lon2, adjust=True, wrap=False):
165 '''Return the angle from North for the direction vector M{(lon2 - lon1,
166 lat2 - lat1)} between two points.
168 Suitable only for short, not near-polar vectors up to a few hundred
169 Km or Miles. Use function L{pygeodesy.bearing} for longer vectors.
171 @arg lat1: From latitude (C{degrees}).
172 @arg lon1: From longitude (C{degrees}).
173 @arg lat2: To latitude (C{degrees}).
174 @arg lon2: To longitude (C{degrees}).
175 @kwarg adjust: Adjust the longitudinal delta by the cosine of the
176 mean latitude (C{bool}).
177 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
178 and B{C{lon2}} (C{bool}).
180 @return: Compass angle from North (C{degrees360}).
182 @note: Courtesy of Martin Schultz.
184 @see: U{Local, flat earth approximation
185 <https://www.EdWilliams.org/avform.htm#flat>}.
186 '''
187 d_lon, lat2, lon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap)
188 if adjust: # scale delta lon
189 d_lon *= _scale_deg(lat1, lat2)
190 return atan2b(d_lon, lat2 - lat1)
193def cosineAndoyerLambert(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):
194 '''Compute the distance between two (ellipsoidal) points using the U{Andoyer-Lambert
195 <https://books.google.com/books?id=x2UiAQAAIAAJ>} correction of the U{Law of
196 Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula.
198 @arg lat1: Start latitude (C{degrees}).
199 @arg lon1: Start longitude (C{degrees}).
200 @arg lat2: End latitude (C{degrees}).
201 @arg lon2: End longitude (C{degrees}).
202 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
203 L{a_f2Tuple}) to use.
204 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} and
205 B{C{lon2}} (C{bool}).
207 @return: Distance (C{meter}, same units as the B{C{datum}}'s ellipsoid axes or
208 C{radians} if C{B{datum} is None}).
210 @raise TypeError: Invalid B{C{datum}}.
212 @see: Functions L{cosineAndoyerLambert_}, L{cosineForsytheAndoyerLambert},
213 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny},
214 L{flatPolar}, L{haversine}, L{thomas} and L{vincentys} and method
215 L{Ellipsoid.distance2}.
216 '''
217 return _dE(cosineAndoyerLambert_, datum, wrap, lat1, lon1, lat2, lon2)
220def cosineAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84):
221 '''Compute the I{angular} distance between two (ellipsoidal) points using the U{Andoyer-Lambert
222 <https://books.google.com/books?id=x2UiAQAAIAAJ>} correction of the U{Law of
223 Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula.
225 @arg phi2: End latitude (C{radians}).
226 @arg phi1: Start latitude (C{radians}).
227 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
228 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
229 L{a_f2Tuple}) to use.
231 @return: Angular distance (C{radians}).
233 @raise TypeError: Invalid B{C{datum}}.
235 @see: Functions L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert_},
236 L{cosineLaw_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_},
237 L{haversine_}, L{thomas_} and L{vincentys_} and U{Geodesy-PHP
238 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/Distance/
239 AndoyerLambert.php>}.
240 '''
241 s2, c2, s1, c1, r, c21 = _sincosa6(phi2, phi1, lam21)
242 if isnon0(c1) and isnon0(c2):
243 E = _ellipsoidal(datum, cosineAndoyerLambert_)
244 if E.f: # ellipsoidal
245 r2 = atan2(E.b_a * s2, c2)
246 r1 = atan2(E.b_a * s1, c1)
247 s2, c2, s1, c1 = sincos2_(r2, r1)
248 r = acos1(s1 * s2 + c1 * c2 * c21)
249 if r:
250 sr, _, sr_2, cr_2 = sincos2_(r, r * _0_5)
251 if isnon0(sr_2) and isnon0(cr_2):
252 s = (sr + r) * ((s1 - s2) / sr_2)**2
253 c = (sr - r) * ((s1 + s2) / cr_2)**2
254 r += (c - s) * E.f * _0_125
255 return r
258def cosineForsytheAndoyerLambert(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):
259 '''Compute the distance between two (ellipsoidal) points using the U{Forsythe-Andoyer-Lambert
260 <https://www2.UNB.Ca/gge/Pubs/TR77.pdf>} correction of the U{Law of Cosines
261 <https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula.
263 @arg lat1: Start latitude (C{degrees}).
264 @arg lon1: Start longitude (C{degrees}).
265 @arg lat2: End latitude (C{degrees}).
266 @arg lon2: End longitude (C{degrees}).
267 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
268 L{a_f2Tuple}) to use.
269 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} and
270 B{C{lon2}} (C{bool}).
272 @return: Distance (C{meter}, same units as the B{C{datum}}'s ellipsoid axes or
273 C{radians} if C{B{datum} is None}).
275 @raise TypeError: Invalid B{C{datum}}.
277 @see: Functions L{cosineForsytheAndoyerLambert_}, L{cosineAndoyerLambert},
278 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny},
279 L{flatPolar}, L{haversine}, L{thomas} and L{vincentys} and method
280 L{Ellipsoid.distance2}.
281 '''
282 return _dE(cosineForsytheAndoyerLambert_, datum, wrap, lat1, lon1, lat2, lon2)
285def cosineForsytheAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84):
286 '''Compute the I{angular} distance between two (ellipsoidal) points using the
287 U{Forsythe-Andoyer-Lambert<https://www2.UNB.Ca/gge/Pubs/TR77.pdf>} correction of
288 the U{Law of Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>}
289 formula.
291 @arg phi2: End latitude (C{radians}).
292 @arg phi1: Start latitude (C{radians}).
293 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
294 @kwarg datum: Datum (L{Datum}) or ellipsoid to use (L{Ellipsoid},
295 L{Ellipsoid2} or L{a_f2Tuple}).
297 @return: Angular distance (C{radians}).
299 @raise TypeError: Invalid B{C{datum}}.
301 @see: Functions L{cosineForsytheAndoyerLambert}, L{cosineAndoyerLambert_},
302 L{cosineLaw_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_},
303 L{haversine_}, L{thomas_} and L{vincentys_} and U{Geodesy-PHP
304 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/
305 Distance/ForsytheCorrection.php>}.
306 '''
307 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21)
308 if r and isnon0(c1) and isnon0(c2):
309 E = _ellipsoidal(datum, cosineForsytheAndoyerLambert_)
310 if E.f: # ellipsoidal
311 sr, cr, s2r, _ = sincos2_(r, r * 2)
312 if isnon0(sr) and fabs(cr) < EPS1:
313 s = (s1 + s2)**2 / (1 + cr)
314 t = (s1 - s2)**2 / (1 - cr)
315 x = s + t
316 y = s - t
318 s = 8 * r**2 / sr
319 a = 64 * r + s * cr * 2 # 16 * r**2 / tan(r)
320 d = 48 * sr + s # 8 * r**2 / tan(r)
321 b = -2 * d
322 e = 30 * s2r
323 c = fsumf_(30 * r, e * _0_5, s * cr) # 8 * r**2 / tan(r)
324 t = fsumf_( a * x, e * y**2, b * y, -c * x**2, d * x * y)
326 r += fsumf_(-r * x, 3 * y * sr, t * E.f / _32_0) * E.f * _0_25
327 return r
330def cosineLaw(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
331 '''Compute the distance between two points using the U{spherical Law of Cosines
332 <https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula.
334 @arg lat1: Start latitude (C{degrees}).
335 @arg lon1: Start longitude (C{degrees}).
336 @arg lat2: End latitude (C{degrees}).
337 @arg lon2: End longitude (C{degrees}).
338 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
339 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
340 L{a_f2Tuple}) to use.
341 @kwarg wrap: If C{True}, wrap or I{normalize} and B{C{lat2}}
342 and B{C{lon2}} (C{bool}).
344 @return: Distance (C{meter}, same units as B{C{radius}} or the
345 ellipsoid or datum axes).
347 @raise TypeError: Invalid B{C{radius}}.
349 @see: Functions L{cosineLaw_}, L{cosineAndoyerLambert},
350 L{cosineForsytheAndoyerLambert}, L{equirectangular}, L{euclidean},
351 L{flatLocal}/L{hubeny}, L{flatPolar}, L{haversine}, L{thomas} and
352 L{vincentys} and method L{Ellipsoid.distance2}.
354 @note: See note at function L{vincentys_}.
355 '''
356 return _dS(cosineLaw_, radius, wrap, lat1, lon1, lat2, lon2)
359def cosineLaw_(phi2, phi1, lam21):
360 '''Compute the I{angular} distance between two points using the U{spherical Law of
361 Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} 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{euclidean_},
371 L{flatLocal_}/L{hubeny_}, L{flatPolar_}, L{haversine_},
372 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), Phid(lat1=lat1), radians(d_lon)
385 else: # for backward compaibility
386 return Phid(lat2=lat2), Phid(lat1=lat1), Phid(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)).ellipsoid)
426def equirectangular(lat1, lon1, lat2, lon2, radius=R_M, **adjust_limit_wrap):
427 '''Compute the distance between two points using the U{Equirectangular Approximation
428 / Projection<https://www.Movable-Type.co.UK/scripts/latlong.html#equirectangular>}.
430 @arg lat1: Start latitude (C{degrees}).
431 @arg lon1: Start longitude (C{degrees}).
432 @arg lat2: End latitude (C{degrees}).
433 @arg lon2: End longitude (C{degrees}).
434 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum}) or ellipsoid
435 (L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}).
436 @kwarg adjust_limit_wrap: Optional keyword arguments for function L{equirectangular4}.
438 @return: Distance (C{meter}, same units as B{C{radius}} or the ellipsoid or datum axes).
440 @raise TypeError: Invalid B{C{radius}}.
442 @see: Function L{equirectangular4} for more details, the available B{C{options}},
443 errors, restrictions and other, approximate or accurate distance functions.
444 '''
445 d = sqrt(equirectangular4(Lat(lat1=lat1), Lon(lon1=lon1),
446 Lat(lat2=lat2), Lon(lon2=lon2),
447 **adjust_limit_wrap).distance2) # PYCHOK 4 vs 2-3
448 return degrees2m(d, radius=_mean_radius(radius, lat1, lat2))
451def _equirectangular(lat1, lon1, lat2, lon2, **adjust_limit_wrap):
452 '''(INTERNAL) Helper for the L{frechet._FrechetMeterRadians}
453 and L{hausdorff._HausdorffMeterRedians} classes.
454 '''
455 return equirectangular4(lat1, lon1, lat2, lon2, **adjust_limit_wrap).distance2 * _RADIANS2
458def equirectangular4(lat1, lon1, lat2, lon2, adjust=True, limit=45, wrap=False):
459 '''Compute the distance between two points using the U{Equirectangular Approximation
460 / Projection<https://www.Movable-Type.co.UK/scripts/latlong.html#equirectangular>}.
462 This approximation is valid for short distance of several hundred Km or Miles, see
463 the B{C{limit}} keyword argument and L{LimitError}.
465 @arg lat1: Start latitude (C{degrees}).
466 @arg lon1: Start longitude (C{degrees}).
467 @arg lat2: End latitude (C{degrees}).
468 @arg lon2: End longitude (C{degrees}).
469 @kwarg adjust: Adjust the wrapped, unrolled longitudinal delta by the cosine of the mean
470 latitude (C{bool}).
471 @kwarg limit: Optional limit for lat- and longitudinal deltas (C{degrees}) or C{None}
472 or C{0} for unlimited.
473 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}} and B{C{lon2}}
474 (C{bool}).
476 @return: A L{Distance4Tuple}C{(distance2, delta_lat, delta_lon, unroll_lon2)}
477 in C{degrees squared}.
479 @raise LimitError: If the lat- and/or longitudinal delta exceeds the B{C{-limit..limit}}
480 range and L{limiterrors<pygeodesy.limiterrors>} is C{True}.
482 @see: U{Local, flat earth approximation<https://www.EdWilliams.org/avform.htm#flat>},
483 functions L{equirectangular}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
484 L{cosineLaw}, L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar}, L{haversine},
485 L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2}, C{LatLon.distanceTo*}
486 and C{LatLon.equirectangularTo}.
487 '''
488 d_lon, lat2, ulon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap)
489 d_lat = lat2 - lat1
491 if limit and limit > 0 and limiterrors():
492 d = max(fabs(d_lat), fabs(d_lon))
493 if d > limit:
494 t = _SPACE_(_delta_, Fmt.PAREN_g(d), Fmt.exceeds_limit(limit))
495 s = unstr(equirectangular4, lat1, lon1, lat2, lon2,
496 limit=limit, wrap=wrap)
497 raise LimitError(s, txt=t)
499 if adjust: # scale delta lon
500 d_lon *= _scale_deg(lat1, lat2)
502 d2 = hypot2(d_lat, d_lon) # degrees squared!
503 return Distance4Tuple(d2, d_lat, d_lon, ulon2 - lon2)
506def euclidean(lat1, lon1, lat2, lon2, radius=R_M, adjust=True, wrap=False):
507 '''Approximate the C{Euclidean} distance between two (spherical) points.
509 @arg lat1: Start latitude (C{degrees}).
510 @arg lon1: Start longitude (C{degrees}).
511 @arg lat2: End latitude (C{degrees}).
512 @arg lon2: End longitude (C{degrees}).
513 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
514 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
515 L{a_f2Tuple}) to use.
516 @kwarg adjust: Adjust the longitudinal delta by the cosine of
517 the mean latitude (C{bool}).
518 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
519 and B{C{lon2}} (C{bool}).
521 @return: Distance (C{meter}, same units as B{C{radius}} or the
522 ellipsoid or datum axes).
524 @raise TypeError: Invalid B{C{radius}}.
526 @see: U{Distance between two (spherical) points
527 <https://www.EdWilliams.org/avform.htm#Dist>}, functions L{euclid},
528 L{euclidean_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
529 L{cosineLaw}, L{equirectangular}, L{flatLocal}/L{hubeny}, L{flatPolar},
530 L{haversine}, L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2},
531 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
532 '''
533 return _dS(euclidean_, radius, wrap, lat1, lon1, lat2, lon2, adjust=adjust)
536def euclidean_(phi2, phi1, lam21, adjust=True):
537 '''Approximate the I{angular} C{Euclidean} distance between two (spherical) points.
539 @arg phi2: End latitude (C{radians}).
540 @arg phi1: Start latitude (C{radians}).
541 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
542 @kwarg adjust: Adjust the longitudinal delta by the cosine
543 of the mean latitude (C{bool}).
545 @return: Angular distance (C{radians}).
547 @see: Functions L{euclid}, L{euclidean}, L{cosineAndoyerLambert_},
548 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
549 L{flatLocal_}/L{hubeny_}, L{flatPolar_}, L{haversine_},
550 L{thomas_} and L{vincentys_}.
551 '''
552 if adjust:
553 lam21 *= _scale_rad(phi2, phi1)
554 return euclid(phi2 - phi1, lam21)
557def excessAbc_(A, b, c):
558 '''Compute the I{spherical excess} C{E} of a (spherical) triangle from two sides
559 and the included (small) angle.
561 @arg A: An interior triangle angle (C{radians}).
562 @arg b: Frist adjacent triangle side (C{radians}).
563 @arg c: Second adjacent triangle side (C{radians}).
565 @return: Spherical excess (C{radians}).
567 @raise UnitError: Invalid B{C{A}}, B{C{b}} or B{C{c}}.
569 @see: Functions L{excessGirard_}, L{excessLHuilier_} and U{Spherical
570 trigonometry<https://WikiPedia.org/wiki/Spherical_trigonometry>}.
571 '''
572 A = Radians_(A=A)
573 b = Radians_(b=b) * _0_5
574 c = Radians_(c=c) * _0_5
576 sA, cA, sb, cb, sc, cc = sincos2_(A, b, c)
577 return atan2(sA * sb * sc, cb * cc + cA * sb * sc) * _2_0
580def excessCagnoli_(a, b, c):
581 '''Compute the I{spherical excess} C{E} of a (spherical) triangle using U{Cagnoli's
582 <https://Zenodo.org/record/35392>} (D.34) formula.
584 @arg a: First triangle side (C{radians}).
585 @arg b: Second triangle side (C{radians}).
586 @arg c: Third triangle side (C{radians}).
588 @return: Spherical excess (C{radians}).
590 @raise UnitError: Invalid B{C{a}}, B{C{b}} or B{C{c}}.
592 @see: Function L{excessLHuilier_} and U{Spherical trigonometry
593 <https://WikiPedia.org/wiki/Spherical_trigonometry>}.
594 '''
595 a = Radians_(a=a)
596 b = Radians_(b=b)
597 c = Radians_(c=c)
599 s = fsumf_(a, b, c) * _0_5
600 _s = sin
601 r = _s(s) * _s(s - a) * _s(s - b) * _s(s - c)
602 c = cos(a * _0_5) * cos(b * _0_5) * cos(c * _0_5)
603 r = asin(sqrt(r) * _0_5 / c) if c and r > 0 else _0_0
604 return Radians(Cagnoli=r * _2_0)
607def excessGirard_(A, B, C):
608 '''Compute the I{spherical excess} C{E} of a (spherical) triangle using U{Girard's
609 <https://MathWorld.Wolfram.com/GirardsSphericalExcessFormula.html>} formula.
611 @arg A: First interior triangle angle (C{radians}).
612 @arg B: Second interior triangle angle (C{radians}).
613 @arg C: Third interior triangle angle (C{radians}).
615 @return: Spherical excess (C{radians}).
617 @raise UnitError: Invalid B{C{A}}, B{C{B}} or B{C{C}}.
619 @see: Function L{excessLHuilier_} and U{Spherical trigonometry
620 <https://WikiPedia.org/wiki/Spherical_trigonometry>}.
621 '''
622 return Radians(Girard=fsumf_(Radians_(A=A),
623 Radians_(B=B),
624 Radians_(C=C), -PI))
627def excessLHuilier_(a, b, c):
628 '''Compute the I{spherical excess} C{E} of a (spherical) triangle using U{L'Huilier's
629 <https://MathWorld.Wolfram.com/LHuiliersTheorem.html>}'s Theorem.
631 @arg a: First triangle side (C{radians}).
632 @arg b: Second triangle side (C{radians}).
633 @arg c: Third triangle side (C{radians}).
635 @return: Spherical excess (C{radians}).
637 @raise UnitError: Invalid B{C{a}}, B{C{b}} or B{C{c}}.
639 @see: Function L{excessCagnoli_}, L{excessGirard_} and U{Spherical
640 trigonometry<https://WikiPedia.org/wiki/Spherical_trigonometry>}.
641 '''
642 a = Radians_(a=a)
643 b = Radians_(b=b)
644 c = Radians_(c=c)
646 s = fsumf_(a, b, c) * _0_5
647 _t = tan_2
648 r = _t(s) * _t(s - a) * _t(s - b) * _t(s - c)
649 r = atan(sqrt(r)) if r > 0 else _0_0
650 return Radians(LHuilier=r * _4_0)
653def excessKarney(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
654 '''Compute the surface area of a (spherical) quadrilateral bounded by a
655 segment of a great circle, two meridians and the equator using U{Karney's
656 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>}
657 method.
659 @arg lat1: Start latitude (C{degrees}).
660 @arg lon1: Start longitude (C{degrees}).
661 @arg lat2: End latitude (C{degrees}).
662 @arg lon2: End longitude (C{degrees}).
663 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
664 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
665 L{a_f2Tuple}) or C{None}.
666 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
667 B{C{lat2}} and B{C{lon2}} (C{bool}).
669 @return: Surface area, I{signed} (I{square} C{meter} or the same units as
670 B{C{radius}} I{squared}) or the I{spherical excess} (C{radians})
671 if C{B{radius}=0} or C{None}.
673 @raise TypeError: Invalid B{C{radius}}.
675 @raise UnitError: Invalid B{C{lat2}} or B{C{lat1}}.
677 @raise ValueError: Semi-circular longitudinal delta.
679 @see: Functions L{excessKarney_} and L{excessQuad}.
680 '''
681 return _eA(excessKarney_, radius, wrap, lat1, lon1, lat2, lon2)
684def excessKarney_(phi2, phi1, lam21):
685 '''Compute the I{spherical excess} C{E} of a (spherical) quadrilateral bounded by
686 a segment of a great circle, two meridians and the equator using U{Karney's
687 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>}
688 method.
690 @arg phi2: End latitude (C{radians}).
691 @arg phi1: Start latitude (C{radians}).
692 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
694 @return: Spherical excess, I{signed} (C{radians}).
696 @raise ValueError: Semi-circular longitudinal delta B{C{lam21}}.
698 @see: Function L{excessKarney} and U{Area of a spherical polygon
699 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>}.
700 '''
701 # from: Veness <https://www.Movable-Type.co.UK/scripts/latlong.html> Area
702 # method due to Karney: for each edge of the polygon,
703 #
704 # tan(Δλ / 2) · (tan(φ1 / 2) + tan(φ2 / 2))
705 # tan(E / 2) = -----------------------------------------
706 # 1 + tan(φ1 / 2) · tan(φ2 / 2)
707 #
708 # where E is the spherical excess of the trapezium obtained by extending
709 # the edge to the equator-circle vector for each edge (see also ***).
710 _t = tan_2
711 t2 = _t(phi2)
712 t1 = _t(phi1)
713 t = _t(lam21, lam21=None)
714 return Radians(Karney=atan2(t * (t1 + t2),
715 _1_0 + (t1 * t2)) * _2_0)
718# ***) Original post no longer available, following is a copy of the main part
719# <http://OSGeo-org.1560.x6.Nabble.com/Area-of-a-spherical-polygon-td3841625.html>
720#
721# The area of a polygon on a (unit) sphere is given by the spherical excess
722#
723# A = 2 * pi - sum(exterior angles)
724#
725# However this is badly conditioned if the polygon is small. In this case, use
726#
727# A = sum(S12{i, i+1}) over the edges of the polygon
728#
729# where S12 is the area of the quadrilateral bounded by an edge of the polygon,
730# two meridians and the equator, i.e. with vertices (phi1, lambda1), (phi2,
731# lambda2), (0, lambda1) and (0, lambda2). S12 is given by
732#
733# tan(S12 / 2) = tan(lambda21 / 2) * (tan(phi1 / 2) + tan(phi2 / 2)) /
734# (tan(phi1 / 2) * tan(phi2 / 2) + 1)
735#
736# = tan(lambda21 / 2) * tanh((Lamb(phi1) + Lamb(phi2)) / 2)
737#
738# where lambda21 = lambda2 - lambda1 and Lamb(x) is the Lambertian (or the
739# inverse Gudermannian) function
740#
741# Lambertian(x) = asinh(tan(x)) = atanh(sin(x)) = 2 * atanh(tan(x / 2))
742#
743# Notes: The formula for S12 is exact, except that...
744# - it is indeterminate if an edge is a semi-circle
745# - the formula for A applies only if the polygon does not include a pole
746# (if it does, then add +/- 2 * pi to the result)
747# - in the limit of small phi and lambda, S12 reduces to the trapezoidal
748# formula, S12 = (lambda2 - lambda1) * (phi1 + phi2) / 2
749# - I derived this result from the equation for the area of a spherical
750# triangle in terms of two edges and the included angle given by, e.g.
751# U{Todhunter, I. - Spherical Trigonometry (1871), Sec. 103, Eq. (2)
752# <http://Books.Google.com/books?id=3uBHAAAAIAAJ&pg=PA71>}
753# - I would be interested to know if this formula for S12 is already known
754# - Charles Karney
757def excessQuad(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
758 '''Compute the surface area of a (spherical) quadrilateral bounded by a segment
759 of a great circle, two meridians and the equator.
761 @arg lat1: Start latitude (C{degrees}).
762 @arg lon1: Start longitude (C{degrees}).
763 @arg lat2: End latitude (C{degrees}).
764 @arg lon2: End longitude (C{degrees}).
765 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
766 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
767 L{a_f2Tuple}) or C{None}.
768 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
769 B{C{lat2}} and B{C{lon2}} (C{bool}).
771 @return: Surface area, I{signed} (I{square} C{meter} or the same units as
772 B{C{radius}} I{squared}) or the I{spherical excess} (C{radians})
773 if C{B{radius}=0} or C{None}.
775 @raise TypeError: Invalid B{C{radius}}.
777 @raise UnitError: Invalid B{C{lat2}} or B{C{lat1}}.
779 @see: Function L{excessQuad_} and L{excessKarney}.
780 '''
781 return _eA(excessQuad_, radius, wrap, lat1, lon1, lat2, lon2)
784def excessQuad_(phi2, phi1, lam21):
785 '''Compute the I{spherical excess} C{E} of a (spherical) quadrilateral bounded
786 by a segment of a great circle, two meridians and the equator.
788 @arg phi2: End latitude (C{radians}).
789 @arg phi1: Start latitude (C{radians}).
790 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
792 @return: Spherical excess, I{signed} (C{radians}).
794 @see: Function L{excessQuad} and U{Spherical trigonometry
795 <https://WikiPedia.org/wiki/Spherical_trigonometry>}.
796 '''
797 s = sin((phi2 + phi1) * _0_5)
798 c = cos((phi2 - phi1) * _0_5)
799 return Radians(Quad=atan2(tan_2(lam21) * s, c) * _2_0)
802def flatLocal(lat1, lon1, lat2, lon2, datum=_WGS84, scaled=True, wrap=False):
803 '''Compute the distance between two (ellipsoidal) points using
804 the U{ellipsoidal Earth to plane projection<https://WikiPedia.org/
805 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>}
806 aka U{Hubeny<https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula.
808 @arg lat1: Start latitude (C{degrees}).
809 @arg lon1: Start longitude (C{degrees}).
810 @arg lat2: End latitude (C{degrees}).
811 @arg lon2: End longitude (C{degrees}).
812 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
813 L{Ellipsoid2} or L{a_f2Tuple}) to use.
814 @kwarg scaled: Scale prime_vertical by C{cos(B{phi})} (C{bool}),
815 see method L{pygeodesy.Ellipsoid.roc2_}.
816 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
817 B{C{lat2}} and B{C{lon2}} (C{bool}).
819 @return: Distance (C{meter}, same units as the B{C{datum}}'s
820 ellipsoid axes).
822 @raise TypeError: Invalid B{C{datum}}.
824 @note: The meridional and prime_vertical radii of curvature
825 are taken and scaled at the mean of both latitude.
827 @see: Functions L{flatLocal_} or L{hubeny_}, L{cosineLaw}, L{flatPolar},
828 L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
829 L{equirectangular}, L{euclidean}, L{haversine}, L{thomas},
830 L{vincentys}, method L{Ellipsoid.distance2} and U{local, flat
831 earth approximation<https://www.EdWilliams.org/avform.htm#flat>}.
832 '''
833 E = _ellipsoidal(datum, flatLocal)
834 return E._hubeny_2(*_d3(wrap, lat1, lon1, lat2, lon2),
835 scaled=scaled, squared=False) * E.a
837hubeny = flatLocal # PYCHOK for Karl Hubeny
840def flatLocal_(phi2, phi1, lam21, datum=_WGS84, scaled=True):
841 '''Compute the I{angular} distance between two (ellipsoidal) points using
842 the U{ellipsoidal Earth to plane projection<https://WikiPedia.org/
843 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>}
844 aka U{Hubeny<https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula.
846 @arg phi2: End latitude (C{radians}).
847 @arg phi1: Start latitude (C{radians}).
848 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
849 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
850 L{Ellipsoid2} or L{a_f2Tuple}) to use.
851 @kwarg scaled: Scale prime_vertical by C{cos(B{phi})} (C{bool}),
852 see method L{pygeodesy.Ellipsoid.roc2_}.
854 @return: Angular distance (C{radians}).
856 @raise TypeError: Invalid B{C{datum}}.
858 @note: The meridional and prime_vertical radii of curvature
859 are taken and scaled I{at the mean of both latitude}.
861 @see: Functions L{flatLocal} or L{hubeny}, L{cosineAndoyerLambert_},
862 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{flatPolar_},
863 L{euclidean_}, L{haversine_}, L{thomas_} and L{vincentys_} and
864 U{local, flat earth approximation
865 <https://www.EdWilliams.org/avform.htm#flat>}.
866 '''
867 E = _ellipsoidal(datum, flatLocal_)
868 return E._hubeny_2(phi2, phi1, lam21, scaled=scaled, squared=False)
870hubeny_ = flatLocal_ # PYCHOK for Karl Hubeny
873def flatPolar(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
874 '''Compute the distance between two (spherical) points using
875 the U{polar coordinate flat-Earth <https://WikiPedia.org/wiki/
876 Geographical_distance#Polar_coordinate_flat-Earth_formula>}
877 formula.
879 @arg lat1: Start latitude (C{degrees}).
880 @arg lon1: Start longitude (C{degrees}).
881 @arg lat2: End latitude (C{degrees}).
882 @arg lon2: End longitude (C{degrees}).
883 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
884 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
885 L{a_f2Tuple}) to use.
886 @kwarg wrap: If C{True}, wrap or I{normalize} and B{C{lat2}}
887 and B{C{lon2}} (C{bool}).
889 @return: Distance (C{meter}, same units as B{C{radius}} or the
890 ellipsoid or datum axes).
892 @raise TypeError: Invalid B{C{radius}}.
894 @see: Functions L{flatPolar_}, L{cosineAndoyerLambert},
895 L{cosineForsytheAndoyerLambert},L{cosineLaw},
896 L{flatLocal}/L{hubeny}, L{equirectangular},
897 L{euclidean}, L{haversine}, L{thomas} and
898 L{vincentys}.
899 '''
900 return _dS(flatPolar_, radius, wrap, lat1, lon1, lat2, lon2)
903def flatPolar_(phi2, phi1, lam21):
904 '''Compute the I{angular} distance between two (spherical) points
905 using the U{polar coordinate flat-Earth<https://WikiPedia.org/wiki/
906 Geographical_distance#Polar_coordinate_flat-Earth_formula>}
907 formula.
909 @arg phi2: End latitude (C{radians}).
910 @arg phi1: Start latitude (C{radians}).
911 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
913 @return: Angular distance (C{radians}).
915 @see: Functions L{flatPolar}, L{cosineAndoyerLambert_},
916 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
917 L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{haversine_},
918 L{thomas_} and L{vincentys_}.
919 '''
920 a = fabs(PI_2 - phi1) # co-latitude
921 b = fabs(PI_2 - phi2) # co-latitude
922 if a < b:
923 a, b = b, a
924 if a < EPS0:
925 a = _0_0
926 elif b > 0:
927 b = b / a # /= chokes PyChecker
928 c = b * cos(lam21) * _2_0
929 c = fsumf_(_1_0, b**2, -fabs(c))
930 a *= sqrt0(c)
931 return a
934def _hartzell(pov, los, earth, **kwds):
935 '''(INTERNAL) Helper for C{CartesianBase.hartzell} and C{LatLonBase.hartzell}.
936 '''
937 if earth is None:
938 earth = pov.datum
939 else:
940 earth = _spherical_datum(earth, name__=hartzell)
941 pov = pov.toDatum(earth)
942 h = pov.height
943 if h < 0: # EPS0
944 t = _SPACE_(Fmt.PARENSPACED(height=h), _inside_)
945 raise IntersectionError(pov=pov, earth=earth, txt=t)
946 return hartzell(pov, los=los, earth=earth, **kwds) if h > 0 else pov # EPS0
949def hartzell(pov, los=False, earth=_WGS84, **name_LatLon_and_kwds):
950 '''Compute the intersection of the earth's surface and a Line-Of-Sight from
951 a Point-Of-View in space.
953 @arg pov: Point-Of-View outside the earth (C{LatLon}, C{Cartesian},
954 L{Ecef9Tuple} or L{Vector3d}).
955 @kwarg los: Line-Of-Sight, I{direction} to earth (L{Los}, L{Vector3d}),
956 C{True} for the I{normal, plumb} onto the surface or C{False}
957 or C{None} to point to the center of the earth.
958 @kwarg earth: The earth model (L{Datum}, L{Ellipsoid}, L{Ellipsoid2},
959 L{a_f2Tuple} or a C{scalar} earth radius in C{meter}).
960 @kwarg name_LatLon_and_kwds: Optional, overriding C{B{name}="hartzell"}
961 (C{str}), class C{B{LatLon}=None} to return the intersection
962 plus additional C{LatLon} keyword arguments, include the
963 B{C{datum}} if different and to convert from B{C{earth}}.
965 @return: The intersection (L{Vector3d}, B{C{pov}}'s C{cartesian type} or the
966 given B{C{LatLon}} instance) with attribute C{height} set to the
967 distance to the B{C{pov}}.
969 @raise IntersectionError: Invalid B{C{pov}} or B{C{pov}} inside the earth or
970 invalid B{C{los}} or B{C{los}} points outside or
971 away from the earth.
973 @raise TypeError: Invalid B{C{earth}}, C{ellipsoid} or C{datum}.
975 @see: Class L{Los}, functions L{tyr3d} and L{hartzell4} and methods
976 L{Ellipsoid.hartzell4} and any C{Cartesian.hartzell} and C{LatLon.hartzell}.
977 '''
978 n, LatLon_and_kwds = _name2__(name_LatLon_and_kwds, name__=hartzell)
979 try:
980 D = _spherical_datum(earth, name__=hartzell)
981 r, h, i = _MODS.triaxials._hartzell3(pov, los, D.ellipsoid._triaxial)
983 C = _MODS.cartesianBase.CartesianBase
984 if LatLon_and_kwds:
985 c = C(r, datum=D)
986 r = c.toLatLon(**_xkwds(LatLon_and_kwds, height=h))
987 elif isinstance(r, C):
988 r.height = h
989 if i:
990 r._iteration = i
991 except Exception as x:
992 raise IntersectionError(pov=pov, los=los, earth=earth, cause=x,
993 **LatLon_and_kwds)
994 return _xnamed(r, n) if n else r
997def haversine(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
998 '''Compute the distance between two (spherical) points using the
999 U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>}
1000 formula.
1002 @arg lat1: Start latitude (C{degrees}).
1003 @arg lon1: Start longitude (C{degrees}).
1004 @arg lat2: End latitude (C{degrees}).
1005 @arg lon2: End longitude (C{degrees}).
1006 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
1007 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
1008 L{a_f2Tuple}) to use.
1009 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
1010 B{C{lat2}} and B{C{lon2}} (C{bool}).
1012 @return: Distance (C{meter}, same units as B{C{radius}}).
1014 @raise TypeError: Invalid B{C{radius}}.
1016 @see: U{Distance between two (spherical) points
1017 <https://www.EdWilliams.org/avform.htm#Dist>}, functions
1018 L{cosineLaw}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
1019 L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar},
1020 L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2},
1021 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
1023 @note: See note at function L{vincentys_}.
1024 '''
1025 return _dS(haversine_, radius, wrap, lat1, lon1, lat2, lon2)
1028def haversine_(phi2, phi1, lam21):
1029 '''Compute the I{angular} distance between two (spherical) points
1030 using the U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>}
1031 formula.
1033 @arg phi2: End latitude (C{radians}).
1034 @arg phi1: Start latitude (C{radians}).
1035 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
1037 @return: Angular distance (C{radians}).
1039 @see: Functions L{haversine}, L{cosineAndoyerLambert_},
1040 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
1041 L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_},
1042 L{thomas_} and L{vincentys_}.
1044 @note: See note at function L{vincentys_}.
1045 '''
1046 def _hsin(rad):
1047 return sin(rad * _0_5)**2
1049 h = _hsin(phi2 - phi1) + cos(phi1) * cos(phi2) * _hsin(lam21) # haversine
1050 return atan2(sqrt0(h), sqrt0(_1_0 - h)) * _2_0 # == asin(sqrt(h)) * 2
1053def heightOf(angle, distance, radius=R_M):
1054 '''Determine the height above the (spherical) earth' surface after
1055 traveling along a straight line at a given tilt.
1057 @arg angle: Tilt angle above horizontal (C{degrees}).
1058 @arg distance: Distance along the line (C{meter} or same units as
1059 B{C{radius}}).
1060 @kwarg radius: Optional mean earth radius (C{meter}).
1062 @return: Height (C{meter}, same units as B{C{distance}} and B{C{radius}}).
1064 @raise ValueError: Invalid B{C{angle}}, B{C{distance}} or B{C{radius}}.
1066 @see: U{MultiDop geog_lib.GeogBeamHt<https://GitHub.com/NASA/MultiDop>}
1067 (U{Shapiro et al. 2009, JTECH
1068 <https://Journals.AMetSoc.org/doi/abs/10.1175/2009JTECHA1256.1>}
1069 and U{Potvin et al. 2012, JTECH
1070 <https://Journals.AMetSoc.org/doi/abs/10.1175/JTECH-D-11-00019.1>}).
1071 '''
1072 r = h = Radius(radius)
1073 d = fabs(Distance(distance))
1074 if d > h:
1075 d, h = h, d
1077 if d > EPS0: # and h > EPS0
1078 d = d / h # /= h chokes PyChecker
1079 s = sin(Phid(angle=angle, clip=_180_0))
1080 s = fsumf_(_1_0, s * d * _2_0, d**2)
1081 if s > 0:
1082 return h * sqrt(s) - r
1084 raise _ValueError(angle=angle, distance=distance, radius=radius)
1087def heightOrthometric(h_ll, N):
1088 '''Get the I{orthometric} height B{H}, the height above the geoid, earth surface.
1090 @arg h_ll: The height above the ellipsoid (C{meter}) or an I{ellipsoidal}
1091 location (C{LatLon} with a C{height} or C{h} attribute).
1092 @arg N: The I{geoid} height (C{meter}), the height of the geoid above the
1093 ellipsoid at the same B{C{h_ll}} location.
1095 @return: I{Orthometric} height C{B{H} = B{h} - B{N}} (C{meter}, same units
1096 as B{C{h}} and B{C{N}}).
1098 @see: U{Ellipsoid, Geoid, and Othometric Heights<https://www.NGS.NOAA.gov/
1099 GEOID/PRESENTATIONS/2007_02_24_CCPS/Roman_A_PLSC2007notes.pdf>}, page
1100 6 and module L{pygeodesy.geoids}.
1101 '''
1102 h = h_ll if _isHeight(h_ll) else _xattr(h_ll, height=_xattr(h_ll, h=0))
1103 return Height(H=Height(h=h) - Height(N=N))
1106def horizon(height, radius=R_M, refraction=False):
1107 '''Determine the distance to the horizon from a given altitude above the
1108 (spherical) earth.
1110 @arg height: Altitude (C{meter} or same units as B{C{radius}}).
1111 @kwarg radius: Optional mean earth radius (C{meter}).
1112 @kwarg refraction: Consider atmospheric refraction (C{bool}).
1114 @return: Distance (C{meter}, same units as B{C{height}} and B{C{radius}}).
1116 @raise ValueError: Invalid B{C{height}} or B{C{radius}}.
1118 @see: U{Distance to horizon<https://www.EdWilliams.org/avform.htm#Horizon>}.
1119 '''
1120 h, r = Height(height), Radius(radius)
1121 if min(h, r) < 0:
1122 raise _ValueError(height=height, radius=radius)
1124 d2 = ((r * 2.415750694528) if refraction else # 2.0 / 0.8279
1125 fsumf_(r, r, h)) * h
1126 return sqrt0(d2)
1129class _idllmn6(object): # see also .geodesicw._wargs, .latlonBase._toCartesian3, .vector2d._numpy
1130 '''(INTERNAL) Helper for C{intersection2} and C{intersections2}.
1131 '''
1132 @contextmanager # <https://www.Python.org/dev/peps/pep-0343/> Examples
1133 def __call__(self, datum, lat1, lon1, lat2, lon2, small, wrap, s, **kwds):
1134 try:
1135 if wrap:
1136 _, lat2, lon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap)
1137 kwds = _xkwds(kwds, wrap=wrap) # for _xError
1138 m = small if small is _100km else Meter_(small=small)
1139 n = _DUNDER_nameof(intersections2 if s else intersection2)
1140 if datum is None or euclidean(lat1, lon1, lat2, lon2) < m:
1141 d, m = None, _MODS.vector3d
1142 _i = m._intersects2 if s else m._intersect3d3
1143 elif _isRadius(datum) and datum < 0 and not s:
1144 d = _spherical_datum(-datum, name=n)
1145 m = _MODS.sphericalNvector
1146 _i = m.intersection
1147 else:
1148 d = _spherical_datum(datum, name=n)
1149 if d.isSpherical:
1150 m = _MODS.sphericalTrigonometry
1151 _i = m._intersects2 if s else m._intersect
1152 elif d.isEllipsoidal:
1153 try:
1154 if d.ellipsoid.geodesic:
1155 pass
1156 m = _MODS.ellipsoidalKarney
1157 except ImportError:
1158 m = _MODS.ellipsoidalExact
1159 _i = m._intersections2 if s else m._intersection3 # ellipsoidalBaseDI
1160 else:
1161 raise _TypeError(datum=datum)
1162 yield _i, d, lat2, lon2, m, n
1164 except (TypeError, ValueError) as x:
1165 raise _xError(x, lat1=lat1, lon1=lon1, datum=datum,
1166 lat2=lat2, lon2=lon2, small=small, **kwds)
1168_idllmn6 = _idllmn6() # PYCHOK singleton
1171def intersection2(lat1, lon1, bearing1,
1172 lat2, lon2, bearing2, datum=None, wrap=False, small=_100km): # was=True
1173 '''I{Conveniently} compute the intersection of two lines each defined
1174 by a (geodetic) point and a bearing from North, using either ...
1176 1) L{vector3d.intersection3d3} for B{C{small}} distances (below 100 Km
1177 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ...
1179 2) L{sphericalTrigonometry.intersection} for a spherical B{C{datum}}
1180 or a C{scalar B{datum}} representing the earth radius, conventionally
1181 in C{meter} or ...
1183 3) L{sphericalNvector.intersection} if B{C{datum}} is a I{negative}
1184 C{scalar}, (negative) earth radius, conventionally in C{meter} or ...
1186 4) L{ellipsoidalKarney.intersection3} for an ellipsoidal B{C{datum}}
1187 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>}
1188 is installed, otherwise ...
1190 5) L{ellipsoidalExact.intersection3}, provided B{C{datum}} is ellipsoidal.
1192 @arg lat1: Latitude of the first point (C{degrees}).
1193 @arg lon1: Longitude of the first point (C{degrees}).
1194 @arg bearing1: Bearing at the first point (compass C{degrees}).
1195 @arg lat2: Latitude of the second point (C{degrees}).
1196 @arg lon2: Longitude of the second point (C{degrees}).
1197 @arg bearing2: Bearing at the second point (compass C{degrees}).
1198 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid},
1199 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth
1200 radius (C{meter}, same units as B{C{radius1}} and
1201 B{C{radius2}}) or C{None}.
1202 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
1203 and B{C{lon2}} (C{bool}).
1204 @kwarg small: Upper limit for small distances (C{meter}).
1206 @return: A L{LatLon2Tuple}C{(lat, lon)} with the lat- and
1207 longitude of the intersection point.
1209 @raise IntersectionError: Ambiguous or infinite intersection
1210 or colinear, parallel or otherwise
1211 non-intersecting lines.
1213 @raise TypeError: Invalid B{C{datum}}.
1215 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{bearing1}},
1216 B{C{lat2}}, B{C{lon2}} or B{C{bearing2}}.
1218 @see: Method L{RhumbLine.intersection2}.
1220 @note: The returned intersections may be near-antipodal.
1221 '''
1222 b1 = Bearing(bearing1=bearing1)
1223 b2 = Bearing(bearing2=bearing2)
1224 with _idllmn6(datum, lat1, lon1, lat2, lon2,
1225 small, wrap, False, bearing1=b1, bearing2=b2) as t:
1226 _i, d, lat2, lon2, m, n = t
1227 if d is None:
1228 t, _, _ = _i(m.Vector3d(lon1, lat1, 0), b1,
1229 m.Vector3d(lon2, lat2, 0), b2, useZ=False)
1230 t = LatLon2Tuple(t.y, t.x, name=n)
1232 else:
1233 t = _i(m.LatLon(lat1, lon1, datum=d), b1,
1234 m.LatLon(lat2, lon2, datum=d), b2,
1235 LatLon=None, height=0, wrap=False)
1236 if isinstance(t, Intersection3Tuple): # ellipsoidal
1237 t, _, _ = t
1238 t = LatLon2Tuple(t.lat, t.lon, name=n)
1239 return t
1242def intersections2(lat1, lon1, radius1,
1243 lat2, lon2, radius2, datum=None, wrap=False, small=_100km): # was=True
1244 '''I{Conveniently} compute the intersections of two circles each defined
1245 by a (geodetic) center point and a radius, using either ...
1247 1) L{vector3d.intersections2} for B{C{small}} distances (below 100 Km
1248 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ...
1250 2) L{sphericalTrigonometry.intersections2} for a spherical B{C{datum}}
1251 or a C{scalar B{datum}} representing the earth radius, conventionally
1252 in C{meter} or ...
1254 3) L{ellipsoidalKarney.intersections2} for an ellipsoidal B{C{datum}}
1255 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>}
1256 is installed, otherwise ...
1258 4) L{ellipsoidalExact.intersections2}, provided B{C{datum}} is ellipsoidal.
1260 @arg lat1: Latitude of the first circle center (C{degrees}).
1261 @arg lon1: Longitude of the first circle center (C{degrees}).
1262 @arg radius1: Radius of the first circle (C{meter}, conventionally).
1263 @arg lat2: Latitude of the second circle center (C{degrees}).
1264 @arg lon2: Longitude of the second circle center (C{degrees}).
1265 @arg radius2: Radius of the second circle (C{meter}, same units as B{C{radius1}}).
1266 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid},
1267 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth
1268 radius (C{meter}, same units as B{C{radius1}} and
1269 B{C{radius2}}) or C{None}.
1270 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
1271 and B{C{lon2}} (C{bool}).
1272 @kwarg small: Upper limit for small distances (C{meter}).
1274 @return: 2-Tuple of the intersection points, each a
1275 L{LatLon2Tuple}C{(lat, lon)}. For abutting circles, the
1276 points are the same instance, aka the I{radical center}.
1278 @raise IntersectionError: Concentric, antipodal, invalid or
1279 non-intersecting circles or no
1280 convergence.
1282 @raise TypeError: Invalid B{C{datum}}.
1284 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{radius1}},
1285 B{C{lat2}}, B{C{lon2}} or B{C{radius2}}.
1286 '''
1287 r1 = Radius_(radius1=radius1)
1288 r2 = Radius_(radius2=radius2)
1289 with _idllmn6(datum, lat1, lon1, lat2, lon2,
1290 small, wrap, True, radius1=r1, radius2=r2) as t:
1291 _i, d, lat2, lon2, m, n = t
1292 if d is None:
1293 r1 = m2degrees(r1, radius=R_M, lat=lat1)
1294 r2 = m2degrees(r2, radius=R_M, lat=lat2)
1296 def _V2T(x, y, _, **unused): # _ == z unused
1297 return LatLon2Tuple(y, x, name=n)
1299 t = _i(m.Vector3d(lon1, lat1, 0), r1,
1300 m.Vector3d(lon2, lat2, 0), r2, sphere=False,
1301 Vector=_V2T)
1302 else:
1303 def _LL2T(lat, lon, **unused):
1304 return LatLon2Tuple(lat, lon, name=n)
1306 t = _i(m.LatLon(lat1, lon1, datum=d), r1,
1307 m.LatLon(lat2, lon2, datum=d), r2,
1308 LatLon=_LL2T, height=0, wrap=False)
1309 return t
1312def isantipode(lat1, lon1, lat2, lon2, eps=EPS):
1313 '''Check whether two points are I{antipodal}, on diametrically
1314 opposite sides of the earth.
1316 @arg lat1: Latitude of one point (C{degrees}).
1317 @arg lon1: Longitude of one point (C{degrees}).
1318 @arg lat2: Latitude of the other point (C{degrees}).
1319 @arg lon2: Longitude of the other point (C{degrees}).
1320 @kwarg eps: Tolerance for near-equality (C{degrees}).
1322 @return: C{True} if points are antipodal within the
1323 B{C{eps}} tolerance, C{False} otherwise.
1325 @see: Functions L{isantipode_} and L{antipode}.
1326 '''
1327 return (fabs(lat1 + lat2) <= eps and
1328 fabs(lon1 + lon2) <= eps) or _isequalTo(
1329 normal(lat1, lon1), antipode(lat2, lon2), eps)
1332def isantipode_(phi1, lam1, phi2, lam2, eps=EPS):
1333 '''Check whether two points are I{antipodal}, on diametrically
1334 opposite sides of the earth.
1336 @arg phi1: Latitude of one point (C{radians}).
1337 @arg lam1: Longitude of one point (C{radians}).
1338 @arg phi2: Latitude of the other point (C{radians}).
1339 @arg lam2: Longitude of the other point (C{radians}).
1340 @kwarg eps: Tolerance for near-equality (C{radians}).
1342 @return: C{True} if points are antipodal within the
1343 B{C{eps}} tolerance, C{False} otherwise.
1345 @see: Functions L{isantipode} and L{antipode_}.
1346 '''
1347 return (fabs(phi1 + phi2) <= eps and
1348 fabs(lam1 + lam2) <= eps) or _isequalTo_(
1349 normal_(phi1, lam1), antipode_(phi2, lam2), eps)
1352def _isequalTo(p1, p2, eps=EPS):
1353 '''Compare 2 point lat-/lons ignoring C{class}.
1354 '''
1355 return (fabs(p1.lat - p2.lat) <= eps and
1356 fabs(p1.lon - p2.lon) <= eps) if eps else (p1.latlon == p2.latlon)
1359def _isequalTo_(p1, p2, eps=EPS):
1360 '''(INTERNAL) Compare 2 point phi-/lams ignoring C{class}.
1361 '''
1362 return (fabs(p1.phi - p2.phi) <= eps and
1363 fabs(p1.lam - p2.lam) <= eps) if eps else (p1.philam == p2.philam)
1366def isnormal(lat, lon, eps=0):
1367 '''Check whether B{C{lat}} I{and} B{C{lon}} are within their
1368 respective I{normal} range in C{degrees}.
1370 @arg lat: Latitude (C{degrees}).
1371 @arg lon: Longitude (C{degrees}).
1372 @kwarg eps: Optional tolerance C{degrees}).
1374 @return: C{True} if C{(abs(B{lat}) + B{eps}) <= 90} and
1375 C{(abs(B{lon}) + B{eps}) <= 180}, C{False} othwerwise.
1377 @see: Functions L{isnormal_} and L{normal}.
1378 '''
1379 return (_90_0 - fabs(lat)) >= eps and _loneg(fabs(lon)) >= eps
1382def isnormal_(phi, lam, eps=0):
1383 '''Check whether B{C{phi}} I{and} B{C{lam}} are within their
1384 respective I{normal} range in C{radians}.
1386 @arg phi: Latitude (C{radians}).
1387 @arg lam: Longitude (C{radians}).
1388 @kwarg eps: Optional tolerance C{radians}).
1390 @return: C{True} if C{(abs(B{phi}) + B{eps}) <= PI/2} and
1391 C{(abs(B{lam}) + B{eps}) <= PI}, C{False} othwerwise.
1393 @see: Functions L{isnormal} and L{normal_}.
1394 '''
1395 return (PI_2 - fabs(phi)) >= eps and (PI - fabs(lam)) >= eps
1398def latlon2n_xyz(lat, lon, **name):
1399 '''Convert lat-, longitude to C{n-vector} (I{normal} to the
1400 earth's surface) X, Y and Z components.
1402 @arg lat: Latitude (C{degrees}).
1403 @arg lon: Longitude (C{degrees}).
1404 @kwarg name: Optional C{B{name}=NN} (C{str}).
1406 @return: A L{Vector3Tuple}C{(x, y, z)}.
1408 @see: Function L{philam2n_xyz}.
1410 @note: These are C{n-vector} x, y and z components,
1411 I{NOT} geocentric ECEF x, y and z coordinates!
1412 '''
1413 return _2n_xyz(name, *sincos2d_(lat, lon))
1416def _normal2(a, b, n_2, n, n2):
1417 '''(INTERNAL) Helper for C{normal} and C{normal_}.
1418 '''
1419 if fabs(b) > n:
1420 b = remainder(b, n2)
1421 if fabs(a) > n_2:
1422 r = remainder(a, n)
1423 if r != a:
1424 a = -r
1425 b -= n if b > 0 else -n
1426 return float0_(a, b)
1429def normal(lat, lon, **name):
1430 '''Normalize a lat- I{and} longitude pair in C{degrees}.
1432 @arg lat: Latitude (C{degrees}).
1433 @arg lon: Longitude (C{degrees}).
1434 @kwarg name: Optional, overriding C{B{name}="normal"} (C{str}).
1436 @return: L{LatLon2Tuple}C{(lat, lon)} with C{abs(lat) <= 90}
1437 and C{abs(lon) <= 180}.
1439 @see: Functions L{normal_} and L{isnormal}.
1440 '''
1441 return LatLon2Tuple(*_normal2(lat, lon, _90_0, _180_0, _360_0),
1442 name=_name__(name, name__=normal))
1445def normal_(phi, lam, **name):
1446 '''Normalize a lat- I{and} longitude pair in C{radians}.
1448 @arg phi: Latitude (C{radians}).
1449 @arg lam: Longitude (C{radians}).
1450 @kwarg name: Optional, overriding C{B{name}="normal_"} (C{str}).
1452 @return: L{PhiLam2Tuple}C{(phi, lam)} with C{abs(phi) <= PI/2}
1453 and C{abs(lam) <= PI}.
1455 @see: Functions L{normal} and L{isnormal_}.
1456 '''
1457 return PhiLam2Tuple(*_normal2(phi, lam, PI_2, PI, PI2),
1458 name=_name__(name, name__=normal_))
1461def _2n_xyz(name, sa, ca, sb, cb): # name always **name
1462 '''(INTERNAL) Helper for C{latlon2n_xyz} and C{philam2n_xyz}.
1463 '''
1464 # Kenneth Gade eqn 3, but using right-handed
1465 # vector x -> 0°E,0°N, y -> 90°E,0°N, z -> 90°N
1466 return Vector3Tuple(ca * cb, ca * sb, sa, **name)
1469def n_xyz2latlon(x, y, z, **name):
1470 '''Convert C{n-vector} components to lat- and longitude in C{degrees}.
1472 @arg x: X component (C{scalar}).
1473 @arg y: Y component (C{scalar}).
1474 @arg z: Z component (C{scalar}).
1475 @kwarg name: Optional C{B{name}=NN} (C{str}).
1477 @return: A L{LatLon2Tuple}C{(lat, lon)}.
1479 @see: Function L{n_xyz2philam}.
1480 '''
1481 return LatLon2Tuple(atan2d(z, hypot(x, y)), atan2d(y, x), **name)
1484def n_xyz2philam(x, y, z, **name):
1485 '''Convert C{n-vector} components to lat- and longitude in C{radians}.
1487 @arg x: X component (C{scalar}).
1488 @arg y: Y component (C{scalar}).
1489 @arg z: Z component (C{scalar}).
1490 @kwarg name: Optional C{B{name}=NN} (C{str}).
1492 @return: A L{PhiLam2Tuple}C{(phi, lam)}.
1494 @see: Function L{n_xyz2latlon}.
1495 '''
1496 return PhiLam2Tuple(atan2(z, hypot(x, y)), atan2(y, x), **name)
1499def _opposes(d, m, n, n2):
1500 '''(INTERNAL) Helper for C{opposing} and C{opposing_}.
1501 '''
1502 d = d % n2 # -20 % 360 == 340, -1 % PI2 == PI2 - 1
1503 return False if d < m or d > (n2 - m) else (
1504 True if (n - m) < d < (n + m) else None)
1507def opposing(bearing1, bearing2, margin=_90_0):
1508 '''Compare the direction of two bearings given in C{degrees}.
1510 @arg bearing1: First bearing (compass C{degrees}).
1511 @arg bearing2: Second bearing (compass C{degrees}).
1512 @kwarg margin: Optional, interior angle bracket (C{degrees}).
1514 @return: C{True} if both bearings point in opposite, C{False} if
1515 in similar or C{None} if in I{perpendicular} directions.
1517 @see: Function L{opposing_}.
1518 '''
1519 m = Degrees_(margin=margin, low=EPS0, high=_90_0)
1520 return _opposes(bearing2 - bearing1, m, _180_0, _360_0)
1523def opposing_(radians1, radians2, margin=PI_2):
1524 '''Compare the direction of two bearings given in C{radians}.
1526 @arg radians1: First bearing (C{radians}).
1527 @arg radians2: Second bearing (C{radians}).
1528 @kwarg margin: Optional, interior angle bracket (C{radians}).
1530 @return: C{True} if both bearings point in opposite, C{False} if
1531 in similar or C{None} if in perpendicular directions.
1533 @see: Function L{opposing}.
1534 '''
1535 m = Radians_(margin=margin, low=EPS0, high=PI_2)
1536 return _opposes(radians2 - radians1, m, PI, PI2)
1539def philam2n_xyz(phi, lam, **name):
1540 '''Convert lat-, longitude to C{n-vector} (I{normal} to the
1541 earth's surface) X, Y and Z components.
1543 @arg phi: Latitude (C{radians}).
1544 @arg lam: Longitude (C{radians}).
1545 @kwarg name: Optional name (C{str}).
1547 @return: A L{Vector3Tuple}C{(x, y, z)}.
1549 @see: Function L{latlon2n_xyz}.
1551 @note: These are C{n-vector} x, y and z components,
1552 I{NOT} geocentric ECEF x, y and z coordinates!
1553 '''
1554 return _2n_xyz(name, *sincos2_(phi, lam))
1557def _Propy(func, nargs, kwds):
1558 '''(INTERNAL) Helper for the C{frechet.[-]Frechet**} and
1559 C{hausdorff.[-]Hausdorff*} classes.
1560 '''
1561 try:
1562 _xcallable(distance=func)
1563 # assert _args_kwds_count2(func)[0] == nargs + int(ismethod(func))
1564 _ = func(*_0_0s(nargs), **kwds)
1565 except Exception as x:
1566 t = unstr(func, **kwds)
1567 raise _TypeError(t, cause=x)
1568 return func
1571def _radical2(d, r1, r2, **name): # in .ellipsoidalBaseDI, .sphericalTrigonometry, .vector3d
1572 # (INTERNAL) See C{radical2} below
1573 # assert d > EPS0
1574 r = fsumf_(_1_0, (r1 / d)**2, -(r2 / d)**2) * _0_5
1575 n = _name__(name, name__=radical2)
1576 return Radical2Tuple(max(_0_0, min(_1_0, r)), r * d, name=n)
1579def radical2(distance, radius1, radius2, **name):
1580 '''Compute the I{radical ratio} and I{radical line} of two
1581 U{intersecting circles<https://MathWorld.Wolfram.com/
1582 Circle-CircleIntersection.html>}.
1584 The I{radical line} is perpendicular to the axis thru the
1585 centers of the circles at C{(0, 0)} and C{(B{distance}, 0)}.
1587 @arg distance: Distance between the circle centers (C{scalar}).
1588 @arg radius1: Radius of the first circle (C{scalar}).
1589 @arg radius2: Radius of the second circle (C{scalar}).
1590 @kwarg name: Optional C{B{name}=NN} (C{str}).
1592 @return: A L{Radical2Tuple}C{(ratio, xline)} where C{0.0 <=
1593 ratio <= 1.0} and C{xline} is along the B{C{distance}}.
1595 @raise IntersectionError: The B{C{distance}} exceeds the sum
1596 of B{C{radius1}} and B{C{radius2}}.
1598 @raise UnitError: Invalid B{C{distance}}, B{C{radius1}} or
1599 B{C{radius2}}.
1601 @see: U{Circle-Circle Intersection
1602 <https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>}.
1603 '''
1604 d = Distance_(distance, low=_0_0)
1605 r1 = Radius_(radius1=radius1)
1606 r2 = Radius_(radius2=radius2)
1607 if d > (r1 + r2):
1608 raise IntersectionError(distance=d, radius1=r1, radius2=r2,
1609 txt=_too_(_distant_))
1610 return _radical2(d, r1, r2, **name) if d > EPS0 else \
1611 Radical2Tuple(_0_5, _0_0, **name)
1614class Radical2Tuple(_NamedTuple):
1615 '''2-Tuple C{(ratio, xline)} of the I{radical} C{ratio} and
1616 I{radical} C{xline}, both C{scalar} and C{0.0 <= ratio <= 1.0}
1617 '''
1618 _Names_ = (_ratio_, _xline_)
1619 _Units_ = ( Scalar, Scalar)
1622def _radistance(inst):
1623 '''(INTERNAL) Helper for the L{frechet._FrechetMeterRadians}
1624 and L{hausdorff._HausdorffMeterRedians} classes.
1625 '''
1626 wrap_, kwds_ = _xkwds_pop2(inst._kwds, wrap=False)
1627 func_ = inst._func_
1628 try: # calling lower-overhead C{func_}
1629 func_(0, _0_25, _0_5, **kwds_)
1630 wrap_ = _Wrap._philamop(wrap_)
1631 except TypeError:
1632 return inst.distance
1634 def _philam(p):
1635 try:
1636 return p.phi, p.lam
1637 except AttributeError: # no .phi or .lam
1638 return radians(p.lat), radians(p.lon)
1640 def _func_wrap(point1, point2):
1641 phi1, lam1 = wrap_(*_philam(point1))
1642 phi2, lam2 = wrap_(*_philam(point2))
1643 return func_(phi2, phi1, lam2 - lam1, **kwds_)
1645 inst._units = inst._units_
1646 return _func_wrap
1649def _scale_deg(lat1, lat2): # degrees
1650 # scale factor cos(mean of lats) for delta lon
1651 m = fabs(lat1 + lat2) * _0_5
1652 return cos(radians(m)) if m < 90 else _0_0
1655def _scale_rad(phi1, phi2): # radians, by .frechet, .hausdorff, .heights
1656 # scale factor cos(mean of phis) for delta lam
1657 m = fabs(phi1 + phi2) * _0_5
1658 return cos(m) if m < PI_2 else _0_0
1661def _sincosa6(phi2, phi1, lam21): # [4] in cosineLaw
1662 '''(INTERNAL) C{sin}es, C{cos}ines and C{acos}ine.
1663 '''
1664 s2, c2, s1, c1, _, c21 = sincos2_(phi2, phi1, lam21)
1665 return s2, c2, s1, c1, acos1(s1 * s2 + c1 * c2 * c21), c21
1668def thomas(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):
1669 '''Compute the distance between two (ellipsoidal) points using
1670 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>}
1671 formula.
1673 @arg lat1: Start latitude (C{degrees}).
1674 @arg lon1: Start longitude (C{degrees}).
1675 @arg lat2: End latitude (C{degrees}).
1676 @arg lon2: End longitude (C{degrees}).
1677 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
1678 L{Ellipsoid2} or L{a_f2Tuple}) to use.
1679 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
1680 B{C{lat2}} and B{C{lon2}} (C{bool}).
1682 @return: Distance (C{meter}, same units as the B{C{datum}}'s
1683 ellipsoid axes).
1685 @raise TypeError: Invalid B{C{datum}}.
1687 @see: Functions L{thomas_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
1688 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny},
1689 L{flatPolar}, L{haversine}, L{vincentys} and method L{Ellipsoid.distance2}.
1690 '''
1691 return _dE(thomas_, datum, wrap, lat1, lon1, lat2, lon2)
1694def thomas_(phi2, phi1, lam21, datum=_WGS84):
1695 '''Compute the I{angular} distance between two (ellipsoidal) points using
1696 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>}
1697 formula.
1699 @arg phi2: End latitude (C{radians}).
1700 @arg phi1: Start latitude (C{radians}).
1701 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
1702 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid},
1703 L{Ellipsoid2} or L{a_f2Tuple}).
1705 @return: Angular distance (C{radians}).
1707 @raise TypeError: Invalid B{C{datum}}.
1709 @see: Functions L{thomas}, L{cosineAndoyerLambert_},
1710 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
1711 L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_},
1712 L{haversine_} and L{vincentys_} and U{Geodesy-PHP
1713 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/
1714 Distance/ThomasFormula.php>}.
1715 '''
1716 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21)
1717 if r and isnon0(c1) and isnon0(c2):
1718 E = _ellipsoidal(datum, thomas_)
1719 if E.f:
1720 r1 = atan2(E.b_a * s1, c1)
1721 r2 = atan2(E.b_a * s2, c2)
1723 j = (r2 + r1) * _0_5
1724 k = (r2 - r1) * _0_5
1725 sj, cj, sk, ck, h, _ = sincos2_(j, k, lam21 * _0_5)
1727 h = fsumf_(sk**2, (ck * h)**2, -(sj * h)**2)
1728 u = _1_0 - h
1729 if isnon0(u) and isnon0(h):
1730 r = atan(sqrt0(h / u)) * 2 # == acos(1 - 2 * h)
1731 sr, cr = sincos2(r)
1732 if isnon0(sr):
1733 u = 2 * (sj * ck)**2 / u
1734 h = 2 * (sk * cj)**2 / h
1735 x = u + h
1736 y = u - h
1738 s = r / sr
1739 e = 4 * s**2
1740 d = 2 * cr
1741 a = e * d
1742 b = 2 * r
1743 c = s - (a - d) * _0_5
1744 f = E.f * _0_25
1746 t = fsumf_(a * x, -b * y, c * x**2, -d * y**2, e * x * y)
1747 r -= fsumf_(s * x, -y, -t * f * _0_25) * f * sr
1748 return r
1751def vincentys(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
1752 '''Compute the distance between two (spherical) points using
1753 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>}
1754 spherical formula.
1756 @arg lat1: Start latitude (C{degrees}).
1757 @arg lon1: Start longitude (C{degrees}).
1758 @arg lat2: End latitude (C{degrees}).
1759 @arg lon2: End longitude (C{degrees}).
1760 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
1761 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
1762 L{a_f2Tuple}) to use.
1763 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
1764 B{C{lat2}} and B{C{lon2}} (C{bool}).
1766 @return: Distance (C{meter}, same units as B{C{radius}}).
1768 @raise UnitError: Invalid B{C{radius}}.
1770 @see: Functions L{vincentys_}, L{cosineAndoyerLambert},
1771 L{cosineForsytheAndoyerLambert},L{cosineLaw}, L{equirectangular},
1772 L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar},
1773 L{haversine} and L{thomas} and methods L{Ellipsoid.distance2},
1774 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
1776 @note: See note at function L{vincentys_}.
1777 '''
1778 return _dS(vincentys_, radius, wrap, lat1, lon1, lat2, lon2)
1781def vincentys_(phi2, phi1, lam21):
1782 '''Compute the I{angular} distance between two (spherical) points using
1783 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>}
1784 spherical formula.
1786 @arg phi2: End latitude (C{radians}).
1787 @arg phi1: Start latitude (C{radians}).
1788 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
1790 @return: Angular distance (C{radians}).
1792 @see: Functions L{vincentys}, L{cosineAndoyerLambert_},
1793 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
1794 L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_},
1795 L{haversine_} and L{thomas_}.
1797 @note: Functions L{vincentys_}, L{haversine_} and L{cosineLaw_}
1798 produce equivalent results, but L{vincentys_} is suitable
1799 for antipodal points and slightly more expensive (M{3 cos,
1800 3 sin, 1 hypot, 1 atan2, 6 mul, 2 add}) than L{haversine_}
1801 (M{2 cos, 2 sin, 2 sqrt, 1 atan2, 5 mul, 1 add}) and
1802 L{cosineLaw_} (M{3 cos, 3 sin, 1 acos, 3 mul, 1 add}).
1803 '''
1804 s1, c1, s2, c2, s21, c21 = sincos2_(phi1, phi2, lam21)
1806 c = c2 * c21
1807 x = s1 * s2 + c1 * c
1808 y = c1 * s2 - s1 * c
1809 return atan2(hypot(c2 * s21, y), x)
1811# **) MIT License
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