Coverage for pygeodesy/formy.py: 98%
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« prev ^ index » next coverage.py v7.2.2, created at 2023-10-04 12:08 -0400
2# -*- coding: utf-8 -*-
4u'''Formulary of basic geodesy functions and approximations.
5'''
6# make sure int/int division yields float quotient, see .basics
7from __future__ import division as _; del _ # PYCHOK semicolon
9# from pygeodesy.basics import isscalar # from .fsums
10# from pygeodesy.cartesianBase import CartesianBase # _MODS
11from pygeodesy.constants import EPS, EPS0, EPS1, PI, PI2, PI3, PI_2, R_M, \
12 _umod_PI2, float0_, isnon0, remainder, \
13 _0_0, _0_125, _0_25, _0_5, _1_0, _2_0, \
14 _4_0, _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 _xkwds, _xkwds_pop
21from pygeodesy.fmath import euclid, hypot, hypot2, sqrt0
22from pygeodesy.fsums import fsumf_, isscalar
23from pygeodesy.interns import NN, _delta_, _distant_, _inside_, _SPACE_, _too_
24from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS
25from pygeodesy.named import _NamedTuple, _xnamed, Fmt, unstr
26from pygeodesy.namedTuples import Bearing2Tuple, Distance4Tuple, \
27 Intersection3Tuple, LatLon2Tuple, \
28 PhiLam2Tuple, Vector3Tuple
29# from pygeodesy.streprs import Fmt, unstr # from .named
30# from pygeodesy.triaxials import _hartzell3d2 # _MODS
31from pygeodesy.units import Bearing, Degrees_, Distance, Distance_, Height, \
32 Lam_, Lat, Lon, Meter_, Phi_, Radians, Radians_, \
33 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 import ellipsoidalExact, ellipsoidalKarney, vector3d, \
38# sphericalNvector, sphericalTrigonometry # _MODS
40from contextlib import contextmanager
41from math import asin, atan, atan2, cos, degrees, fabs, radians, sin, sqrt # pow
43__all__ = _ALL_LAZY.formy
44__version__ = '23.10.04'
46_D2_R2 = (PI / _180_0)**2 # degrees- to radians-squared
47_ratio_ = 'ratio'
48_xline_ = 'xline'
51def _anti2(a, b, n_2, n, n2):
52 '''(INTERNAL) Helper for C{antipode} and C{antipode_}.
53 '''
54 r = remainder(a, n) if fabs(a) > n_2 else a
55 if r == a:
56 r = -r
57 b += n
58 if fabs(b) > n:
59 b = remainder(b, n2)
60 return float0_(r, b)
63def antipode(lat, lon, name=NN):
64 '''Return the antipode, the point diametrically opposite
65 to a given point in C{degrees}.
67 @arg lat: Latitude (C{degrees}).
68 @arg lon: Longitude (C{degrees}).
69 @kwarg name: Optional name (C{str}).
71 @return: A L{LatLon2Tuple}C{(lat, lon)}.
73 @see: Functions L{antipode_} and L{normal} and U{Geosphere
74 <https://CRAN.R-Project.org/web/packages/geosphere/geosphere.pdf>}.
75 '''
76 return LatLon2Tuple(*_anti2(lat, lon, _90_0, _180_0, _360_0), name=name)
79def antipode_(phi, lam, name=NN):
80 '''Return the antipode, the point diametrically opposite
81 to a given point in C{radians}.
83 @arg phi: Latitude (C{radians}).
84 @arg lam: Longitude (C{radians}).
85 @kwarg name: Optional name (C{str}).
87 @return: A L{PhiLam2Tuple}C{(phi, lam)}.
89 @see: Functions L{antipode} and L{normal_} and U{Geosphere
90 <https://CRAN.R-Project.org/web/packages/geosphere/geosphere.pdf>}.
91 '''
92 return PhiLam2Tuple(*_anti2(phi, lam, PI_2, PI, PI2), name=name)
95def bearing(lat1, lon1, lat2, lon2, **final_wrap):
96 '''Compute the initial or final bearing (forward or reverse
97 azimuth) between a (spherical) start and end point.
99 @arg lat1: Start latitude (C{degrees}).
100 @arg lon1: Start longitude (C{degrees}).
101 @arg lat2: End latitude (C{degrees}).
102 @arg lon2: End longitude (C{degrees}).
103 @kwarg final_wrap: Optional keyword arguments for function
104 L{pygeodesy.bearing_}.
106 @return: Initial or final bearing (compass C{degrees360}) or
107 zero if start and end point coincide.
108 '''
109 r = bearing_(Phi_(lat1=lat1), Lam_(lon1=lon1),
110 Phi_(lat2=lat2), Lam_(lon2=lon2), **final_wrap)
111 return degrees(r)
114def bearing_(phi1, lam1, phi2, lam2, final=False, wrap=False):
115 '''Compute the initial or final bearing (forward or reverse azimuth)
116 between a (spherical) start and end point.
118 @arg phi1: Start latitude (C{radians}).
119 @arg lam1: Start longitude (C{radians}).
120 @arg phi2: End latitude (C{radians}).
121 @arg lam2: End longitude (C{radians}).
122 @kwarg final: Return final bearing if C{True}, initial otherwise (C{bool}).
123 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{phi2}} and
124 B{C{lam2}} (C{bool}).
126 @return: Initial or final bearing (compass C{radiansPI2}) or zero if start
127 and end point coincide.
129 @see: U{Bearing<https://www.Movable-Type.co.UK/scripts/latlong.html>}, U{Course
130 between two points<https://www.EdWilliams.org/avform147.htm#Crs>} and
131 U{Bearing Between Two Points<https://web.Archive.org/web/20020630205931/
132 https://MathForum.org/library/drmath/view/55417.html>}.
133 '''
134 db, phi2, lam2 = _Wrap.philam3(lam1, phi2, lam2, wrap)
135 if final: # swap plus PI
136 phi1, lam1, phi2, lam2, db = phi2, lam2, phi1, lam1, -db
137 r = PI3
138 else:
139 r = PI2
140 sa1, ca1, sa2, ca2, sdb, cdb = sincos2_(phi1, phi2, db)
142 x = ca1 * sa2 - sa1 * ca2 * cdb
143 y = sdb * ca2
144 return _umod_PI2(atan2(y, x) + r) # .utily.wrapPI2
147def _bearingTo2(p1, p2, wrap=False): # for points.ispolar, sphericalTrigonometry.areaOf
148 '''(INTERNAL) Compute initial and final bearing.
149 '''
150 try: # for LatLon_ and ellipsoidal LatLon
151 return p1.bearingTo2(p2, wrap=wrap)
152 except AttributeError:
153 pass
154 # XXX spherical version, OK for ellipsoidal ispolar?
155 a1, b1 = p1.philam
156 a2, b2 = p2.philam
157 return Bearing2Tuple(degrees(bearing_(a1, b1, a2, b2, final=False, wrap=wrap)),
158 degrees(bearing_(a1, b1, a2, b2, final=True, wrap=wrap)),
159 name=_bearingTo2.__name__)
162def compassAngle(lat1, lon1, lat2, lon2, adjust=True, wrap=False):
163 '''Return the angle from North for the direction vector M{(lon2 - lon1,
164 lat2 - lat1)} between two points.
166 Suitable only for short, not near-polar vectors up to a few hundred
167 Km or Miles. Use function L{pygeodesy.bearing} for longer vectors.
169 @arg lat1: From latitude (C{degrees}).
170 @arg lon1: From longitude (C{degrees}).
171 @arg lat2: To latitude (C{degrees}).
172 @arg lon2: To longitude (C{degrees}).
173 @kwarg adjust: Adjust the longitudinal delta by the cosine of the
174 mean latitude (C{bool}).
175 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
176 and B{C{lon2}} (C{bool}).
178 @return: Compass angle from North (C{degrees360}).
180 @note: Courtesy of Martin Schultz.
182 @see: U{Local, flat earth approximation
183 <https://www.EdWilliams.org/avform.htm#flat>}.
184 '''
185 d_lon, lat2, lon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap)
186 if adjust: # scale delta lon
187 d_lon *= _scale_deg(lat1, lat2)
188 return atan2b(d_lon, lat2 - lat1)
191def cosineAndoyerLambert(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):
192 '''Compute the distance between two (ellipsoidal) points using the U{Andoyer-Lambert
193 <https://books.google.com/books?id=x2UiAQAAIAAJ>} correction of the U{Law of
194 Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula.
196 @arg lat1: Start latitude (C{degrees}).
197 @arg lon1: Start longitude (C{degrees}).
198 @arg lat2: End latitude (C{degrees}).
199 @arg lon2: End longitude (C{degrees}).
200 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
201 L{Ellipsoid2} or L{a_f2Tuple}) to use.
202 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
203 B{C{lat2}} and B{C{lon2}} (C{bool}).
205 @return: Distance (C{meter}, same units as the B{C{datum}}'s
206 ellipsoid axes or C{radians} if B{C{datum}} is C{None}).
208 @raise TypeError: Invalid B{C{datum}}.
210 @see: Functions L{cosineAndoyerLambert_}, L{cosineForsytheAndoyerLambert},
211 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny},
212 L{flatPolar}, L{haversine}, L{thomas} and L{vincentys} and method
213 L{Ellipsoid.distance2}.
214 '''
215 return _dE(cosineAndoyerLambert_, datum, wrap, lat1, lon1, lat2, lon2)
218def cosineAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84):
219 '''Compute the I{angular} distance between two (ellipsoidal) points using the U{Andoyer-Lambert
220 <https://books.google.com/books?id=x2UiAQAAIAAJ>} correction of the U{Law of
221 Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula.
223 @arg phi2: End latitude (C{radians}).
224 @arg phi1: Start latitude (C{radians}).
225 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
226 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
227 L{Ellipsoid2} or L{a_f2Tuple}) to use.
229 @return: Angular distance (C{radians}).
231 @raise TypeError: Invalid B{C{datum}}.
233 @see: Functions L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert_},
234 L{cosineLaw_}, L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
235 L{flatPolar_}, L{haversine_}, L{thomas_} and L{vincentys_} and U{Geodesy-PHP
236 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/Distance/
237 AndoyerLambert.php>}.
238 '''
239 s2, c2, s1, c1, r, c21 = _sincosa6(phi2, phi1, lam21)
240 if isnon0(c1) and isnon0(c2):
241 E = _ellipsoidal(datum, cosineAndoyerLambert_)
242 if E.f: # ellipsoidal
243 r2 = atan2(E.b_a * s2, c2)
244 r1 = atan2(E.b_a * s1, c1)
245 s2, c2, s1, c1 = sincos2_(r2, r1)
246 r = acos1(s1 * s2 + c1 * c2 * c21)
247 if r:
248 sr, _, sr_2, cr_2 = sincos2_(r, r * _0_5)
249 if isnon0(sr_2) and isnon0(cr_2):
250 s = (sr + r) * ((s1 - s2) / sr_2)**2
251 c = (sr - r) * ((s1 + s2) / cr_2)**2
252 r += (c - s) * E.f * _0_125
253 return r
256def cosineForsytheAndoyerLambert(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):
257 '''Compute the distance between two (ellipsoidal) points using the U{Forsythe-Andoyer-Lambert
258 <https://www2.UNB.Ca/gge/Pubs/TR77.pdf>} correction of the U{Law of Cosines
259 <https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula.
261 @arg lat1: Start latitude (C{degrees}).
262 @arg lon1: Start longitude (C{degrees}).
263 @arg lat2: End latitude (C{degrees}).
264 @arg lon2: End longitude (C{degrees}).
265 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
266 L{Ellipsoid2} or L{a_f2Tuple}) to use.
267 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
268 B{C{lat2}} and B{C{lon2}} (C{bool}).
270 @return: Distance (C{meter}, same units as the B{C{datum}}'s
271 ellipsoid axes or C{radians} if B{C{datum}} is C{None}).
273 @raise TypeError: Invalid B{C{datum}}.
275 @see: Functions L{cosineForsytheAndoyerLambert_}, L{cosineAndoyerLambert},
276 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny},
277 L{flatPolar}, L{haversine}, L{thomas} and L{vincentys} and method
278 L{Ellipsoid.distance2}.
279 '''
280 return _dE(cosineForsytheAndoyerLambert_, datum, wrap, lat1, lon1, lat2, lon2)
283def cosineForsytheAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84):
284 '''Compute the I{angular} distance between two (ellipsoidal) points using the
285 U{Forsythe-Andoyer-Lambert<https://www2.UNB.Ca/gge/Pubs/TR77.pdf>} correction of
286 the U{Law of Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>}
287 formula.
289 @arg phi2: End latitude (C{radians}).
290 @arg phi1: Start latitude (C{radians}).
291 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
292 @kwarg datum: Datum (L{Datum}) or ellipsoid to use (L{Ellipsoid},
293 L{Ellipsoid2} or L{a_f2Tuple}).
295 @return: Angular distance (C{radians}).
297 @raise TypeError: Invalid B{C{datum}}.
299 @see: Functions L{cosineForsytheAndoyerLambert}, L{cosineAndoyerLambert_},
300 L{cosineLaw_}, L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
301 L{flatPolar_}, L{haversine_}, L{thomas_} and L{vincentys_} and U{Geodesy-PHP
302 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/
303 Distance/ForsytheCorrection.php>}.
304 '''
305 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21)
306 if r and isnon0(c1) and isnon0(c2):
307 E = _ellipsoidal(datum, cosineForsytheAndoyerLambert_)
308 if E.f: # ellipsoidal
309 sr, cr, s2r, _ = sincos2_(r, r * 2)
310 if isnon0(sr) and fabs(cr) < EPS1:
311 s = (s1 + s2)**2 / (1 + cr)
312 t = (s1 - s2)**2 / (1 - cr)
313 x = s + t
314 y = s - t
316 s = 8 * r**2 / sr
317 a = 64 * r + s * cr * 2 # 16 * r**2 / tan(r)
318 d = 48 * sr + s # 8 * r**2 / tan(r)
319 b = -2 * d
320 e = 30 * s2r
321 c = fsumf_(30 * r, e * _0_5, s * cr) # 8 * r**2 / tan(r)
322 t = fsumf_( a * x, e * y**2, b * y, -c * x**2, d * x * y)
324 r += fsumf_(-r * x, 3 * y * sr, t * E.f / _32_0) * E.f * _0_25
325 return r
328def cosineLaw(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
329 '''Compute the distance between two points using the U{spherical Law of Cosines
330 <https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula.
332 @arg lat1: Start latitude (C{degrees}).
333 @arg lon1: Start longitude (C{degrees}).
334 @arg lat2: End latitude (C{degrees}).
335 @arg lon2: End longitude (C{degrees}).
336 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
337 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
338 L{a_f2Tuple}) to use.
339 @kwarg wrap: If C{True}, wrap or I{normalize} and B{C{lat2}}
340 and B{C{lon2}} (C{bool}).
342 @return: Distance (C{meter}, same units as B{C{radius}} or the
343 ellipsoid or datum axes).
345 @raise TypeError: Invalid B{C{radius}}.
347 @see: Functions L{cosineLaw_}, L{cosineAndoyerLambert},
348 L{cosineForsytheAndoyerLambert}, L{equirectangular}, L{euclidean},
349 L{flatLocal}/L{hubeny}, L{flatPolar}, L{haversine}, L{thomas} and
350 L{vincentys} and method L{Ellipsoid.distance2}.
352 @note: See note at function L{vincentys_}.
353 '''
354 return _dS(cosineLaw_, radius, wrap, lat1, lon1, lat2, lon2)
357def cosineLaw_(phi2, phi1, lam21):
358 '''Compute the I{angular} distance between two points using the U{spherical Law of
359 Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula.
361 @arg phi2: End latitude (C{radians}).
362 @arg phi1: Start latitude (C{radians}).
363 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
365 @return: Angular distance (C{radians}).
367 @see: Functions L{cosineLaw}, L{cosineAndoyerLambert_},
368 L{cosineForsytheAndoyerLambert_}, L{equirectangular_},
369 L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_},
370 L{haversine_}, L{thomas_} and L{vincentys_}.
372 @note: See note at function L{vincentys_}.
373 '''
374 return _sincosa6(phi2, phi1, lam21)[4]
377def _d3(wrap, lat1, lon1, lat2, lon2):
378 '''(INTERNAL) Helper for _dE, _dS and _eA.
379 '''
380 if wrap:
381 d_lon, lat2, _ = _Wrap.latlon3(lon1, lat2, lon2, wrap)
382 return radians(lat2), Phi_(lat1=lat1), radians(d_lon)
383 else: # for backward compaibility
384 return Phi_(lat2=lat2), Phi_(lat1=lat1), Phi_(d_lon=lon2 - lon1)
387def _dE(func_, earth, *wrap_lls):
388 '''(INTERNAL) Helper for ellipsoidal distances.
389 '''
390 E = _ellipsoidal(earth, func_)
391 r = func_(*_d3(*wrap_lls), datum=E)
392 return r * E.a
395def _dS(func_, radius, *wrap_lls, **adjust):
396 '''(INTERNAL) Helper for spherical distances.
397 '''
398 r = func_(*_d3(*wrap_lls), **adjust)
399 if radius is not R_M:
400 _, lat1, _, lat2, _ = wrap_lls
401 radius = _mean_radius(radius, lat1, lat2)
402 return r * radius
405def _eA(excess_, radius, *wrap_lls):
406 '''(INTERNAL) Helper for spherical excess or area.
407 '''
408 r = excess_(*_d3(*wrap_lls))
409 if radius:
410 _, lat1, _, lat2, _ = wrap_lls
411 r *= _mean_radius(radius, lat1, lat2)**2
412 return r
415def _ellipsoidal(earth, where):
416 '''(INTERNAL) Helper for distances.
417 '''
418 return _EWGS84 if earth in (_WGS84, _EWGS84) else (
419 earth if isinstance(earth, Ellipsoid) else
420 (earth if isinstance(earth, Datum) else # PYCHOK indent
421 _ellipsoidal_datum(earth, name=where.__name__)).ellipsoid)
424def equirectangular(lat1, lon1, lat2, lon2, radius=R_M, **adjust_limit_wrap):
425 '''Compute the distance between two points using the U{Equirectangular Approximation
426 / Projection<https://www.Movable-Type.co.UK/scripts/latlong.html#equirectangular>}.
428 @arg lat1: Start latitude (C{degrees}).
429 @arg lon1: Start longitude (C{degrees}).
430 @arg lat2: End latitude (C{degrees}).
431 @arg lon2: End longitude (C{degrees}).
432 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
433 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
434 L{a_f2Tuple}).
435 @kwarg adjust_limit_wrap: Optional keyword arguments for
436 function L{equirectangular_}.
438 @return: Distance (C{meter}, same units as B{C{radius}} or
439 the ellipsoid or datum axes).
441 @raise TypeError: Invalid B{C{radius}}.
443 @see: Function L{equirectangular_} for more details, the
444 available B{C{options}}, errors, restrictions and other,
445 approximate or accurate distance functions.
446 '''
447 d = sqrt(equirectangular_(Lat(lat1=lat1), Lon(lon1=lon1),
448 Lat(lat2=lat2), Lon(lon2=lon2),
449 **adjust_limit_wrap).distance2) # PYCHOK 4 vs 2-3
450 return degrees2m(d, radius=_mean_radius(radius, lat1, lat2))
453def _equirectangular(lat1, lon1, lat2, lon2, **adjust_limit_wrap):
454 '''(INTERNAL) Helper for the L{frechet._FrecherMeterRadians}
455 and L{hausdorff._HausdorffMeterRedians} classes.
456 '''
457 return equirectangular_(lat1, lon1, lat2, lon2, **adjust_limit_wrap).distance2 * _D2_R2
460def equirectangular_(lat1, lon1, lat2, lon2, adjust=True, limit=45, wrap=False):
461 '''Compute the distance between two points using the U{Equirectangular Approximation
462 / Projection<https://www.Movable-Type.co.UK/scripts/latlong.html#equirectangular>}.
464 This approximation is valid for short distance of several hundred Km
465 or Miles, see the B{C{limit}} keyword argument and L{LimitError}.
467 @arg lat1: Start latitude (C{degrees}).
468 @arg lon1: Start longitude (C{degrees}).
469 @arg lat2: End latitude (C{degrees}).
470 @arg lon2: End longitude (C{degrees}).
471 @kwarg adjust: Adjust the wrapped, unrolled longitudinal delta
472 by the cosine of the mean latitude (C{bool}).
473 @kwarg limit: Optional limit for lat- and longitudinal deltas
474 (C{degrees}) or C{None} or C{0} for unlimited.
475 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
476 and B{C{lon2}} (C{bool}).
478 @return: A L{Distance4Tuple}C{(distance2, delta_lat, delta_lon,
479 unroll_lon2)} in C{degrees squared}.
481 @raise LimitError: If the lat- and/or longitudinal delta exceeds the
482 B{C{-limit..limit}} range and L{pygeodesy.limiterrors}
483 set to C{True}.
485 @see: U{Local, flat earth approximation
486 <https://www.EdWilliams.org/avform.htm#flat>}, functions
487 L{equirectangular}, L{cosineAndoyerLambert},
488 L{cosineForsytheAndoyerLambert}, L{cosineLaw}, L{euclidean},
489 L{flatLocal}/L{hubeny}, L{flatPolar}, L{haversine}, L{thomas}
490 and L{vincentys} and methods L{Ellipsoid.distance2},
491 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
492 '''
493 d_lon, lat2, ulon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap)
494 d_lat = lat2 - lat1
496 if limit and limit > 0 and limiterrors():
497 d = max(fabs(d_lat), fabs(d_lon))
498 if d > limit:
499 t = _SPACE_(_delta_, Fmt.PAREN_g(d), Fmt.exceeds_limit(limit))
500 s = unstr(equirectangular_, lat1, lon1, lat2, lon2,
501 limit=limit, wrap=wrap)
502 raise LimitError(s, txt=t)
504 if adjust: # scale delta lon
505 d_lon *= _scale_deg(lat1, lat2)
507 d2 = hypot2(d_lat, d_lon) # degrees squared!
508 return Distance4Tuple(d2, d_lat, d_lon, ulon2 - lon2)
511def euclidean(lat1, lon1, lat2, lon2, radius=R_M, adjust=True, wrap=False):
512 '''Approximate the C{Euclidean} distance between two (spherical) points.
514 @arg lat1: Start latitude (C{degrees}).
515 @arg lon1: Start longitude (C{degrees}).
516 @arg lat2: End latitude (C{degrees}).
517 @arg lon2: End longitude (C{degrees}).
518 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
519 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
520 L{a_f2Tuple}) to use.
521 @kwarg adjust: Adjust the longitudinal delta by the cosine of
522 the mean latitude (C{bool}).
523 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
524 and B{C{lon2}} (C{bool}).
526 @return: Distance (C{meter}, same units as B{C{radius}} or the
527 ellipsoid or datum axes).
529 @raise TypeError: Invalid B{C{radius}}.
531 @see: U{Distance between two (spherical) points
532 <https://www.EdWilliams.org/avform.htm#Dist>}, functions L{euclid},
533 L{euclidean_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
534 L{cosineLaw}, L{equirectangular}, L{flatLocal}/L{hubeny}, L{flatPolar},
535 L{haversine}, L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2},
536 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
537 '''
538 return _dS(euclidean_, radius, wrap, lat1, lon1, lat2, lon2, adjust=adjust)
541def euclidean_(phi2, phi1, lam21, adjust=True):
542 '''Approximate the I{angular} C{Euclidean} distance between two (spherical) points.
544 @arg phi2: End latitude (C{radians}).
545 @arg phi1: Start latitude (C{radians}).
546 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
547 @kwarg adjust: Adjust the longitudinal delta by the cosine
548 of the mean latitude (C{bool}).
550 @return: Angular distance (C{radians}).
552 @see: Functions L{euclid}, L{euclidean}, L{cosineAndoyerLambert_},
553 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{equirectangular_},
554 L{flatLocal_}/L{hubeny_}, L{flatPolar_}, L{haversine_}, L{thomas_}
555 and L{vincentys_}.
556 '''
557 if adjust:
558 lam21 *= _scale_rad(phi2, phi1)
559 return euclid(phi2 - phi1, lam21)
562def excessAbc_(A, b, c):
563 '''Compute the I{spherical excess} C{E} of a (spherical) triangle from two sides
564 and the included (small) angle.
566 @arg A: An interior triangle angle (C{radians}).
567 @arg b: Frist adjacent triangle side (C{radians}).
568 @arg c: Second adjacent triangle side (C{radians}).
570 @return: Spherical excess (C{radians}).
572 @raise UnitError: Invalid B{C{A}}, B{C{b}} or B{C{c}}.
574 @see: Functions L{excessGirard_}, L{excessLHuilier_} and U{Spherical
575 trigonometry<https://WikiPedia.org/wiki/Spherical_trigonometry>}.
576 '''
577 A = Radians_(A=A)
578 b = Radians_(b=b) * _0_5
579 c = Radians_(c=c) * _0_5
581 sA, cA, sb, cb, sc, cc = sincos2_(A, b, c)
582 return atan2(sA * sb * sc, cb * cc + cA * sb * sc) * _2_0
585def excessCagnoli_(a, b, c):
586 '''Compute the I{spherical excess} C{E} of a (spherical) triangle using U{Cagnoli's
587 <https://Zenodo.org/record/35392>} (D.34) formula.
589 @arg a: First triangle side (C{radians}).
590 @arg b: Second triangle side (C{radians}).
591 @arg c: Third triangle side (C{radians}).
593 @return: Spherical excess (C{radians}).
595 @raise UnitError: Invalid B{C{a}}, B{C{b}} or B{C{c}}.
597 @see: Function L{excessLHuilier_} and U{Spherical trigonometry
598 <https://WikiPedia.org/wiki/Spherical_trigonometry>}.
599 '''
600 a = Radians_(a=a)
601 b = Radians_(b=b)
602 c = Radians_(c=c)
604 s = fsumf_(a, b, c) * _0_5
605 r = sin(s) * sin(s - a) * sin(s - b) * sin(s - c)
606 c = cos(a * _0_5) * cos(b * _0_5) * cos(c * _0_5)
607 r = asin(sqrt(r) * _0_5 / c) if c and r > 0 else _0_0
608 return Radians(Cagnoli=r * _2_0)
611def excessGirard_(A, B, C):
612 '''Compute the I{spherical excess} C{E} of a (spherical) triangle using U{Girard's
613 <https://MathWorld.Wolfram.com/GirardsSphericalExcessFormula.html>} formula.
615 @arg A: First interior triangle angle (C{radians}).
616 @arg B: Second interior triangle angle (C{radians}).
617 @arg C: Third interior triangle angle (C{radians}).
619 @return: Spherical excess (C{radians}).
621 @raise UnitError: Invalid B{C{A}}, B{C{B}} or B{C{C}}.
623 @see: Function L{excessLHuilier_} and U{Spherical trigonometry
624 <https://WikiPedia.org/wiki/Spherical_trigonometry>}.
625 '''
626 return Radians(Girard=fsumf_(Radians_(A=A),
627 Radians_(B=B),
628 Radians_(C=C), -PI))
631def excessLHuilier_(a, b, c):
632 '''Compute the I{spherical excess} C{E} of a (spherical) triangle using U{L'Huilier's
633 <https://MathWorld.Wolfram.com/LHuiliersTheorem.html>} Theorem.
635 @arg a: First triangle side (C{radians}).
636 @arg b: Second triangle side (C{radians}).
637 @arg c: Third triangle side (C{radians}).
639 @return: Spherical excess (C{radians}).
641 @raise UnitError: Invalid B{C{a}}, B{C{b}} or B{C{c}}.
643 @see: Function L{excessCagnoli_}, L{excessGirard_} and U{Spherical
644 trigonometry<https://WikiPedia.org/wiki/Spherical_trigonometry>}.
645 '''
646 a = Radians_(a=a)
647 b = Radians_(b=b)
648 c = Radians_(c=c)
650 s = fsumf_(a, b, c) * _0_5
651 r = tan_2(s) * tan_2(s - a) * tan_2(s - b) * tan_2(s - c)
652 r = atan(sqrt(r)) if r > 0 else _0_0
653 return Radians(LHuilier=r * _4_0)
656def excessKarney(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
657 '''Compute the surface area of a (spherical) quadrilateral bounded by a
658 segment of a great circle, two meridians and the equator using U{Karney's
659 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>}
660 method.
662 @arg lat1: Start latitude (C{degrees}).
663 @arg lon1: Start longitude (C{degrees}).
664 @arg lat2: End latitude (C{degrees}).
665 @arg lon2: End longitude (C{degrees}).
666 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
667 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
668 L{a_f2Tuple}) or C{None}.
669 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
670 B{C{lat2}} and B{C{lon2}} (C{bool}).
672 @return: Surface area, I{signed} (I{square} C{meter} or the same units as
673 B{C{radius}} I{squared}) or the I{spherical excess} (C{radians})
674 if C{B{radius}=0} or C{None}.
676 @raise TypeError: Invalid B{C{radius}}.
678 @raise UnitError: Invalid B{C{lat2}} or B{C{lat1}}.
680 @raise ValueError: Semi-circular longitudinal delta.
682 @see: Functions L{excessKarney_} and L{excessQuad}.
683 '''
684 return _eA(excessKarney_, radius, wrap, lat1, lon1, lat2, lon2)
687def excessKarney_(phi2, phi1, lam21):
688 '''Compute the I{spherical excess} C{E} of a (spherical) quadrilateral bounded by
689 a segment of a great circle, two meridians and the equator using U{Karney's
690 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>}
691 method.
693 @arg phi2: End latitude (C{radians}).
694 @arg phi1: Start latitude (C{radians}).
695 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
697 @return: Spherical excess, I{signed} (C{radians}).
699 @raise ValueError: Semi-circular longitudinal delta B{C{lam21}}.
701 @see: Function L{excessKarney} and U{Area of a spherical polygon
702 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>}.
703 '''
704 # from: Veness <https://www.Movable-Type.co.UK/scripts/latlong.html> Area
705 # method due to Karney: for each edge of the polygon,
706 #
707 # tan(Δλ / 2) · (tan(φ1 / 2) + tan(φ2 / 2))
708 # tan(E / 2) = -----------------------------------------
709 # 1 + tan(φ1 / 2) · tan(φ2 / 2)
710 #
711 # where E is the spherical excess of the trapezium obtained by extending
712 # the edge to the equator-circle vector for each edge (see also ***).
713 t2 = tan_2(phi2)
714 t1 = tan_2(phi1)
715 t = tan_2(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{equirectangular_}, L{euclidean_}, L{haversine_}, L{thomas_}
866 and L{vincentys_} and 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{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
920 L{haversine_}, 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(inst, los, earth, **kwds):
937 '''(INTERNAL) Helper for C{CartesianBase.hartzell} and C{LatLonBase.hartzell}.
938 '''
939 if earth is not None:
940 earth = _spherical_datum(earth, name=hartzell.__name__)
941 inst = inst.toDatum(earth)
942 h = inst.height
943 if h > 0: # EPS0
944 r = hartzell(inst, los=los, earth=earth or inst.datum, **kwds)
945 elif h < 0: # EPS0
946 raise IntersectionError(pov=inst, los=los, height=h, txt=_inside_)
947 else:
948 r = inst
949 return r
952def hartzell(pov, los=None, earth=_WGS84, name=NN, **LatLon_and_kwds):
953 '''Compute the intersection of the earth's surface and a Line-Of-Sight
954 from a Point-Of-View in space.
956 @arg pov: Point-Of-View outside the earth (C{Cartesian}, L{Ecef9Tuple}
957 C{LatLon} or L{Vector3d}).
958 @kwarg los: Line-Of-Sight, I{direction} to earth (L{Los}, L{Vector3d})
959 or C{None} to point to the earth' center.
960 @kwarg earth: The earth model (L{Datum}, L{Ellipsoid}, L{Ellipsoid2},
961 L{a_f2Tuple} or C{scalar} radius in C{meter}).
962 @kwarg name: Optional name (C{str}).
963 @kwarg LatLon_and_kwds: Optional C{LatLon} class for the intersection
964 point plus C{LatLon} keyword arguments, include
965 B{C{datum}} if different from B{C{earth}}.
967 @return: The intersection point (L{Vector3d}, the C{Cartesian type} of
968 B{C{pov}} or the given C{B{LatLon}_and_kwds}) with C{.heigth}
969 set to the distance to the B{C{pov}}.
971 @raise IntersectionError: Null B{C{pov}} or B{C{los}} vector, B{C{pov}}
972 is inside the earth or B{C{los}} points outside
973 the earth or points in an opposite direction.
975 @raise TypeError: Invalid B{C{pov}}, B{C{los}} or B{C{earth}}.
977 @see: Function L{pygeodesy.hartzell4}, L{pygeodesy.tyr3d} for B{C{los}},
978 methods L{Ellipsoid.hartzell4}, C{Cartesian.hartzell}, C{LatLon.hartzell}
979 and U{I{Satellite Line-of-Sight Intersection with Earth}<https://
980 StephenHartzell.Medium.com/satellite-line-of-sight-intersection-with-earth-d786b4a6a9b6>}.
981 '''
982 n = hartzell.__name__
983 D = earth if isinstance(earth, Datum) else \
984 _spherical_datum(earth, name=n)
985 try:
986 r, h = _MODS.triaxials._hartzell3d2(pov, los, D.ellipsoid._triaxial)
987 except Exception as x:
988 raise IntersectionError(pov=pov, los=los, earth=earth, cause=x)
990 r = _xnamed(r, name or n)
991 C = _MODS.cartesianBase.CartesianBase
992 if LatLon_and_kwds:
993 c = C(r, datum=D, name=r.name)
994 r = c.toLatLon(**_xkwds(LatLon_and_kwds, height=h))
995 elif isinstance(r, C):
996 r.height = h
997 return r
1000def haversine(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
1001 '''Compute the distance between two (spherical) points using the
1002 U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>}
1003 formula.
1005 @arg lat1: Start latitude (C{degrees}).
1006 @arg lon1: Start longitude (C{degrees}).
1007 @arg lat2: End latitude (C{degrees}).
1008 @arg lon2: End longitude (C{degrees}).
1009 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
1010 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
1011 L{a_f2Tuple}) to use.
1012 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
1013 B{C{lat2}} and B{C{lon2}} (C{bool}).
1015 @return: Distance (C{meter}, same units as B{C{radius}}).
1017 @raise TypeError: Invalid B{C{radius}}.
1019 @see: U{Distance between two (spherical) points
1020 <https://www.EdWilliams.org/avform.htm#Dist>}, functions
1021 L{cosineLaw}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
1022 L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar},
1023 L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2},
1024 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
1026 @note: See note at function L{vincentys_}.
1027 '''
1028 return _dS(haversine_, radius, wrap, lat1, lon1, lat2, lon2)
1031def haversine_(phi2, phi1, lam21):
1032 '''Compute the I{angular} distance between two (spherical) points
1033 using the U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>}
1034 formula.
1036 @arg phi2: End latitude (C{radians}).
1037 @arg phi1: Start latitude (C{radians}).
1038 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
1040 @return: Angular distance (C{radians}).
1042 @see: Functions L{haversine}, L{cosineAndoyerLambert_},
1043 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
1044 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
1045 L{flatPolar_}, L{thomas_} and L{vincentys_}.
1047 @note: See note at function L{vincentys_}.
1048 '''
1049 def _hsin(rad):
1050 return sin(rad * _0_5)**2
1052 h = _hsin(phi2 - phi1) + cos(phi1) * cos(phi2) * _hsin(lam21) # haversine
1053 return atan2(sqrt0(h), sqrt0(_1_0 - h)) * _2_0 # == asin(sqrt(h)) * 2
1056def heightOf(angle, distance, radius=R_M):
1057 '''Determine the height above the (spherical) earth' surface after
1058 traveling along a straight line at a given tilt.
1060 @arg angle: Tilt angle above horizontal (C{degrees}).
1061 @arg distance: Distance along the line (C{meter} or same units as
1062 B{C{radius}}).
1063 @kwarg radius: Optional mean earth radius (C{meter}).
1065 @return: Height (C{meter}, same units as B{C{distance}} and B{C{radius}}).
1067 @raise ValueError: Invalid B{C{angle}}, B{C{distance}} or B{C{radius}}.
1069 @see: U{MultiDop geog_lib.GeogBeamHt<https://GitHub.com/NASA/MultiDop>}
1070 (U{Shapiro et al. 2009, JTECH
1071 <https://Journals.AMetSoc.org/doi/abs/10.1175/2009JTECHA1256.1>}
1072 and U{Potvin et al. 2012, JTECH
1073 <https://Journals.AMetSoc.org/doi/abs/10.1175/JTECH-D-11-00019.1>}).
1074 '''
1075 r = h = Radius(radius)
1076 d = fabs(Distance(distance))
1077 if d > h:
1078 d, h = h, d
1080 if d > EPS0: # and h > EPS0
1081 d = d / h # /= h chokes PyChecker
1082 s = sin(Phi_(angle=angle, clip=_180_0))
1083 s = fsumf_(_1_0, _2_0 * s * d, d**2)
1084 if s > 0:
1085 return h * sqrt(s) - r
1087 raise _ValueError(angle=angle, distance=distance, radius=radius)
1090def heightOrthometric(h_ll, N):
1091 '''Get the I{orthometric} height B{H}, the height above the geoid, earth surface.
1093 @arg h_ll: The height above the ellipsoid (C{meter}) or an I{ellipsoidal}
1094 location (C{LatLon} with a C{height} or C{h} attribute).
1095 @arg N: The I{geoid} height (C{meter}), the height of the geoid above the
1096 ellipsoid at the same B{C{h_ll}} location.
1098 @return: I{Orthometric} height C{B{H} = B{h} - B{N}} (C{meter}, same units
1099 as B{C{h}} and B{C{N}}).
1101 @see: U{Ellipsoid, Geoid, and Othometric Heights<https://www.NGS.NOAA.gov/
1102 GEOID/PRESENTATIONS/2007_02_24_CCPS/Roman_A_PLSC2007notes.pdf>}, page
1103 6 and module L{pygeodesy.geoids}.
1104 '''
1105 h = h_ll if isscalar(h_ll) else _xattr(h_ll, height=_xattr(h_ll, h=0))
1106 return Height(H=Height(h=h) - Height(N=N))
1109def horizon(height, radius=R_M, refraction=False):
1110 '''Determine the distance to the horizon from a given altitude
1111 above the (spherical) earth.
1113 @arg height: Altitude (C{meter} or same units as B{C{radius}}).
1114 @kwarg radius: Optional mean earth radius (C{meter}).
1115 @kwarg refraction: Consider atmospheric refraction (C{bool}).
1117 @return: Distance (C{meter}, same units as B{C{height}} and B{C{radius}}).
1119 @raise ValueError: Invalid B{C{height}} or B{C{radius}}.
1121 @see: U{Distance to horizon<https://www.EdWilliams.org/avform.htm#Horizon>}.
1122 '''
1123 h, r = Height(height), Radius(radius)
1124 if min(h, r) < 0:
1125 raise _ValueError(height=height, radius=radius)
1127 if refraction:
1128 d2 = 2.415750694528 * h * r # 2.0 / 0.8279
1129 else:
1130 d2 = h * fsumf_(r, r, h)
1131 return sqrt0(d2)
1134class _idllmn6(object): # see also .geodesicw._wargs, .vector2d._numpy
1135 '''(INTERNAL) Helper for C{intersection2} and C{intersections2}.
1136 '''
1137 @contextmanager # <https://www.python.org/dev/peps/pep-0343/> Examples
1138 def __call__(self, datum, lat1, lon1, lat2, lon2, small, wrap, s, **kwds):
1139 try:
1140 if wrap:
1141 _, lat2, lon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap)
1142 kwds = _xkwds(kwds, wrap=wrap) # for _xError
1143 m = small if small is _100km else Meter_(small=small)
1144 n = (intersections2 if s else intersection2).__name__
1145 if datum is None or euclidean(lat1, lon1, lat2, lon2) < m:
1146 d, m = None, _MODS.vector3d
1147 _i = m._intersects2 if s else m._intersect3d3
1148 elif isscalar(datum) and datum < 0 and not s:
1149 d = _spherical_datum(-datum, name=n)
1150 m = _MODS.sphericalNvector
1151 _i = m.intersection
1152 else:
1153 d = _spherical_datum(datum, name=n)
1154 if d.isSpherical:
1155 m = _MODS.sphericalTrigonometry
1156 _i = m._intersects2 if s else m._intersect
1157 elif d.isEllipsoidal:
1158 try:
1159 if d.ellipsoid.geodesic:
1160 pass
1161 m = _MODS.ellipsoidalKarney
1162 except ImportError:
1163 m = _MODS.ellipsoidalExact
1164 _i = m._intersections2 if s else m._intersection3 # ellispoidalBaseDI
1165 else:
1166 raise _TypeError(datum=datum)
1167 yield _i, d, lat2, lon2, m, n
1169 except (TypeError, ValueError) as x:
1170 raise _xError(x, lat1=lat1, lon1=lon1, datum=datum,
1171 lat2=lat2, lon2=lon2, small=small, **kwds)
1173_idllmn6 = _idllmn6() # PYCHOK singleton
1176def intersection2(lat1, lon1, bearing1,
1177 lat2, lon2, bearing2, datum=None, wrap=False, small=_100km): # was=True
1178 '''I{Conveniently} compute the intersection of two lines each defined
1179 by a (geodetic) point and a bearing from North, using either ...
1181 1) L{vector3d.intersection3d3} for B{C{small}} distances (below 100 Km
1182 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ...
1184 2) L{sphericalTrigonometry.intersection} for a spherical B{C{datum}}
1185 or a C{scalar B{datum}} representing the earth radius, conventionally
1186 in C{meter} or ...
1188 3) L{sphericalNvector.intersection} if B{C{datum}} is a I{negative}
1189 C{scalar}, (negative) earth radius, conventionally in C{meter} or ...
1191 4) L{ellipsoidalKarney.intersection3} for an ellipsoidal B{C{datum}}
1192 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>}
1193 is installed, otherwise ...
1195 5) L{ellipsoidalExact.intersection3}, provided B{C{datum}} is ellipsoidal.
1197 @arg lat1: Latitude of the first point (C{degrees}).
1198 @arg lon1: Longitude of the first point (C{degrees}).
1199 @arg bearing1: Bearing at the first point (compass C{degrees}).
1200 @arg lat2: Latitude of the second point (C{degrees}).
1201 @arg lon2: Longitude of the second point (C{degrees}).
1202 @arg bearing2: Bearing at the second point (compass C{degrees}).
1203 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid},
1204 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth
1205 radius (C{meter}, same units as B{C{radius1}} and
1206 B{C{radius2}}) or C{None}.
1207 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
1208 and B{C{lon2}} (C{bool}).
1209 @kwarg small: Upper limit for small distances (C{meter}).
1211 @return: A L{LatLon2Tuple}C{(lat, lon)} with the lat- and
1212 longitude of the intersection point.
1214 @raise IntersectionError: Ambiguous or infinite intersection
1215 or colinear, parallel or otherwise
1216 non-intersecting lines.
1218 @raise TypeError: Invalid B{C{datum}}.
1220 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{bearing1}},
1221 B{C{lat2}}, B{C{lon2}} or B{C{bearing2}}.
1223 @see: Method L{RhumbLine.intersection2}.
1225 @note: The returned intersections may be near-antipodal.
1226 '''
1227 b1 = Bearing(bearing1=bearing1)
1228 b2 = Bearing(bearing2=bearing2)
1229 with _idllmn6(datum, lat1, lon1, lat2, lon2,
1230 small, wrap, False, bearing1=b1, bearing2=b2) as t:
1231 _i, d, lat2, lon2, m, n = t
1232 if d is None:
1233 t, _, _ = _i(m.Vector3d(lon1, lat1, 0), b1,
1234 m.Vector3d(lon2, lat2, 0), b2, useZ=False)
1235 t = LatLon2Tuple(t.y, t.x, name=n)
1237 else:
1238 t = _i(m.LatLon(lat1, lon1, datum=d), b1,
1239 m.LatLon(lat2, lon2, datum=d), b2, height=0, wrap=False)
1240 if isinstance(t, Intersection3Tuple): # ellipsoidal
1241 t, _, _ = t
1242 t = LatLon2Tuple(t.lat, t.lon, name=n)
1243 return t
1246def intersections2(lat1, lon1, radius1,
1247 lat2, lon2, radius2, datum=None, wrap=False, small=_100km): # was=True
1248 '''I{Conveniently} compute the intersections of two circles each defined
1249 by a (geodetic) center point and a radius, using either ...
1251 1) L{vector3d.intersections2} for B{C{small}} distances (below 100 Km
1252 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ...
1254 2) L{sphericalTrigonometry.intersections2} for a spherical B{C{datum}}
1255 or a C{scalar B{datum}} representing the earth radius, conventionally
1256 in C{meter} or ...
1258 3) L{ellipsoidalKarney.intersections2} for an ellipsoidal B{C{datum}}
1259 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>}
1260 is installed, otherwise ...
1262 4) L{ellipsoidalExact.intersections2}, provided B{C{datum}} is ellipsoidal.
1264 @arg lat1: Latitude of the first circle center (C{degrees}).
1265 @arg lon1: Longitude of the first circle center (C{degrees}).
1266 @arg radius1: Radius of the first circle (C{meter}, conventionally).
1267 @arg lat2: Latitude of the second circle center (C{degrees}).
1268 @arg lon2: Longitude of the second circle center (C{degrees}).
1269 @arg radius2: Radius of the second circle (C{meter}, same units as B{C{radius1}}).
1270 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid},
1271 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth
1272 radius (C{meter}, same units as B{C{radius1}} and
1273 B{C{radius2}}) or C{None}.
1274 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
1275 and B{C{lon2}} (C{bool}).
1276 @kwarg small: Upper limit for small distances (C{meter}).
1278 @return: 2-Tuple of the intersection points, each a
1279 L{LatLon2Tuple}C{(lat, lon)}. For abutting circles, the
1280 points are the same instance, aka the I{radical center}.
1282 @raise IntersectionError: Concentric, antipodal, invalid or
1283 non-intersecting circles or no
1284 convergence.
1286 @raise TypeError: Invalid B{C{datum}}.
1288 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{radius1}},
1289 B{C{lat2}}, B{C{lon2}} or B{C{radius2}}.
1290 '''
1291 r1 = Radius_(radius1=radius1)
1292 r2 = Radius_(radius2=radius2)
1293 with _idllmn6(datum, lat1, lon1, lat2, lon2,
1294 small, wrap, True, radius1=r1, radius2=r2) as t:
1295 _i, d, lat2, lon2, m, n = t
1296 if d is None:
1297 r1 = m2degrees(r1, radius=R_M, lat=lat1)
1298 r2 = m2degrees(r2, radius=R_M, lat=lat2)
1300 def _V2T(x, y, _, **unused): # _ == z unused
1301 return LatLon2Tuple(y, x, name=n)
1303 t = _i(m.Vector3d(lon1, lat1, 0), r1,
1304 m.Vector3d(lon2, lat2, 0), r2, sphere=False,
1305 Vector=_V2T)
1306 else:
1307 def _LL2T(lat, lon, **unused):
1308 return LatLon2Tuple(lat, lon, name=n)
1310 t = _i(m.LatLon(lat1, lon1, datum=d), r1,
1311 m.LatLon(lat2, lon2, datum=d), r2,
1312 LatLon=_LL2T, height=0, wrap=False)
1313 return t
1316def isantipode(lat1, lon1, lat2, lon2, eps=EPS):
1317 '''Check whether two points are I{antipodal}, on diametrically
1318 opposite sides of the earth.
1320 @arg lat1: Latitude of one point (C{degrees}).
1321 @arg lon1: Longitude of one point (C{degrees}).
1322 @arg lat2: Latitude of the other point (C{degrees}).
1323 @arg lon2: Longitude of the other point (C{degrees}).
1324 @kwarg eps: Tolerance for near-equality (C{degrees}).
1326 @return: C{True} if points are antipodal within the
1327 B{C{eps}} tolerance, C{False} otherwise.
1329 @see: Functions L{isantipode_} and L{antipode}.
1330 '''
1331 return (fabs(lat1 + lat2) <= eps and
1332 fabs(lon1 + lon2) <= eps) or _isequalTo(
1333 normal(lat1, lon1), antipode(lat2, lon2), eps)
1336def isantipode_(phi1, lam1, phi2, lam2, eps=EPS):
1337 '''Check whether two points are I{antipodal}, on diametrically
1338 opposite sides of the earth.
1340 @arg phi1: Latitude of one point (C{radians}).
1341 @arg lam1: Longitude of one point (C{radians}).
1342 @arg phi2: Latitude of the other point (C{radians}).
1343 @arg lam2: Longitude of the other point (C{radians}).
1344 @kwarg eps: Tolerance for near-equality (C{radians}).
1346 @return: C{True} if points are antipodal within the
1347 B{C{eps}} tolerance, C{False} otherwise.
1349 @see: Functions L{isantipode} and L{antipode_}.
1350 '''
1351 return (fabs(phi1 + phi2) <= eps and
1352 fabs(lam1 + lam2) <= eps) or _isequalTo_(
1353 normal_(phi1, lam1), antipode_(phi2, lam2), eps)
1356def _isequalTo(p1, p2, eps=EPS):
1357 '''Compare 2 point lat-/lons ignoring C{class}.
1358 '''
1359 return (fabs(p1.lat - p2.lat) <= eps and
1360 fabs(p1.lon - p2.lon) <= eps) if eps else (p1.latlon == p2.latlon)
1363def _isequalTo_(p1, p2, eps=EPS):
1364 '''(INTERNAL) Compare 2 point phi-/lams ignoring C{class}.
1365 '''
1366 return (fabs(p1.phi - p2.phi) <= eps and
1367 fabs(p1.lam - p2.lam) <= eps) if eps else (p1.philam == p2.philam)
1370def isnormal(lat, lon, eps=0):
1371 '''Check whether B{C{lat}} I{and} B{C{lon}} are within their
1372 respective I{normal} range in C{degrees}.
1374 @arg lat: Latitude (C{degrees}).
1375 @arg lon: Longitude (C{degrees}).
1376 @kwarg eps: Optional tolerance C{degrees}).
1378 @return: C{True} if C{(abs(B{lat}) + B{eps}) <= 90} and
1379 C{(abs(B{lon}) + B{eps}) <= 180}, C{False} othwerwise.
1381 @see: Functions L{isnormal_} and L{normal}.
1382 '''
1383 return (_90_0 - fabs(lat)) >= eps and _loneg(fabs(lon)) >= eps
1386def isnormal_(phi, lam, eps=0):
1387 '''Check whether B{C{phi}} I{and} B{C{lam}} are within their
1388 respective I{normal} range in C{radians}.
1390 @arg phi: Latitude (C{radians}).
1391 @arg lam: Longitude (C{radians}).
1392 @kwarg eps: Optional tolerance C{radians}).
1394 @return: C{True} if C{(abs(B{phi}) + B{eps}) <= PI/2} and
1395 C{(abs(B{lam}) + B{eps}) <= PI}, C{False} othwerwise.
1397 @see: Functions L{isnormal} and L{normal_}.
1398 '''
1399 return (PI_2 - fabs(phi)) >= eps and (PI - fabs(lam)) >= eps
1402def latlon2n_xyz(lat, lon, name=NN):
1403 '''Convert lat-, longitude to C{n-vector} (I{normal} to the
1404 earth's surface) X, Y and Z components.
1406 @arg lat: Latitude (C{degrees}).
1407 @arg lon: Longitude (C{degrees}).
1408 @kwarg name: Optional name (C{str}).
1410 @return: A L{Vector3Tuple}C{(x, y, z)}.
1412 @see: Function L{philam2n_xyz}.
1414 @note: These are C{n-vector} x, y and z components,
1415 I{NOT} geocentric ECEF x, y and z coordinates!
1416 '''
1417 return _2n_xyz(name, *sincos2d_(lat, lon))
1420def _normal2(a, b, n_2, n, n2):
1421 '''(INTERNAL) Helper for C{normal} and C{normal_}.
1422 '''
1423 if fabs(b) > n:
1424 b = remainder(b, n2)
1425 if fabs(a) > n_2:
1426 r = remainder(a, n)
1427 if r != a:
1428 a = -r
1429 b -= n if b > 0 else -n
1430 return float0_(a, b)
1433def normal(lat, lon, name=NN):
1434 '''Normalize a lat- I{and} longitude pair in C{degrees}.
1436 @arg lat: Latitude (C{degrees}).
1437 @arg lon: Longitude (C{degrees}).
1438 @kwarg name: Optional name (C{str}).
1440 @return: L{LatLon2Tuple}C{(lat, lon)} with C{abs(lat) <= 90}
1441 and C{abs(lon) <= 180}.
1443 @see: Functions L{normal_} and L{isnormal}.
1444 '''
1445 return LatLon2Tuple(*_normal2(lat, lon, _90_0, _180_0, _360_0),
1446 name=name or normal.__name__)
1449def normal_(phi, lam, name=NN):
1450 '''Normalize a lat- I{and} longitude pair in C{radians}.
1452 @arg phi: Latitude (C{radians}).
1453 @arg lam: Longitude (C{radians}).
1454 @kwarg name: Optional name (C{str}).
1456 @return: L{PhiLam2Tuple}C{(phi, lam)} with C{abs(phi) <= PI/2}
1457 and C{abs(lam) <= PI}.
1459 @see: Functions L{normal} and L{isnormal_}.
1460 '''
1461 return PhiLam2Tuple(*_normal2(phi, lam, PI_2, PI, PI2),
1462 name=name or normal_.__name__)
1465def _2n_xyz(name, sa, ca, sb, cb):
1466 '''(INTERNAL) Helper for C{latlon2n_xyz} and C{philam2n_xyz}.
1467 '''
1468 # Kenneth Gade eqn 3, but using right-handed
1469 # vector x -> 0°E,0°N, y -> 90°E,0°N, z -> 90°N
1470 return Vector3Tuple(ca * cb, ca * sb, sa, name=name)
1473def n_xyz2latlon(x, y, z, name=NN):
1474 '''Convert C{n-vector} components to lat- and longitude in C{degrees}.
1476 @arg x: X component (C{scalar}).
1477 @arg y: Y component (C{scalar}).
1478 @arg z: Z component (C{scalar}).
1479 @kwarg name: Optional name (C{str}).
1481 @return: A L{LatLon2Tuple}C{(lat, lon)}.
1483 @see: Function L{n_xyz2philam}.
1484 '''
1485 return LatLon2Tuple(atan2d(z, hypot(x, y)), atan2d(y, x), name=name)
1488def n_xyz2philam(x, y, z, name=NN):
1489 '''Convert C{n-vector} components to lat- and longitude in C{radians}.
1491 @arg x: X component (C{scalar}).
1492 @arg y: Y component (C{scalar}).
1493 @arg z: Z component (C{scalar}).
1494 @kwarg name: Optional name (C{str}).
1496 @return: A L{PhiLam2Tuple}C{(phi, lam)}.
1498 @see: Function L{n_xyz2latlon}.
1499 '''
1500 return PhiLam2Tuple(atan2(z, hypot(x, y)), atan2(y, x), name=name)
1503def _opposes(d, m, n, n2):
1504 '''(INTERNAL) Helper for C{opposing} and C{opposing_}.
1505 '''
1506 d = d % n2 # -20 % 360 == 340, -1 % PI2 == PI2 - 1
1507 return False if d < m or d > (n2 - m) else (
1508 True if (n - m) < d < (n + m) else None)
1511def opposing(bearing1, bearing2, margin=_90_0):
1512 '''Compare the direction of two bearings given in C{degrees}.
1514 @arg bearing1: First bearing (compass C{degrees}).
1515 @arg bearing2: Second bearing (compass C{degrees}).
1516 @kwarg margin: Optional, interior angle bracket (C{degrees}).
1518 @return: C{True} if both bearings point in opposite, C{False} if
1519 in similar or C{None} if in I{perpendicular} directions.
1521 @see: Function L{opposing_}.
1522 '''
1523 m = Degrees_(margin=margin, low=EPS0, high=_90_0)
1524 return _opposes(bearing2 - bearing1, m, _180_0, _360_0)
1527def opposing_(radians1, radians2, margin=PI_2):
1528 '''Compare the direction of two bearings given in C{radians}.
1530 @arg radians1: First bearing (C{radians}).
1531 @arg radians2: Second bearing (C{radians}).
1532 @kwarg margin: Optional, interior angle bracket (C{radians}).
1534 @return: C{True} if both bearings point in opposite, C{False} if
1535 in similar or C{None} if in perpendicular directions.
1537 @see: Function L{opposing}.
1538 '''
1539 m = Radians_(margin=margin, low=EPS0, high=PI_2)
1540 return _opposes(radians2 - radians1, m, PI, PI2)
1543def philam2n_xyz(phi, lam, name=NN):
1544 '''Convert lat-, longitude to C{n-vector} (I{normal} to the
1545 earth's surface) X, Y and Z components.
1547 @arg phi: Latitude (C{radians}).
1548 @arg lam: Longitude (C{radians}).
1549 @kwarg name: Optional name (C{str}).
1551 @return: A L{Vector3Tuple}C{(x, y, z)}.
1553 @see: Function L{latlon2n_xyz}.
1555 @note: These are C{n-vector} x, y and z components,
1556 I{NOT} geocentric ECEF x, y and z coordinates!
1557 '''
1558 return _2n_xyz(name, *sincos2_(phi, lam))
1561def _radical2(d, r1, r2): # in .ellipsoidalBaseDI, .sphericalTrigonometry, .vector3d
1562 # (INTERNAL) See C{radical2} below
1563 # assert d > EPS0
1564 r = fsumf_(_1_0, (r1 / d)**2, -(r2 / d)**2) * _0_5
1565 return Radical2Tuple(max(_0_0, min(_1_0, r)), r * d)
1568def radical2(distance, radius1, radius2):
1569 '''Compute the I{radical ratio} and I{radical line} of two
1570 U{intersecting circles<https://MathWorld.Wolfram.com/
1571 Circle-CircleIntersection.html>}.
1573 The I{radical line} is perpendicular to the axis thru the
1574 centers of the circles at C{(0, 0)} and C{(B{distance}, 0)}.
1576 @arg distance: Distance between the circle centers (C{scalar}).
1577 @arg radius1: Radius of the first circle (C{scalar}).
1578 @arg radius2: Radius of the second circle (C{scalar}).
1580 @return: A L{Radical2Tuple}C{(ratio, xline)} where C{0.0 <=
1581 ratio <= 1.0} and C{xline} is along the B{C{distance}}.
1583 @raise IntersectionError: The B{C{distance}} exceeds the sum
1584 of B{C{radius1}} and B{C{radius2}}.
1586 @raise UnitError: Invalid B{C{distance}}, B{C{radius1}} or
1587 B{C{radius2}}.
1589 @see: U{Circle-Circle Intersection
1590 <https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>}.
1591 '''
1592 d = Distance_(distance, low=_0_0)
1593 r1 = Radius_(radius1=radius1)
1594 r2 = Radius_(radius2=radius2)
1595 if d > (r1 + r2):
1596 raise IntersectionError(distance=d, radius1=r1, radius2=r2,
1597 txt=_too_(_distant_))
1598 return _radical2(d, r1, r2) if d > EPS0 else \
1599 Radical2Tuple(_0_5, _0_0)
1602class Radical2Tuple(_NamedTuple):
1603 '''2-Tuple C{(ratio, xline)} of the I{radical} C{ratio} and
1604 I{radical} C{xline}, both C{scalar} and C{0.0 <= ratio <= 1.0}
1605 '''
1606 _Names_ = (_ratio_, _xline_)
1607 _Units_ = ( Scalar, Scalar)
1610def _radistance(inst):
1611 '''(INTERNAL) Helper for the L{frechet._FrecherMeterRadians}
1612 and L{hausdorff._HausdorffMeterRedians} classes.
1613 '''
1614 kwds_ = _xkwds(inst._kwds, wrap=False)
1615 wrap_ = _xkwds_pop(kwds_, wrap=False)
1616 func_ = inst._func_
1617 try: # calling lower-overhead C{func_}
1618 func_(0, _0_25, _0_5, **kwds_)
1619 wrap_ = _Wrap._philamop(wrap_)
1620 except TypeError:
1621 return inst.distance
1623 def _philam(p):
1624 try:
1625 return p.phi, p.lam
1626 except AttributeError: # no .phi or .lam
1627 return radians(p.lat), radians(p.lon)
1629 def _func_wrap(point1, point2):
1630 phi1, lam1 = wrap_(*_philam(point1))
1631 phi2, lam2 = wrap_(*_philam(point2))
1632 return func_(phi2, phi1, lam2 - lam1, **kwds_)
1634 inst._units = inst._units_
1635 return _func_wrap
1638def _scale_deg(lat1, lat2): # degrees
1639 # scale factor cos(mean of lats) for delta lon
1640 m = fabs(lat1 + lat2) * _0_5
1641 return cos(radians(m)) if m < 90 else _0_0
1644def _scale_rad(phi1, phi2): # radians, by .frechet, .hausdorff, .heights
1645 # scale factor cos(mean of phis) for delta lam
1646 m = fabs(phi1 + phi2) * _0_5
1647 return cos(m) if m < PI_2 else _0_0
1650def _sincosa6(phi2, phi1, lam21): # [4] in cosineLaw
1651 '''(INTERNAL) C{sin}es, C{cos}ines and C{acos}ine.
1652 '''
1653 s2, c2, s1, c1, _, c21 = sincos2_(phi2, phi1, lam21)
1654 return s2, c2, s1, c1, acos1(s1 * s2 + c1 * c2 * c21), c21
1657def thomas(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):
1658 '''Compute the distance between two (ellipsoidal) points using
1659 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>}
1660 formula.
1662 @arg lat1: Start latitude (C{degrees}).
1663 @arg lon1: Start longitude (C{degrees}).
1664 @arg lat2: End latitude (C{degrees}).
1665 @arg lon2: End longitude (C{degrees}).
1666 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
1667 L{Ellipsoid2} or L{a_f2Tuple}) to use.
1668 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
1669 B{C{lat2}} and B{C{lon2}} (C{bool}).
1671 @return: Distance (C{meter}, same units as the B{C{datum}}'s
1672 ellipsoid axes).
1674 @raise TypeError: Invalid B{C{datum}}.
1676 @see: Functions L{thomas_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
1677 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny},
1678 L{flatPolar}, L{haversine}, L{vincentys} and method L{Ellipsoid.distance2}.
1679 '''
1680 return _dE(thomas_, datum, wrap, lat1, lon1, lat2, lon2)
1683def thomas_(phi2, phi1, lam21, datum=_WGS84):
1684 '''Compute the I{angular} distance between two (ellipsoidal) points using
1685 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>}
1686 formula.
1688 @arg phi2: End latitude (C{radians}).
1689 @arg phi1: Start latitude (C{radians}).
1690 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
1691 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid},
1692 L{Ellipsoid2} or L{a_f2Tuple}).
1694 @return: Angular distance (C{radians}).
1696 @raise TypeError: Invalid B{C{datum}}.
1698 @see: Functions L{thomas}, L{cosineAndoyerLambert_},
1699 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
1700 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
1701 L{flatPolar_}, L{haversine_} and L{vincentys_} and U{Geodesy-PHP
1702 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/
1703 Distance/ThomasFormula.php>}.
1704 '''
1705 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21)
1706 if r and isnon0(c1) and isnon0(c2):
1707 E = _ellipsoidal(datum, thomas_)
1708 if E.f:
1709 r1 = atan2(E.b_a * s1, c1)
1710 r2 = atan2(E.b_a * s2, c2)
1712 j = (r2 + r1) * _0_5
1713 k = (r2 - r1) * _0_5
1714 sj, cj, sk, ck, h, _ = sincos2_(j, k, lam21 * _0_5)
1716 h = fsumf_(sk**2, (ck * h)**2, -(sj * h)**2)
1717 u = _1_0 - h
1718 if isnon0(u) and isnon0(h):
1719 r = atan(sqrt0(h / u)) * 2 # == acos(1 - 2 * h)
1720 sr, cr = sincos2(r)
1721 if isnon0(sr):
1722 u = 2 * (sj * ck)**2 / u
1723 h = 2 * (sk * cj)**2 / h
1724 x = u + h
1725 y = u - h
1727 s = r / sr
1728 e = 4 * s**2
1729 d = 2 * cr
1730 a = e * d
1731 b = 2 * r
1732 c = s - (a - d) * _0_5
1733 f = E.f * _0_25
1735 t = fsumf_(a * x, -b * y, c * x**2, -d * y**2, e * x * y)
1736 r -= fsumf_(s * x, -y, -t * f * _0_25) * f * sr
1737 return r
1740def vincentys(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
1741 '''Compute the distance between two (spherical) points using
1742 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>}
1743 spherical formula.
1745 @arg lat1: Start latitude (C{degrees}).
1746 @arg lon1: Start longitude (C{degrees}).
1747 @arg lat2: End latitude (C{degrees}).
1748 @arg lon2: End longitude (C{degrees}).
1749 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
1750 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
1751 L{a_f2Tuple}) to use.
1752 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
1753 B{C{lat2}} and B{C{lon2}} (C{bool}).
1755 @return: Distance (C{meter}, same units as B{C{radius}}).
1757 @raise UnitError: Invalid B{C{radius}}.
1759 @see: Functions L{vincentys_}, L{cosineAndoyerLambert},
1760 L{cosineForsytheAndoyerLambert},L{cosineLaw}, L{equirectangular},
1761 L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar},
1762 L{haversine} and L{thomas} and methods L{Ellipsoid.distance2},
1763 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
1765 @note: See note at function L{vincentys_}.
1766 '''
1767 return _dS(vincentys_, radius, wrap, lat1, lon1, lat2, lon2)
1770def vincentys_(phi2, phi1, lam21):
1771 '''Compute the I{angular} distance between two (spherical) points using
1772 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>}
1773 spherical formula.
1775 @arg phi2: End latitude (C{radians}).
1776 @arg phi1: Start latitude (C{radians}).
1777 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
1779 @return: Angular distance (C{radians}).
1781 @see: Functions L{vincentys}, L{cosineAndoyerLambert_},
1782 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
1783 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
1784 L{flatPolar_}, L{haversine_} and L{thomas_}.
1786 @note: Functions L{vincentys_}, L{haversine_} and L{cosineLaw_}
1787 produce equivalent results, but L{vincentys_} is suitable
1788 for antipodal points and slightly more expensive (M{3 cos,
1789 3 sin, 1 hypot, 1 atan2, 6 mul, 2 add}) than L{haversine_}
1790 (M{2 cos, 2 sin, 2 sqrt, 1 atan2, 5 mul, 1 add}) and
1791 L{cosineLaw_} (M{3 cos, 3 sin, 1 acos, 3 mul, 1 add}).
1792 '''
1793 s1, c1, s2, c2, s21, c21 = sincos2_(phi1, phi2, lam21)
1795 c = c2 * c21
1796 x = s1 * s2 + c1 * c
1797 y = c1 * s2 - s1 * c
1798 return atan2(hypot(c2 * s21, y), x)
1800# **) MIT License
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