Coverage for pygeodesy/formy.py: 99%
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« prev ^ index » next coverage.py v7.2.2, created at 2023-08-12 12:31 -0400
2# -*- coding: utf-8 -*-
4u'''Formulary of basic geodesy functions and approximations.
5'''
6# make sure int/int division yields float quotient, see .basics
7from __future__ import division as _; del _ # PYCHOK semicolon
9# from pygeodesy.basics import isscalar # from .fsums
10from pygeodesy.constants import EPS, EPS0, EPS1, PI, PI2, PI3, PI_2, R_M, \
11 _umod_PI2, float0_, isnon0, remainder, \
12 _0_0, _0_125, _0_25, _0_5, _1_0, _2_0, \
13 _4_0, _32_0, _90_0, _180_0, _360_0
14from pygeodesy.datums import Datum, Ellipsoid, _ellipsoidal_datum, \
15 _mean_radius, _spherical_datum, _WGS84
16# from pygeodesy.ellipsoids import Ellipsoid # from .datums
17from pygeodesy.errors import IntersectionError, LimitError, limiterrors, \
18 _TypeError, _ValueError, \
19 _xError, _xkwds, _xkwds_pop
20from pygeodesy.fmath import euclid, hypot, hypot2, sqrt0
21from pygeodesy.fsums import fsumf_, isscalar
22from pygeodesy.interns import NN, _delta_, _distant_, _SPACE_, _too_
23from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS
24from pygeodesy.named import _NamedTuple, _xnamed, Fmt, unstr
25from pygeodesy.namedTuples import Bearing2Tuple, Distance4Tuple, \
26 Intersection3Tuple, LatLon2Tuple, \
27 PhiLam2Tuple, Vector3Tuple
28# from pygeodesy.streprs import Fmt, unstr # from .named
29from pygeodesy.units import Bearing, Degrees_, Distance, Distance_, Height, \
30 Lam_, Lat, Lon, Meter_, Phi_, Radians, Radians_, \
31 Radius, Radius_, Scalar, _100km
32from pygeodesy.utily import acos1, atan2b, atan2d, degrees2m, m2degrees, \
33 tan_2, sincos2, sincos2_, sincos2d_, _Wrap
35from contextlib import contextmanager
36from math import asin, atan, atan2, cos, degrees, fabs, radians, sin, sqrt # pow
38__all__ = _ALL_LAZY.formy
39__version__ = '23.08.11'
41_D2_R2 = (PI / _180_0)**2 # degrees- to radians-squared
42_EWGS84 = _WGS84.ellipsoid
43_ratio_ = 'ratio'
44_xline_ = 'xline'
47def _anti2(a, b, n_2, n, n2):
48 '''(INTERNAL) Helper for C{antipode} and C{antipode_}.
49 '''
50 r = remainder(a, n) if fabs(a) > n_2 else a
51 if r == a:
52 r = -r
53 b += n
54 if fabs(b) > n:
55 b = remainder(b, n2)
56 return float0_(r, b)
59def antipode(lat, lon, name=NN):
60 '''Return the antipode, the point diametrically opposite
61 to a given point in C{degrees}.
63 @arg lat: Latitude (C{degrees}).
64 @arg lon: Longitude (C{degrees}).
65 @kwarg name: Optional name (C{str}).
67 @return: A L{LatLon2Tuple}C{(lat, lon)}.
69 @see: Functions L{antipode_} and L{normal} and U{Geosphere
70 <https://CRAN.R-Project.org/web/packages/geosphere/geosphere.pdf>}.
71 '''
72 return LatLon2Tuple(*_anti2(lat, lon, _90_0, _180_0, _360_0), name=name)
75def antipode_(phi, lam, name=NN):
76 '''Return the antipode, the point diametrically opposite
77 to a given point in C{radians}.
79 @arg phi: Latitude (C{radians}).
80 @arg lam: Longitude (C{radians}).
81 @kwarg name: Optional name (C{str}).
83 @return: A L{PhiLam2Tuple}C{(phi, lam)}.
85 @see: Functions L{antipode} and L{normal_} and U{Geosphere
86 <https://CRAN.R-Project.org/web/packages/geosphere/geosphere.pdf>}.
87 '''
88 return PhiLam2Tuple(*_anti2(phi, lam, PI_2, PI, PI2), name=name)
91def bearing(lat1, lon1, lat2, lon2, **final_wrap):
92 '''Compute the initial or final bearing (forward or reverse
93 azimuth) between a (spherical) start and end point.
95 @arg lat1: Start latitude (C{degrees}).
96 @arg lon1: Start longitude (C{degrees}).
97 @arg lat2: End latitude (C{degrees}).
98 @arg lon2: End longitude (C{degrees}).
99 @kwarg final_wrap: Optional keyword arguments for function
100 L{pygeodesy.bearing_}.
102 @return: Initial or final bearing (compass C{degrees360}) or
103 zero if start and end point coincide.
104 '''
105 r = bearing_(Phi_(lat1=lat1), Lam_(lon1=lon1),
106 Phi_(lat2=lat2), Lam_(lon2=lon2), **final_wrap)
107 return degrees(r)
110def bearing_(phi1, lam1, phi2, lam2, final=False, wrap=False):
111 '''Compute the initial or final bearing (forward or reverse azimuth)
112 between a (spherical) start and end point.
114 @arg phi1: Start latitude (C{radians}).
115 @arg lam1: Start longitude (C{radians}).
116 @arg phi2: End latitude (C{radians}).
117 @arg lam2: End longitude (C{radians}).
118 @kwarg final: Return final bearing if C{True}, initial otherwise (C{bool}).
119 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{phi2}} and
120 B{C{lam2}} (C{bool}).
122 @return: Initial or final bearing (compass C{radiansPI2}) or zero if start
123 and end point coincide.
125 @see: U{Bearing<https://www.Movable-Type.co.UK/scripts/latlong.html>}, U{Course
126 between two points<https://www.EdWilliams.org/avform147.htm#Crs>} and
127 U{Bearing Between Two Points<https://web.Archive.org/web/20020630205931/
128 https://MathForum.org/library/drmath/view/55417.html>}.
129 '''
130 db, phi2, lam2 = _Wrap.philam3(lam1, phi2, lam2, wrap)
131 if final: # swap plus PI
132 phi1, lam1, phi2, lam2, db = phi2, lam2, phi1, lam1, -db
133 r = PI3
134 else:
135 r = PI2
136 sa1, ca1, sa2, ca2, sdb, cdb = sincos2_(phi1, phi2, db)
138 x = ca1 * sa2 - sa1 * ca2 * cdb
139 y = sdb * ca2
140 return _umod_PI2(atan2(y, x) + r) # .utily.wrapPI2
143def _bearingTo2(p1, p2, wrap=False): # for points.ispolar, sphericalTrigonometry.areaOf
144 '''(INTERNAL) Compute initial and final bearing.
145 '''
146 try: # for LatLon_ and ellipsoidal LatLon
147 return p1.bearingTo2(p2, wrap=wrap)
148 except AttributeError:
149 pass
150 # XXX spherical version, OK for ellipsoidal ispolar?
151 a1, b1 = p1.philam
152 a2, b2 = p2.philam
153 return Bearing2Tuple(degrees(bearing_(a1, b1, a2, b2, final=False, wrap=wrap)),
154 degrees(bearing_(a1, b1, a2, b2, final=True, wrap=wrap)),
155 name=_bearingTo2.__name__)
158def compassAngle(lat1, lon1, lat2, lon2, adjust=True, wrap=False):
159 '''Return the angle from North for the direction vector M{(lon2 - lon1,
160 lat2 - lat1)} between two points.
162 Suitable only for short, not near-polar vectors up to a few hundred
163 Km or Miles. Use function L{pygeodesy.bearing} for longer vectors.
165 @arg lat1: From latitude (C{degrees}).
166 @arg lon1: From longitude (C{degrees}).
167 @arg lat2: To latitude (C{degrees}).
168 @arg lon2: To longitude (C{degrees}).
169 @kwarg adjust: Adjust the longitudinal delta by the cosine of the
170 mean latitude (C{bool}).
171 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
172 and B{C{lon2}} (C{bool}).
174 @return: Compass angle from North (C{degrees360}).
176 @note: Courtesy of Martin Schultz.
178 @see: U{Local, flat earth approximation
179 <https://www.EdWilliams.org/avform.htm#flat>}.
180 '''
181 d_lon, lat2, lon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap)
182 if adjust: # scale delta lon
183 d_lon *= _scale_deg(lat1, lat2)
184 return atan2b(d_lon, lat2 - lat1)
187def cosineAndoyerLambert(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):
188 '''Compute the distance between two (ellipsoidal) points using the
189 U{Andoyer-Lambert correction<https://NavLib.net/wp-content/uploads/2013/10/
190 admiralty-manual-of-navigation-vol-1-1964-english501c.pdf>} of the U{Law of
191 Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula.
193 @arg lat1: Start latitude (C{degrees}).
194 @arg lon1: Start longitude (C{degrees}).
195 @arg lat2: End latitude (C{degrees}).
196 @arg lon2: End longitude (C{degrees}).
197 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
198 L{Ellipsoid2} or L{a_f2Tuple}) to use.
199 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
200 B{C{lat2}} and B{C{lon2}} (C{bool}).
202 @return: Distance (C{meter}, same units as the B{C{datum}}'s
203 ellipsoid axes or C{radians} if B{C{datum}} is C{None}).
205 @raise TypeError: Invalid B{C{datum}}.
207 @see: Functions L{cosineAndoyerLambert_}, L{cosineForsytheAndoyerLambert},
208 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny},
209 L{flatPolar}, L{haversine}, L{thomas} and L{vincentys} and method
210 L{Ellipsoid.distance2}.
211 '''
212 return _dE(cosineAndoyerLambert_, datum, wrap, lat1, lon1, lat2, lon2)
215def cosineAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84):
216 '''Compute the I{angular} distance between two (ellipsoidal) points using the
217 U{Andoyer-Lambert correction<https://NavLib.net/wp-content/uploads/2013/10/
218 admiralty-manual-of-navigation-vol-1-1964-english501c.pdf>} of the U{Law of
219 Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula.
221 @arg phi2: End latitude (C{radians}).
222 @arg phi1: Start latitude (C{radians}).
223 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
224 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
225 L{Ellipsoid2} or L{a_f2Tuple}) to use.
227 @return: Angular distance (C{radians}).
229 @raise TypeError: Invalid B{C{datum}}.
231 @see: Functions L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert_},
232 L{cosineLaw_}, L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
233 L{flatPolar_}, L{haversine_}, L{thomas_} and L{vincentys_} and U{Geodesy-PHP
234 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/Distance/
235 AndoyerLambert.php>}.
236 '''
237 s2, c2, s1, c1, r, c21 = _sincosa6(phi2, phi1, lam21)
238 if isnon0(c1) and isnon0(c2):
239 E = _ellipsoidal(datum, cosineAndoyerLambert_)
240 if E.f: # ellipsoidal
241 r2 = atan2(E.b_a * s2, c2)
242 r1 = atan2(E.b_a * s1, c1)
243 s2, c2, s1, c1 = sincos2_(r2, r1)
244 r = acos1(s1 * s2 + c1 * c2 * c21)
245 if r:
246 sr, _, sr_2, cr_2 = sincos2_(r, r * _0_5)
247 if isnon0(sr_2) and isnon0(cr_2):
248 s = (sr + r) * ((s1 - s2) / sr_2)**2
249 c = (sr - r) * ((s1 + s2) / cr_2)**2
250 r += (c - s) * E.f * _0_125
251 return r
254def cosineForsytheAndoyerLambert(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):
255 '''Compute the distance between two (ellipsoidal) points using the
256 U{Forsythe-Andoyer-Lambert correction<https://www2.UNB.Ca/gge/Pubs/TR77.pdf>} of
257 the U{Law of Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>}
258 formula.
260 @arg lat1: Start latitude (C{degrees}).
261 @arg lon1: Start longitude (C{degrees}).
262 @arg lat2: End latitude (C{degrees}).
263 @arg lon2: End longitude (C{degrees}).
264 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
265 L{Ellipsoid2} or L{a_f2Tuple}) to use.
266 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
267 B{C{lat2}} and B{C{lon2}} (C{bool}).
269 @return: Distance (C{meter}, same units as the B{C{datum}}'s
270 ellipsoid axes or C{radians} if B{C{datum}} is C{None}).
272 @raise TypeError: Invalid B{C{datum}}.
274 @see: Functions L{cosineForsytheAndoyerLambert_}, L{cosineAndoyerLambert},
275 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny},
276 L{flatPolar}, L{haversine}, L{thomas} and L{vincentys} and method
277 L{Ellipsoid.distance2}.
278 '''
279 return _dE(cosineForsytheAndoyerLambert_, datum, wrap, lat1, lon1, lat2, lon2)
282def cosineForsytheAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84):
283 '''Compute the I{angular} distance between two (ellipsoidal) points using the
284 U{Forsythe-Andoyer-Lambert correction<https://www2.UNB.Ca/gge/Pubs/TR77.pdf>} of
285 the U{Law of Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>}
286 formula.
288 @arg phi2: End latitude (C{radians}).
289 @arg phi1: Start latitude (C{radians}).
290 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
291 @kwarg datum: Datum (L{Datum}) or ellipsoid to use (L{Ellipsoid},
292 L{Ellipsoid2} or L{a_f2Tuple}).
294 @return: Angular distance (C{radians}).
296 @raise TypeError: Invalid B{C{datum}}.
298 @see: Functions L{cosineForsytheAndoyerLambert}, L{cosineAndoyerLambert_},
299 L{cosineLaw_}, L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
300 L{flatPolar_}, L{haversine_}, L{thomas_} and L{vincentys_} and U{Geodesy-PHP
301 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/
302 Distance/ForsytheCorrection.php>}.
303 '''
304 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21)
305 if r and isnon0(c1) and isnon0(c2):
306 E = _ellipsoidal(datum, cosineForsytheAndoyerLambert_)
307 if E.f: # ellipsoidal
308 sr, cr, s2r, _ = sincos2_(r, r * 2)
309 if isnon0(sr) and fabs(cr) < EPS1:
310 s = (s1 + s2)**2 / (1 + cr)
311 t = (s1 - s2)**2 / (1 - cr)
312 x = s + t
313 y = s - t
315 s = 8 * r**2 / sr
316 a = 64 * r + s * cr * 2 # 16 * r**2 / tan(r)
317 d = 48 * sr + s # 8 * r**2 / tan(r)
318 b = -2 * d
319 e = 30 * s2r
320 c = fsumf_(30 * r, e * _0_5, s * cr) # 8 * r**2 / tan(r)
322 t = fsumf_( a * x, b * y, -c * x**2, d * x * y, e * y**2)
323 r += fsumf_(-r * x, 3 * y * sr, t * E.f / _32_0) * E.f * _0_25
324 return r
327def cosineLaw(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
328 '''Compute the distance between two points using the U{spherical Law of
329 Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>}
330 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
359 Law of Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>}
360 formula.
362 @arg phi2: End latitude (C{radians}).
363 @arg phi1: Start latitude (C{radians}).
364 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
366 @return: Angular distance (C{radians}).
368 @see: Functions L{cosineLaw}, L{cosineAndoyerLambert_},
369 L{cosineForsytheAndoyerLambert_}, L{equirectangular_},
370 L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_},
371 L{haversine_}, L{thomas_} and L{vincentys_}.
373 @note: See note at function L{vincentys_}.
374 '''
375 return _sincosa6(phi2, phi1, lam21)[4]
378def _d3(wrap, lat1, lon1, lat2, lon2):
379 '''(INTERNAL) Helper for _dE, _dS and _eA.
380 '''
381 if wrap:
382 d_lon, lat2, _ = _Wrap.latlon3(lon1, lat2, lon2, wrap)
383 return radians(lat2), Phi_(lat1=lat1), radians(d_lon)
384 else: # for backward compaibility
385 return Phi_(lat2=lat2), Phi_(lat1=lat1), Phi_(d_lon=lon2 - lon1)
388def _dE(func_, earth, *wrap_lls):
389 '''(INTERNAL) Helper for ellipsoidal distances.
390 '''
391 E = _ellipsoidal(earth, func_)
392 r = func_(*_d3(*wrap_lls), datum=E)
393 return r * E.a
396def _dS(func_, radius, *wrap_lls, **adjust):
397 '''(INTERNAL) Helper for spherical distances.
398 '''
399 r = func_(*_d3(*wrap_lls), **adjust)
400 if radius is not R_M:
401 _, lat1, _, lat2, _ = wrap_lls
402 radius = _mean_radius(radius, lat1, lat2)
403 return r * radius
406def _eA(excess_, radius, *wrap_lls):
407 '''(INTERNAL) Helper for spherical excess or area.
408 '''
409 r = excess_(*_d3(*wrap_lls))
410 if radius:
411 _, lat1, _, lat2, _ = wrap_lls
412 r *= _mean_radius(radius, lat1, lat2)**2
413 return r
416def _ellipsoidal(earth, where):
417 '''(INTERNAL) Helper for distances.
418 '''
419 return _EWGS84 if earth in (_WGS84, _EWGS84) else (
420 earth if isinstance(earth, Ellipsoid) else
421 (earth if isinstance(earth, Datum) else # PYCHOK indent
422 _ellipsoidal_datum(earth, name=where.__name__)).ellipsoid)
425def equirectangular(lat1, lon1, lat2, lon2, radius=R_M, **adjust_limit_wrap):
426 '''Compute the distance between two points using
427 the U{Equirectangular Approximation / Projection
428 <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})
435 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
436 L{a_f2Tuple}).
437 @kwarg adjust_limit_wrap: Optional keyword arguments for
438 function L{equirectangular_}.
440 @return: Distance (C{meter}, same units as B{C{radius}} or
441 the ellipsoid or datum axes).
443 @raise TypeError: Invalid B{C{radius}}.
445 @see: Function L{equirectangular_} for more details, the
446 available B{C{options}}, errors, restrictions and other,
447 approximate or accurate distance functions.
448 '''
449 d = sqrt(equirectangular_(Lat(lat1=lat1), Lon(lon1=lon1),
450 Lat(lat2=lat2), Lon(lon2=lon2),
451 **adjust_limit_wrap).distance2) # PYCHOK 4 vs 2-3
452 return degrees2m(d, radius=_mean_radius(radius, lat1, lat2))
455def _equirectangular(lat1, lon1, lat2, lon2, **adjust_limit_wrap):
456 '''(INTERNAL) Helper for the L{frechet._FrecherMeterRadians}
457 and L{hausdorff._HausdorffMeterRedians} classes.
458 '''
459 return equirectangular_(lat1, lon1, lat2, lon2, **adjust_limit_wrap).distance2 * _D2_R2
462def equirectangular_(lat1, lon1, lat2, lon2, adjust=True, limit=45, wrap=False):
463 '''Compute the distance between two points using the U{Equirectangular
464 Approximation / Projection
465 <https://www.Movable-Type.co.UK/scripts/latlong.html#equirectangular>}.
467 This approximation is valid for short distance of several hundred Km
468 or Miles, see the B{C{limit}} keyword argument and L{LimitError}.
470 @arg lat1: Start latitude (C{degrees}).
471 @arg lon1: Start longitude (C{degrees}).
472 @arg lat2: End latitude (C{degrees}).
473 @arg lon2: End longitude (C{degrees}).
474 @kwarg adjust: Adjust the wrapped, unrolled longitudinal delta
475 by the cosine of the mean latitude (C{bool}).
476 @kwarg limit: Optional limit for lat- and longitudinal deltas
477 (C{degrees}) or C{None} or C{0} for unlimited.
478 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
479 and B{C{lon2}} (C{bool}).
481 @return: A L{Distance4Tuple}C{(distance2, delta_lat, delta_lon,
482 unroll_lon2)} in C{degrees squared}.
484 @raise LimitError: If the lat- and/or longitudinal delta exceeds the
485 B{C{-limit..limit}} range and L{pygeodesy.limiterrors}
486 set to C{True}.
488 @see: U{Local, flat earth approximation
489 <https://www.EdWilliams.org/avform.htm#flat>}, functions
490 L{equirectangular}, L{cosineAndoyerLambert},
491 L{cosineForsytheAndoyerLambert}, L{cosineLaw}, L{euclidean},
492 L{flatLocal}/L{hubeny}, L{flatPolar}, L{haversine}, L{thomas}
493 and L{vincentys} and methods L{Ellipsoid.distance2},
494 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
495 '''
496 d_lon, lat2, ulon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap)
497 d_lat = lat2 - lat1
499 if limit and limit > 0 and limiterrors():
500 d = max(fabs(d_lat), fabs(d_lon))
501 if d > limit:
502 t = _SPACE_(_delta_, Fmt.PAREN_g(d), Fmt.exceeds_limit(limit))
503 s = unstr(equirectangular_, lat1, lon1, lat2, lon2,
504 limit=limit, wrap=wrap)
505 raise LimitError(s, txt=t)
507 if adjust: # scale delta lon
508 d_lon *= _scale_deg(lat1, lat2)
510 d2 = hypot2(d_lat, d_lon) # degrees squared!
511 return Distance4Tuple(d2, d_lat, d_lon, ulon2 - lon2)
514def euclidean(lat1, lon1, lat2, lon2, radius=R_M, adjust=True, wrap=False):
515 '''Approximate the C{Euclidean} distance between two (spherical) points.
517 @arg lat1: Start latitude (C{degrees}).
518 @arg lon1: Start longitude (C{degrees}).
519 @arg lat2: End latitude (C{degrees}).
520 @arg lon2: End longitude (C{degrees}).
521 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
522 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
523 L{a_f2Tuple}) to use.
524 @kwarg adjust: Adjust the longitudinal delta by the cosine of
525 the mean latitude (C{bool}).
526 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
527 and B{C{lon2}} (C{bool}).
529 @return: Distance (C{meter}, same units as B{C{radius}} or the
530 ellipsoid or datum axes).
532 @raise TypeError: Invalid B{C{radius}}.
534 @see: U{Distance between two (spherical) points
535 <https://www.EdWilliams.org/avform.htm#Dist>}, functions L{euclid},
536 L{euclidean_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
537 L{cosineLaw}, L{equirectangular}, L{flatLocal}/L{hubeny}, L{flatPolar},
538 L{haversine}, L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2},
539 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
540 '''
541 return _dS(euclidean_, radius, wrap, lat1, lon1, lat2, lon2, adjust=adjust)
544def euclidean_(phi2, phi1, lam21, adjust=True):
545 '''Approximate the I{angular} C{Euclidean} distance between two
546 (spherical) points.
548 @arg phi2: End latitude (C{radians}).
549 @arg phi1: Start latitude (C{radians}).
550 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
551 @kwarg adjust: Adjust the longitudinal delta by the cosine
552 of the mean latitude (C{bool}).
554 @return: Angular distance (C{radians}).
556 @see: Functions L{euclid}, L{euclidean}, L{cosineAndoyerLambert_},
557 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{equirectangular_},
558 L{flatLocal_}/L{hubeny_}, L{flatPolar_}, L{haversine_}, L{thomas_}
559 and L{vincentys_}.
560 '''
561 if adjust:
562 lam21 *= _scale_rad(phi2, phi1)
563 return euclid(phi2 - phi1, lam21)
566def excessAbc_(A, b, c):
567 '''Compute the I{spherical excess} C{E} of a (spherical) triangle
568 from two sides and the included (small) angle.
570 @arg A: An interior triangle angle (C{radians}).
571 @arg b: Frist adjacent triangle side (C{radians}).
572 @arg c: Second adjacent triangle side (C{radians}).
574 @return: Spherical excess (C{radians}).
576 @raise UnitError: Invalid B{C{A}}, B{C{b}} or B{C{c}}.
578 @see: Functions L{excessGirard_}, L{excessLHuilier_} and U{Spherical
579 trigonometry<https://WikiPedia.org/wiki/Spherical_trigonometry>}.
580 '''
581 sA, cA, sb, cb, sc, cc = sincos2_(Radians_(A=A), Radians_(b=b) * _0_5,
582 Radians_(c=c) * _0_5)
583 return atan2(sA * sb * sc, cb * cc + cA * sb * sc) * _2_0
586def excessCagnoli_(a, b, c):
587 '''Compute the I{spherical excess} C{E} of a (spherical) triangle using
588 U{Cagnoli's<https://Zenodo.org/record/35392>} (D.34) formula.
590 @arg a: First triangle side (C{radians}).
591 @arg b: Second triangle side (C{radians}).
592 @arg c: Third triangle side (C{radians}).
594 @return: Spherical excess (C{radians}).
596 @raise UnitError: Invalid B{C{a}}, B{C{b}} or B{C{c}}.
598 @see: Function L{excessLHuilier_} and U{Spherical trigonometry
599 <https://WikiPedia.org/wiki/Spherical_trigonometry>}.
600 '''
601 a = Radians_(a=a)
602 b = Radians_(b=b)
603 c = Radians_(c=c)
605 s = fsumf_(a, b, c) * _0_5
606 r = sin(s) * sin(s - a) * sin(s - b) * sin(s - c)
607 c = cos(a * _0_5) * cos(b * _0_5) * cos(c * _0_5)
608 r = asin(sqrt(r) * _0_5 / c) if c and r > 0 else _0_0
609 return Radians(Cagnoli=r * _2_0)
612def excessGirard_(A, B, C):
613 '''Compute the I{spherical excess} C{E} of a (spherical) triangle using
614 U{Girard's<https://MathWorld.Wolfram.com/GirardsSphericalExcessFormula.html>}
615 formula.
617 @arg A: First interior triangle angle (C{radians}).
618 @arg B: Second interior triangle angle (C{radians}).
619 @arg C: Third interior triangle angle (C{radians}).
621 @return: Spherical excess (C{radians}).
623 @raise UnitError: Invalid B{C{A}}, B{C{B}} or B{C{C}}.
625 @see: Function L{excessLHuilier_} and U{Spherical trigonometry
626 <https://WikiPedia.org/wiki/Spherical_trigonometry>}.
627 '''
628 return Radians(Girard=fsumf_(Radians_(A=A),
629 Radians_(B=B),
630 Radians_(C=C), -PI))
633def excessLHuilier_(a, b, c):
634 '''Compute the I{spherical excess} C{E} of a (spherical) triangle using
635 U{L'Huilier's<https://MathWorld.Wolfram.com/LHuiliersTheorem.html>}
636 Theorem.
638 @arg a: First triangle side (C{radians}).
639 @arg b: Second triangle side (C{radians}).
640 @arg c: Third triangle side (C{radians}).
642 @return: Spherical excess (C{radians}).
644 @raise UnitError: Invalid B{C{a}}, B{C{b}} or B{C{c}}.
646 @see: Function L{excessCagnoli_}, L{excessGirard_} and U{Spherical
647 trigonometry<https://WikiPedia.org/wiki/Spherical_trigonometry>}.
648 '''
649 a = Radians_(a=a)
650 b = Radians_(b=b)
651 c = Radians_(c=c)
653 s = fsumf_(a, b, c) * _0_5
654 r = tan_2(s) * tan_2(s - a) * tan_2(s - b) * tan_2(s - c)
655 r = atan(sqrt(r)) if r > 0 else _0_0
656 return Radians(LHuilier=r * _4_0)
659def excessKarney(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
660 '''Compute the surface area of a (spherical) quadrilateral bounded by a
661 segment of a great circle, two meridians and the equator using U{Karney's
662 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>}
663 method.
665 @arg lat1: Start latitude (C{degrees}).
666 @arg lon1: Start longitude (C{degrees}).
667 @arg lat2: End latitude (C{degrees}).
668 @arg lon2: End longitude (C{degrees}).
669 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
670 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
671 L{a_f2Tuple}) or C{None}.
672 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
673 B{C{lat2}} and B{C{lon2}} (C{bool}).
675 @return: Surface area, I{signed} (I{square} C{meter} or the same units as
676 B{C{radius}} I{squared}) or the I{spherical excess} (C{radians})
677 if C{B{radius}=0} or C{None}.
679 @raise TypeError: Invalid B{C{radius}}.
681 @raise UnitError: Invalid B{C{lat2}} or B{C{lat1}}.
683 @raise ValueError: Semi-circular longitudinal delta.
685 @see: Functions L{excessKarney_} and L{excessQuad}.
686 '''
687 return _eA(excessKarney_, radius, wrap, lat1, lon1, lat2, lon2)
690def excessKarney_(phi2, phi1, lam21):
691 '''Compute the I{spherical excess} C{E} of a (spherical) quadrilateral bounded
692 by a segment of a great circle, two meridians and the equator using U{Karney's
693 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>}
694 method.
696 @arg phi2: End latitude (C{radians}).
697 @arg phi1: Start latitude (C{radians}).
698 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
700 @return: Spherical excess, I{signed} (C{radians}).
702 @raise ValueError: Semi-circular longitudinal delta B{C{lam21}}.
704 @see: Function L{excessKarney} and U{Area of a spherical polygon
705 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>}.
706 '''
707 # from: Veness <https://www.Movable-Type.co.UK/scripts/latlong.html> Area
708 # method due to Karney: for each edge of the polygon,
709 #
710 # tan(Δλ / 2) · (tan(φ1 / 2) + tan(φ2 / 2))
711 # tan(E / 2) = -----------------------------------------
712 # 1 + tan(φ1 / 2) · tan(φ2 / 2)
713 #
714 # where E is the spherical excess of the trapezium obtained by extending
715 # the edge to the equator-circle vector for each edge (see also ***).
716 t2 = tan_2(phi2)
717 t1 = tan_2(phi1)
718 t = tan_2(lam21, lam21=None)
719 return Radians(Karney=atan2(t * (t1 + t2),
720 _1_0 + (t1 * t2)) * _2_0)
723# ***) Original post no longer available, following is a copy of the main part
724# <http://OSGeo-org.1560.x6.Nabble.com/Area-of-a-spherical-polygon-td3841625.html>
725#
726# The area of a polygon on a (unit) sphere is given by the spherical excess
727#
728# A = 2 * pi - sum(exterior angles)
729#
730# However this is badly conditioned if the polygon is small. In this case, use
731#
732# A = sum(S12{i, i+1}) over the edges of the polygon
733#
734# where S12 is the area of the quadrilateral bounded by an edge of the polygon,
735# two meridians and the equator, i.e. with vertices (phi1, lambda1), (phi2,
736# lambda2), (0, lambda1) and (0, lambda2). S12 is given by
737#
738# tan(S12 / 2) = tan(lambda21 / 2) * (tan(phi1 / 2) + tan(phi2 / 2)) /
739# (tan(phi1 / 2) * tan(phi2 / 2) + 1)
740#
741# = tan(lambda21 / 2) * tanh((Lambertian(phi1) +
742# Lambertian(phi2)) / 2)
743#
744# where lambda21 = lambda2 - lambda1 and lamb(x) is the Lambertian (or
745# inverse Gudermannian) function
746#
747# Lambertian(x) = asinh(tan(x)) = atanh(sin(x)) = 2 * atanh(tan(x / 2))
748#
749# Notes: The formula for S12 is exact, except that...
750# - it is indeterminate if an edge is a semi-circle
751# - the formula for A applies only if the polygon does not include a pole
752# (if it does, then add +/- 2 * pi to the result)
753# - in the limit of small phi and lambda, S12 reduces to the trapezoidal
754# formula, S12 = (lambda2 - lambda1) * (phi1 + phi2) / 2
755# - I derived this result from the equation for the area of a spherical
756# triangle in terms of two edges and the included angle given by, e.g.
757# U{Todhunter, I. - Spherical Trigonometry (1871), Sec. 103, Eq. (2)
758# <http://Books.Google.com/books?id=3uBHAAAAIAAJ&pg=PA71>}
759# - I would be interested to know if this formula for S12 is already known
760# - Charles Karney
763def excessQuad(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
764 '''Compute the surface area of a (spherical) quadrilateral bounded by a segment
765 of a great circle, two meridians and the equator.
767 @arg lat1: Start latitude (C{degrees}).
768 @arg lon1: Start longitude (C{degrees}).
769 @arg lat2: End latitude (C{degrees}).
770 @arg lon2: End longitude (C{degrees}).
771 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
772 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
773 L{a_f2Tuple}) or C{None}.
774 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
775 B{C{lat2}} and B{C{lon2}} (C{bool}).
777 @return: Surface area, I{signed} (I{square} C{meter} or the same units as
778 B{C{radius}} I{squared}) or the I{spherical excess} (C{radians})
779 if C{B{radius}=0} or C{None}.
781 @raise TypeError: Invalid B{C{radius}}.
783 @raise UnitError: Invalid B{C{lat2}} or B{C{lat1}}.
785 @see: Function L{excessQuad_} and L{excessKarney}.
786 '''
787 return _eA(excessQuad_, radius, wrap, lat1, lon1, lat2, lon2)
790def excessQuad_(phi2, phi1, lam21):
791 '''Compute the I{spherical excess} C{E} of a (spherical) quadrilateral bounded
792 by a segment of a great circle, two meridians and the equator.
794 @arg phi2: End latitude (C{radians}).
795 @arg phi1: Start latitude (C{radians}).
796 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
798 @return: Spherical excess, I{signed} (C{radians}).
800 @see: Function L{excessQuad}, U{Spherical trigonometry
801 <https://WikiPedia.org/wiki/Spherical_trigonometry>}.
802 '''
803 s = sin((phi2 + phi1) * _0_5)
804 c = cos((phi2 - phi1) * _0_5)
805 return Radians(Quad=atan2(tan_2(lam21) * s, c) * _2_0)
808def flatLocal(lat1, lon1, lat2, lon2, datum=_WGS84, scaled=True, wrap=False):
809 '''Compute the distance between two (ellipsoidal) points using
810 the U{ellipsoidal Earth to plane projection<https://WikiPedia.org/
811 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>}
812 aka U{Hubeny<https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula.
814 @arg lat1: Start latitude (C{degrees}).
815 @arg lon1: Start longitude (C{degrees}).
816 @arg lat2: End latitude (C{degrees}).
817 @arg lon2: End longitude (C{degrees}).
818 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
819 L{Ellipsoid2} or L{a_f2Tuple}) to use.
820 @kwarg scaled: Scale prime_vertical by C{cos(B{phi})} (C{bool}),
821 see method L{pygeodesy.Ellipsoid.roc2_}.
822 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
823 B{C{lat2}} and B{C{lon2}} (C{bool}).
825 @return: Distance (C{meter}, same units as the B{C{datum}}'s
826 ellipsoid axes).
828 @raise TypeError: Invalid B{C{datum}}.
830 @note: The meridional and prime_vertical radii of curvature
831 are taken and scaled at the mean of both latitude.
833 @see: Functions L{flatLocal_} or L{hubeny_}, L{cosineLaw}, L{flatPolar},
834 L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
835 L{equirectangular}, L{euclidean}, L{haversine}, L{thomas},
836 L{vincentys}, method L{Ellipsoid.distance2} and U{local, flat
837 earth approximation<https://www.EdWilliams.org/avform.htm#flat>}.
838 '''
839 E = _ellipsoidal(datum, flatLocal)
840 return E._hubeny_2(*_d3(wrap, lat1, lon1, lat2, lon2),
841 scaled=scaled, squared=False) * E.a
843hubeny = flatLocal # PYCHOK for Karl Hubeny
846def flatLocal_(phi2, phi1, lam21, datum=_WGS84, scaled=True):
847 '''Compute the I{angular} distance between two (ellipsoidal) points using
848 the U{ellipsoidal Earth to plane projection<https://WikiPedia.org/
849 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>}
850 aka U{Hubeny<https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula.
852 @arg phi2: End latitude (C{radians}).
853 @arg phi1: Start latitude (C{radians}).
854 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
855 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
856 L{Ellipsoid2} or L{a_f2Tuple}) to use.
857 @kwarg scaled: Scale prime_vertical by C{cos(B{phi})} (C{bool}),
858 see method L{pygeodesy.Ellipsoid.roc2_}.
860 @return: Angular distance (C{radians}).
862 @raise TypeError: Invalid B{C{datum}}.
864 @note: The meridional and prime_vertical radii of curvature
865 are taken and scaled I{at the mean of both latitude}.
867 @see: Functions L{flatLocal} or L{hubeny}, L{cosineAndoyerLambert_},
868 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{flatPolar_},
869 L{equirectangular_}, L{euclidean_}, L{haversine_}, L{thomas_}
870 and L{vincentys_} and U{local, flat earth approximation
871 <https://www.EdWilliams.org/avform.htm#flat>}.
872 '''
873 E = _ellipsoidal(datum, flatLocal_)
874 return E._hubeny_2(phi2, phi1, lam21, scaled=scaled, squared=False)
876hubeny_ = flatLocal_ # PYCHOK for Karl Hubeny
879def flatPolar(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
880 '''Compute the distance between two (spherical) points using
881 the U{polar coordinate flat-Earth <https://WikiPedia.org/wiki/
882 Geographical_distance#Polar_coordinate_flat-Earth_formula>}
883 formula.
885 @arg lat1: Start latitude (C{degrees}).
886 @arg lon1: Start longitude (C{degrees}).
887 @arg lat2: End latitude (C{degrees}).
888 @arg lon2: End longitude (C{degrees}).
889 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
890 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
891 L{a_f2Tuple}) to use.
892 @kwarg wrap: If C{True}, wrap or I{normalize} and B{C{lat2}}
893 and B{C{lon2}} (C{bool}).
895 @return: Distance (C{meter}, same units as B{C{radius}} or the
896 ellipsoid or datum axes).
898 @raise TypeError: Invalid B{C{radius}}.
900 @see: Functions L{flatPolar_}, L{cosineAndoyerLambert},
901 L{cosineForsytheAndoyerLambert},L{cosineLaw},
902 L{flatLocal}/L{hubeny}, L{equirectangular},
903 L{euclidean}, L{haversine}, L{thomas} and
904 L{vincentys}.
905 '''
906 return _dS(flatPolar_, radius, wrap, lat1, lon1, lat2, lon2)
909def flatPolar_(phi2, phi1, lam21):
910 '''Compute the I{angular} distance between two (spherical) points
911 using the U{polar coordinate flat-Earth<https://WikiPedia.org/wiki/
912 Geographical_distance#Polar_coordinate_flat-Earth_formula>}
913 formula.
915 @arg phi2: End latitude (C{radians}).
916 @arg phi1: Start latitude (C{radians}).
917 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
919 @return: Angular distance (C{radians}).
921 @see: Functions L{flatPolar}, L{cosineAndoyerLambert_},
922 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
923 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
924 L{haversine_}, L{thomas_} and L{vincentys_}.
925 '''
926 a = fabs(PI_2 - phi1) # co-latitude
927 b = fabs(PI_2 - phi2) # co-latitude
928 if a < b:
929 a, b = b, a
930 if a < EPS0:
931 a = _0_0
932 elif b > 0:
933 b = b / a # /= chokes PyChecker
934 c = b * cos(lam21) * _2_0
935 c = fsumf_(_1_0, b**2, -fabs(c))
936 a *= sqrt0(c)
937 return a
940def hartzell(pov, los=None, earth=_WGS84, name=NN, **LatLon_and_kwds):
941 '''Compute the intersection of the earth's surface and a Line-Of-Sight
942 from a Point-Of-View in space.
944 @arg pov: Point-Of-View outside the earth (C{Cartesian}, L{Ecef9Tuple}
945 or L{Vector3d}).
946 @kwarg los: Line-Of-Sight, I{direction} to earth (L{Vector3d}) or
947 C{None} to point to the earth' center.
948 @kwarg earth: The earth model (L{Datum}, L{Ellipsoid}, L{Ellipsoid2},
949 L{a_f2Tuple} or C{scalar} radius in C{meter}).
950 @kwarg name: Optional name (C{str}).
951 @kwarg LatLon_and_kwds: Optional C{LatLon} class for the intersection
952 point plus C{LatLon} keyword arguments, include
953 B{C{datum}} if different from B{C{earth}}.
955 @return: The earth intersection (L{Vector3d}, C{Cartesian type} of
956 B{C{pov}} or B{C{LatLon}}).
958 @raise IntersectionError: Null B{C{pov}} or B{C{los}} vector, B{C{pov}}
959 is inside the earth or B{C{los}} points outside
960 the earth or points in an opposite direction.
962 @raise TypeError: Invalid B{C{pov}}, B{C{los}} or B{C{earth}}.
964 @see: Function L{pygeodesy.hartzell4}, L{pygeodesy.tyr3d} for B{C{los}},
965 method L{Ellipsoid.hartzell4} and U{I{Satellite Line-of-Sight
966 Intersection with Earth}<https://StephenHartzell.Medium.com/
967 satellite-line-of-sight-intersection-with-earth-d786b4a6a9b6>}.
968 '''
969 D = earth if isinstance(earth, Datum) else \
970 _spherical_datum(earth, name=hartzell.__name__)
971 try:
972 r, _ = _MODS.triaxials._hartzell3d2(pov, los, D.ellipsoid._triaxial)
973 except Exception as x:
974 raise IntersectionError(pov=pov, los=los, earth=earth, cause=x)
976# else:
977# E = D.ellipsoid
978# # Triaxial(a, b, c) == (E.a, E.a, E.b)
979#
980# def _Error(txt):
981# return IntersectionError(pov=pov, los=los, earth=earth, txt=txt)
982#
983# a2 = b2 = E.a2 # earth' x, y, ...
984# c2 = E.b2 # ... z semi-axis squared
985# q2 = E.b2_a2 # == c2 / a2
986# bc = E.a * E.b # == b * c
987#
988# V3 = _MODS.vector3d._otherV3d
989# p3 = V3(pov=pov)
990# u3 = V3(los=los) if los else p3.negate()
991# u3 = u3.unit() # unit vector, opposing signs
992#
993# x2, y2, z2 = p3.x2y2z2 # p3.times_(p3).xyz
994# ux, vy, wz = u3.times_(p3).xyz
995# u2, v2, w2 = u3.x2y2z2 # u3.times_(u3).xyz
996#
997# t = c2, c2, b2
998# m = fdot(t, u2, v2, w2) # a2 factored out
999# if m < EPS0: # zero or near-null LOS vector
1000# raise _Error(_near_(_null_))
1001#
1002# # a2 and b2 factored out, b2 == a2 and b2 / a2 == 1
1003# r = fsumf_(b2 * w2, c2 * v2, -v2 * z2, vy * wz * 2,
1004# c2 * u2, -u2 * z2, -w2 * x2, ux * wz * 2,
1005# -w2 * y2, -u2 * y2 * q2, -v2 * x2 * q2, ux * vy * 2 * q2)
1006# if r > 0:
1007# r = sqrt(r) * bc # == a * a * b * c / a2
1008# elif r < 0: # LOS pointing away from or missing the earth
1009# raise _Error(_opposite_ if max(ux, vy, wz) > 0 else _outside_)
1010#
1011# d = Fdot(t, ux, vy, wz).fadd_(r).fover(m) # -r for antipode, a2 factored out
1012# if d > 0: # POV inside or LOS missing, outside the earth
1013# s = fsumf_(_1_0, x2 / a2, y2 / b2, z2 / c2, _N_2_0) # like _sideOf
1014# raise _Error(_outside_ if s > 0 else _inside_)
1015# elif fsumf_(x2, y2, z2) < d**2: # d past earth center
1016# raise _Error(_too_(_distant_))
1017#
1018# r = p3.minus(u3.times(d))
1019# # h = p3.minus(r).length # distance to ellipsoid
1021 r = _xnamed(r, name or hartzell.__name__)
1022 if LatLon_and_kwds:
1023 c = _MODS.cartesianBase.CartesianBase(r, datum=D, name=r.name)
1024 r = c.toLatLon(**LatLon_and_kwds)
1025 return r
1028def haversine(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
1029 '''Compute the distance between two (spherical) points using the
1030 U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>}
1031 formula.
1033 @arg lat1: Start latitude (C{degrees}).
1034 @arg lon1: Start longitude (C{degrees}).
1035 @arg lat2: End latitude (C{degrees}).
1036 @arg lon2: End longitude (C{degrees}).
1037 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
1038 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
1039 L{a_f2Tuple}) to use.
1040 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
1041 B{C{lat2}} and B{C{lon2}} (C{bool}).
1043 @return: Distance (C{meter}, same units as B{C{radius}}).
1045 @raise TypeError: Invalid B{C{radius}}.
1047 @see: U{Distance between two (spherical) points
1048 <https://www.EdWilliams.org/avform.htm#Dist>}, functions
1049 L{cosineLaw}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
1050 L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar},
1051 L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2},
1052 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
1054 @note: See note at function L{vincentys_}.
1055 '''
1056 return _dS(haversine_, radius, wrap, lat1, lon1, lat2, lon2)
1059def haversine_(phi2, phi1, lam21):
1060 '''Compute the I{angular} distance between two (spherical) points
1061 using the U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>}
1062 formula.
1064 @arg phi2: End latitude (C{radians}).
1065 @arg phi1: Start latitude (C{radians}).
1066 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
1068 @return: Angular distance (C{radians}).
1070 @see: Functions L{haversine}, L{cosineAndoyerLambert_},
1071 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
1072 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
1073 L{flatPolar_}, L{thomas_} and L{vincentys_}.
1075 @note: See note at function L{vincentys_}.
1076 '''
1077 def _hsin(rad):
1078 return sin(rad * _0_5)**2
1080 h = _hsin(phi2 - phi1) + cos(phi1) * cos(phi2) * _hsin(lam21) # haversine
1081 return atan2(sqrt0(h), sqrt0(_1_0 - h)) * _2_0 # == asin(sqrt(h)) * 2
1084def heightOf(angle, distance, radius=R_M):
1085 '''Determine the height above the (spherical) earth' surface after
1086 traveling along a straight line at a given tilt.
1088 @arg angle: Tilt angle above horizontal (C{degrees}).
1089 @arg distance: Distance along the line (C{meter} or same units as
1090 B{C{radius}}).
1091 @kwarg radius: Optional mean earth radius (C{meter}).
1093 @return: Height (C{meter}, same units as B{C{distance}} and B{C{radius}}).
1095 @raise ValueError: Invalid B{C{angle}}, B{C{distance}} or B{C{radius}}.
1097 @see: U{MultiDop geog_lib.GeogBeamHt<https://GitHub.com/NASA/MultiDop>}
1098 (U{Shapiro et al. 2009, JTECH
1099 <https://Journals.AMetSoc.org/doi/abs/10.1175/2009JTECHA1256.1>}
1100 and U{Potvin et al. 2012, JTECH
1101 <https://Journals.AMetSoc.org/doi/abs/10.1175/JTECH-D-11-00019.1>}).
1102 '''
1103 r = h = Radius(radius)
1104 d = fabs(Distance(distance))
1105 if d > h:
1106 d, h = h, d
1108 if d > EPS0: # and h > EPS0
1109 d = d / h # /= h chokes PyChecker
1110 s = sin(Phi_(angle=angle, clip=_180_0))
1111 s = fsumf_(_1_0, _2_0 * s * d, d**2)
1112 if s > 0:
1113 return h * sqrt(s) - r
1115 raise _ValueError(angle=angle, distance=distance, radius=radius)
1118def horizon(height, radius=R_M, refraction=False):
1119 '''Determine the distance to the horizon from a given altitude
1120 above the (spherical) earth.
1122 @arg height: Altitude (C{meter} or same units as B{C{radius}}).
1123 @kwarg radius: Optional mean earth radius (C{meter}).
1124 @kwarg refraction: Consider atmospheric refraction (C{bool}).
1126 @return: Distance (C{meter}, same units as B{C{height}} and B{C{radius}}).
1128 @raise ValueError: Invalid B{C{height}} or B{C{radius}}.
1130 @see: U{Distance to horizon<https://www.EdWilliams.org/avform.htm#Horizon>}.
1131 '''
1132 h, r = Height(height), Radius(radius)
1133 if min(h, r) < 0:
1134 raise _ValueError(height=height, radius=radius)
1136 if refraction:
1137 d2 = 2.415750694528 * h * r # 2.0 / 0.8279
1138 else:
1139 d2 = h * fsumf_(r, r, h)
1140 return sqrt0(d2)
1143class _idllmn6(object): # see also .geodesicw._wargs, .vector2d._numpy
1144 '''(INTERNAL) Helper for C{intersection2} and C{intersections2}.
1145 '''
1146 @contextmanager # <https://www.python.org/dev/peps/pep-0343/> Examples
1147 def __call__(self, datum, lat1, lon1, lat2, lon2, small, wrap, s, **kwds):
1148 try:
1149 if wrap:
1150 _, lat2, lon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap)
1151 kwds = _xkwds(kwds, wrap=wrap) # for _xError
1152 m = small if small is _100km else Meter_(small=small)
1153 n = (intersections2 if s else intersection2).__name__
1154 if datum is None or euclidean(lat1, lon1, lat2, lon2) < m:
1155 d, m = None, _MODS.vector3d
1156 _i = m._intersects2 if s else m._intersect3d3
1157 elif isscalar(datum) and datum < 0 and not s:
1158 d = _spherical_datum(-datum, name=n)
1159 m = _MODS.sphericalNvector
1160 _i = m.intersection
1161 else:
1162 d = _spherical_datum(datum, name=n)
1163 if d.isSpherical:
1164 m = _MODS.sphericalTrigonometry
1165 _i = m._intersects2 if s else m._intersect
1166 elif d.isEllipsoidal:
1167 try:
1168 if d.ellipsoid.geodesic:
1169 pass
1170 m = _MODS.ellipsoidalKarney
1171 except ImportError:
1172 m = _MODS.ellipsoidalExact
1173 _i = m._intersections2 if s else m._intersection3 # ellispoidalBaseDI
1174 else:
1175 raise _TypeError(datum=datum)
1176 yield _i, d, lat2, lon2, m, n
1178 except (TypeError, ValueError) as x:
1179 raise _xError(x, lat1=lat1, lon1=lon1, datum=datum,
1180 lat2=lat2, lon2=lon2, small=small, **kwds)
1182_idllmn6 = _idllmn6() # PYCHOK singleton
1185def intersection2(lat1, lon1, bearing1,
1186 lat2, lon2, bearing2, datum=None, wrap=False, small=_100km): # was=True
1187 '''I{Conveniently} compute the intersection of two lines each defined
1188 by a (geodetic) point and a bearing from North, using either ...
1190 1) L{vector3d.intersection3d3} for B{C{small}} distances (below 100 Km
1191 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ...
1193 2) L{sphericalTrigonometry.intersection} for a spherical B{C{datum}}
1194 or a C{scalar B{datum}} representing the earth radius, conventionally
1195 in C{meter} or ...
1197 3) L{sphericalNvector.intersection} if B{C{datum}} is a I{negative}
1198 C{scalar}, (negative) earth radius, conventionally in C{meter} or ...
1200 4) L{ellipsoidalKarney.intersection3} for an ellipsoidal B{C{datum}}
1201 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>}
1202 is installed, otherwise ...
1204 5) L{ellipsoidalExact.intersection3}, provided B{C{datum}} is ellipsoidal.
1206 @arg lat1: Latitude of the first point (C{degrees}).
1207 @arg lon1: Longitude of the first point (C{degrees}).
1208 @arg bearing1: Bearing at the first point (compass C{degrees}).
1209 @arg lat2: Latitude of the second point (C{degrees}).
1210 @arg lon2: Longitude of the second point (C{degrees}).
1211 @arg bearing2: Bearing at the second point (compass C{degrees}).
1212 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid},
1213 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth
1214 radius (C{meter}, same units as B{C{radius1}} and
1215 B{C{radius2}}) or C{None}.
1216 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
1217 and B{C{lon2}} (C{bool}).
1218 @kwarg small: Upper limit for small distances (C{meter}).
1220 @return: A L{LatLon2Tuple}C{(lat, lon)} with the lat- and
1221 longitude of the intersection point.
1223 @raise IntersectionError: Ambiguous or infinite intersection
1224 or colinear, parallel or otherwise
1225 non-intersecting lines.
1227 @raise TypeError: Invalid B{C{datum}}.
1229 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{bearing1}},
1230 B{C{lat2}}, B{C{lon2}} or B{C{bearing2}}.
1232 @see: Method L{RhumbLine.intersection2}.
1234 @note: The returned intersections may be near-antipodal.
1235 '''
1236 b1 = Bearing(bearing1=bearing1)
1237 b2 = Bearing(bearing2=bearing2)
1238 with _idllmn6(datum, lat1, lon1, lat2, lon2,
1239 small, wrap, False, bearing1=b1, bearing2=b2) as t:
1240 _i, d, lat2, lon2, m, n = t
1241 if d is None:
1242 t, _, _ = _i(m.Vector3d(lon1, lat1, 0), b1,
1243 m.Vector3d(lon2, lat2, 0), b2, useZ=False)
1244 t = LatLon2Tuple(t.y, t.x, name=n)
1246 else:
1247 t = _i(m.LatLon(lat1, lon1, datum=d), b1,
1248 m.LatLon(lat2, lon2, datum=d), b2, height=0, wrap=False)
1249 if isinstance(t, Intersection3Tuple): # ellipsoidal
1250 t, _, _ = t
1251 t = LatLon2Tuple(t.lat, t.lon, name=n)
1252 return t
1255def intersections2(lat1, lon1, radius1,
1256 lat2, lon2, radius2, datum=None, wrap=False, small=_100km): # was=True
1257 '''I{Conveniently} compute the intersections of two circles each defined
1258 by a (geodetic) center point and a radius, using either ...
1260 1) L{vector3d.intersections2} for B{C{small}} distances (below 100 Km
1261 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ...
1263 2) L{sphericalTrigonometry.intersections2} for a spherical B{C{datum}}
1264 or a C{scalar B{datum}} representing the earth radius, conventionally
1265 in C{meter} or ...
1267 3) L{ellipsoidalKarney.intersections2} for an ellipsoidal B{C{datum}}
1268 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>}
1269 is installed, otherwise ...
1271 4) L{ellipsoidalExact.intersections2}, provided B{C{datum}} is ellipsoidal.
1273 @arg lat1: Latitude of the first circle center (C{degrees}).
1274 @arg lon1: Longitude of the first circle center (C{degrees}).
1275 @arg radius1: Radius of the first circle (C{meter}, conventionally).
1276 @arg lat2: Latitude of the second circle center (C{degrees}).
1277 @arg lon2: Longitude of the second circle center (C{degrees}).
1278 @arg radius2: Radius of the second circle (C{meter}, same units as B{C{radius1}}).
1279 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid},
1280 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth
1281 radius (C{meter}, same units as B{C{radius1}} and
1282 B{C{radius2}}) or C{None}.
1283 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
1284 and B{C{lon2}} (C{bool}).
1285 @kwarg small: Upper limit for small distances (C{meter}).
1287 @return: 2-Tuple of the intersection points, each a
1288 L{LatLon2Tuple}C{(lat, lon)}. For abutting circles, the
1289 points are the same instance, aka the I{radical center}.
1291 @raise IntersectionError: Concentric, antipodal, invalid or
1292 non-intersecting circles or no
1293 convergence.
1295 @raise TypeError: Invalid B{C{datum}}.
1297 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{radius1}},
1298 B{C{lat2}}, B{C{lon2}} or B{C{radius2}}.
1299 '''
1300 r1 = Radius_(radius1=radius1)
1301 r2 = Radius_(radius2=radius2)
1302 with _idllmn6(datum, lat1, lon1, lat2, lon2,
1303 small, wrap, True, radius1=r1, radius2=r2) as t:
1304 _i, d, lat2, lon2, m, n = t
1305 if d is None:
1306 r1 = m2degrees(r1, radius=R_M, lat=lat1)
1307 r2 = m2degrees(r2, radius=R_M, lat=lat2)
1309 def _V2T(x, y, _, **unused): # _ == z unused
1310 return LatLon2Tuple(y, x, name=n)
1312 t = _i(m.Vector3d(lon1, lat1, 0), r1,
1313 m.Vector3d(lon2, lat2, 0), r2, sphere=False,
1314 Vector=_V2T)
1315 else:
1316 def _LL2T(lat, lon, **unused):
1317 return LatLon2Tuple(lat, lon, name=n)
1319 t = _i(m.LatLon(lat1, lon1, datum=d), r1,
1320 m.LatLon(lat2, lon2, datum=d), r2,
1321 LatLon=_LL2T, height=0, wrap=False)
1322 return t
1325def isantipode(lat1, lon1, lat2, lon2, eps=EPS):
1326 '''Check whether two points are I{antipodal}, on diametrically
1327 opposite sides of the earth.
1329 @arg lat1: Latitude of one point (C{degrees}).
1330 @arg lon1: Longitude of one point (C{degrees}).
1331 @arg lat2: Latitude of the other point (C{degrees}).
1332 @arg lon2: Longitude of the other point (C{degrees}).
1333 @kwarg eps: Tolerance for near-equality (C{degrees}).
1335 @return: C{True} if points are antipodal within the
1336 B{C{eps}} tolerance, C{False} otherwise.
1338 @see: Functions L{isantipode_} and L{antipode}.
1339 '''
1340 return (fabs(lat1 + lat2) <= eps and
1341 fabs(lon1 + lon2) <= eps) or _isequalTo(
1342 normal(lat1, lon1), antipode(lat2, lon2), eps)
1345def isantipode_(phi1, lam1, phi2, lam2, eps=EPS):
1346 '''Check whether two points are I{antipodal}, on diametrically
1347 opposite sides of the earth.
1349 @arg phi1: Latitude of one point (C{radians}).
1350 @arg lam1: Longitude of one point (C{radians}).
1351 @arg phi2: Latitude of the other point (C{radians}).
1352 @arg lam2: Longitude of the other point (C{radians}).
1353 @kwarg eps: Tolerance for near-equality (C{radians}).
1355 @return: C{True} if points are antipodal within the
1356 B{C{eps}} tolerance, C{False} otherwise.
1358 @see: Functions L{isantipode} and L{antipode_}.
1359 '''
1360 return (fabs(phi1 + phi2) <= eps and
1361 fabs(lam1 + lam2) <= eps) or _isequalTo_(
1362 normal_(phi1, lam1), antipode_(phi2, lam2), eps)
1365def _isequalTo(p1, p2, eps=EPS):
1366 '''Compare 2 point lat-/lons ignoring C{class}.
1367 '''
1368 return (fabs(p1.lat - p2.lat) <= eps and
1369 fabs(p1.lon - p2.lon) <= eps) if eps else (p1.latlon == p2.latlon)
1372def _isequalTo_(p1, p2, eps=EPS):
1373 '''(INTERNAL) Compare 2 point phi-/lams ignoring C{class}.
1374 '''
1375 return (fabs(p1.phi - p2.phi) <= eps and
1376 fabs(p1.lam - p2.lam) <= eps) if eps else (p1.philam == p2.philam)
1379def isnormal(lat, lon, eps=0):
1380 '''Check whether B{C{lat}} I{and} B{C{lon}} are within their
1381 respective I{normal} range in C{degrees}.
1383 @arg lat: Latitude (C{degrees}).
1384 @arg lon: Longitude (C{degrees}).
1385 @kwarg eps: Optional tolerance C{degrees}).
1387 @return: C{True} if C{(abs(B{lat}) + B{eps}) <= 90} and
1388 C{(abs(B{lon}) + B{eps}) <= 180}, C{False} othwerwise.
1390 @see: Functions L{isnormal_} and L{normal}.
1391 '''
1392 return (_90_0 - fabs(lat)) >= eps and (_180_0 - fabs(lon)) >= eps
1395def isnormal_(phi, lam, eps=0):
1396 '''Check whether B{C{phi}} I{and} B{C{lam}} are within their
1397 respective I{normal} range in C{radians}.
1399 @arg phi: Latitude (C{radians}).
1400 @arg lam: Longitude (C{radians}).
1401 @kwarg eps: Optional tolerance C{radians}).
1403 @return: C{True} if C{(abs(B{phi}) + B{eps}) <= PI/2} and
1404 C{(abs(B{lam}) + B{eps}) <= PI}, C{False} othwerwise.
1406 @see: Functions L{isnormal} and L{normal_}.
1407 '''
1408 return (PI_2 - fabs(phi)) >= eps and (PI - fabs(lam)) >= eps
1411def latlon2n_xyz(lat, lon, name=NN):
1412 '''Convert lat-, longitude to C{n-vector} (I{normal} to the
1413 earth's surface) X, Y and Z components.
1415 @arg lat: Latitude (C{degrees}).
1416 @arg lon: Longitude (C{degrees}).
1417 @kwarg name: Optional name (C{str}).
1419 @return: A L{Vector3Tuple}C{(x, y, z)}.
1421 @see: Function L{philam2n_xyz}.
1423 @note: These are C{n-vector} x, y and z components,
1424 I{NOT} geocentric ECEF x, y and z coordinates!
1425 '''
1426 return _2n_xyz(name, *sincos2d_(lat, lon))
1429def _normal2(a, b, n_2, n, n2):
1430 '''(INTERNAL) Helper for C{normal} and C{normal_}.
1431 '''
1432 if fabs(b) > n:
1433 b = remainder(b, n2)
1434 if fabs(a) > n_2:
1435 r = remainder(a, n)
1436 if r != a:
1437 a = -r
1438 b -= n if b > 0 else -n
1439 return float0_(a, b)
1442def normal(lat, lon, name=NN):
1443 '''Normalize a lat- I{and} longitude pair in C{degrees}.
1445 @arg lat: Latitude (C{degrees}).
1446 @arg lon: Longitude (C{degrees}).
1447 @kwarg name: Optional name (C{str}).
1449 @return: L{LatLon2Tuple}C{(lat, lon)} with C{abs(lat) <= 90}
1450 and C{abs(lon) <= 180}.
1452 @see: Functions L{normal_} and L{isnormal}.
1453 '''
1454 return LatLon2Tuple(*_normal2(lat, lon, _90_0, _180_0, _360_0),
1455 name=name or normal.__name__)
1458def normal_(phi, lam, name=NN):
1459 '''Normalize a lat- I{and} longitude pair in C{radians}.
1461 @arg phi: Latitude (C{radians}).
1462 @arg lam: Longitude (C{radians}).
1463 @kwarg name: Optional name (C{str}).
1465 @return: L{PhiLam2Tuple}C{(phi, lam)} with C{abs(phi) <= PI/2}
1466 and C{abs(lam) <= PI}.
1468 @see: Functions L{normal} and L{isnormal_}.
1469 '''
1470 return PhiLam2Tuple(*_normal2(phi, lam, PI_2, PI, PI2),
1471 name=name or normal_.__name__)
1474def _2n_xyz(name, sa, ca, sb, cb):
1475 '''(INTERNAL) Helper for C{latlon2n_xyz} and C{philam2n_xyz}.
1476 '''
1477 # Kenneth Gade eqn 3, but using right-handed
1478 # vector x -> 0°E,0°N, y -> 90°E,0°N, z -> 90°N
1479 return Vector3Tuple(ca * cb, ca * sb, sa, name=name)
1482def n_xyz2latlon(x, y, z, name=NN):
1483 '''Convert C{n-vector} components to lat- and longitude in C{degrees}.
1485 @arg x: X component (C{scalar}).
1486 @arg y: Y component (C{scalar}).
1487 @arg z: Z component (C{scalar}).
1488 @kwarg name: Optional name (C{str}).
1490 @return: A L{LatLon2Tuple}C{(lat, lon)}.
1492 @see: Function L{n_xyz2philam}.
1493 '''
1494 return LatLon2Tuple(atan2d(z, hypot(x, y)), atan2d(y, x), name=name)
1497def n_xyz2philam(x, y, z, name=NN):
1498 '''Convert C{n-vector} components to lat- and longitude in C{radians}.
1500 @arg x: X component (C{scalar}).
1501 @arg y: Y component (C{scalar}).
1502 @arg z: Z component (C{scalar}).
1503 @kwarg name: Optional name (C{str}).
1505 @return: A L{PhiLam2Tuple}C{(phi, lam)}.
1507 @see: Function L{n_xyz2latlon}.
1508 '''
1509 return PhiLam2Tuple(atan2(z, hypot(x, y)), atan2(y, x), name=name)
1512def _opposes(d, m, n, n2):
1513 '''(INTERNAL) Helper for C{opposing} and C{opposing_}.
1514 '''
1515 d = d % n2 # -20 % 360 == 340, -1 % PI2 == PI2 - 1
1516 return False if d < m or d > (n2 - m) else (
1517 True if (n - m) < d < (n + m) else None)
1520def opposing(bearing1, bearing2, margin=_90_0):
1521 '''Compare the direction of two bearings given in C{degrees}.
1523 @arg bearing1: First bearing (compass C{degrees}).
1524 @arg bearing2: Second bearing (compass C{degrees}).
1525 @kwarg margin: Optional, interior angle bracket (C{degrees}).
1527 @return: C{True} if both bearings point in opposite, C{False} if
1528 in similar or C{None} if in perpendicular directions.
1530 @see: Function L{opposing_}.
1531 '''
1532 m = Degrees_(margin=margin, low=EPS0, high=_90_0)
1533 return _opposes(bearing2 - bearing1, m, _180_0, _360_0)
1536def opposing_(radians1, radians2, margin=PI_2):
1537 '''Compare the direction of two bearings given in C{radians}.
1539 @arg radians1: First bearing (C{radians}).
1540 @arg radians2: Second bearing (C{radians}).
1541 @kwarg margin: Optional, interior angle bracket (C{radians}).
1543 @return: C{True} if both bearings point in opposite, C{False} if
1544 in similar or C{None} if in perpendicular directions.
1546 @see: Function L{opposing}.
1547 '''
1548 m = Radians_(margin=margin, low=EPS0, high=PI_2)
1549 return _opposes(radians2 - radians1, m, PI, PI2)
1552def philam2n_xyz(phi, lam, name=NN):
1553 '''Convert lat-, longitude to C{n-vector} (I{normal} to the
1554 earth's surface) X, Y and Z components.
1556 @arg phi: Latitude (C{radians}).
1557 @arg lam: Longitude (C{radians}).
1558 @kwarg name: Optional name (C{str}).
1560 @return: A L{Vector3Tuple}C{(x, y, z)}.
1562 @see: Function L{latlon2n_xyz}.
1564 @note: These are C{n-vector} x, y and z components,
1565 I{NOT} geocentric ECEF x, y and z coordinates!
1566 '''
1567 return _2n_xyz(name, *sincos2_(phi, lam))
1570def _radical2(d, r1, r2): # in .ellipsoidalBaseDI, .sphericalTrigonometry, .vector3d
1571 # (INTERNAL) See C{radical2} below
1572 # assert d > EPS0
1573 r = fsumf_(_1_0, (r1 / d)**2, -(r2 / d)**2) * _0_5
1574 return Radical2Tuple(max(_0_0, min(_1_0, r)), r * d)
1577def radical2(distance, radius1, radius2):
1578 '''Compute the I{radical ratio} and I{radical line} of two
1579 U{intersecting circles<https://MathWorld.Wolfram.com/
1580 Circle-CircleIntersection.html>}.
1582 The I{radical line} is perpendicular to the axis thru the
1583 centers of the circles at C{(0, 0)} and C{(B{distance}, 0)}.
1585 @arg distance: Distance between the circle centers (C{scalar}).
1586 @arg radius1: Radius of the first circle (C{scalar}).
1587 @arg radius2: Radius of the second circle (C{scalar}).
1589 @return: A L{Radical2Tuple}C{(ratio, xline)} where C{0.0 <=
1590 ratio <= 1.0} and C{xline} is along the B{C{distance}}.
1592 @raise IntersectionError: The B{C{distance}} exceeds the sum
1593 of B{C{radius1}} and B{C{radius2}}.
1595 @raise UnitError: Invalid B{C{distance}}, B{C{radius1}} or
1596 B{C{radius2}}.
1598 @see: U{Circle-Circle Intersection
1599 <https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>}.
1600 '''
1601 d = Distance_(distance, low=_0_0)
1602 r1 = Radius_(radius1=radius1)
1603 r2 = Radius_(radius2=radius2)
1604 if d > (r1 + r2):
1605 raise IntersectionError(distance=d, radius1=r1, radius2=r2,
1606 txt=_too_(_distant_))
1607 return _radical2(d, r1, r2) if d > EPS0 else \
1608 Radical2Tuple(_0_5, _0_0)
1611class Radical2Tuple(_NamedTuple):
1612 '''2-Tuple C{(ratio, xline)} of the I{radical} C{ratio} and
1613 I{radical} C{xline}, both C{scalar} and C{0.0 <= ratio <= 1.0}
1614 '''
1615 _Names_ = (_ratio_, _xline_)
1616 _Units_ = ( Scalar, Scalar)
1619def _radistance(inst):
1620 '''(INTERNAL) Helper for the L{frechet._FrecherMeterRadians}
1621 and L{hausdorff._HausdorffMeterRedians} classes.
1622 '''
1623 kwds_ = _xkwds(inst._kwds, wrap=False)
1624 wrap_ = _xkwds_pop(kwds_, wrap=False)
1625 func_ = inst._func_
1626 try: # calling lower-overhead C{func_}
1627 func_(0, _0_25, _0_5, **kwds_)
1628 wrap_ = _Wrap._philamop(wrap_)
1629 except TypeError:
1630 return inst.distance
1632 def _philam(p):
1633 try:
1634 return p.phi, p.lam
1635 except AttributeError: # no .phi or .lam
1636 return radians(p.lat), radians(p.lon)
1638 def _func_wrap(point1, point2):
1639 phi1, lam1 = wrap_(*_philam(point1))
1640 phi2, lam2 = wrap_(*_philam(point2))
1641 return func_(phi2, phi1, lam2 - lam1, **kwds_)
1643 inst._units = inst._units_
1644 return _func_wrap
1647def _scale_deg(lat1, lat2): # degrees
1648 # scale factor cos(mean of lats) for delta lon
1649 m = fabs(lat1 + lat2) * _0_5
1650 return cos(radians(m)) if m < 90 else _0_0
1653def _scale_rad(phi1, phi2): # radians, by .frechet, .hausdorff, .heights
1654 # scale factor cos(mean of phis) for delta lam
1655 m = fabs(phi1 + phi2) * _0_5
1656 return cos(m) if m < PI_2 else _0_0
1659def _sincosa6(phi2, phi1, lam21): # [4] in cosineLaw
1660 '''(INTERNAL) C{sin}es, C{cos}ines and C{acos}ine.
1661 '''
1662 s2, c2, s1, c1, _, c21 = sincos2_(phi2, phi1, lam21)
1663 return s2, c2, s1, c1, acos1(s1 * s2 + c1 * c2 * c21), c21
1666def thomas(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):
1667 '''Compute the distance between two (ellipsoidal) points using
1668 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>}
1669 formula.
1671 @arg lat1: Start latitude (C{degrees}).
1672 @arg lon1: Start longitude (C{degrees}).
1673 @arg lat2: End latitude (C{degrees}).
1674 @arg lon2: End longitude (C{degrees}).
1675 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
1676 L{Ellipsoid2} or L{a_f2Tuple}) to use.
1677 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
1678 B{C{lat2}} and B{C{lon2}} (C{bool}).
1680 @return: Distance (C{meter}, same units as the B{C{datum}}'s
1681 ellipsoid axes).
1683 @raise TypeError: Invalid B{C{datum}}.
1685 @see: Functions L{thomas_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
1686 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny},
1687 L{flatPolar}, L{haversine}, L{vincentys} and method L{Ellipsoid.distance2}.
1688 '''
1689 return _dE(thomas_, datum, wrap, lat1, lon1, lat2, lon2)
1692def thomas_(phi2, phi1, lam21, datum=_WGS84):
1693 '''Compute the I{angular} distance between two (ellipsoidal) points using
1694 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>}
1695 formula.
1697 @arg phi2: End latitude (C{radians}).
1698 @arg phi1: Start latitude (C{radians}).
1699 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
1700 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid},
1701 L{Ellipsoid2} or L{a_f2Tuple}).
1703 @return: Angular distance (C{radians}).
1705 @raise TypeError: Invalid B{C{datum}}.
1707 @see: Functions L{thomas}, L{cosineAndoyerLambert_},
1708 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
1709 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
1710 L{flatPolar_}, L{haversine_} and L{vincentys_} and U{Geodesy-PHP
1711 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/
1712 Distance/ThomasFormula.php>}.
1713 '''
1714 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21)
1715 if r and isnon0(c1) and isnon0(c2):
1716 E = _ellipsoidal(datum, thomas_)
1717 if E.f:
1718 r1 = atan2(E.b_a * s1, c1)
1719 r2 = atan2(E.b_a * s2, c2)
1721 j = (r2 + r1) * _0_5
1722 k = (r2 - r1) * _0_5
1723 sj, cj, sk, ck, h, _ = sincos2_(j, k, lam21 * _0_5)
1725 h = fsumf_(sk**2, (ck * h)**2, -(sj * h)**2)
1726 u = _1_0 - h
1727 if isnon0(u) and isnon0(h):
1728 r = atan(sqrt0(h / u)) * 2 # == acos(1 - 2 * h)
1729 sr, cr = sincos2(r)
1730 if isnon0(sr):
1731 u = 2 * (sj * ck)**2 / u
1732 h = 2 * (sk * cj)**2 / h
1733 x = u + h
1734 y = u - h
1736 s = r / sr
1737 e = 4 * s**2
1738 d = 2 * cr
1739 a = e * d
1740 b = 2 * r
1741 c = s - (a - d) * _0_5
1742 f = E.f * _0_25
1744 t = fsumf_(a * x, -b * y, c * x**2, -d * y**2, e * x * y)
1745 r -= fsumf_(s * x, -y, -t * f * _0_25) * f * sr
1746 return r
1749def vincentys(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
1750 '''Compute the distance between two (spherical) points using
1751 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>}
1752 spherical formula.
1754 @arg lat1: Start latitude (C{degrees}).
1755 @arg lon1: Start longitude (C{degrees}).
1756 @arg lat2: End latitude (C{degrees}).
1757 @arg lon2: End longitude (C{degrees}).
1758 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
1759 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
1760 L{a_f2Tuple}) to use.
1761 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
1762 B{C{lat2}} and B{C{lon2}} (C{bool}).
1764 @return: Distance (C{meter}, same units as B{C{radius}}).
1766 @raise UnitError: Invalid B{C{radius}}.
1768 @see: Functions L{vincentys_}, L{cosineAndoyerLambert},
1769 L{cosineForsytheAndoyerLambert},L{cosineLaw}, L{equirectangular},
1770 L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar},
1771 L{haversine} and L{thomas} and methods L{Ellipsoid.distance2},
1772 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
1774 @note: See note at function L{vincentys_}.
1775 '''
1776 return _dS(vincentys_, radius, wrap, lat1, lon1, lat2, lon2)
1779def vincentys_(phi2, phi1, lam21):
1780 '''Compute the I{angular} distance between two (spherical) points using
1781 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>}
1782 spherical formula.
1784 @arg phi2: End latitude (C{radians}).
1785 @arg phi1: Start latitude (C{radians}).
1786 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
1788 @return: Angular distance (C{radians}).
1790 @see: Functions L{vincentys}, L{cosineAndoyerLambert_},
1791 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
1792 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
1793 L{flatPolar_}, L{haversine_} and L{thomas_}.
1795 @note: Functions L{vincentys_}, L{haversine_} and L{cosineLaw_}
1796 produce equivalent results, but L{vincentys_} is suitable
1797 for antipodal points and slightly more expensive (M{3 cos,
1798 3 sin, 1 hypot, 1 atan2, 6 mul, 2 add}) than L{haversine_}
1799 (M{2 cos, 2 sin, 2 sqrt, 1 atan2, 5 mul, 1 add}) and
1800 L{cosineLaw_} (M{3 cos, 3 sin, 1 acos, 3 mul, 1 add}).
1801 '''
1802 s1, c1, s2, c2, s21, c21 = sincos2_(phi1, phi2, lam21)
1804 c = c2 * c21
1805 x = s1 * s2 + c1 * c
1806 y = c1 * s2 - s1 * c
1807 return atan2(hypot(c2 * s21, y), x)
1809# **) MIT License
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