Coverage for pygeodesy/formy.py: 99%
421 statements
« prev ^ index » next coverage.py v7.2.2, created at 2023-09-15 09:43 -0400
« prev ^ index » next coverage.py v7.2.2, created at 2023-09-15 09:43 -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_, _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__ = '23.09.06'
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
193 U{Andoyer-Lambert correction<https://NavLib.net/wp-content/uploads/2013/10/
194 admiralty-manual-of-navigation-vol-1-1964-english501c.pdf>} of the U{Law of
195 Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula.
197 @arg lat1: Start latitude (C{degrees}).
198 @arg lon1: Start longitude (C{degrees}).
199 @arg lat2: End latitude (C{degrees}).
200 @arg lon2: End longitude (C{degrees}).
201 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
202 L{Ellipsoid2} or L{a_f2Tuple}) to use.
203 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
204 B{C{lat2}} and B{C{lon2}} (C{bool}).
206 @return: Distance (C{meter}, same units as the B{C{datum}}'s
207 ellipsoid axes or C{radians} if B{C{datum}} is C{None}).
209 @raise TypeError: Invalid B{C{datum}}.
211 @see: Functions L{cosineAndoyerLambert_}, L{cosineForsytheAndoyerLambert},
212 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny},
213 L{flatPolar}, L{haversine}, L{thomas} and L{vincentys} and method
214 L{Ellipsoid.distance2}.
215 '''
216 return _dE(cosineAndoyerLambert_, datum, wrap, lat1, lon1, lat2, lon2)
219def cosineAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84):
220 '''Compute the I{angular} distance between two (ellipsoidal) points using the
221 U{Andoyer-Lambert correction<https://NavLib.net/wp-content/uploads/2013/10/
222 admiralty-manual-of-navigation-vol-1-1964-english501c.pdf>} of the U{Law of
223 Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula.
225 @arg phi2: End latitude (C{radians}).
226 @arg phi1: Start latitude (C{radians}).
227 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
228 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
229 L{Ellipsoid2} or L{a_f2Tuple}) to use.
231 @return: Angular distance (C{radians}).
233 @raise TypeError: Invalid B{C{datum}}.
235 @see: Functions L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert_},
236 L{cosineLaw_}, L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
237 L{flatPolar_}, L{haversine_}, L{thomas_} and L{vincentys_} and U{Geodesy-PHP
238 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/Distance/
239 AndoyerLambert.php>}.
240 '''
241 s2, c2, s1, c1, r, c21 = _sincosa6(phi2, phi1, lam21)
242 if isnon0(c1) and isnon0(c2):
243 E = _ellipsoidal(datum, cosineAndoyerLambert_)
244 if E.f: # ellipsoidal
245 r2 = atan2(E.b_a * s2, c2)
246 r1 = atan2(E.b_a * s1, c1)
247 s2, c2, s1, c1 = sincos2_(r2, r1)
248 r = acos1(s1 * s2 + c1 * c2 * c21)
249 if r:
250 sr, _, sr_2, cr_2 = sincos2_(r, r * _0_5)
251 if isnon0(sr_2) and isnon0(cr_2):
252 s = (sr + r) * ((s1 - s2) / sr_2)**2
253 c = (sr - r) * ((s1 + s2) / cr_2)**2
254 r += (c - s) * E.f * _0_125
255 return r
258def cosineForsytheAndoyerLambert(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):
259 '''Compute the distance between two (ellipsoidal) points using the
260 U{Forsythe-Andoyer-Lambert correction<https://www2.UNB.Ca/gge/Pubs/TR77.pdf>} of
261 the U{Law of Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>}
262 formula.
264 @arg lat1: Start latitude (C{degrees}).
265 @arg lon1: Start longitude (C{degrees}).
266 @arg lat2: End latitude (C{degrees}).
267 @arg lon2: End longitude (C{degrees}).
268 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
269 L{Ellipsoid2} or L{a_f2Tuple}) to use.
270 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
271 B{C{lat2}} and B{C{lon2}} (C{bool}).
273 @return: Distance (C{meter}, same units as the B{C{datum}}'s
274 ellipsoid axes or C{radians} if B{C{datum}} is C{None}).
276 @raise TypeError: Invalid B{C{datum}}.
278 @see: Functions L{cosineForsytheAndoyerLambert_}, L{cosineAndoyerLambert},
279 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny},
280 L{flatPolar}, L{haversine}, L{thomas} and L{vincentys} and method
281 L{Ellipsoid.distance2}.
282 '''
283 return _dE(cosineForsytheAndoyerLambert_, datum, wrap, lat1, lon1, lat2, lon2)
286def cosineForsytheAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84):
287 '''Compute the I{angular} distance between two (ellipsoidal) points using the
288 U{Forsythe-Andoyer-Lambert correction<https://www2.UNB.Ca/gge/Pubs/TR77.pdf>} of
289 the U{Law of Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>}
290 formula.
292 @arg phi2: End latitude (C{radians}).
293 @arg phi1: Start latitude (C{radians}).
294 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
295 @kwarg datum: Datum (L{Datum}) or ellipsoid to use (L{Ellipsoid},
296 L{Ellipsoid2} or L{a_f2Tuple}).
298 @return: Angular distance (C{radians}).
300 @raise TypeError: Invalid B{C{datum}}.
302 @see: Functions L{cosineForsytheAndoyerLambert}, L{cosineAndoyerLambert_},
303 L{cosineLaw_}, L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
304 L{flatPolar_}, L{haversine_}, L{thomas_} and L{vincentys_} and U{Geodesy-PHP
305 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/
306 Distance/ForsytheCorrection.php>}.
307 '''
308 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21)
309 if r and isnon0(c1) and isnon0(c2):
310 E = _ellipsoidal(datum, cosineForsytheAndoyerLambert_)
311 if E.f: # ellipsoidal
312 sr, cr, s2r, _ = sincos2_(r, r * 2)
313 if isnon0(sr) and fabs(cr) < EPS1:
314 s = (s1 + s2)**2 / (1 + cr)
315 t = (s1 - s2)**2 / (1 - cr)
316 x = s + t
317 y = s - t
319 s = 8 * r**2 / sr
320 a = 64 * r + s * cr * 2 # 16 * r**2 / tan(r)
321 d = 48 * sr + s # 8 * r**2 / tan(r)
322 b = -2 * d
323 e = 30 * s2r
324 c = fsumf_(30 * r, e * _0_5, s * cr) # 8 * r**2 / tan(r)
325 t = fsumf_( a * x, e * y**2, b * y, -c * x**2, d * x * y)
327 r += fsumf_(-r * x, 3 * y * sr, t * E.f / _32_0) * E.f * _0_25
328 return r
331def cosineLaw(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
332 '''Compute the distance between two points using the U{spherical Law of
333 Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>}
334 formula.
336 @arg lat1: Start latitude (C{degrees}).
337 @arg lon1: Start longitude (C{degrees}).
338 @arg lat2: End latitude (C{degrees}).
339 @arg lon2: End longitude (C{degrees}).
340 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
341 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
342 L{a_f2Tuple}) to use.
343 @kwarg wrap: If C{True}, wrap or I{normalize} and B{C{lat2}}
344 and B{C{lon2}} (C{bool}).
346 @return: Distance (C{meter}, same units as B{C{radius}} or the
347 ellipsoid or datum axes).
349 @raise TypeError: Invalid B{C{radius}}.
351 @see: Functions L{cosineLaw_}, L{cosineAndoyerLambert},
352 L{cosineForsytheAndoyerLambert}, L{equirectangular}, L{euclidean},
353 L{flatLocal}/L{hubeny}, L{flatPolar}, L{haversine}, L{thomas} and
354 L{vincentys} and method L{Ellipsoid.distance2}.
356 @note: See note at function L{vincentys_}.
357 '''
358 return _dS(cosineLaw_, radius, wrap, lat1, lon1, lat2, lon2)
361def cosineLaw_(phi2, phi1, lam21):
362 '''Compute the I{angular} distance between two points using the U{spherical
363 Law of Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>}
364 formula.
366 @arg phi2: End latitude (C{radians}).
367 @arg phi1: Start latitude (C{radians}).
368 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
370 @return: Angular distance (C{radians}).
372 @see: Functions L{cosineLaw}, L{cosineAndoyerLambert_},
373 L{cosineForsytheAndoyerLambert_}, L{equirectangular_},
374 L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_},
375 L{haversine_}, L{thomas_} and L{vincentys_}.
377 @note: See note at function L{vincentys_}.
378 '''
379 return _sincosa6(phi2, phi1, lam21)[4]
382def _d3(wrap, lat1, lon1, lat2, lon2):
383 '''(INTERNAL) Helper for _dE, _dS and _eA.
384 '''
385 if wrap:
386 d_lon, lat2, _ = _Wrap.latlon3(lon1, lat2, lon2, wrap)
387 return radians(lat2), Phi_(lat1=lat1), radians(d_lon)
388 else: # for backward compaibility
389 return Phi_(lat2=lat2), Phi_(lat1=lat1), Phi_(d_lon=lon2 - lon1)
392def _dE(func_, earth, *wrap_lls):
393 '''(INTERNAL) Helper for ellipsoidal distances.
394 '''
395 E = _ellipsoidal(earth, func_)
396 r = func_(*_d3(*wrap_lls), datum=E)
397 return r * E.a
400def _dS(func_, radius, *wrap_lls, **adjust):
401 '''(INTERNAL) Helper for spherical distances.
402 '''
403 r = func_(*_d3(*wrap_lls), **adjust)
404 if radius is not R_M:
405 _, lat1, _, lat2, _ = wrap_lls
406 radius = _mean_radius(radius, lat1, lat2)
407 return r * radius
410def _eA(excess_, radius, *wrap_lls):
411 '''(INTERNAL) Helper for spherical excess or area.
412 '''
413 r = excess_(*_d3(*wrap_lls))
414 if radius:
415 _, lat1, _, lat2, _ = wrap_lls
416 r *= _mean_radius(radius, lat1, lat2)**2
417 return r
420def _ellipsoidal(earth, where):
421 '''(INTERNAL) Helper for distances.
422 '''
423 return _EWGS84 if earth in (_WGS84, _EWGS84) else (
424 earth if isinstance(earth, Ellipsoid) else
425 (earth if isinstance(earth, Datum) else # PYCHOK indent
426 _ellipsoidal_datum(earth, name=where.__name__)).ellipsoid)
429def equirectangular(lat1, lon1, lat2, lon2, radius=R_M, **adjust_limit_wrap):
430 '''Compute the distance between two points using
431 the U{Equirectangular Approximation / Projection
432 <https://www.Movable-Type.co.UK/scripts/latlong.html#equirectangular>}.
434 @arg lat1: Start latitude (C{degrees}).
435 @arg lon1: Start longitude (C{degrees}).
436 @arg lat2: End latitude (C{degrees}).
437 @arg lon2: End longitude (C{degrees}).
438 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
439 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
440 L{a_f2Tuple}).
441 @kwarg adjust_limit_wrap: Optional keyword arguments for
442 function L{equirectangular_}.
444 @return: Distance (C{meter}, same units as B{C{radius}} or
445 the ellipsoid or datum axes).
447 @raise TypeError: Invalid B{C{radius}}.
449 @see: Function L{equirectangular_} for more details, the
450 available B{C{options}}, errors, restrictions and other,
451 approximate or accurate distance functions.
452 '''
453 d = sqrt(equirectangular_(Lat(lat1=lat1), Lon(lon1=lon1),
454 Lat(lat2=lat2), Lon(lon2=lon2),
455 **adjust_limit_wrap).distance2) # PYCHOK 4 vs 2-3
456 return degrees2m(d, radius=_mean_radius(radius, lat1, lat2))
459def _equirectangular(lat1, lon1, lat2, lon2, **adjust_limit_wrap):
460 '''(INTERNAL) Helper for the L{frechet._FrecherMeterRadians}
461 and L{hausdorff._HausdorffMeterRedians} classes.
462 '''
463 return equirectangular_(lat1, lon1, lat2, lon2, **adjust_limit_wrap).distance2 * _D2_R2
466def equirectangular_(lat1, lon1, lat2, lon2, adjust=True, limit=45, wrap=False):
467 '''Compute the distance between two points using the U{Equirectangular
468 Approximation / Projection
469 <https://www.Movable-Type.co.UK/scripts/latlong.html#equirectangular>}.
471 This approximation is valid for short distance of several hundred Km
472 or Miles, see the B{C{limit}} keyword argument and L{LimitError}.
474 @arg lat1: Start latitude (C{degrees}).
475 @arg lon1: Start longitude (C{degrees}).
476 @arg lat2: End latitude (C{degrees}).
477 @arg lon2: End longitude (C{degrees}).
478 @kwarg adjust: Adjust the wrapped, unrolled longitudinal delta
479 by the cosine of the mean latitude (C{bool}).
480 @kwarg limit: Optional limit for lat- and longitudinal deltas
481 (C{degrees}) or C{None} or C{0} for unlimited.
482 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
483 and B{C{lon2}} (C{bool}).
485 @return: A L{Distance4Tuple}C{(distance2, delta_lat, delta_lon,
486 unroll_lon2)} in C{degrees squared}.
488 @raise LimitError: If the lat- and/or longitudinal delta exceeds the
489 B{C{-limit..limit}} range and L{pygeodesy.limiterrors}
490 set to C{True}.
492 @see: U{Local, flat earth approximation
493 <https://www.EdWilliams.org/avform.htm#flat>}, functions
494 L{equirectangular}, L{cosineAndoyerLambert},
495 L{cosineForsytheAndoyerLambert}, L{cosineLaw}, L{euclidean},
496 L{flatLocal}/L{hubeny}, L{flatPolar}, L{haversine}, L{thomas}
497 and L{vincentys} and methods L{Ellipsoid.distance2},
498 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
499 '''
500 d_lon, lat2, ulon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap)
501 d_lat = lat2 - lat1
503 if limit and limit > 0 and limiterrors():
504 d = max(fabs(d_lat), fabs(d_lon))
505 if d > limit:
506 t = _SPACE_(_delta_, Fmt.PAREN_g(d), Fmt.exceeds_limit(limit))
507 s = unstr(equirectangular_, lat1, lon1, lat2, lon2,
508 limit=limit, wrap=wrap)
509 raise LimitError(s, txt=t)
511 if adjust: # scale delta lon
512 d_lon *= _scale_deg(lat1, lat2)
514 d2 = hypot2(d_lat, d_lon) # degrees squared!
515 return Distance4Tuple(d2, d_lat, d_lon, ulon2 - lon2)
518def euclidean(lat1, lon1, lat2, lon2, radius=R_M, adjust=True, wrap=False):
519 '''Approximate the C{Euclidean} distance between two (spherical) points.
521 @arg lat1: Start latitude (C{degrees}).
522 @arg lon1: Start longitude (C{degrees}).
523 @arg lat2: End latitude (C{degrees}).
524 @arg lon2: End longitude (C{degrees}).
525 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
526 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
527 L{a_f2Tuple}) to use.
528 @kwarg adjust: Adjust the longitudinal delta by the cosine of
529 the mean latitude (C{bool}).
530 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
531 and B{C{lon2}} (C{bool}).
533 @return: Distance (C{meter}, same units as B{C{radius}} or the
534 ellipsoid or datum axes).
536 @raise TypeError: Invalid B{C{radius}}.
538 @see: U{Distance between two (spherical) points
539 <https://www.EdWilliams.org/avform.htm#Dist>}, functions L{euclid},
540 L{euclidean_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
541 L{cosineLaw}, L{equirectangular}, L{flatLocal}/L{hubeny}, L{flatPolar},
542 L{haversine}, L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2},
543 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
544 '''
545 return _dS(euclidean_, radius, wrap, lat1, lon1, lat2, lon2, adjust=adjust)
548def euclidean_(phi2, phi1, lam21, adjust=True):
549 '''Approximate the I{angular} C{Euclidean} distance between two
550 (spherical) points.
552 @arg phi2: End latitude (C{radians}).
553 @arg phi1: Start latitude (C{radians}).
554 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
555 @kwarg adjust: Adjust the longitudinal delta by the cosine
556 of the mean latitude (C{bool}).
558 @return: Angular distance (C{radians}).
560 @see: Functions L{euclid}, L{euclidean}, L{cosineAndoyerLambert_},
561 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{equirectangular_},
562 L{flatLocal_}/L{hubeny_}, L{flatPolar_}, L{haversine_}, L{thomas_}
563 and L{vincentys_}.
564 '''
565 if adjust:
566 lam21 *= _scale_rad(phi2, phi1)
567 return euclid(phi2 - phi1, lam21)
570def excessAbc_(A, b, c):
571 '''Compute the I{spherical excess} C{E} of a (spherical) triangle
572 from two sides and the included (small) angle.
574 @arg A: An interior triangle angle (C{radians}).
575 @arg b: Frist adjacent triangle side (C{radians}).
576 @arg c: Second adjacent triangle side (C{radians}).
578 @return: Spherical excess (C{radians}).
580 @raise UnitError: Invalid B{C{A}}, B{C{b}} or B{C{c}}.
582 @see: Functions L{excessGirard_}, L{excessLHuilier_} and U{Spherical
583 trigonometry<https://WikiPedia.org/wiki/Spherical_trigonometry>}.
584 '''
585 A = Radians_(A=A)
586 b = Radians_(b=b) * _0_5
587 c = Radians_(c=c) * _0_5
589 sA, cA, sb, cb, sc, cc = sincos2_(A, b, c)
590 return atan2(sA * sb * sc, cb * cc + cA * sb * sc) * _2_0
593def excessCagnoli_(a, b, c):
594 '''Compute the I{spherical excess} C{E} of a (spherical) triangle using
595 U{Cagnoli's<https://Zenodo.org/record/35392>} (D.34) formula.
597 @arg a: First triangle side (C{radians}).
598 @arg b: Second triangle side (C{radians}).
599 @arg c: Third triangle side (C{radians}).
601 @return: Spherical excess (C{radians}).
603 @raise UnitError: Invalid B{C{a}}, B{C{b}} or B{C{c}}.
605 @see: Function L{excessLHuilier_} and U{Spherical trigonometry
606 <https://WikiPedia.org/wiki/Spherical_trigonometry>}.
607 '''
608 a = Radians_(a=a)
609 b = Radians_(b=b)
610 c = Radians_(c=c)
612 s = fsumf_(a, b, c) * _0_5
613 r = sin(s) * sin(s - a) * sin(s - b) * sin(s - c)
614 c = cos(a * _0_5) * cos(b * _0_5) * cos(c * _0_5)
615 r = asin(sqrt(r) * _0_5 / c) if c and r > 0 else _0_0
616 return Radians(Cagnoli=r * _2_0)
619def excessGirard_(A, B, C):
620 '''Compute the I{spherical excess} C{E} of a (spherical) triangle using
621 U{Girard's<https://MathWorld.Wolfram.com/GirardsSphericalExcessFormula.html>}
622 formula.
624 @arg A: First interior triangle angle (C{radians}).
625 @arg B: Second interior triangle angle (C{radians}).
626 @arg C: Third interior triangle angle (C{radians}).
628 @return: Spherical excess (C{radians}).
630 @raise UnitError: Invalid B{C{A}}, B{C{B}} or B{C{C}}.
632 @see: Function L{excessLHuilier_} and U{Spherical trigonometry
633 <https://WikiPedia.org/wiki/Spherical_trigonometry>}.
634 '''
635 return Radians(Girard=fsumf_(Radians_(A=A),
636 Radians_(B=B),
637 Radians_(C=C), -PI))
640def excessLHuilier_(a, b, c):
641 '''Compute the I{spherical excess} C{E} of a (spherical) triangle using
642 U{L'Huilier's<https://MathWorld.Wolfram.com/LHuiliersTheorem.html>}
643 Theorem.
645 @arg a: First triangle side (C{radians}).
646 @arg b: Second triangle side (C{radians}).
647 @arg c: Third triangle side (C{radians}).
649 @return: Spherical excess (C{radians}).
651 @raise UnitError: Invalid B{C{a}}, B{C{b}} or B{C{c}}.
653 @see: Function L{excessCagnoli_}, L{excessGirard_} and U{Spherical
654 trigonometry<https://WikiPedia.org/wiki/Spherical_trigonometry>}.
655 '''
656 a = Radians_(a=a)
657 b = Radians_(b=b)
658 c = Radians_(c=c)
660 s = fsumf_(a, b, c) * _0_5
661 r = tan_2(s) * tan_2(s - a) * tan_2(s - b) * tan_2(s - c)
662 r = atan(sqrt(r)) if r > 0 else _0_0
663 return Radians(LHuilier=r * _4_0)
666def excessKarney(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
667 '''Compute the surface area of a (spherical) quadrilateral bounded by a
668 segment of a great circle, two meridians and the equator using U{Karney's
669 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>}
670 method.
672 @arg lat1: Start latitude (C{degrees}).
673 @arg lon1: Start longitude (C{degrees}).
674 @arg lat2: End latitude (C{degrees}).
675 @arg lon2: End longitude (C{degrees}).
676 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
677 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
678 L{a_f2Tuple}) or C{None}.
679 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
680 B{C{lat2}} and B{C{lon2}} (C{bool}).
682 @return: Surface area, I{signed} (I{square} C{meter} or the same units as
683 B{C{radius}} I{squared}) or the I{spherical excess} (C{radians})
684 if C{B{radius}=0} or C{None}.
686 @raise TypeError: Invalid B{C{radius}}.
688 @raise UnitError: Invalid B{C{lat2}} or B{C{lat1}}.
690 @raise ValueError: Semi-circular longitudinal delta.
692 @see: Functions L{excessKarney_} and L{excessQuad}.
693 '''
694 return _eA(excessKarney_, radius, wrap, lat1, lon1, lat2, lon2)
697def excessKarney_(phi2, phi1, lam21):
698 '''Compute the I{spherical excess} C{E} of a (spherical) quadrilateral bounded
699 by a segment of a great circle, two meridians and the equator using U{Karney's
700 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>}
701 method.
703 @arg phi2: End latitude (C{radians}).
704 @arg phi1: Start latitude (C{radians}).
705 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
707 @return: Spherical excess, I{signed} (C{radians}).
709 @raise ValueError: Semi-circular longitudinal delta B{C{lam21}}.
711 @see: Function L{excessKarney} and U{Area of a spherical polygon
712 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>}.
713 '''
714 # from: Veness <https://www.Movable-Type.co.UK/scripts/latlong.html> Area
715 # method due to Karney: for each edge of the polygon,
716 #
717 # tan(Δλ / 2) · (tan(φ1 / 2) + tan(φ2 / 2))
718 # tan(E / 2) = -----------------------------------------
719 # 1 + tan(φ1 / 2) · tan(φ2 / 2)
720 #
721 # where E is the spherical excess of the trapezium obtained by extending
722 # the edge to the equator-circle vector for each edge (see also ***).
723 t2 = tan_2(phi2)
724 t1 = tan_2(phi1)
725 t = tan_2(lam21, lam21=None)
726 return Radians(Karney=atan2(t * (t1 + t2),
727 _1_0 + (t1 * t2)) * _2_0)
730# ***) Original post no longer available, following is a copy of the main part
731# <http://OSGeo-org.1560.x6.Nabble.com/Area-of-a-spherical-polygon-td3841625.html>
732#
733# The area of a polygon on a (unit) sphere is given by the spherical excess
734#
735# A = 2 * pi - sum(exterior angles)
736#
737# However this is badly conditioned if the polygon is small. In this case, use
738#
739# A = sum(S12{i, i+1}) over the edges of the polygon
740#
741# where S12 is the area of the quadrilateral bounded by an edge of the polygon,
742# two meridians and the equator, i.e. with vertices (phi1, lambda1), (phi2,
743# lambda2), (0, lambda1) and (0, lambda2). S12 is given by
744#
745# tan(S12 / 2) = tan(lambda21 / 2) * (tan(phi1 / 2) + tan(phi2 / 2)) /
746# (tan(phi1 / 2) * tan(phi2 / 2) + 1)
747#
748# = tan(lambda21 / 2) * tanh((Lamb(phi1) + Lamb(phi2)) / 2)
749#
750# where lambda21 = lambda2 - lambda1 and Lamb(x) is the Lambertian (or the
751# inverse Gudermannian) function
752#
753# Lambertian(x) = asinh(tan(x)) = atanh(sin(x)) = 2 * atanh(tan(x / 2))
754#
755# Notes: The formula for S12 is exact, except that...
756# - it is indeterminate if an edge is a semi-circle
757# - the formula for A applies only if the polygon does not include a pole
758# (if it does, then add +/- 2 * pi to the result)
759# - in the limit of small phi and lambda, S12 reduces to the trapezoidal
760# formula, S12 = (lambda2 - lambda1) * (phi1 + phi2) / 2
761# - I derived this result from the equation for the area of a spherical
762# triangle in terms of two edges and the included angle given by, e.g.
763# U{Todhunter, I. - Spherical Trigonometry (1871), Sec. 103, Eq. (2)
764# <http://Books.Google.com/books?id=3uBHAAAAIAAJ&pg=PA71>}
765# - I would be interested to know if this formula for S12 is already known
766# - Charles Karney
769def excessQuad(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
770 '''Compute the surface area of a (spherical) quadrilateral bounded by a segment
771 of a great circle, two meridians and the equator.
773 @arg lat1: Start latitude (C{degrees}).
774 @arg lon1: Start longitude (C{degrees}).
775 @arg lat2: End latitude (C{degrees}).
776 @arg lon2: End longitude (C{degrees}).
777 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
778 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
779 L{a_f2Tuple}) or C{None}.
780 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
781 B{C{lat2}} and B{C{lon2}} (C{bool}).
783 @return: Surface area, I{signed} (I{square} C{meter} or the same units as
784 B{C{radius}} I{squared}) or the I{spherical excess} (C{radians})
785 if C{B{radius}=0} or C{None}.
787 @raise TypeError: Invalid B{C{radius}}.
789 @raise UnitError: Invalid B{C{lat2}} or B{C{lat1}}.
791 @see: Function L{excessQuad_} and L{excessKarney}.
792 '''
793 return _eA(excessQuad_, radius, wrap, lat1, lon1, lat2, lon2)
796def excessQuad_(phi2, phi1, lam21):
797 '''Compute the I{spherical excess} C{E} of a (spherical) quadrilateral bounded
798 by a segment of a great circle, two meridians and the equator.
800 @arg phi2: End latitude (C{radians}).
801 @arg phi1: Start latitude (C{radians}).
802 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
804 @return: Spherical excess, I{signed} (C{radians}).
806 @see: Function L{excessQuad} and U{Spherical trigonometry
807 <https://WikiPedia.org/wiki/Spherical_trigonometry>}.
808 '''
809 s = sin((phi2 + phi1) * _0_5)
810 c = cos((phi2 - phi1) * _0_5)
811 return Radians(Quad=atan2(tan_2(lam21) * s, c) * _2_0)
814def flatLocal(lat1, lon1, lat2, lon2, datum=_WGS84, scaled=True, wrap=False):
815 '''Compute the distance between two (ellipsoidal) points using
816 the U{ellipsoidal Earth to plane projection<https://WikiPedia.org/
817 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>}
818 aka U{Hubeny<https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula.
820 @arg lat1: Start latitude (C{degrees}).
821 @arg lon1: Start longitude (C{degrees}).
822 @arg lat2: End latitude (C{degrees}).
823 @arg lon2: End longitude (C{degrees}).
824 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
825 L{Ellipsoid2} or L{a_f2Tuple}) to use.
826 @kwarg scaled: Scale prime_vertical by C{cos(B{phi})} (C{bool}),
827 see method L{pygeodesy.Ellipsoid.roc2_}.
828 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
829 B{C{lat2}} and B{C{lon2}} (C{bool}).
831 @return: Distance (C{meter}, same units as the B{C{datum}}'s
832 ellipsoid axes).
834 @raise TypeError: Invalid B{C{datum}}.
836 @note: The meridional and prime_vertical radii of curvature
837 are taken and scaled at the mean of both latitude.
839 @see: Functions L{flatLocal_} or L{hubeny_}, L{cosineLaw}, L{flatPolar},
840 L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
841 L{equirectangular}, L{euclidean}, L{haversine}, L{thomas},
842 L{vincentys}, method L{Ellipsoid.distance2} and U{local, flat
843 earth approximation<https://www.EdWilliams.org/avform.htm#flat>}.
844 '''
845 E = _ellipsoidal(datum, flatLocal)
846 return E._hubeny_2(*_d3(wrap, lat1, lon1, lat2, lon2),
847 scaled=scaled, squared=False) * E.a
849hubeny = flatLocal # PYCHOK for Karl Hubeny
852def flatLocal_(phi2, phi1, lam21, datum=_WGS84, scaled=True):
853 '''Compute the I{angular} distance between two (ellipsoidal) points using
854 the U{ellipsoidal Earth to plane projection<https://WikiPedia.org/
855 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>}
856 aka U{Hubeny<https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula.
858 @arg phi2: End latitude (C{radians}).
859 @arg phi1: Start latitude (C{radians}).
860 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
861 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
862 L{Ellipsoid2} or L{a_f2Tuple}) to use.
863 @kwarg scaled: Scale prime_vertical by C{cos(B{phi})} (C{bool}),
864 see method L{pygeodesy.Ellipsoid.roc2_}.
866 @return: Angular distance (C{radians}).
868 @raise TypeError: Invalid B{C{datum}}.
870 @note: The meridional and prime_vertical radii of curvature
871 are taken and scaled I{at the mean of both latitude}.
873 @see: Functions L{flatLocal} or L{hubeny}, L{cosineAndoyerLambert_},
874 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{flatPolar_},
875 L{equirectangular_}, L{euclidean_}, L{haversine_}, L{thomas_}
876 and L{vincentys_} and U{local, flat earth approximation
877 <https://www.EdWilliams.org/avform.htm#flat>}.
878 '''
879 E = _ellipsoidal(datum, flatLocal_)
880 return E._hubeny_2(phi2, phi1, lam21, scaled=scaled, squared=False)
882hubeny_ = flatLocal_ # PYCHOK for Karl Hubeny
885def flatPolar(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
886 '''Compute the distance between two (spherical) points using
887 the U{polar coordinate flat-Earth <https://WikiPedia.org/wiki/
888 Geographical_distance#Polar_coordinate_flat-Earth_formula>}
889 formula.
891 @arg lat1: Start latitude (C{degrees}).
892 @arg lon1: Start longitude (C{degrees}).
893 @arg lat2: End latitude (C{degrees}).
894 @arg lon2: End longitude (C{degrees}).
895 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
896 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
897 L{a_f2Tuple}) to use.
898 @kwarg wrap: If C{True}, wrap or I{normalize} and B{C{lat2}}
899 and B{C{lon2}} (C{bool}).
901 @return: Distance (C{meter}, same units as B{C{radius}} or the
902 ellipsoid or datum axes).
904 @raise TypeError: Invalid B{C{radius}}.
906 @see: Functions L{flatPolar_}, L{cosineAndoyerLambert},
907 L{cosineForsytheAndoyerLambert},L{cosineLaw},
908 L{flatLocal}/L{hubeny}, L{equirectangular},
909 L{euclidean}, L{haversine}, L{thomas} and
910 L{vincentys}.
911 '''
912 return _dS(flatPolar_, radius, wrap, lat1, lon1, lat2, lon2)
915def flatPolar_(phi2, phi1, lam21):
916 '''Compute the I{angular} distance between two (spherical) points
917 using the U{polar coordinate flat-Earth<https://WikiPedia.org/wiki/
918 Geographical_distance#Polar_coordinate_flat-Earth_formula>}
919 formula.
921 @arg phi2: End latitude (C{radians}).
922 @arg phi1: Start latitude (C{radians}).
923 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
925 @return: Angular distance (C{radians}).
927 @see: Functions L{flatPolar}, L{cosineAndoyerLambert_},
928 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
929 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
930 L{haversine_}, L{thomas_} and L{vincentys_}.
931 '''
932 a = fabs(PI_2 - phi1) # co-latitude
933 b = fabs(PI_2 - phi2) # co-latitude
934 if a < b:
935 a, b = b, a
936 if a < EPS0:
937 a = _0_0
938 elif b > 0:
939 b = b / a # /= chokes PyChecker
940 c = b * cos(lam21) * _2_0
941 c = fsumf_(_1_0, b**2, -fabs(c))
942 a *= sqrt0(c)
943 return a
946def hartzell(pov, los=None, earth=_WGS84, name=NN, **LatLon_and_kwds):
947 '''Compute the intersection of the earth's surface and a Line-Of-Sight
948 from a Point-Of-View in space.
950 @arg pov: Point-Of-View outside the earth (C{Cartesian}, L{Ecef9Tuple}
951 or L{Vector3d}).
952 @kwarg los: Line-Of-Sight, I{direction} to earth (L{Vector3d}) or
953 C{None} to point to the earth' center.
954 @kwarg earth: The earth model (L{Datum}, L{Ellipsoid}, L{Ellipsoid2},
955 L{a_f2Tuple} or C{scalar} radius in C{meter}).
956 @kwarg name: Optional name (C{str}).
957 @kwarg LatLon_and_kwds: Optional C{LatLon} class for the intersection
958 point plus C{LatLon} keyword arguments, include
959 B{C{datum}} if different from B{C{earth}}.
961 @return: The intersection point (L{Vector3d}, C{Cartesian type} of
962 B{C{pov}} or B{C{LatLon}}).
964 @raise IntersectionError: Null B{C{pov}} or B{C{los}} vector, B{C{pov}}
965 is inside the earth or B{C{los}} points outside
966 the earth or points in an opposite direction.
968 @raise TypeError: Invalid B{C{pov}}, B{C{los}} or B{C{earth}}.
970 @see: Function L{pygeodesy.hartzell4}, L{pygeodesy.tyr3d} for B{C{los}},
971 method L{Ellipsoid.hartzell4} and U{I{Satellite Line-of-Sight
972 Intersection with Earth}<https://StephenHartzell.Medium.com/
973 satellite-line-of-sight-intersection-with-earth-d786b4a6a9b6>}.
974 '''
975 D = earth if isinstance(earth, Datum) else \
976 _spherical_datum(earth, name=hartzell.__name__)
977 try:
978 r, _ = _MODS.triaxials._hartzell3d2(pov, los, D.ellipsoid._triaxial)
979 except Exception as x:
980 raise IntersectionError(pov=pov, los=los, earth=earth, cause=x)
982# else:
983# E = D.ellipsoid
984# # Triaxial(a, b, c) == (E.a, E.a, E.b)
985#
986# def _Error(txt):
987# return IntersectionError(pov=pov, los=los, earth=earth, txt=txt)
988#
989# a2 = b2 = E.a2 # earth' x, y, ...
990# c2 = E.b2 # ... z semi-axis squared
991# q2 = E.b2_a2 # == c2 / a2
992# bc = E.a * E.b # == b * c
993#
994# V3 = _MODS.vector3d._otherV3d
995# p3 = V3(pov=pov)
996# u3 = V3(los=los) if los else p3.negate()
997# u3 = u3.unit() # unit vector, opposing signs
998#
999# x2, y2, z2 = p3.x2y2z2 # p3.times_(p3).xyz
1000# ux, vy, wz = u3.times_(p3).xyz
1001# u2, v2, w2 = u3.x2y2z2 # u3.times_(u3).xyz
1002#
1003# t = c2, c2, b2
1004# m = fdot(t, u2, v2, w2) # a2 factored out
1005# if m < EPS0: # zero or near-null LOS vector
1006# raise _Error(_near_(_null_))
1007#
1008# # a2 and b2 factored out, b2 == a2 and b2 / a2 == 1
1009# r = fsumf_(b2 * w2, c2 * v2, -v2 * z2, vy * wz * 2,
1010# c2 * u2, -u2 * z2, -w2 * x2, ux * wz * 2,
1011# -w2 * y2, -u2 * y2 * q2, -v2 * x2 * q2, ux * vy * 2 * q2)
1012# if r > 0:
1013# r = sqrt(r) * bc # == a * a * b * c / a2
1014# elif r < 0: # LOS pointing away from or missing the earth
1015# raise _Error(_opposite_ if max(ux, vy, wz) > 0 else _outside_)
1016#
1017# d = Fdot(t, ux, vy, wz).fadd_(r).fover(m) # -r for antipode, a2 factored out
1018# if d > 0: # POV inside or LOS missing, outside the earth
1019# s = fsumf_(_1_0, x2 / a2, y2 / b2, z2 / c2, _N_2_0) # like _sideOf
1020# raise _Error(_outside_ if s > 0 else _inside_)
1021# elif fsumf_(x2, y2, z2) < d**2: # d past earth center
1022# raise _Error(_too_(_distant_))
1023#
1024# r = p3.minus(u3.times(d))
1025# # h = p3.minus(r).length # distance to ellipsoid
1027 r = _xnamed(r, name or hartzell.__name__)
1028 if LatLon_and_kwds:
1029 c = _MODS.cartesianBase.CartesianBase(r, datum=D, name=r.name)
1030 r = c.toLatLon(**LatLon_and_kwds)
1031 return r
1034def haversine(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
1035 '''Compute the distance between two (spherical) points using the
1036 U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>}
1037 formula.
1039 @arg lat1: Start latitude (C{degrees}).
1040 @arg lon1: Start longitude (C{degrees}).
1041 @arg lat2: End latitude (C{degrees}).
1042 @arg lon2: End longitude (C{degrees}).
1043 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
1044 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
1045 L{a_f2Tuple}) to use.
1046 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
1047 B{C{lat2}} and B{C{lon2}} (C{bool}).
1049 @return: Distance (C{meter}, same units as B{C{radius}}).
1051 @raise TypeError: Invalid B{C{radius}}.
1053 @see: U{Distance between two (spherical) points
1054 <https://www.EdWilliams.org/avform.htm#Dist>}, functions
1055 L{cosineLaw}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
1056 L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar},
1057 L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2},
1058 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
1060 @note: See note at function L{vincentys_}.
1061 '''
1062 return _dS(haversine_, radius, wrap, lat1, lon1, lat2, lon2)
1065def haversine_(phi2, phi1, lam21):
1066 '''Compute the I{angular} distance between two (spherical) points
1067 using the U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>}
1068 formula.
1070 @arg phi2: End latitude (C{radians}).
1071 @arg phi1: Start latitude (C{radians}).
1072 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
1074 @return: Angular distance (C{radians}).
1076 @see: Functions L{haversine}, L{cosineAndoyerLambert_},
1077 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
1078 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
1079 L{flatPolar_}, L{thomas_} and L{vincentys_}.
1081 @note: See note at function L{vincentys_}.
1082 '''
1083 def _hsin(rad):
1084 return sin(rad * _0_5)**2
1086 h = _hsin(phi2 - phi1) + cos(phi1) * cos(phi2) * _hsin(lam21) # haversine
1087 return atan2(sqrt0(h), sqrt0(_1_0 - h)) * _2_0 # == asin(sqrt(h)) * 2
1090def heightOf(angle, distance, radius=R_M):
1091 '''Determine the height above the (spherical) earth' surface after
1092 traveling along a straight line at a given tilt.
1094 @arg angle: Tilt angle above horizontal (C{degrees}).
1095 @arg distance: Distance along the line (C{meter} or same units as
1096 B{C{radius}}).
1097 @kwarg radius: Optional mean earth radius (C{meter}).
1099 @return: Height (C{meter}, same units as B{C{distance}} and B{C{radius}}).
1101 @raise ValueError: Invalid B{C{angle}}, B{C{distance}} or B{C{radius}}.
1103 @see: U{MultiDop geog_lib.GeogBeamHt<https://GitHub.com/NASA/MultiDop>}
1104 (U{Shapiro et al. 2009, JTECH
1105 <https://Journals.AMetSoc.org/doi/abs/10.1175/2009JTECHA1256.1>}
1106 and U{Potvin et al. 2012, JTECH
1107 <https://Journals.AMetSoc.org/doi/abs/10.1175/JTECH-D-11-00019.1>}).
1108 '''
1109 r = h = Radius(radius)
1110 d = fabs(Distance(distance))
1111 if d > h:
1112 d, h = h, d
1114 if d > EPS0: # and h > EPS0
1115 d = d / h # /= h chokes PyChecker
1116 s = sin(Phi_(angle=angle, clip=_180_0))
1117 s = fsumf_(_1_0, _2_0 * s * d, d**2)
1118 if s > 0:
1119 return h * sqrt(s) - r
1121 raise _ValueError(angle=angle, distance=distance, radius=radius)
1124def heightOrthometric(h_ll, N):
1125 '''Get the I{orthometric} height B{H}, the height above the geoid, earth surface.
1127 @arg h_ll: The height above the ellipsoid (C{meter}) or an I{ellipsoidal}
1128 location (C{LatLon} with a C{height} or C{h} attribute).
1129 @arg N: The I{geoid} height (C{meter}), the height of the geoid above the
1130 ellipsoid at the same B{C{h_ll}} location.
1132 @return: I{Orthometric} height C{B{H} = B{h} - B{N}} (C{meter}, same units
1133 as B{C{h}} and B{C{N}}).
1135 @see: U{Ellipsoid, Geoid, and Othometric Heights<https://www.NGS.NOAA.gov/
1136 GEOID/PRESENTATIONS/2007_02_24_CCPS/Roman_A_PLSC2007notes.pdf>}, page
1137 6 and module L{pygeodesy.geoids}.
1138 '''
1139 h = h_ll if isscalar(h_ll) else _xattr(h_ll, height=_xattr(h_ll, h=0))
1140 return Height(H=Height(h=h) - Height(N=N))
1143def horizon(height, radius=R_M, refraction=False):
1144 '''Determine the distance to the horizon from a given altitude
1145 above the (spherical) earth.
1147 @arg height: Altitude (C{meter} or same units as B{C{radius}}).
1148 @kwarg radius: Optional mean earth radius (C{meter}).
1149 @kwarg refraction: Consider atmospheric refraction (C{bool}).
1151 @return: Distance (C{meter}, same units as B{C{height}} and B{C{radius}}).
1153 @raise ValueError: Invalid B{C{height}} or B{C{radius}}.
1155 @see: U{Distance to horizon<https://www.EdWilliams.org/avform.htm#Horizon>}.
1156 '''
1157 h, r = Height(height), Radius(radius)
1158 if min(h, r) < 0:
1159 raise _ValueError(height=height, radius=radius)
1161 if refraction:
1162 d2 = 2.415750694528 * h * r # 2.0 / 0.8279
1163 else:
1164 d2 = h * fsumf_(r, r, h)
1165 return sqrt0(d2)
1168class _idllmn6(object): # see also .geodesicw._wargs, .vector2d._numpy
1169 '''(INTERNAL) Helper for C{intersection2} and C{intersections2}.
1170 '''
1171 @contextmanager # <https://www.python.org/dev/peps/pep-0343/> Examples
1172 def __call__(self, datum, lat1, lon1, lat2, lon2, small, wrap, s, **kwds):
1173 try:
1174 if wrap:
1175 _, lat2, lon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap)
1176 kwds = _xkwds(kwds, wrap=wrap) # for _xError
1177 m = small if small is _100km else Meter_(small=small)
1178 n = (intersections2 if s else intersection2).__name__
1179 if datum is None or euclidean(lat1, lon1, lat2, lon2) < m:
1180 d, m = None, _MODS.vector3d
1181 _i = m._intersects2 if s else m._intersect3d3
1182 elif isscalar(datum) and datum < 0 and not s:
1183 d = _spherical_datum(-datum, name=n)
1184 m = _MODS.sphericalNvector
1185 _i = m.intersection
1186 else:
1187 d = _spherical_datum(datum, name=n)
1188 if d.isSpherical:
1189 m = _MODS.sphericalTrigonometry
1190 _i = m._intersects2 if s else m._intersect
1191 elif d.isEllipsoidal:
1192 try:
1193 if d.ellipsoid.geodesic:
1194 pass
1195 m = _MODS.ellipsoidalKarney
1196 except ImportError:
1197 m = _MODS.ellipsoidalExact
1198 _i = m._intersections2 if s else m._intersection3 # ellispoidalBaseDI
1199 else:
1200 raise _TypeError(datum=datum)
1201 yield _i, d, lat2, lon2, m, n
1203 except (TypeError, ValueError) as x:
1204 raise _xError(x, lat1=lat1, lon1=lon1, datum=datum,
1205 lat2=lat2, lon2=lon2, small=small, **kwds)
1207_idllmn6 = _idllmn6() # PYCHOK singleton
1210def intersection2(lat1, lon1, bearing1,
1211 lat2, lon2, bearing2, datum=None, wrap=False, small=_100km): # was=True
1212 '''I{Conveniently} compute the intersection of two lines each defined
1213 by a (geodetic) point and a bearing from North, using either ...
1215 1) L{vector3d.intersection3d3} for B{C{small}} distances (below 100 Km
1216 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ...
1218 2) L{sphericalTrigonometry.intersection} for a spherical B{C{datum}}
1219 or a C{scalar B{datum}} representing the earth radius, conventionally
1220 in C{meter} or ...
1222 3) L{sphericalNvector.intersection} if B{C{datum}} is a I{negative}
1223 C{scalar}, (negative) earth radius, conventionally in C{meter} or ...
1225 4) L{ellipsoidalKarney.intersection3} for an ellipsoidal B{C{datum}}
1226 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>}
1227 is installed, otherwise ...
1229 5) L{ellipsoidalExact.intersection3}, provided B{C{datum}} is ellipsoidal.
1231 @arg lat1: Latitude of the first point (C{degrees}).
1232 @arg lon1: Longitude of the first point (C{degrees}).
1233 @arg bearing1: Bearing at the first point (compass C{degrees}).
1234 @arg lat2: Latitude of the second point (C{degrees}).
1235 @arg lon2: Longitude of the second point (C{degrees}).
1236 @arg bearing2: Bearing at the second point (compass C{degrees}).
1237 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid},
1238 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth
1239 radius (C{meter}, same units as B{C{radius1}} and
1240 B{C{radius2}}) or C{None}.
1241 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
1242 and B{C{lon2}} (C{bool}).
1243 @kwarg small: Upper limit for small distances (C{meter}).
1245 @return: A L{LatLon2Tuple}C{(lat, lon)} with the lat- and
1246 longitude of the intersection point.
1248 @raise IntersectionError: Ambiguous or infinite intersection
1249 or colinear, parallel or otherwise
1250 non-intersecting lines.
1252 @raise TypeError: Invalid B{C{datum}}.
1254 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{bearing1}},
1255 B{C{lat2}}, B{C{lon2}} or B{C{bearing2}}.
1257 @see: Method L{RhumbLine.intersection2}.
1259 @note: The returned intersections may be near-antipodal.
1260 '''
1261 b1 = Bearing(bearing1=bearing1)
1262 b2 = Bearing(bearing2=bearing2)
1263 with _idllmn6(datum, lat1, lon1, lat2, lon2,
1264 small, wrap, False, bearing1=b1, bearing2=b2) as t:
1265 _i, d, lat2, lon2, m, n = t
1266 if d is None:
1267 t, _, _ = _i(m.Vector3d(lon1, lat1, 0), b1,
1268 m.Vector3d(lon2, lat2, 0), b2, useZ=False)
1269 t = LatLon2Tuple(t.y, t.x, name=n)
1271 else:
1272 t = _i(m.LatLon(lat1, lon1, datum=d), b1,
1273 m.LatLon(lat2, lon2, datum=d), b2, height=0, wrap=False)
1274 if isinstance(t, Intersection3Tuple): # ellipsoidal
1275 t, _, _ = t
1276 t = LatLon2Tuple(t.lat, t.lon, name=n)
1277 return t
1280def intersections2(lat1, lon1, radius1,
1281 lat2, lon2, radius2, datum=None, wrap=False, small=_100km): # was=True
1282 '''I{Conveniently} compute the intersections of two circles each defined
1283 by a (geodetic) center point and a radius, using either ...
1285 1) L{vector3d.intersections2} for B{C{small}} distances (below 100 Km
1286 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ...
1288 2) L{sphericalTrigonometry.intersections2} for a spherical B{C{datum}}
1289 or a C{scalar B{datum}} representing the earth radius, conventionally
1290 in C{meter} or ...
1292 3) L{ellipsoidalKarney.intersections2} for an ellipsoidal B{C{datum}}
1293 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>}
1294 is installed, otherwise ...
1296 4) L{ellipsoidalExact.intersections2}, provided B{C{datum}} is ellipsoidal.
1298 @arg lat1: Latitude of the first circle center (C{degrees}).
1299 @arg lon1: Longitude of the first circle center (C{degrees}).
1300 @arg radius1: Radius of the first circle (C{meter}, conventionally).
1301 @arg lat2: Latitude of the second circle center (C{degrees}).
1302 @arg lon2: Longitude of the second circle center (C{degrees}).
1303 @arg radius2: Radius of the second circle (C{meter}, same units as B{C{radius1}}).
1304 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid},
1305 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth
1306 radius (C{meter}, same units as B{C{radius1}} and
1307 B{C{radius2}}) or C{None}.
1308 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
1309 and B{C{lon2}} (C{bool}).
1310 @kwarg small: Upper limit for small distances (C{meter}).
1312 @return: 2-Tuple of the intersection points, each a
1313 L{LatLon2Tuple}C{(lat, lon)}. For abutting circles, the
1314 points are the same instance, aka the I{radical center}.
1316 @raise IntersectionError: Concentric, antipodal, invalid or
1317 non-intersecting circles or no
1318 convergence.
1320 @raise TypeError: Invalid B{C{datum}}.
1322 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{radius1}},
1323 B{C{lat2}}, B{C{lon2}} or B{C{radius2}}.
1324 '''
1325 r1 = Radius_(radius1=radius1)
1326 r2 = Radius_(radius2=radius2)
1327 with _idllmn6(datum, lat1, lon1, lat2, lon2,
1328 small, wrap, True, radius1=r1, radius2=r2) as t:
1329 _i, d, lat2, lon2, m, n = t
1330 if d is None:
1331 r1 = m2degrees(r1, radius=R_M, lat=lat1)
1332 r2 = m2degrees(r2, radius=R_M, lat=lat2)
1334 def _V2T(x, y, _, **unused): # _ == z unused
1335 return LatLon2Tuple(y, x, name=n)
1337 t = _i(m.Vector3d(lon1, lat1, 0), r1,
1338 m.Vector3d(lon2, lat2, 0), r2, sphere=False,
1339 Vector=_V2T)
1340 else:
1341 def _LL2T(lat, lon, **unused):
1342 return LatLon2Tuple(lat, lon, name=n)
1344 t = _i(m.LatLon(lat1, lon1, datum=d), r1,
1345 m.LatLon(lat2, lon2, datum=d), r2,
1346 LatLon=_LL2T, height=0, wrap=False)
1347 return t
1350def isantipode(lat1, lon1, lat2, lon2, eps=EPS):
1351 '''Check whether two points are I{antipodal}, on diametrically
1352 opposite sides of the earth.
1354 @arg lat1: Latitude of one point (C{degrees}).
1355 @arg lon1: Longitude of one point (C{degrees}).
1356 @arg lat2: Latitude of the other point (C{degrees}).
1357 @arg lon2: Longitude of the other point (C{degrees}).
1358 @kwarg eps: Tolerance for near-equality (C{degrees}).
1360 @return: C{True} if points are antipodal within the
1361 B{C{eps}} tolerance, C{False} otherwise.
1363 @see: Functions L{isantipode_} and L{antipode}.
1364 '''
1365 return (fabs(lat1 + lat2) <= eps and
1366 fabs(lon1 + lon2) <= eps) or _isequalTo(
1367 normal(lat1, lon1), antipode(lat2, lon2), eps)
1370def isantipode_(phi1, lam1, phi2, lam2, eps=EPS):
1371 '''Check whether two points are I{antipodal}, on diametrically
1372 opposite sides of the earth.
1374 @arg phi1: Latitude of one point (C{radians}).
1375 @arg lam1: Longitude of one point (C{radians}).
1376 @arg phi2: Latitude of the other point (C{radians}).
1377 @arg lam2: Longitude of the other point (C{radians}).
1378 @kwarg eps: Tolerance for near-equality (C{radians}).
1380 @return: C{True} if points are antipodal within the
1381 B{C{eps}} tolerance, C{False} otherwise.
1383 @see: Functions L{isantipode} and L{antipode_}.
1384 '''
1385 return (fabs(phi1 + phi2) <= eps and
1386 fabs(lam1 + lam2) <= eps) or _isequalTo_(
1387 normal_(phi1, lam1), antipode_(phi2, lam2), eps)
1390def _isequalTo(p1, p2, eps=EPS):
1391 '''Compare 2 point lat-/lons ignoring C{class}.
1392 '''
1393 return (fabs(p1.lat - p2.lat) <= eps and
1394 fabs(p1.lon - p2.lon) <= eps) if eps else (p1.latlon == p2.latlon)
1397def _isequalTo_(p1, p2, eps=EPS):
1398 '''(INTERNAL) Compare 2 point phi-/lams ignoring C{class}.
1399 '''
1400 return (fabs(p1.phi - p2.phi) <= eps and
1401 fabs(p1.lam - p2.lam) <= eps) if eps else (p1.philam == p2.philam)
1404def isnormal(lat, lon, eps=0):
1405 '''Check whether B{C{lat}} I{and} B{C{lon}} are within their
1406 respective I{normal} range in C{degrees}.
1408 @arg lat: Latitude (C{degrees}).
1409 @arg lon: Longitude (C{degrees}).
1410 @kwarg eps: Optional tolerance C{degrees}).
1412 @return: C{True} if C{(abs(B{lat}) + B{eps}) <= 90} and
1413 C{(abs(B{lon}) + B{eps}) <= 180}, C{False} othwerwise.
1415 @see: Functions L{isnormal_} and L{normal}.
1416 '''
1417 return (_90_0 - fabs(lat)) >= eps and _loneg(fabs(lon)) >= eps
1420def isnormal_(phi, lam, eps=0):
1421 '''Check whether B{C{phi}} I{and} B{C{lam}} are within their
1422 respective I{normal} range in C{radians}.
1424 @arg phi: Latitude (C{radians}).
1425 @arg lam: Longitude (C{radians}).
1426 @kwarg eps: Optional tolerance C{radians}).
1428 @return: C{True} if C{(abs(B{phi}) + B{eps}) <= PI/2} and
1429 C{(abs(B{lam}) + B{eps}) <= PI}, C{False} othwerwise.
1431 @see: Functions L{isnormal} and L{normal_}.
1432 '''
1433 return (PI_2 - fabs(phi)) >= eps and (PI - fabs(lam)) >= eps
1436def latlon2n_xyz(lat, lon, name=NN):
1437 '''Convert lat-, longitude to C{n-vector} (I{normal} to the
1438 earth's surface) X, Y and Z components.
1440 @arg lat: Latitude (C{degrees}).
1441 @arg lon: Longitude (C{degrees}).
1442 @kwarg name: Optional name (C{str}).
1444 @return: A L{Vector3Tuple}C{(x, y, z)}.
1446 @see: Function L{philam2n_xyz}.
1448 @note: These are C{n-vector} x, y and z components,
1449 I{NOT} geocentric ECEF x, y and z coordinates!
1450 '''
1451 return _2n_xyz(name, *sincos2d_(lat, lon))
1454def _normal2(a, b, n_2, n, n2):
1455 '''(INTERNAL) Helper for C{normal} and C{normal_}.
1456 '''
1457 if fabs(b) > n:
1458 b = remainder(b, n2)
1459 if fabs(a) > n_2:
1460 r = remainder(a, n)
1461 if r != a:
1462 a = -r
1463 b -= n if b > 0 else -n
1464 return float0_(a, b)
1467def normal(lat, lon, name=NN):
1468 '''Normalize a lat- I{and} longitude pair in C{degrees}.
1470 @arg lat: Latitude (C{degrees}).
1471 @arg lon: Longitude (C{degrees}).
1472 @kwarg name: Optional name (C{str}).
1474 @return: L{LatLon2Tuple}C{(lat, lon)} with C{abs(lat) <= 90}
1475 and C{abs(lon) <= 180}.
1477 @see: Functions L{normal_} and L{isnormal}.
1478 '''
1479 return LatLon2Tuple(*_normal2(lat, lon, _90_0, _180_0, _360_0),
1480 name=name or normal.__name__)
1483def normal_(phi, lam, name=NN):
1484 '''Normalize a lat- I{and} longitude pair in C{radians}.
1486 @arg phi: Latitude (C{radians}).
1487 @arg lam: Longitude (C{radians}).
1488 @kwarg name: Optional name (C{str}).
1490 @return: L{PhiLam2Tuple}C{(phi, lam)} with C{abs(phi) <= PI/2}
1491 and C{abs(lam) <= PI}.
1493 @see: Functions L{normal} and L{isnormal_}.
1494 '''
1495 return PhiLam2Tuple(*_normal2(phi, lam, PI_2, PI, PI2),
1496 name=name or normal_.__name__)
1499def _2n_xyz(name, sa, ca, sb, cb):
1500 '''(INTERNAL) Helper for C{latlon2n_xyz} and C{philam2n_xyz}.
1501 '''
1502 # Kenneth Gade eqn 3, but using right-handed
1503 # vector x -> 0°E,0°N, y -> 90°E,0°N, z -> 90°N
1504 return Vector3Tuple(ca * cb, ca * sb, sa, name=name)
1507def n_xyz2latlon(x, y, z, name=NN):
1508 '''Convert C{n-vector} components to lat- and longitude in C{degrees}.
1510 @arg x: X component (C{scalar}).
1511 @arg y: Y component (C{scalar}).
1512 @arg z: Z component (C{scalar}).
1513 @kwarg name: Optional name (C{str}).
1515 @return: A L{LatLon2Tuple}C{(lat, lon)}.
1517 @see: Function L{n_xyz2philam}.
1518 '''
1519 return LatLon2Tuple(atan2d(z, hypot(x, y)), atan2d(y, x), name=name)
1522def n_xyz2philam(x, y, z, name=NN):
1523 '''Convert C{n-vector} components to lat- and longitude in C{radians}.
1525 @arg x: X component (C{scalar}).
1526 @arg y: Y component (C{scalar}).
1527 @arg z: Z component (C{scalar}).
1528 @kwarg name: Optional name (C{str}).
1530 @return: A L{PhiLam2Tuple}C{(phi, lam)}.
1532 @see: Function L{n_xyz2latlon}.
1533 '''
1534 return PhiLam2Tuple(atan2(z, hypot(x, y)), atan2(y, x), name=name)
1537def _opposes(d, m, n, n2):
1538 '''(INTERNAL) Helper for C{opposing} and C{opposing_}.
1539 '''
1540 d = d % n2 # -20 % 360 == 340, -1 % PI2 == PI2 - 1
1541 return False if d < m or d > (n2 - m) else (
1542 True if (n - m) < d < (n + m) else None)
1545def opposing(bearing1, bearing2, margin=_90_0):
1546 '''Compare the direction of two bearings given in C{degrees}.
1548 @arg bearing1: First bearing (compass C{degrees}).
1549 @arg bearing2: Second bearing (compass C{degrees}).
1550 @kwarg margin: Optional, interior angle bracket (C{degrees}).
1552 @return: C{True} if both bearings point in opposite, C{False} if
1553 in similar or C{None} if in I{perpendicular} directions.
1555 @see: Function L{opposing_}.
1556 '''
1557 m = Degrees_(margin=margin, low=EPS0, high=_90_0)
1558 return _opposes(bearing2 - bearing1, m, _180_0, _360_0)
1561def opposing_(radians1, radians2, margin=PI_2):
1562 '''Compare the direction of two bearings given in C{radians}.
1564 @arg radians1: First bearing (C{radians}).
1565 @arg radians2: Second bearing (C{radians}).
1566 @kwarg margin: Optional, interior angle bracket (C{radians}).
1568 @return: C{True} if both bearings point in opposite, C{False} if
1569 in similar or C{None} if in perpendicular directions.
1571 @see: Function L{opposing}.
1572 '''
1573 m = Radians_(margin=margin, low=EPS0, high=PI_2)
1574 return _opposes(radians2 - radians1, m, PI, PI2)
1577def philam2n_xyz(phi, lam, name=NN):
1578 '''Convert lat-, longitude to C{n-vector} (I{normal} to the
1579 earth's surface) X, Y and Z components.
1581 @arg phi: Latitude (C{radians}).
1582 @arg lam: Longitude (C{radians}).
1583 @kwarg name: Optional name (C{str}).
1585 @return: A L{Vector3Tuple}C{(x, y, z)}.
1587 @see: Function L{latlon2n_xyz}.
1589 @note: These are C{n-vector} x, y and z components,
1590 I{NOT} geocentric ECEF x, y and z coordinates!
1591 '''
1592 return _2n_xyz(name, *sincos2_(phi, lam))
1595def _radical2(d, r1, r2): # in .ellipsoidalBaseDI, .sphericalTrigonometry, .vector3d
1596 # (INTERNAL) See C{radical2} below
1597 # assert d > EPS0
1598 r = fsumf_(_1_0, (r1 / d)**2, -(r2 / d)**2) * _0_5
1599 return Radical2Tuple(max(_0_0, min(_1_0, r)), r * d)
1602def radical2(distance, radius1, radius2):
1603 '''Compute the I{radical ratio} and I{radical line} of two
1604 U{intersecting circles<https://MathWorld.Wolfram.com/
1605 Circle-CircleIntersection.html>}.
1607 The I{radical line} is perpendicular to the axis thru the
1608 centers of the circles at C{(0, 0)} and C{(B{distance}, 0)}.
1610 @arg distance: Distance between the circle centers (C{scalar}).
1611 @arg radius1: Radius of the first circle (C{scalar}).
1612 @arg radius2: Radius of the second circle (C{scalar}).
1614 @return: A L{Radical2Tuple}C{(ratio, xline)} where C{0.0 <=
1615 ratio <= 1.0} and C{xline} is along the B{C{distance}}.
1617 @raise IntersectionError: The B{C{distance}} exceeds the sum
1618 of B{C{radius1}} and B{C{radius2}}.
1620 @raise UnitError: Invalid B{C{distance}}, B{C{radius1}} or
1621 B{C{radius2}}.
1623 @see: U{Circle-Circle Intersection
1624 <https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>}.
1625 '''
1626 d = Distance_(distance, low=_0_0)
1627 r1 = Radius_(radius1=radius1)
1628 r2 = Radius_(radius2=radius2)
1629 if d > (r1 + r2):
1630 raise IntersectionError(distance=d, radius1=r1, radius2=r2,
1631 txt=_too_(_distant_))
1632 return _radical2(d, r1, r2) if d > EPS0 else \
1633 Radical2Tuple(_0_5, _0_0)
1636class Radical2Tuple(_NamedTuple):
1637 '''2-Tuple C{(ratio, xline)} of the I{radical} C{ratio} and
1638 I{radical} C{xline}, both C{scalar} and C{0.0 <= ratio <= 1.0}
1639 '''
1640 _Names_ = (_ratio_, _xline_)
1641 _Units_ = ( Scalar, Scalar)
1644def _radistance(inst):
1645 '''(INTERNAL) Helper for the L{frechet._FrecherMeterRadians}
1646 and L{hausdorff._HausdorffMeterRedians} classes.
1647 '''
1648 kwds_ = _xkwds(inst._kwds, wrap=False)
1649 wrap_ = _xkwds_pop(kwds_, wrap=False)
1650 func_ = inst._func_
1651 try: # calling lower-overhead C{func_}
1652 func_(0, _0_25, _0_5, **kwds_)
1653 wrap_ = _Wrap._philamop(wrap_)
1654 except TypeError:
1655 return inst.distance
1657 def _philam(p):
1658 try:
1659 return p.phi, p.lam
1660 except AttributeError: # no .phi or .lam
1661 return radians(p.lat), radians(p.lon)
1663 def _func_wrap(point1, point2):
1664 phi1, lam1 = wrap_(*_philam(point1))
1665 phi2, lam2 = wrap_(*_philam(point2))
1666 return func_(phi2, phi1, lam2 - lam1, **kwds_)
1668 inst._units = inst._units_
1669 return _func_wrap
1672def _scale_deg(lat1, lat2): # degrees
1673 # scale factor cos(mean of lats) for delta lon
1674 m = fabs(lat1 + lat2) * _0_5
1675 return cos(radians(m)) if m < 90 else _0_0
1678def _scale_rad(phi1, phi2): # radians, by .frechet, .hausdorff, .heights
1679 # scale factor cos(mean of phis) for delta lam
1680 m = fabs(phi1 + phi2) * _0_5
1681 return cos(m) if m < PI_2 else _0_0
1684def _sincosa6(phi2, phi1, lam21): # [4] in cosineLaw
1685 '''(INTERNAL) C{sin}es, C{cos}ines and C{acos}ine.
1686 '''
1687 s2, c2, s1, c1, _, c21 = sincos2_(phi2, phi1, lam21)
1688 return s2, c2, s1, c1, acos1(s1 * s2 + c1 * c2 * c21), c21
1691def thomas(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):
1692 '''Compute the distance between two (ellipsoidal) points using
1693 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>}
1694 formula.
1696 @arg lat1: Start latitude (C{degrees}).
1697 @arg lon1: Start longitude (C{degrees}).
1698 @arg lat2: End latitude (C{degrees}).
1699 @arg lon2: End longitude (C{degrees}).
1700 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
1701 L{Ellipsoid2} or L{a_f2Tuple}) to use.
1702 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
1703 B{C{lat2}} and B{C{lon2}} (C{bool}).
1705 @return: Distance (C{meter}, same units as the B{C{datum}}'s
1706 ellipsoid axes).
1708 @raise TypeError: Invalid B{C{datum}}.
1710 @see: Functions L{thomas_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
1711 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny},
1712 L{flatPolar}, L{haversine}, L{vincentys} and method L{Ellipsoid.distance2}.
1713 '''
1714 return _dE(thomas_, datum, wrap, lat1, lon1, lat2, lon2)
1717def thomas_(phi2, phi1, lam21, datum=_WGS84):
1718 '''Compute the I{angular} distance between two (ellipsoidal) points using
1719 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>}
1720 formula.
1722 @arg phi2: End latitude (C{radians}).
1723 @arg phi1: Start latitude (C{radians}).
1724 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
1725 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid},
1726 L{Ellipsoid2} or L{a_f2Tuple}).
1728 @return: Angular distance (C{radians}).
1730 @raise TypeError: Invalid B{C{datum}}.
1732 @see: Functions L{thomas}, L{cosineAndoyerLambert_},
1733 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
1734 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
1735 L{flatPolar_}, L{haversine_} and L{vincentys_} and U{Geodesy-PHP
1736 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/
1737 Distance/ThomasFormula.php>}.
1738 '''
1739 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21)
1740 if r and isnon0(c1) and isnon0(c2):
1741 E = _ellipsoidal(datum, thomas_)
1742 if E.f:
1743 r1 = atan2(E.b_a * s1, c1)
1744 r2 = atan2(E.b_a * s2, c2)
1746 j = (r2 + r1) * _0_5
1747 k = (r2 - r1) * _0_5
1748 sj, cj, sk, ck, h, _ = sincos2_(j, k, lam21 * _0_5)
1750 h = fsumf_(sk**2, (ck * h)**2, -(sj * h)**2)
1751 u = _1_0 - h
1752 if isnon0(u) and isnon0(h):
1753 r = atan(sqrt0(h / u)) * 2 # == acos(1 - 2 * h)
1754 sr, cr = sincos2(r)
1755 if isnon0(sr):
1756 u = 2 * (sj * ck)**2 / u
1757 h = 2 * (sk * cj)**2 / h
1758 x = u + h
1759 y = u - h
1761 s = r / sr
1762 e = 4 * s**2
1763 d = 2 * cr
1764 a = e * d
1765 b = 2 * r
1766 c = s - (a - d) * _0_5
1767 f = E.f * _0_25
1769 t = fsumf_(a * x, -b * y, c * x**2, -d * y**2, e * x * y)
1770 r -= fsumf_(s * x, -y, -t * f * _0_25) * f * sr
1771 return r
1774def vincentys(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
1775 '''Compute the distance between two (spherical) points using
1776 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>}
1777 spherical formula.
1779 @arg lat1: Start latitude (C{degrees}).
1780 @arg lon1: Start longitude (C{degrees}).
1781 @arg lat2: End latitude (C{degrees}).
1782 @arg lon2: End longitude (C{degrees}).
1783 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
1784 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
1785 L{a_f2Tuple}) to use.
1786 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
1787 B{C{lat2}} and B{C{lon2}} (C{bool}).
1789 @return: Distance (C{meter}, same units as B{C{radius}}).
1791 @raise UnitError: Invalid B{C{radius}}.
1793 @see: Functions L{vincentys_}, L{cosineAndoyerLambert},
1794 L{cosineForsytheAndoyerLambert},L{cosineLaw}, L{equirectangular},
1795 L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar},
1796 L{haversine} and L{thomas} and methods L{Ellipsoid.distance2},
1797 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
1799 @note: See note at function L{vincentys_}.
1800 '''
1801 return _dS(vincentys_, radius, wrap, lat1, lon1, lat2, lon2)
1804def vincentys_(phi2, phi1, lam21):
1805 '''Compute the I{angular} distance between two (spherical) points using
1806 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>}
1807 spherical formula.
1809 @arg phi2: End latitude (C{radians}).
1810 @arg phi1: Start latitude (C{radians}).
1811 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
1813 @return: Angular distance (C{radians}).
1815 @see: Functions L{vincentys}, L{cosineAndoyerLambert_},
1816 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
1817 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
1818 L{flatPolar_}, L{haversine_} and L{thomas_}.
1820 @note: Functions L{vincentys_}, L{haversine_} and L{cosineLaw_}
1821 produce equivalent results, but L{vincentys_} is suitable
1822 for antipodal points and slightly more expensive (M{3 cos,
1823 3 sin, 1 hypot, 1 atan2, 6 mul, 2 add}) than L{haversine_}
1824 (M{2 cos, 2 sin, 2 sqrt, 1 atan2, 5 mul, 1 add}) and
1825 L{cosineLaw_} (M{3 cos, 3 sin, 1 acos, 3 mul, 1 add}).
1826 '''
1827 s1, c1, s2, c2, s21, c21 = sincos2_(phi1, phi2, lam21)
1829 c = c2 * c21
1830 x = s1 * s2 + c1 * c
1831 y = c1 * s2 - s1 * c
1832 return atan2(hypot(c2 * s21, y), x)
1834# **) MIT License
1835#
1836# Copyright (C) 2016-2023 -- mrJean1 at Gmail -- All Rights Reserved.
1837#
1838# Permission is hereby granted, free of charge, to any person obtaining a
1839# copy of this software and associated documentation files (the "Software"),
1840# to deal in the Software without restriction, including without limitation
1841# the rights to use, copy, modify, merge, publish, distribute, sublicense,
1842# and/or sell copies of the Software, and to permit persons to whom the
1843# Software is furnished to do so, subject to the following conditions:
1844#
1845# The above copyright notice and this permission notice shall be included
1846# in all copies or substantial portions of the Software.
1847#
1848# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
1849# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
1850# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
1851# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
1852# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
1853# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
1854# OTHER DEALINGS IN THE SOFTWARE.