Coverage for pygeodesy/formy.py: 97%
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2# -*- coding: utf-8 -*-
4u'''Formulary of basic geodesy functions and approximations.
5'''
6# make sure int/int division yields float quotient, see .basics
7from __future__ import division as _; del _ # PYCHOK semicolon
9# from pygeodesy.cartesianBase import CartesianBase # _MODS
10from pygeodesy.constants import EPS, EPS0, EPS1, PI, PI2, PI3, PI_2, R_M, \
11 _isfinite, float0_, isnon0, remainder, _umod_PI2, \
12 _0_0, _0_125, _0_25, _0_5, _1_0, _2_0, _4_0, \
13 _32_0, _90_0, _180_0, _360_0
14from pygeodesy.datums import Datum, Ellipsoid, _ellipsoidal_datum, \
15 _mean_radius, _spherical_datum, _WGS84, _EWGS84
16# from pygeodesy.ellipsoids import Ellipsoid, _EWGS84 # from .datums
17from pygeodesy.errors import IntersectionError, LimitError, limiterrors, \
18 _TypeError, _ValueError, _xattr, _xError, \
19 _xkwds, _xkwds_pop
20from pygeodesy.fmath import euclid, hypot, hypot_, hypot2, sqrt0
21from pygeodesy.fsums import fsumf_
22from pygeodesy.interns import NN, _delta_, _distant_, _inside_, _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
29# from pygeodesy.triaxials import _hartzell2 # _MODS
30from pygeodesy.units import _isHeight, _isRadius, Bearing, Degrees, Degrees_, \
31 Distance, Distance_, Height, Lam_, Lat, Lon, \
32 Meter_, Phi_, Radians, Radians_, Radius, Radius_, \
33 Scalar, _100km
34from pygeodesy.utily import acos1, atan2b, atan2d, degrees2m, _loneg, m2degrees, \
35 tan_2, sincos2, sincos2_, sincos2d_, _Wrap
36# from pygeodesy.vector3d import _otherV3d # _MODS
37# from pygeodesy.vector3dBase import _xyz_y_z3 # _MODS
38# from pygeodesy import ellipsoidalExact, ellipsoidalKarney, vector3d, \
39# sphericalNvector, sphericalTrigonometry # _MODS
41from contextlib import contextmanager
42from math import asin, atan, atan2, cos, degrees, fabs, radians, sin, sqrt # pow
44__all__ = _ALL_LAZY.formy
45__version__ = '23.12.03'
47_D2_R2 = (PI / _180_0)**2 # degrees- to radians-squared
48_ratio_ = 'ratio'
49_xline_ = 'xline'
52def _anti2(a, b, n_2, n, n2):
53 '''(INTERNAL) Helper for C{antipode} and C{antipode_}.
54 '''
55 r = remainder(a, n) if fabs(a) > n_2 else a
56 if r == a:
57 r = -r
58 b += n
59 if fabs(b) > n:
60 b = remainder(b, n2)
61 return float0_(r, b)
64def antipode(lat, lon, name=NN):
65 '''Return the antipode, the point diametrically opposite
66 to a given point in C{degrees}.
68 @arg lat: Latitude (C{degrees}).
69 @arg lon: Longitude (C{degrees}).
70 @kwarg name: Optional name (C{str}).
72 @return: A L{LatLon2Tuple}C{(lat, lon)}.
74 @see: Functions L{antipode_} and L{normal} and U{Geosphere
75 <https://CRAN.R-Project.org/web/packages/geosphere/geosphere.pdf>}.
76 '''
77 return LatLon2Tuple(*_anti2(lat, lon, _90_0, _180_0, _360_0), name=name)
80def antipode_(phi, lam, name=NN):
81 '''Return the antipode, the point diametrically opposite
82 to a given point in C{radians}.
84 @arg phi: Latitude (C{radians}).
85 @arg lam: Longitude (C{radians}).
86 @kwarg name: Optional name (C{str}).
88 @return: A L{PhiLam2Tuple}C{(phi, lam)}.
90 @see: Functions L{antipode} and L{normal_} and U{Geosphere
91 <https://CRAN.R-Project.org/web/packages/geosphere/geosphere.pdf>}.
92 '''
93 return PhiLam2Tuple(*_anti2(phi, lam, PI_2, PI, PI2), name=name)
96def bearing(lat1, lon1, lat2, lon2, **final_wrap):
97 '''Compute the initial or final bearing (forward or reverse
98 azimuth) between a (spherical) start and end point.
100 @arg lat1: Start latitude (C{degrees}).
101 @arg lon1: Start longitude (C{degrees}).
102 @arg lat2: End latitude (C{degrees}).
103 @arg lon2: End longitude (C{degrees}).
104 @kwarg final_wrap: Optional keyword arguments for function
105 L{pygeodesy.bearing_}.
107 @return: Initial or final bearing (compass C{degrees360}) or
108 zero if start and end point coincide.
109 '''
110 r = bearing_(Phi_(lat1=lat1), Lam_(lon1=lon1),
111 Phi_(lat2=lat2), Lam_(lon2=lon2), **final_wrap)
112 return degrees(r)
115def bearing_(phi1, lam1, phi2, lam2, final=False, wrap=False):
116 '''Compute the initial or final bearing (forward or reverse azimuth)
117 between a (spherical) start and end point.
119 @arg phi1: Start latitude (C{radians}).
120 @arg lam1: Start longitude (C{radians}).
121 @arg phi2: End latitude (C{radians}).
122 @arg lam2: End longitude (C{radians}).
123 @kwarg final: Return final bearing if C{True}, initial otherwise (C{bool}).
124 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{phi2}} and
125 B{C{lam2}} (C{bool}).
127 @return: Initial or final bearing (compass C{radiansPI2}) or zero if start
128 and end point coincide.
130 @see: U{Bearing<https://www.Movable-Type.co.UK/scripts/latlong.html>}, U{Course
131 between two points<https://www.EdWilliams.org/avform147.htm#Crs>} and
132 U{Bearing Between Two Points<https://web.Archive.org/web/20020630205931/
133 https://MathForum.org/library/drmath/view/55417.html>}.
134 '''
135 db, phi2, lam2 = _Wrap.philam3(lam1, phi2, lam2, wrap)
136 if final: # swap plus PI
137 phi1, lam1, phi2, lam2, db = phi2, lam2, phi1, lam1, -db
138 r = PI3
139 else:
140 r = PI2
141 sa1, ca1, sa2, ca2, sdb, cdb = sincos2_(phi1, phi2, db)
143 x = ca1 * sa2 - sa1 * ca2 * cdb
144 y = sdb * ca2
145 return _umod_PI2(atan2(y, x) + r) # .utily.wrapPI2
148def _bearingTo2(p1, p2, wrap=False): # for points.ispolar, sphericalTrigonometry.areaOf
149 '''(INTERNAL) Compute initial and final bearing.
150 '''
151 try: # for LatLon_ and ellipsoidal LatLon
152 return p1.bearingTo2(p2, wrap=wrap)
153 except AttributeError:
154 pass
155 # XXX spherical version, OK for ellipsoidal ispolar?
156 t = p1.philam + p2.philam
157 return Bearing2Tuple(degrees(bearing_(*t, final=False, wrap=wrap)),
158 degrees(bearing_(*t, final=True, wrap=wrap)),
159 name=_bearingTo2.__name__)
162def compassAngle(lat1, lon1, lat2, lon2, adjust=True, wrap=False):
163 '''Return the angle from North for the direction vector M{(lon2 - lon1,
164 lat2 - lat1)} between two points.
166 Suitable only for short, not near-polar vectors up to a few hundred
167 Km or Miles. Use function L{pygeodesy.bearing} for longer vectors.
169 @arg lat1: From latitude (C{degrees}).
170 @arg lon1: From longitude (C{degrees}).
171 @arg lat2: To latitude (C{degrees}).
172 @arg lon2: To longitude (C{degrees}).
173 @kwarg adjust: Adjust the longitudinal delta by the cosine of the
174 mean latitude (C{bool}).
175 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
176 and B{C{lon2}} (C{bool}).
178 @return: Compass angle from North (C{degrees360}).
180 @note: Courtesy of Martin Schultz.
182 @see: U{Local, flat earth approximation
183 <https://www.EdWilliams.org/avform.htm#flat>}.
184 '''
185 d_lon, lat2, lon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap)
186 if adjust: # scale delta lon
187 d_lon *= _scale_deg(lat1, lat2)
188 return atan2b(d_lon, lat2 - lat1)
191def cosineAndoyerLambert(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):
192 '''Compute the distance between two (ellipsoidal) points using the U{Andoyer-Lambert
193 <https://books.google.com/books?id=x2UiAQAAIAAJ>} correction of the U{Law of
194 Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula.
196 @arg lat1: Start latitude (C{degrees}).
197 @arg lon1: Start longitude (C{degrees}).
198 @arg lat2: End latitude (C{degrees}).
199 @arg lon2: End longitude (C{degrees}).
200 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
201 L{Ellipsoid2} or L{a_f2Tuple}) to use.
202 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
203 B{C{lat2}} and B{C{lon2}} (C{bool}).
205 @return: Distance (C{meter}, same units as the B{C{datum}}'s
206 ellipsoid axes or C{radians} if B{C{datum}} is C{None}).
208 @raise TypeError: Invalid B{C{datum}}.
210 @see: Functions L{cosineAndoyerLambert_}, L{cosineForsytheAndoyerLambert},
211 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny},
212 L{flatPolar}, L{haversine}, L{thomas} and L{vincentys} and method
213 L{Ellipsoid.distance2}.
214 '''
215 return _dE(cosineAndoyerLambert_, datum, wrap, lat1, lon1, lat2, lon2)
218def cosineAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84):
219 '''Compute the I{angular} distance between two (ellipsoidal) points using the U{Andoyer-Lambert
220 <https://books.google.com/books?id=x2UiAQAAIAAJ>} correction of the U{Law of
221 Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula.
223 @arg phi2: End latitude (C{radians}).
224 @arg phi1: Start latitude (C{radians}).
225 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
226 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
227 L{Ellipsoid2} or L{a_f2Tuple}) to use.
229 @return: Angular distance (C{radians}).
231 @raise TypeError: Invalid B{C{datum}}.
233 @see: Functions L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert_},
234 L{cosineLaw_}, L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
235 L{flatPolar_}, L{haversine_}, L{thomas_} and L{vincentys_} and U{Geodesy-PHP
236 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/Distance/
237 AndoyerLambert.php>}.
238 '''
239 s2, c2, s1, c1, r, c21 = _sincosa6(phi2, phi1, lam21)
240 if isnon0(c1) and isnon0(c2):
241 E = _ellipsoidal(datum, cosineAndoyerLambert_)
242 if E.f: # ellipsoidal
243 r2 = atan2(E.b_a * s2, c2)
244 r1 = atan2(E.b_a * s1, c1)
245 s2, c2, s1, c1 = sincos2_(r2, r1)
246 r = acos1(s1 * s2 + c1 * c2 * c21)
247 if r:
248 sr, _, sr_2, cr_2 = sincos2_(r, r * _0_5)
249 if isnon0(sr_2) and isnon0(cr_2):
250 s = (sr + r) * ((s1 - s2) / sr_2)**2
251 c = (sr - r) * ((s1 + s2) / cr_2)**2
252 r += (c - s) * E.f * _0_125
253 return r
256def cosineForsytheAndoyerLambert(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):
257 '''Compute the distance between two (ellipsoidal) points using the U{Forsythe-Andoyer-Lambert
258 <https://www2.UNB.Ca/gge/Pubs/TR77.pdf>} correction of the U{Law of Cosines
259 <https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula.
261 @arg lat1: Start latitude (C{degrees}).
262 @arg lon1: Start longitude (C{degrees}).
263 @arg lat2: End latitude (C{degrees}).
264 @arg lon2: End longitude (C{degrees}).
265 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
266 L{Ellipsoid2} or L{a_f2Tuple}) to use.
267 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
268 B{C{lat2}} and B{C{lon2}} (C{bool}).
270 @return: Distance (C{meter}, same units as the B{C{datum}}'s
271 ellipsoid axes or C{radians} if B{C{datum}} is C{None}).
273 @raise TypeError: Invalid B{C{datum}}.
275 @see: Functions L{cosineForsytheAndoyerLambert_}, L{cosineAndoyerLambert},
276 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny},
277 L{flatPolar}, L{haversine}, L{thomas} and L{vincentys} and method
278 L{Ellipsoid.distance2}.
279 '''
280 return _dE(cosineForsytheAndoyerLambert_, datum, wrap, lat1, lon1, lat2, lon2)
283def cosineForsytheAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84):
284 '''Compute the I{angular} distance between two (ellipsoidal) points using the
285 U{Forsythe-Andoyer-Lambert<https://www2.UNB.Ca/gge/Pubs/TR77.pdf>} correction of
286 the U{Law of Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>}
287 formula.
289 @arg phi2: End latitude (C{radians}).
290 @arg phi1: Start latitude (C{radians}).
291 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
292 @kwarg datum: Datum (L{Datum}) or ellipsoid to use (L{Ellipsoid},
293 L{Ellipsoid2} or L{a_f2Tuple}).
295 @return: Angular distance (C{radians}).
297 @raise TypeError: Invalid B{C{datum}}.
299 @see: Functions L{cosineForsytheAndoyerLambert}, L{cosineAndoyerLambert_},
300 L{cosineLaw_}, L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
301 L{flatPolar_}, L{haversine_}, L{thomas_} and L{vincentys_} and U{Geodesy-PHP
302 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/
303 Distance/ForsytheCorrection.php>}.
304 '''
305 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21)
306 if r and isnon0(c1) and isnon0(c2):
307 E = _ellipsoidal(datum, cosineForsytheAndoyerLambert_)
308 if E.f: # ellipsoidal
309 sr, cr, s2r, _ = sincos2_(r, r * 2)
310 if isnon0(sr) and fabs(cr) < EPS1:
311 s = (s1 + s2)**2 / (1 + cr)
312 t = (s1 - s2)**2 / (1 - cr)
313 x = s + t
314 y = s - t
316 s = 8 * r**2 / sr
317 a = 64 * r + s * cr * 2 # 16 * r**2 / tan(r)
318 d = 48 * sr + s # 8 * r**2 / tan(r)
319 b = -2 * d
320 e = 30 * s2r
321 c = fsumf_(30 * r, e * _0_5, s * cr) # 8 * r**2 / tan(r)
322 t = fsumf_( a * x, e * y**2, b * y, -c * x**2, d * x * y)
324 r += fsumf_(-r * x, 3 * y * sr, t * E.f / _32_0) * E.f * _0_25
325 return r
328def cosineLaw(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
329 '''Compute the distance between two points using the U{spherical Law of Cosines
330 <https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula.
332 @arg lat1: Start latitude (C{degrees}).
333 @arg lon1: Start longitude (C{degrees}).
334 @arg lat2: End latitude (C{degrees}).
335 @arg lon2: End longitude (C{degrees}).
336 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
337 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
338 L{a_f2Tuple}) to use.
339 @kwarg wrap: If C{True}, wrap or I{normalize} and B{C{lat2}}
340 and B{C{lon2}} (C{bool}).
342 @return: Distance (C{meter}, same units as B{C{radius}} or the
343 ellipsoid or datum axes).
345 @raise TypeError: Invalid B{C{radius}}.
347 @see: Functions L{cosineLaw_}, L{cosineAndoyerLambert},
348 L{cosineForsytheAndoyerLambert}, L{equirectangular}, L{euclidean},
349 L{flatLocal}/L{hubeny}, L{flatPolar}, L{haversine}, L{thomas} and
350 L{vincentys} and method L{Ellipsoid.distance2}.
352 @note: See note at function L{vincentys_}.
353 '''
354 return _dS(cosineLaw_, radius, wrap, lat1, lon1, lat2, lon2)
357def cosineLaw_(phi2, phi1, lam21):
358 '''Compute the I{angular} distance between two points using the U{spherical Law of
359 Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula.
361 @arg phi2: End latitude (C{radians}).
362 @arg phi1: Start latitude (C{radians}).
363 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
365 @return: Angular distance (C{radians}).
367 @see: Functions L{cosineLaw}, L{cosineAndoyerLambert_},
368 L{cosineForsytheAndoyerLambert_}, L{equirectangular_},
369 L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_},
370 L{haversine_}, L{thomas_} and L{vincentys_}.
372 @note: See note at function L{vincentys_}.
373 '''
374 return _sincosa6(phi2, phi1, lam21)[4]
377def _d3(wrap, lat1, lon1, lat2, lon2):
378 '''(INTERNAL) Helper for _dE, _dS and _eA.
379 '''
380 if wrap:
381 d_lon, lat2, _ = _Wrap.latlon3(lon1, lat2, lon2, wrap)
382 return radians(lat2), Phi_(lat1=lat1), radians(d_lon)
383 else: # for backward compaibility
384 return Phi_(lat2=lat2), Phi_(lat1=lat1), Phi_(d_lon=lon2 - lon1)
387def _dE(func_, earth, *wrap_lls):
388 '''(INTERNAL) Helper for ellipsoidal distances.
389 '''
390 E = _ellipsoidal(earth, func_)
391 r = func_(*_d3(*wrap_lls), datum=E)
392 return r * E.a
395def _dS(func_, radius, *wrap_lls, **adjust):
396 '''(INTERNAL) Helper for spherical distances.
397 '''
398 r = func_(*_d3(*wrap_lls), **adjust)
399 if radius is not R_M:
400 _, lat1, _, lat2, _ = wrap_lls
401 radius = _mean_radius(radius, lat1, lat2)
402 return r * radius
405def _eA(excess_, radius, *wrap_lls):
406 '''(INTERNAL) Helper for spherical excess or area.
407 '''
408 r = excess_(*_d3(*wrap_lls))
409 if radius:
410 _, lat1, _, lat2, _ = wrap_lls
411 r *= _mean_radius(radius, lat1, lat2)**2
412 return r
415def _ellipsoidal(earth, where):
416 '''(INTERNAL) Helper for distances.
417 '''
418 return _EWGS84 if earth in (_WGS84, _EWGS84) else (
419 earth if isinstance(earth, Ellipsoid) else
420 (earth if isinstance(earth, Datum) else # PYCHOK indent
421 _ellipsoidal_datum(earth, name=where.__name__)).ellipsoid)
424def equirectangular(lat1, lon1, lat2, lon2, radius=R_M, **adjust_limit_wrap):
425 '''Compute the distance between two points using the U{Equirectangular Approximation
426 / Projection<https://www.Movable-Type.co.UK/scripts/latlong.html#equirectangular>}.
428 @arg lat1: Start latitude (C{degrees}).
429 @arg lon1: Start longitude (C{degrees}).
430 @arg lat2: End latitude (C{degrees}).
431 @arg lon2: End longitude (C{degrees}).
432 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
433 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
434 L{a_f2Tuple}).
435 @kwarg adjust_limit_wrap: Optional keyword arguments for
436 function L{equirectangular_}.
438 @return: Distance (C{meter}, same units as B{C{radius}} or
439 the ellipsoid or datum axes).
441 @raise TypeError: Invalid B{C{radius}}.
443 @see: Function L{equirectangular_} for more details, the
444 available B{C{options}}, errors, restrictions and other,
445 approximate or accurate distance functions.
446 '''
447 d = sqrt(equirectangular_(Lat(lat1=lat1), Lon(lon1=lon1),
448 Lat(lat2=lat2), Lon(lon2=lon2),
449 **adjust_limit_wrap).distance2) # PYCHOK 4 vs 2-3
450 return degrees2m(d, radius=_mean_radius(radius, lat1, lat2))
453def _equirectangular(lat1, lon1, lat2, lon2, **adjust_limit_wrap):
454 '''(INTERNAL) Helper for the L{frechet._FrecherMeterRadians}
455 and L{hausdorff._HausdorffMeterRedians} classes.
456 '''
457 return equirectangular_(lat1, lon1, lat2, lon2, **adjust_limit_wrap).distance2 * _D2_R2
460def equirectangular_(lat1, lon1, lat2, lon2, adjust=True, limit=45, wrap=False):
461 '''Compute the distance between two points using the U{Equirectangular Approximation
462 / Projection<https://www.Movable-Type.co.UK/scripts/latlong.html#equirectangular>}.
464 This approximation is valid for short distance of several hundred Km
465 or Miles, see the B{C{limit}} keyword argument and L{LimitError}.
467 @arg lat1: Start latitude (C{degrees}).
468 @arg lon1: Start longitude (C{degrees}).
469 @arg lat2: End latitude (C{degrees}).
470 @arg lon2: End longitude (C{degrees}).
471 @kwarg adjust: Adjust the wrapped, unrolled longitudinal delta
472 by the cosine of the mean latitude (C{bool}).
473 @kwarg limit: Optional limit for lat- and longitudinal deltas
474 (C{degrees}) or C{None} or C{0} for unlimited.
475 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
476 and B{C{lon2}} (C{bool}).
478 @return: A L{Distance4Tuple}C{(distance2, delta_lat, delta_lon,
479 unroll_lon2)} in C{degrees squared}.
481 @raise LimitError: If the lat- and/or longitudinal delta exceeds the
482 B{C{-limit..limit}} range and L{pygeodesy.limiterrors}
483 set to C{True}.
485 @see: U{Local, flat earth approximation
486 <https://www.EdWilliams.org/avform.htm#flat>}, functions
487 L{equirectangular}, L{cosineAndoyerLambert},
488 L{cosineForsytheAndoyerLambert}, L{cosineLaw}, L{euclidean},
489 L{flatLocal}/L{hubeny}, L{flatPolar}, L{haversine}, L{thomas}
490 and L{vincentys} and methods L{Ellipsoid.distance2},
491 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
492 '''
493 d_lon, lat2, ulon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap)
494 d_lat = lat2 - lat1
496 if limit and limit > 0 and limiterrors():
497 d = max(fabs(d_lat), fabs(d_lon))
498 if d > limit:
499 t = _SPACE_(_delta_, Fmt.PAREN_g(d), Fmt.exceeds_limit(limit))
500 s = unstr(equirectangular_, lat1, lon1, lat2, lon2,
501 limit=limit, wrap=wrap)
502 raise LimitError(s, txt=t)
504 if adjust: # scale delta lon
505 d_lon *= _scale_deg(lat1, lat2)
507 d2 = hypot2(d_lat, d_lon) # degrees squared!
508 return Distance4Tuple(d2, d_lat, d_lon, ulon2 - lon2)
511def euclidean(lat1, lon1, lat2, lon2, radius=R_M, adjust=True, wrap=False):
512 '''Approximate the C{Euclidean} distance between two (spherical) points.
514 @arg lat1: Start latitude (C{degrees}).
515 @arg lon1: Start longitude (C{degrees}).
516 @arg lat2: End latitude (C{degrees}).
517 @arg lon2: End longitude (C{degrees}).
518 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
519 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
520 L{a_f2Tuple}) to use.
521 @kwarg adjust: Adjust the longitudinal delta by the cosine of
522 the mean latitude (C{bool}).
523 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
524 and B{C{lon2}} (C{bool}).
526 @return: Distance (C{meter}, same units as B{C{radius}} or the
527 ellipsoid or datum axes).
529 @raise TypeError: Invalid B{C{radius}}.
531 @see: U{Distance between two (spherical) points
532 <https://www.EdWilliams.org/avform.htm#Dist>}, functions L{euclid},
533 L{euclidean_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
534 L{cosineLaw}, L{equirectangular}, L{flatLocal}/L{hubeny}, L{flatPolar},
535 L{haversine}, L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2},
536 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
537 '''
538 return _dS(euclidean_, radius, wrap, lat1, lon1, lat2, lon2, adjust=adjust)
541def euclidean_(phi2, phi1, lam21, adjust=True):
542 '''Approximate the I{angular} C{Euclidean} distance between two (spherical) points.
544 @arg phi2: End latitude (C{radians}).
545 @arg phi1: Start latitude (C{radians}).
546 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
547 @kwarg adjust: Adjust the longitudinal delta by the cosine
548 of the mean latitude (C{bool}).
550 @return: Angular distance (C{radians}).
552 @see: Functions L{euclid}, L{euclidean}, L{cosineAndoyerLambert_},
553 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{equirectangular_},
554 L{flatLocal_}/L{hubeny_}, L{flatPolar_}, L{haversine_}, L{thomas_}
555 and L{vincentys_}.
556 '''
557 if adjust:
558 lam21 *= _scale_rad(phi2, phi1)
559 return euclid(phi2 - phi1, lam21)
562def excessAbc_(A, b, c):
563 '''Compute the I{spherical excess} C{E} of a (spherical) triangle from two sides
564 and the included (small) angle.
566 @arg A: An interior triangle angle (C{radians}).
567 @arg b: Frist adjacent triangle side (C{radians}).
568 @arg c: Second adjacent triangle side (C{radians}).
570 @return: Spherical excess (C{radians}).
572 @raise UnitError: Invalid B{C{A}}, B{C{b}} or B{C{c}}.
574 @see: Functions L{excessGirard_}, L{excessLHuilier_} and U{Spherical
575 trigonometry<https://WikiPedia.org/wiki/Spherical_trigonometry>}.
576 '''
577 A = Radians_(A=A)
578 b = Radians_(b=b) * _0_5
579 c = Radians_(c=c) * _0_5
581 sA, cA, sb, cb, sc, cc = sincos2_(A, b, c)
582 return atan2(sA * sb * sc, cb * cc + cA * sb * sc) * _2_0
585def excessCagnoli_(a, b, c):
586 '''Compute the I{spherical excess} C{E} of a (spherical) triangle using U{Cagnoli's
587 <https://Zenodo.org/record/35392>} (D.34) formula.
589 @arg a: First triangle side (C{radians}).
590 @arg b: Second triangle side (C{radians}).
591 @arg c: Third triangle side (C{radians}).
593 @return: Spherical excess (C{radians}).
595 @raise UnitError: Invalid B{C{a}}, B{C{b}} or B{C{c}}.
597 @see: Function L{excessLHuilier_} and U{Spherical trigonometry
598 <https://WikiPedia.org/wiki/Spherical_trigonometry>}.
599 '''
600 a = Radians_(a=a)
601 b = Radians_(b=b)
602 c = Radians_(c=c)
604 s = fsumf_(a, b, c) * _0_5
605 _s = sin
606 r = _s(s) * _s(s - a) * _s(s - b) * _s(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 U{Girard's
614 <https://MathWorld.Wolfram.com/GirardsSphericalExcessFormula.html>} formula.
616 @arg A: First interior triangle angle (C{radians}).
617 @arg B: Second interior triangle angle (C{radians}).
618 @arg C: Third interior triangle angle (C{radians}).
620 @return: Spherical excess (C{radians}).
622 @raise UnitError: Invalid B{C{A}}, B{C{B}} or B{C{C}}.
624 @see: Function L{excessLHuilier_} and U{Spherical trigonometry
625 <https://WikiPedia.org/wiki/Spherical_trigonometry>}.
626 '''
627 return Radians(Girard=fsumf_(Radians_(A=A),
628 Radians_(B=B),
629 Radians_(C=C), -PI))
632def excessLHuilier_(a, b, c):
633 '''Compute the I{spherical excess} C{E} of a (spherical) triangle using U{L'Huilier's
634 <https://MathWorld.Wolfram.com/LHuiliersTheorem.html>}'s Theorem.
636 @arg a: First triangle side (C{radians}).
637 @arg b: Second triangle side (C{radians}).
638 @arg c: Third triangle side (C{radians}).
640 @return: Spherical excess (C{radians}).
642 @raise UnitError: Invalid B{C{a}}, B{C{b}} or B{C{c}}.
644 @see: Function L{excessCagnoli_}, L{excessGirard_} and U{Spherical
645 trigonometry<https://WikiPedia.org/wiki/Spherical_trigonometry>}.
646 '''
647 a = Radians_(a=a)
648 b = Radians_(b=b)
649 c = Radians_(c=c)
651 s = fsumf_(a, b, c) * _0_5
652 _t = tan_2
653 r = _t(s) * _t(s - a) * _t(s - b) * _t(s - c)
654 r = atan(sqrt(r)) if r > 0 else _0_0
655 return Radians(LHuilier=r * _4_0)
658def excessKarney(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
659 '''Compute the surface area of a (spherical) quadrilateral bounded by a
660 segment of a great circle, two meridians and the equator using U{Karney's
661 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>}
662 method.
664 @arg lat1: Start latitude (C{degrees}).
665 @arg lon1: Start longitude (C{degrees}).
666 @arg lat2: End latitude (C{degrees}).
667 @arg lon2: End longitude (C{degrees}).
668 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
669 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
670 L{a_f2Tuple}) or C{None}.
671 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
672 B{C{lat2}} and B{C{lon2}} (C{bool}).
674 @return: Surface area, I{signed} (I{square} C{meter} or the same units as
675 B{C{radius}} I{squared}) or the I{spherical excess} (C{radians})
676 if C{B{radius}=0} or C{None}.
678 @raise TypeError: Invalid B{C{radius}}.
680 @raise UnitError: Invalid B{C{lat2}} or B{C{lat1}}.
682 @raise ValueError: Semi-circular longitudinal delta.
684 @see: Functions L{excessKarney_} and L{excessQuad}.
685 '''
686 return _eA(excessKarney_, radius, wrap, lat1, lon1, lat2, lon2)
689def excessKarney_(phi2, phi1, lam21):
690 '''Compute the I{spherical excess} C{E} of a (spherical) quadrilateral bounded by
691 a segment of a great circle, two meridians and the equator using U{Karney's
692 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>}
693 method.
695 @arg phi2: End latitude (C{radians}).
696 @arg phi1: Start latitude (C{radians}).
697 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
699 @return: Spherical excess, I{signed} (C{radians}).
701 @raise ValueError: Semi-circular longitudinal delta B{C{lam21}}.
703 @see: Function L{excessKarney} and U{Area of a spherical polygon
704 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>}.
705 '''
706 # from: Veness <https://www.Movable-Type.co.UK/scripts/latlong.html> Area
707 # method due to Karney: for each edge of the polygon,
708 #
709 # tan(Δλ / 2) · (tan(φ1 / 2) + tan(φ2 / 2))
710 # tan(E / 2) = -----------------------------------------
711 # 1 + tan(φ1 / 2) · tan(φ2 / 2)
712 #
713 # where E is the spherical excess of the trapezium obtained by extending
714 # the edge to the equator-circle vector for each edge (see also ***).
715 _t = tan_2
716 t2 = _t(phi2)
717 t1 = _t(phi1)
718 t = _t(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((Lamb(phi1) + Lamb(phi2)) / 2)
742#
743# where lambda21 = lambda2 - lambda1 and Lamb(x) is the Lambertian (or the
744# inverse Gudermannian) function
745#
746# Lambertian(x) = asinh(tan(x)) = atanh(sin(x)) = 2 * atanh(tan(x / 2))
747#
748# Notes: The formula for S12 is exact, except that...
749# - it is indeterminate if an edge is a semi-circle
750# - the formula for A applies only if the polygon does not include a pole
751# (if it does, then add +/- 2 * pi to the result)
752# - in the limit of small phi and lambda, S12 reduces to the trapezoidal
753# formula, S12 = (lambda2 - lambda1) * (phi1 + phi2) / 2
754# - I derived this result from the equation for the area of a spherical
755# triangle in terms of two edges and the included angle given by, e.g.
756# U{Todhunter, I. - Spherical Trigonometry (1871), Sec. 103, Eq. (2)
757# <http://Books.Google.com/books?id=3uBHAAAAIAAJ&pg=PA71>}
758# - I would be interested to know if this formula for S12 is already known
759# - Charles Karney
762def excessQuad(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
763 '''Compute the surface area of a (spherical) quadrilateral bounded by a segment
764 of a great circle, two meridians and the equator.
766 @arg lat1: Start latitude (C{degrees}).
767 @arg lon1: Start longitude (C{degrees}).
768 @arg lat2: End latitude (C{degrees}).
769 @arg lon2: End longitude (C{degrees}).
770 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
771 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
772 L{a_f2Tuple}) or C{None}.
773 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
774 B{C{lat2}} and B{C{lon2}} (C{bool}).
776 @return: Surface area, I{signed} (I{square} C{meter} or the same units as
777 B{C{radius}} I{squared}) or the I{spherical excess} (C{radians})
778 if C{B{radius}=0} or C{None}.
780 @raise TypeError: Invalid B{C{radius}}.
782 @raise UnitError: Invalid B{C{lat2}} or B{C{lat1}}.
784 @see: Function L{excessQuad_} and L{excessKarney}.
785 '''
786 return _eA(excessQuad_, radius, wrap, lat1, lon1, lat2, lon2)
789def excessQuad_(phi2, phi1, lam21):
790 '''Compute the I{spherical excess} C{E} of a (spherical) quadrilateral bounded
791 by a segment of a great circle, two meridians and the equator.
793 @arg phi2: End latitude (C{radians}).
794 @arg phi1: Start latitude (C{radians}).
795 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
797 @return: Spherical excess, I{signed} (C{radians}).
799 @see: Function L{excessQuad} and U{Spherical trigonometry
800 <https://WikiPedia.org/wiki/Spherical_trigonometry>}.
801 '''
802 s = sin((phi2 + phi1) * _0_5)
803 c = cos((phi2 - phi1) * _0_5)
804 return Radians(Quad=atan2(tan_2(lam21) * s, c) * _2_0)
807def flatLocal(lat1, lon1, lat2, lon2, datum=_WGS84, scaled=True, wrap=False):
808 '''Compute the distance between two (ellipsoidal) points using
809 the U{ellipsoidal Earth to plane projection<https://WikiPedia.org/
810 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>}
811 aka U{Hubeny<https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula.
813 @arg lat1: Start latitude (C{degrees}).
814 @arg lon1: Start longitude (C{degrees}).
815 @arg lat2: End latitude (C{degrees}).
816 @arg lon2: End longitude (C{degrees}).
817 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
818 L{Ellipsoid2} or L{a_f2Tuple}) to use.
819 @kwarg scaled: Scale prime_vertical by C{cos(B{phi})} (C{bool}),
820 see method L{pygeodesy.Ellipsoid.roc2_}.
821 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
822 B{C{lat2}} and B{C{lon2}} (C{bool}).
824 @return: Distance (C{meter}, same units as the B{C{datum}}'s
825 ellipsoid axes).
827 @raise TypeError: Invalid B{C{datum}}.
829 @note: The meridional and prime_vertical radii of curvature
830 are taken and scaled at the mean of both latitude.
832 @see: Functions L{flatLocal_} or L{hubeny_}, L{cosineLaw}, L{flatPolar},
833 L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
834 L{equirectangular}, L{euclidean}, L{haversine}, L{thomas},
835 L{vincentys}, method L{Ellipsoid.distance2} and U{local, flat
836 earth approximation<https://www.EdWilliams.org/avform.htm#flat>}.
837 '''
838 E = _ellipsoidal(datum, flatLocal)
839 return E._hubeny_2(*_d3(wrap, lat1, lon1, lat2, lon2),
840 scaled=scaled, squared=False) * E.a
842hubeny = flatLocal # PYCHOK for Karl Hubeny
845def flatLocal_(phi2, phi1, lam21, datum=_WGS84, scaled=True):
846 '''Compute the I{angular} distance between two (ellipsoidal) points using
847 the U{ellipsoidal Earth to plane projection<https://WikiPedia.org/
848 wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>}
849 aka U{Hubeny<https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula.
851 @arg phi2: End latitude (C{radians}).
852 @arg phi1: Start latitude (C{radians}).
853 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
854 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
855 L{Ellipsoid2} or L{a_f2Tuple}) to use.
856 @kwarg scaled: Scale prime_vertical by C{cos(B{phi})} (C{bool}),
857 see method L{pygeodesy.Ellipsoid.roc2_}.
859 @return: Angular distance (C{radians}).
861 @raise TypeError: Invalid B{C{datum}}.
863 @note: The meridional and prime_vertical radii of curvature
864 are taken and scaled I{at the mean of both latitude}.
866 @see: Functions L{flatLocal} or L{hubeny}, L{cosineAndoyerLambert_},
867 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{flatPolar_},
868 L{equirectangular_}, L{euclidean_}, L{haversine_}, L{thomas_}
869 and L{vincentys_} and U{local, flat earth approximation
870 <https://www.EdWilliams.org/avform.htm#flat>}.
871 '''
872 E = _ellipsoidal(datum, flatLocal_)
873 return E._hubeny_2(phi2, phi1, lam21, scaled=scaled, squared=False)
875hubeny_ = flatLocal_ # PYCHOK for Karl Hubeny
878def flatPolar(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
879 '''Compute the distance between two (spherical) points using
880 the U{polar coordinate flat-Earth <https://WikiPedia.org/wiki/
881 Geographical_distance#Polar_coordinate_flat-Earth_formula>}
882 formula.
884 @arg lat1: Start latitude (C{degrees}).
885 @arg lon1: Start longitude (C{degrees}).
886 @arg lat2: End latitude (C{degrees}).
887 @arg lon2: End longitude (C{degrees}).
888 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
889 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
890 L{a_f2Tuple}) to use.
891 @kwarg wrap: If C{True}, wrap or I{normalize} and B{C{lat2}}
892 and B{C{lon2}} (C{bool}).
894 @return: Distance (C{meter}, same units as B{C{radius}} or the
895 ellipsoid or datum axes).
897 @raise TypeError: Invalid B{C{radius}}.
899 @see: Functions L{flatPolar_}, L{cosineAndoyerLambert},
900 L{cosineForsytheAndoyerLambert},L{cosineLaw},
901 L{flatLocal}/L{hubeny}, L{equirectangular},
902 L{euclidean}, L{haversine}, L{thomas} and
903 L{vincentys}.
904 '''
905 return _dS(flatPolar_, radius, wrap, lat1, lon1, lat2, lon2)
908def flatPolar_(phi2, phi1, lam21):
909 '''Compute the I{angular} distance between two (spherical) points
910 using the U{polar coordinate flat-Earth<https://WikiPedia.org/wiki/
911 Geographical_distance#Polar_coordinate_flat-Earth_formula>}
912 formula.
914 @arg phi2: End latitude (C{radians}).
915 @arg phi1: Start latitude (C{radians}).
916 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
918 @return: Angular distance (C{radians}).
920 @see: Functions L{flatPolar}, L{cosineAndoyerLambert_},
921 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
922 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
923 L{haversine_}, L{thomas_} and L{vincentys_}.
924 '''
925 a = fabs(PI_2 - phi1) # co-latitude
926 b = fabs(PI_2 - phi2) # co-latitude
927 if a < b:
928 a, b = b, a
929 if a < EPS0:
930 a = _0_0
931 elif b > 0:
932 b = b / a # /= chokes PyChecker
933 c = b * cos(lam21) * _2_0
934 c = fsumf_(_1_0, b**2, -fabs(c))
935 a *= sqrt0(c)
936 return a
939def _hartzell(inst, los, earth, **kwds):
940 '''(INTERNAL) Helper for C{CartesianBase.hartzell} and C{LatLonBase.hartzell}.
941 '''
942 if earth is not None:
943 earth = _spherical_datum(earth, name=hartzell.__name__)
944 inst = inst.toDatum(earth)
945 h = inst.height
946 if h > 0: # EPS0
947 r = hartzell(inst, los=los, earth=earth or inst.datum, **kwds)
948 elif h < 0: # EPS0
949 raise IntersectionError(pov=inst, los=los, height=h, txt=_inside_)
950 else:
951 r = inst
952 return r
955def hartzell(pov, los=None, earth=_WGS84, name=NN, **LatLon_and_kwds):
956 '''Compute the intersection of the earth's surface and a Line-Of-Sight
957 from a Point-Of-View in space.
959 @arg pov: Point-Of-View outside the earth (C{Cartesian}, L{Ecef9Tuple}
960 C{LatLon} or L{Vector3d}).
961 @kwarg los: Line-Of-Sight, I{direction} to earth (L{Los}, L{Vector3d})
962 or C{None} to point to the earth' center.
963 @kwarg earth: The earth model (L{Datum}, L{Ellipsoid}, L{Ellipsoid2},
964 L{a_f2Tuple} or C{scalar} radius in C{meter}).
965 @kwarg name: Optional name (C{str}).
966 @kwarg LatLon_and_kwds: Optional C{LatLon} class for the intersection
967 point plus C{LatLon} keyword arguments, include
968 B{C{datum}} if different from B{C{earth}}.
970 @return: The intersection point (L{Vector3d}, the C{Cartesian type} of
971 B{C{pov}} or the given C{B{LatLon}_and_kwds}) with C{.heigth}
972 set to the distance to the B{C{pov}}.
974 @raise IntersectionError: Null B{C{pov}} or B{C{los}} vector, B{C{pov}}
975 is inside the earth or B{C{los}} points outside
976 the earth or points in an opposite direction.
978 @raise TypeError: Invalid B{C{pov}}, B{C{los}} or B{C{earth}}.
980 @see: Function L{pygeodesy.hartzell4}, L{pygeodesy.tyr3d} for B{C{los}},
981 methods L{Ellipsoid.hartzell4}, C{Cartesian.hartzell}, C{LatLon.hartzell}
982 and U{I{Satellite Line-of-Sight Intersection with Earth}<https://
983 StephenHartzell.Medium.com/satellite-line-of-sight-intersection-with-earth-d786b4a6a9b6>}.
984 '''
985 n = hartzell.__name__
986 D = earth if isinstance(earth, Datum) else \
987 _spherical_datum(earth, name=n)
988 try:
989 r, h = _MODS.triaxials._hartzell2(pov, los, D.ellipsoid._triaxial)
990 except Exception as x:
991 raise IntersectionError(pov=pov, los=los, earth=earth, cause=x)
993 r = _xnamed(r, name or n)
994 C = _MODS.cartesianBase.CartesianBase
995 if LatLon_and_kwds:
996 c = C(r, datum=D, name=r.name)
997 r = c.toLatLon(**_xkwds(LatLon_and_kwds, height=h))
998 elif isinstance(r, C):
999 r.height = h
1000 return r
1003def haversine(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
1004 '''Compute the distance between two (spherical) points using the
1005 U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>}
1006 formula.
1008 @arg lat1: Start latitude (C{degrees}).
1009 @arg lon1: Start longitude (C{degrees}).
1010 @arg lat2: End latitude (C{degrees}).
1011 @arg lon2: End longitude (C{degrees}).
1012 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
1013 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
1014 L{a_f2Tuple}) to use.
1015 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
1016 B{C{lat2}} and B{C{lon2}} (C{bool}).
1018 @return: Distance (C{meter}, same units as B{C{radius}}).
1020 @raise TypeError: Invalid B{C{radius}}.
1022 @see: U{Distance between two (spherical) points
1023 <https://www.EdWilliams.org/avform.htm#Dist>}, functions
1024 L{cosineLaw}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
1025 L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar},
1026 L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2},
1027 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
1029 @note: See note at function L{vincentys_}.
1030 '''
1031 return _dS(haversine_, radius, wrap, lat1, lon1, lat2, lon2)
1034def haversine_(phi2, phi1, lam21):
1035 '''Compute the I{angular} distance between two (spherical) points
1036 using the U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>}
1037 formula.
1039 @arg phi2: End latitude (C{radians}).
1040 @arg phi1: Start latitude (C{radians}).
1041 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
1043 @return: Angular distance (C{radians}).
1045 @see: Functions L{haversine}, L{cosineAndoyerLambert_},
1046 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
1047 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
1048 L{flatPolar_}, L{thomas_} and L{vincentys_}.
1050 @note: See note at function L{vincentys_}.
1051 '''
1052 def _hsin(rad):
1053 return sin(rad * _0_5)**2
1055 h = _hsin(phi2 - phi1) + cos(phi1) * cos(phi2) * _hsin(lam21) # haversine
1056 return atan2(sqrt0(h), sqrt0(_1_0 - h)) * _2_0 # == asin(sqrt(h)) * 2
1059def heightOf(angle, distance, radius=R_M):
1060 '''Determine the height above the (spherical) earth' surface after
1061 traveling along a straight line at a given tilt.
1063 @arg angle: Tilt angle above horizontal (C{degrees}).
1064 @arg distance: Distance along the line (C{meter} or same units as
1065 B{C{radius}}).
1066 @kwarg radius: Optional mean earth radius (C{meter}).
1068 @return: Height (C{meter}, same units as B{C{distance}} and B{C{radius}}).
1070 @raise ValueError: Invalid B{C{angle}}, B{C{distance}} or B{C{radius}}.
1072 @see: U{MultiDop geog_lib.GeogBeamHt<https://GitHub.com/NASA/MultiDop>}
1073 (U{Shapiro et al. 2009, JTECH
1074 <https://Journals.AMetSoc.org/doi/abs/10.1175/2009JTECHA1256.1>}
1075 and U{Potvin et al. 2012, JTECH
1076 <https://Journals.AMetSoc.org/doi/abs/10.1175/JTECH-D-11-00019.1>}).
1077 '''
1078 r = h = Radius(radius)
1079 d = fabs(Distance(distance))
1080 if d > h:
1081 d, h = h, d
1083 if d > EPS0: # and h > EPS0
1084 d = d / h # /= h chokes PyChecker
1085 s = sin(Phi_(angle=angle, clip=_180_0))
1086 s = fsumf_(_1_0, s * d * _2_0, d**2)
1087 if s > 0:
1088 return h * sqrt(s) - r
1090 raise _ValueError(angle=angle, distance=distance, radius=radius)
1093def heightOrthometric(h_ll, N):
1094 '''Get the I{orthometric} height B{H}, the height above the geoid, earth surface.
1096 @arg h_ll: The height above the ellipsoid (C{meter}) or an I{ellipsoidal}
1097 location (C{LatLon} with a C{height} or C{h} attribute).
1098 @arg N: The I{geoid} height (C{meter}), the height of the geoid above the
1099 ellipsoid at the same B{C{h_ll}} location.
1101 @return: I{Orthometric} height C{B{H} = B{h} - B{N}} (C{meter}, same units
1102 as B{C{h}} and B{C{N}}).
1104 @see: U{Ellipsoid, Geoid, and Othometric Heights<https://www.NGS.NOAA.gov/
1105 GEOID/PRESENTATIONS/2007_02_24_CCPS/Roman_A_PLSC2007notes.pdf>}, page
1106 6 and module L{pygeodesy.geoids}.
1107 '''
1108 h = h_ll if _isHeight(h_ll) else _xattr(h_ll, height=_xattr(h_ll, h=0))
1109 return Height(H=Height(h=h) - Height(N=N))
1112def horizon(height, radius=R_M, refraction=False):
1113 '''Determine the distance to the horizon from a given altitude
1114 above the (spherical) earth.
1116 @arg height: Altitude (C{meter} or same units as B{C{radius}}).
1117 @kwarg radius: Optional mean earth radius (C{meter}).
1118 @kwarg refraction: Consider atmospheric refraction (C{bool}).
1120 @return: Distance (C{meter}, same units as B{C{height}} and B{C{radius}}).
1122 @raise ValueError: Invalid B{C{height}} or B{C{radius}}.
1124 @see: U{Distance to horizon<https://www.EdWilliams.org/avform.htm#Horizon>}.
1125 '''
1126 h, r = Height(height), Radius(radius)
1127 if min(h, r) < 0:
1128 raise _ValueError(height=height, radius=radius)
1130 d2 = ((r * 2.415750694528) if refraction else # 2.0 / 0.8279
1131 fsumf_(r, r, h)) * h
1132 return sqrt0(d2)
1135class _idllmn6(object): # see also .geodesicw._wargs, .latlonBase._toCartesian3, .vector2d._numpy
1136 '''(INTERNAL) Helper for C{intersection2} and C{intersections2}.
1137 '''
1138 @contextmanager # <https://www.Python.org/dev/peps/pep-0343/> Examples
1139 def __call__(self, datum, lat1, lon1, lat2, lon2, small, wrap, s, **kwds):
1140 try:
1141 if wrap:
1142 _, lat2, lon2 = _Wrap.latlon3(lon1, lat2, lon2, wrap)
1143 kwds = _xkwds(kwds, wrap=wrap) # for _xError
1144 m = small if small is _100km else Meter_(small=small)
1145 n = (intersections2 if s else intersection2).__name__
1146 if datum is None or euclidean(lat1, lon1, lat2, lon2) < m:
1147 d, m = None, _MODS.vector3d
1148 _i = m._intersects2 if s else m._intersect3d3
1149 elif _isRadius(datum) and datum < 0 and not s:
1150 d = _spherical_datum(-datum, name=n)
1151 m = _MODS.sphericalNvector
1152 _i = m.intersection
1153 else:
1154 d = _spherical_datum(datum, name=n)
1155 if d.isSpherical:
1156 m = _MODS.sphericalTrigonometry
1157 _i = m._intersects2 if s else m._intersect
1158 elif d.isEllipsoidal:
1159 try:
1160 if d.ellipsoid.geodesic:
1161 pass
1162 m = _MODS.ellipsoidalKarney
1163 except ImportError:
1164 m = _MODS.ellipsoidalExact
1165 _i = m._intersections2 if s else m._intersection3 # ellispoidalBaseDI
1166 else:
1167 raise _TypeError(datum=datum)
1168 yield _i, d, lat2, lon2, m, n
1170 except (TypeError, ValueError) as x:
1171 raise _xError(x, lat1=lat1, lon1=lon1, datum=datum,
1172 lat2=lat2, lon2=lon2, small=small, **kwds)
1174_idllmn6 = _idllmn6() # PYCHOK singleton
1177def intersection2(lat1, lon1, bearing1,
1178 lat2, lon2, bearing2, datum=None, wrap=False, small=_100km): # was=True
1179 '''I{Conveniently} compute the intersection of two lines each defined
1180 by a (geodetic) point and a bearing from North, using either ...
1182 1) L{vector3d.intersection3d3} for B{C{small}} distances (below 100 Km
1183 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ...
1185 2) L{sphericalTrigonometry.intersection} for a spherical B{C{datum}}
1186 or a C{scalar B{datum}} representing the earth radius, conventionally
1187 in C{meter} or ...
1189 3) L{sphericalNvector.intersection} if B{C{datum}} is a I{negative}
1190 C{scalar}, (negative) earth radius, conventionally in C{meter} or ...
1192 4) L{ellipsoidalKarney.intersection3} for an ellipsoidal B{C{datum}}
1193 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>}
1194 is installed, otherwise ...
1196 5) L{ellipsoidalExact.intersection3}, provided B{C{datum}} is ellipsoidal.
1198 @arg lat1: Latitude of the first point (C{degrees}).
1199 @arg lon1: Longitude of the first point (C{degrees}).
1200 @arg bearing1: Bearing at the first point (compass C{degrees}).
1201 @arg lat2: Latitude of the second point (C{degrees}).
1202 @arg lon2: Longitude of the second point (C{degrees}).
1203 @arg bearing2: Bearing at the second point (compass C{degrees}).
1204 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid},
1205 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth
1206 radius (C{meter}, same units as B{C{radius1}} and
1207 B{C{radius2}}) or C{None}.
1208 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
1209 and B{C{lon2}} (C{bool}).
1210 @kwarg small: Upper limit for small distances (C{meter}).
1212 @return: A L{LatLon2Tuple}C{(lat, lon)} with the lat- and
1213 longitude of the intersection point.
1215 @raise IntersectionError: Ambiguous or infinite intersection
1216 or colinear, parallel or otherwise
1217 non-intersecting lines.
1219 @raise TypeError: Invalid B{C{datum}}.
1221 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{bearing1}},
1222 B{C{lat2}}, B{C{lon2}} or B{C{bearing2}}.
1224 @see: Method L{RhumbLine.intersection2}.
1226 @note: The returned intersections may be near-antipodal.
1227 '''
1228 b1 = Bearing(bearing1=bearing1)
1229 b2 = Bearing(bearing2=bearing2)
1230 with _idllmn6(datum, lat1, lon1, lat2, lon2,
1231 small, wrap, False, bearing1=b1, bearing2=b2) as t:
1232 _i, d, lat2, lon2, m, n = t
1233 if d is None:
1234 t, _, _ = _i(m.Vector3d(lon1, lat1, 0), b1,
1235 m.Vector3d(lon2, lat2, 0), b2, useZ=False)
1236 t = LatLon2Tuple(t.y, t.x, name=n)
1238 else:
1239 t = _i(m.LatLon(lat1, lon1, datum=d), b1,
1240 m.LatLon(lat2, lon2, datum=d), b2, height=0, wrap=False)
1241 if isinstance(t, Intersection3Tuple): # ellipsoidal
1242 t, _, _ = t
1243 t = LatLon2Tuple(t.lat, t.lon, name=n)
1244 return t
1247def intersections2(lat1, lon1, radius1,
1248 lat2, lon2, radius2, datum=None, wrap=False, small=_100km): # was=True
1249 '''I{Conveniently} compute the intersections of two circles each defined
1250 by a (geodetic) center point and a radius, using either ...
1252 1) L{vector3d.intersections2} for B{C{small}} distances (below 100 Km
1253 or about 0.88 degrees) or if I{no} B{C{datum}} is specified, or ...
1255 2) L{sphericalTrigonometry.intersections2} for a spherical B{C{datum}}
1256 or a C{scalar B{datum}} representing the earth radius, conventionally
1257 in C{meter} or ...
1259 3) L{ellipsoidalKarney.intersections2} for an ellipsoidal B{C{datum}}
1260 and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>}
1261 is installed, otherwise ...
1263 4) L{ellipsoidalExact.intersections2}, provided B{C{datum}} is ellipsoidal.
1265 @arg lat1: Latitude of the first circle center (C{degrees}).
1266 @arg lon1: Longitude of the first circle center (C{degrees}).
1267 @arg radius1: Radius of the first circle (C{meter}, conventionally).
1268 @arg lat2: Latitude of the second circle center (C{degrees}).
1269 @arg lon2: Longitude of the second circle center (C{degrees}).
1270 @arg radius2: Radius of the second circle (C{meter}, same units as B{C{radius1}}).
1271 @kwarg datum: Optional datum (L{Datum}) or ellipsoid (L{Ellipsoid},
1272 L{Ellipsoid2} or L{a_f2Tuple}) or C{scalar} earth
1273 radius (C{meter}, same units as B{C{radius1}} and
1274 B{C{radius2}}) or C{None}.
1275 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{lat2}}
1276 and B{C{lon2}} (C{bool}).
1277 @kwarg small: Upper limit for small distances (C{meter}).
1279 @return: 2-Tuple of the intersection points, each a
1280 L{LatLon2Tuple}C{(lat, lon)}. For abutting circles, the
1281 points are the same instance, aka the I{radical center}.
1283 @raise IntersectionError: Concentric, antipodal, invalid or
1284 non-intersecting circles or no
1285 convergence.
1287 @raise TypeError: Invalid B{C{datum}}.
1289 @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{radius1}},
1290 B{C{lat2}}, B{C{lon2}} or B{C{radius2}}.
1291 '''
1292 r1 = Radius_(radius1=radius1)
1293 r2 = Radius_(radius2=radius2)
1294 with _idllmn6(datum, lat1, lon1, lat2, lon2,
1295 small, wrap, True, radius1=r1, radius2=r2) as t:
1296 _i, d, lat2, lon2, m, n = t
1297 if d is None:
1298 r1 = m2degrees(r1, radius=R_M, lat=lat1)
1299 r2 = m2degrees(r2, radius=R_M, lat=lat2)
1301 def _V2T(x, y, _, **unused): # _ == z unused
1302 return LatLon2Tuple(y, x, name=n)
1304 t = _i(m.Vector3d(lon1, lat1, 0), r1,
1305 m.Vector3d(lon2, lat2, 0), r2, sphere=False,
1306 Vector=_V2T)
1307 else:
1308 def _LL2T(lat, lon, **unused):
1309 return LatLon2Tuple(lat, lon, name=n)
1311 t = _i(m.LatLon(lat1, lon1, datum=d), r1,
1312 m.LatLon(lat2, lon2, datum=d), r2,
1313 LatLon=_LL2T, height=0, wrap=False)
1314 return t
1317def isantipode(lat1, lon1, lat2, lon2, eps=EPS):
1318 '''Check whether two points are I{antipodal}, on diametrically
1319 opposite sides of the earth.
1321 @arg lat1: Latitude of one point (C{degrees}).
1322 @arg lon1: Longitude of one point (C{degrees}).
1323 @arg lat2: Latitude of the other point (C{degrees}).
1324 @arg lon2: Longitude of the other point (C{degrees}).
1325 @kwarg eps: Tolerance for near-equality (C{degrees}).
1327 @return: C{True} if points are antipodal within the
1328 B{C{eps}} tolerance, C{False} otherwise.
1330 @see: Functions L{isantipode_} and L{antipode}.
1331 '''
1332 return (fabs(lat1 + lat2) <= eps and
1333 fabs(lon1 + lon2) <= eps) or _isequalTo(
1334 normal(lat1, lon1), antipode(lat2, lon2), eps)
1337def isantipode_(phi1, lam1, phi2, lam2, eps=EPS):
1338 '''Check whether two points are I{antipodal}, on diametrically
1339 opposite sides of the earth.
1341 @arg phi1: Latitude of one point (C{radians}).
1342 @arg lam1: Longitude of one point (C{radians}).
1343 @arg phi2: Latitude of the other point (C{radians}).
1344 @arg lam2: Longitude of the other point (C{radians}).
1345 @kwarg eps: Tolerance for near-equality (C{radians}).
1347 @return: C{True} if points are antipodal within the
1348 B{C{eps}} tolerance, C{False} otherwise.
1350 @see: Functions L{isantipode} and L{antipode_}.
1351 '''
1352 return (fabs(phi1 + phi2) <= eps and
1353 fabs(lam1 + lam2) <= eps) or _isequalTo_(
1354 normal_(phi1, lam1), antipode_(phi2, lam2), eps)
1357def _isequalTo(p1, p2, eps=EPS):
1358 '''Compare 2 point lat-/lons ignoring C{class}.
1359 '''
1360 return (fabs(p1.lat - p2.lat) <= eps and
1361 fabs(p1.lon - p2.lon) <= eps) if eps else (p1.latlon == p2.latlon)
1364def _isequalTo_(p1, p2, eps=EPS):
1365 '''(INTERNAL) Compare 2 point phi-/lams ignoring C{class}.
1366 '''
1367 return (fabs(p1.phi - p2.phi) <= eps and
1368 fabs(p1.lam - p2.lam) <= eps) if eps else (p1.philam == p2.philam)
1371def isnormal(lat, lon, eps=0):
1372 '''Check whether B{C{lat}} I{and} B{C{lon}} are within their
1373 respective I{normal} range in C{degrees}.
1375 @arg lat: Latitude (C{degrees}).
1376 @arg lon: Longitude (C{degrees}).
1377 @kwarg eps: Optional tolerance C{degrees}).
1379 @return: C{True} if C{(abs(B{lat}) + B{eps}) <= 90} and
1380 C{(abs(B{lon}) + B{eps}) <= 180}, C{False} othwerwise.
1382 @see: Functions L{isnormal_} and L{normal}.
1383 '''
1384 return (_90_0 - fabs(lat)) >= eps and _loneg(fabs(lon)) >= eps
1387def isnormal_(phi, lam, eps=0):
1388 '''Check whether B{C{phi}} I{and} B{C{lam}} are within their
1389 respective I{normal} range in C{radians}.
1391 @arg phi: Latitude (C{radians}).
1392 @arg lam: Longitude (C{radians}).
1393 @kwarg eps: Optional tolerance C{radians}).
1395 @return: C{True} if C{(abs(B{phi}) + B{eps}) <= PI/2} and
1396 C{(abs(B{lam}) + B{eps}) <= PI}, C{False} othwerwise.
1398 @see: Functions L{isnormal} and L{normal_}.
1399 '''
1400 return (PI_2 - fabs(phi)) >= eps and (PI - fabs(lam)) >= eps
1403def latlon2n_xyz(lat, lon, name=NN):
1404 '''Convert lat-, longitude to C{n-vector} (I{normal} to the
1405 earth's surface) X, Y and Z components.
1407 @arg lat: Latitude (C{degrees}).
1408 @arg lon: Longitude (C{degrees}).
1409 @kwarg name: Optional name (C{str}).
1411 @return: A L{Vector3Tuple}C{(x, y, z)}.
1413 @see: Function L{philam2n_xyz}.
1415 @note: These are C{n-vector} x, y and z components,
1416 I{NOT} geocentric ECEF x, y and z coordinates!
1417 '''
1418 return _2n_xyz(name, *sincos2d_(lat, lon))
1421def _normal2(a, b, n_2, n, n2):
1422 '''(INTERNAL) Helper for C{normal} and C{normal_}.
1423 '''
1424 if fabs(b) > n:
1425 b = remainder(b, n2)
1426 if fabs(a) > n_2:
1427 r = remainder(a, n)
1428 if r != a:
1429 a = -r
1430 b -= n if b > 0 else -n
1431 return float0_(a, b)
1434def normal(lat, lon, name=NN):
1435 '''Normalize a lat- I{and} longitude pair in C{degrees}.
1437 @arg lat: Latitude (C{degrees}).
1438 @arg lon: Longitude (C{degrees}).
1439 @kwarg name: Optional name (C{str}).
1441 @return: L{LatLon2Tuple}C{(lat, lon)} with C{abs(lat) <= 90}
1442 and C{abs(lon) <= 180}.
1444 @see: Functions L{normal_} and L{isnormal}.
1445 '''
1446 return LatLon2Tuple(*_normal2(lat, lon, _90_0, _180_0, _360_0),
1447 name=name or normal.__name__)
1450def normal_(phi, lam, name=NN):
1451 '''Normalize a lat- I{and} longitude pair in C{radians}.
1453 @arg phi: Latitude (C{radians}).
1454 @arg lam: Longitude (C{radians}).
1455 @kwarg name: Optional name (C{str}).
1457 @return: L{PhiLam2Tuple}C{(phi, lam)} with C{abs(phi) <= PI/2}
1458 and C{abs(lam) <= PI}.
1460 @see: Functions L{normal} and L{isnormal_}.
1461 '''
1462 return PhiLam2Tuple(*_normal2(phi, lam, PI_2, PI, PI2),
1463 name=name or normal_.__name__)
1466def _2n_xyz(name, sa, ca, sb, cb):
1467 '''(INTERNAL) Helper for C{latlon2n_xyz} and C{philam2n_xyz}.
1468 '''
1469 # Kenneth Gade eqn 3, but using right-handed
1470 # vector x -> 0°E,0°N, y -> 90°E,0°N, z -> 90°N
1471 return Vector3Tuple(ca * cb, ca * sb, sa, name=name)
1474def n_xyz2latlon(x, y, z, name=NN):
1475 '''Convert C{n-vector} components to lat- and longitude in C{degrees}.
1477 @arg x: X component (C{scalar}).
1478 @arg y: Y component (C{scalar}).
1479 @arg z: Z component (C{scalar}).
1480 @kwarg name: Optional name (C{str}).
1482 @return: A L{LatLon2Tuple}C{(lat, lon)}.
1484 @see: Function L{n_xyz2philam}.
1485 '''
1486 return LatLon2Tuple(atan2d(z, hypot(x, y)), atan2d(y, x), name=name)
1489def n_xyz2philam(x, y, z, name=NN):
1490 '''Convert C{n-vector} components to lat- and longitude in C{radians}.
1492 @arg x: X component (C{scalar}).
1493 @arg y: Y component (C{scalar}).
1494 @arg z: Z component (C{scalar}).
1495 @kwarg name: Optional name (C{str}).
1497 @return: A L{PhiLam2Tuple}C{(phi, lam)}.
1499 @see: Function L{n_xyz2latlon}.
1500 '''
1501 return PhiLam2Tuple(atan2(z, hypot(x, y)), atan2(y, x), name=name)
1504def _opposes(d, m, n, n2):
1505 '''(INTERNAL) Helper for C{opposing} and C{opposing_}.
1506 '''
1507 d = d % n2 # -20 % 360 == 340, -1 % PI2 == PI2 - 1
1508 return False if d < m or d > (n2 - m) else (
1509 True if (n - m) < d < (n + m) else None)
1512def opposing(bearing1, bearing2, margin=_90_0):
1513 '''Compare the direction of two bearings given in C{degrees}.
1515 @arg bearing1: First bearing (compass C{degrees}).
1516 @arg bearing2: Second bearing (compass C{degrees}).
1517 @kwarg margin: Optional, interior angle bracket (C{degrees}).
1519 @return: C{True} if both bearings point in opposite, C{False} if
1520 in similar or C{None} if in I{perpendicular} directions.
1522 @see: Function L{opposing_}.
1523 '''
1524 m = Degrees_(margin=margin, low=EPS0, high=_90_0)
1525 return _opposes(bearing2 - bearing1, m, _180_0, _360_0)
1528def opposing_(radians1, radians2, margin=PI_2):
1529 '''Compare the direction of two bearings given in C{radians}.
1531 @arg radians1: First bearing (C{radians}).
1532 @arg radians2: Second bearing (C{radians}).
1533 @kwarg margin: Optional, interior angle bracket (C{radians}).
1535 @return: C{True} if both bearings point in opposite, C{False} if
1536 in similar or C{None} if in perpendicular directions.
1538 @see: Function L{opposing}.
1539 '''
1540 m = Radians_(margin=margin, low=EPS0, high=PI_2)
1541 return _opposes(radians2 - radians1, m, PI, PI2)
1544def philam2n_xyz(phi, lam, name=NN):
1545 '''Convert lat-, longitude to C{n-vector} (I{normal} to the
1546 earth's surface) X, Y and Z components.
1548 @arg phi: Latitude (C{radians}).
1549 @arg lam: Longitude (C{radians}).
1550 @kwarg name: Optional name (C{str}).
1552 @return: A L{Vector3Tuple}C{(x, y, z)}.
1554 @see: Function L{latlon2n_xyz}.
1556 @note: These are C{n-vector} x, y and z components,
1557 I{NOT} geocentric ECEF x, y and z coordinates!
1558 '''
1559 return _2n_xyz(name, *sincos2_(phi, lam))
1562def _radical2(d, r1, r2): # in .ellipsoidalBaseDI, .sphericalTrigonometry, .vector3d
1563 # (INTERNAL) See C{radical2} below
1564 # assert d > EPS0
1565 r = fsumf_(_1_0, (r1 / d)**2, -(r2 / d)**2) * _0_5
1566 return Radical2Tuple(max(_0_0, min(_1_0, r)), r * d)
1569def radical2(distance, radius1, radius2):
1570 '''Compute the I{radical ratio} and I{radical line} of two
1571 U{intersecting circles<https://MathWorld.Wolfram.com/
1572 Circle-CircleIntersection.html>}.
1574 The I{radical line} is perpendicular to the axis thru the
1575 centers of the circles at C{(0, 0)} and C{(B{distance}, 0)}.
1577 @arg distance: Distance between the circle centers (C{scalar}).
1578 @arg radius1: Radius of the first circle (C{scalar}).
1579 @arg radius2: Radius of the second circle (C{scalar}).
1581 @return: A L{Radical2Tuple}C{(ratio, xline)} where C{0.0 <=
1582 ratio <= 1.0} and C{xline} is along the B{C{distance}}.
1584 @raise IntersectionError: The B{C{distance}} exceeds the sum
1585 of B{C{radius1}} and B{C{radius2}}.
1587 @raise UnitError: Invalid B{C{distance}}, B{C{radius1}} or
1588 B{C{radius2}}.
1590 @see: U{Circle-Circle Intersection
1591 <https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>}.
1592 '''
1593 d = Distance_(distance, low=_0_0)
1594 r1 = Radius_(radius1=radius1)
1595 r2 = Radius_(radius2=radius2)
1596 if d > (r1 + r2):
1597 raise IntersectionError(distance=d, radius1=r1, radius2=r2,
1598 txt=_too_(_distant_))
1599 return _radical2(d, r1, r2) if d > EPS0 else \
1600 Radical2Tuple(_0_5, _0_0)
1603class Radical2Tuple(_NamedTuple):
1604 '''2-Tuple C{(ratio, xline)} of the I{radical} C{ratio} and
1605 I{radical} C{xline}, both C{scalar} and C{0.0 <= ratio <= 1.0}
1606 '''
1607 _Names_ = (_ratio_, _xline_)
1608 _Units_ = ( Scalar, Scalar)
1611def _radistance(inst):
1612 '''(INTERNAL) Helper for the L{frechet._FrecherMeterRadians}
1613 and L{hausdorff._HausdorffMeterRedians} classes.
1614 '''
1615 kwds_ = _xkwds(inst._kwds, wrap=False)
1616 wrap_ = _xkwds_pop(kwds_, wrap=False)
1617 func_ = inst._func_
1618 try: # calling lower-overhead C{func_}
1619 func_(0, _0_25, _0_5, **kwds_)
1620 wrap_ = _Wrap._philamop(wrap_)
1621 except TypeError:
1622 return inst.distance
1624 def _philam(p):
1625 try:
1626 return p.phi, p.lam
1627 except AttributeError: # no .phi or .lam
1628 return radians(p.lat), radians(p.lon)
1630 def _func_wrap(point1, point2):
1631 phi1, lam1 = wrap_(*_philam(point1))
1632 phi2, lam2 = wrap_(*_philam(point2))
1633 return func_(phi2, phi1, lam2 - lam1, **kwds_)
1635 inst._units = inst._units_
1636 return _func_wrap
1639def rtp2xyz(r, theta, phi):
1640 '''Convert spherical C{(r, theta, phi)} to cartesian C{(x, y, z)} coordinates.
1642 @arg r: Radial distance (C{scalar}, conventially C{meter}).
1643 @arg theta: Inclination (C{degrees} with respect to the positive z-axis).
1644 @arg phi: Azimuthal angle (C{degrees}).
1646 @return: L{Vector3Tuple}C{(x, y, z)} in C{meter}, same units as C{r}.
1648 @see: Functions L{rtp2xyz_} and L{xyz2rtp}.
1649 '''
1650 return rtp2xyz_(r, radians(theta), radians(phi))
1653def rtp2xyz_(r, theta, phi):
1654 '''Convert spherical C{(r, theta, phi)} to cartesian C{(x, y, z)} coordinates.
1656 @arg r: Radial distance (C{scalar}, conventially C{meter}).
1657 @arg theta: Inclination (C{radians} with respect to the positive z-axis).
1658 @arg phi: Azimuthal angle (C{radians}).
1660 @return: L{Vector3Tuple}C{(x, y, z)} in C{meter}, same units as C{r}.
1662 @see: U{Physics<https://WikiPedia.org/wiki/Spherical_coordinate_system>}
1663 convention (ISO 80000-2:2019).
1664 '''
1665 if r and _isfinite(r):
1666 s, z, y, x = sincos2_(theta, phi)
1667 s *= r
1668 x *= s
1669 y *= s
1670 z *= r
1671 else:
1672 x = y = z = r
1673 return Vector3Tuple(x, y, z)
1676def _scale_deg(lat1, lat2): # degrees
1677 # scale factor cos(mean of lats) for delta lon
1678 m = fabs(lat1 + lat2) * _0_5
1679 return cos(radians(m)) if m < 90 else _0_0
1682def _scale_rad(phi1, phi2): # radians, by .frechet, .hausdorff, .heights
1683 # scale factor cos(mean of phis) for delta lam
1684 m = fabs(phi1 + phi2) * _0_5
1685 return cos(m) if m < PI_2 else _0_0
1688def _sincosa6(phi2, phi1, lam21): # [4] in cosineLaw
1689 '''(INTERNAL) C{sin}es, C{cos}ines and C{acos}ine.
1690 '''
1691 s2, c2, s1, c1, _, c21 = sincos2_(phi2, phi1, lam21)
1692 return s2, c2, s1, c1, acos1(s1 * s2 + c1 * c2 * c21), c21
1695def thomas(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False):
1696 '''Compute the distance between two (ellipsoidal) points using
1697 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>}
1698 formula.
1700 @arg lat1: Start latitude (C{degrees}).
1701 @arg lon1: Start longitude (C{degrees}).
1702 @arg lat2: End latitude (C{degrees}).
1703 @arg lon2: End longitude (C{degrees}).
1704 @kwarg datum: Datum (L{Datum}) or ellipsoid (L{Ellipsoid},
1705 L{Ellipsoid2} or L{a_f2Tuple}) to use.
1706 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
1707 B{C{lat2}} and B{C{lon2}} (C{bool}).
1709 @return: Distance (C{meter}, same units as the B{C{datum}}'s
1710 ellipsoid axes).
1712 @raise TypeError: Invalid B{C{datum}}.
1714 @see: Functions L{thomas_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},
1715 L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny},
1716 L{flatPolar}, L{haversine}, L{vincentys} and method L{Ellipsoid.distance2}.
1717 '''
1718 return _dE(thomas_, datum, wrap, lat1, lon1, lat2, lon2)
1721def thomas_(phi2, phi1, lam21, datum=_WGS84):
1722 '''Compute the I{angular} distance between two (ellipsoidal) points using
1723 U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>}
1724 formula.
1726 @arg phi2: End latitude (C{radians}).
1727 @arg phi1: Start latitude (C{radians}).
1728 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
1729 @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid},
1730 L{Ellipsoid2} or L{a_f2Tuple}).
1732 @return: Angular distance (C{radians}).
1734 @raise TypeError: Invalid B{C{datum}}.
1736 @see: Functions L{thomas}, L{cosineAndoyerLambert_},
1737 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
1738 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
1739 L{flatPolar_}, L{haversine_} and L{vincentys_} and U{Geodesy-PHP
1740 <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/
1741 Distance/ThomasFormula.php>}.
1742 '''
1743 s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21)
1744 if r and isnon0(c1) and isnon0(c2):
1745 E = _ellipsoidal(datum, thomas_)
1746 if E.f:
1747 r1 = atan2(E.b_a * s1, c1)
1748 r2 = atan2(E.b_a * s2, c2)
1750 j = (r2 + r1) * _0_5
1751 k = (r2 - r1) * _0_5
1752 sj, cj, sk, ck, h, _ = sincos2_(j, k, lam21 * _0_5)
1754 h = fsumf_(sk**2, (ck * h)**2, -(sj * h)**2)
1755 u = _1_0 - h
1756 if isnon0(u) and isnon0(h):
1757 r = atan(sqrt0(h / u)) * 2 # == acos(1 - 2 * h)
1758 sr, cr = sincos2(r)
1759 if isnon0(sr):
1760 u = 2 * (sj * ck)**2 / u
1761 h = 2 * (sk * cj)**2 / h
1762 x = u + h
1763 y = u - h
1765 s = r / sr
1766 e = 4 * s**2
1767 d = 2 * cr
1768 a = e * d
1769 b = 2 * r
1770 c = s - (a - d) * _0_5
1771 f = E.f * _0_25
1773 t = fsumf_(a * x, -b * y, c * x**2, -d * y**2, e * x * y)
1774 r -= fsumf_(s * x, -y, -t * f * _0_25) * f * sr
1775 return r
1778def vincentys(lat1, lon1, lat2, lon2, radius=R_M, wrap=False):
1779 '''Compute the distance between two (spherical) points using
1780 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>}
1781 spherical formula.
1783 @arg lat1: Start latitude (C{degrees}).
1784 @arg lon1: Start longitude (C{degrees}).
1785 @arg lat2: End latitude (C{degrees}).
1786 @arg lon2: End longitude (C{degrees}).
1787 @kwarg radius: Mean earth radius (C{meter}), datum (L{Datum})
1788 or ellipsoid (L{Ellipsoid}, L{Ellipsoid2} or
1789 L{a_f2Tuple}) to use.
1790 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
1791 B{C{lat2}} and B{C{lon2}} (C{bool}).
1793 @return: Distance (C{meter}, same units as B{C{radius}}).
1795 @raise UnitError: Invalid B{C{radius}}.
1797 @see: Functions L{vincentys_}, L{cosineAndoyerLambert},
1798 L{cosineForsytheAndoyerLambert},L{cosineLaw}, L{equirectangular},
1799 L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar},
1800 L{haversine} and L{thomas} and methods L{Ellipsoid.distance2},
1801 C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}.
1803 @note: See note at function L{vincentys_}.
1804 '''
1805 return _dS(vincentys_, radius, wrap, lat1, lon1, lat2, lon2)
1808def vincentys_(phi2, phi1, lam21):
1809 '''Compute the I{angular} distance between two (spherical) points using
1810 U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>}
1811 spherical formula.
1813 @arg phi2: End latitude (C{radians}).
1814 @arg phi1: Start latitude (C{radians}).
1815 @arg lam21: Longitudinal delta, M{end-start} (C{radians}).
1817 @return: Angular distance (C{radians}).
1819 @see: Functions L{vincentys}, L{cosineAndoyerLambert_},
1820 L{cosineForsytheAndoyerLambert_}, L{cosineLaw_},
1821 L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_},
1822 L{flatPolar_}, L{haversine_} and L{thomas_}.
1824 @note: Functions L{vincentys_}, L{haversine_} and L{cosineLaw_}
1825 produce equivalent results, but L{vincentys_} is suitable
1826 for antipodal points and slightly more expensive (M{3 cos,
1827 3 sin, 1 hypot, 1 atan2, 6 mul, 2 add}) than L{haversine_}
1828 (M{2 cos, 2 sin, 2 sqrt, 1 atan2, 5 mul, 1 add}) and
1829 L{cosineLaw_} (M{3 cos, 3 sin, 1 acos, 3 mul, 1 add}).
1830 '''
1831 s1, c1, s2, c2, s21, c21 = sincos2_(phi1, phi2, lam21)
1833 c = c2 * c21
1834 x = s1 * s2 + c1 * c
1835 y = c1 * s2 - s1 * c
1836 return atan2(hypot(c2 * s21, y), x)
1839def xyz2rtp(x_xyz, *y_z):
1840 '''Convert cartesian C{(x, y, z)} to spherical C{(r, theta, phi)} coordinates.
1842 @return: 3-Tuple C{(r, theta, phi)} in C{degrees}.
1844 @see: Function L{xyz2rtp_}.
1845 '''
1846 r, t, p = xyz2rtp_(x_xyz, *y_z)
1847 return r, Degrees(theta=degrees(t)), Degrees(phi=degrees(p))
1850def xyz2rtp_(x_xyz, *y_z):
1851 '''Convert cartesian C{(x, y, z)} to spherical C{(r, theta, phi)} coordinates.
1853 @arg x_xyz: X component (C{scalar}) or a cartesian (C{Cartesian}, L{Ecef9Tuple},
1854 C{Nvector}, L{Vector3d}, L{Vector3Tuple}, L{Vector4Tuple} or a
1855 C{tuple} or C{list} of 3+ C{scalar} items) if no C{y_z} specified.
1856 @arg y_z: Y and Z component (C{scalar}s), ignored if C{x_xyz} is not a C{scalar}.
1858 @return: 3-Tuple C{(r, theta, phi)} with radial distance C{r} (C{meter}, same
1859 units as C{x}, C{y} and C{z}), inclination C{theta} (with respect to
1860 the positive z-axis) and azimuthal angle C{phi} in C{radians}.
1862 @see: U{Physics<https://WikiPedia.org/wiki/Spherical_coordinate_system>}
1863 convention (ISO 80000-2:2019).
1864 '''
1865 x, y, z = _MODS.vector3dBase._xyz3(xyz2rtp, x_xyz, *y_z)
1866 r = hypot_(x, y, z)
1867 if r:
1868 t = acos1(z / r)
1869 p = atan2(y, x)
1870 if p < 0:
1871 p += PI2
1872 else:
1873 t = p = _0_0
1874 return r, Radians(theta=t), Radians(phi=p)
1876# **) MIT License
1877#
1878# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved.
1879#
1880# Permission is hereby granted, free of charge, to any person obtaining a
1881# copy of this software and associated documentation files (the "Software"),
1882# to deal in the Software without restriction, including without limitation
1883# the rights to use, copy, modify, merge, publish, distribute, sublicense,
1884# and/or sell copies of the Software, and to permit persons to whom the
1885# Software is furnished to do so, subject to the following conditions:
1886#
1887# The above copyright notice and this permission notice shall be included
1888# in all copies or substantial portions of the Software.
1889#
1890# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
1891# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
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1893# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
1894# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
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