# Coverage for pygeodesy/formy.py : 98%

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1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328 1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344 1345 1346 1347 1348 1349 1350 1351 1352 1353 1354 1355 1356 1357 | # -*- coding: utf-8 -*- u'''Formulary of basic geodesy functions and approximations. @newfield example: Example, Examples ''' # make sure int/int division yields float quotient, see .basics from __future__ import division from pygeodesy.datums import Datum, _ellipsoidal_datum, _mean_radius, \ _spherical_datum, _WGS84 from pygeodesy.ellipsoids import Ellipsoid from pygeodesy.errors import _AssertionError, IntersectionError, \ LimitError, _limiterrors, _ValueError from pygeodesy.fmath import euclid, fsum_, hypot, hypot2, sqrt0 from pygeodesy.interns import EPS, EPS0, EPS1, NN, PI, PI2, PI3, PI_2, R_M, \ _distant_, _too_, _0_0, _0_125, _0_25, _0_5, \ _1_0, _2_0, _4_0, _32_0, _90_0, _180_0, _360_0 from pygeodesy.lazily import _ALL_LAZY from pygeodesy.named import _NamedTuple from pygeodesy.namedTuples import Bearing2Tuple, Distance4Tuple, \ LatLon2Tuple, PhiLam2Tuple, Vector3Tuple from pygeodesy.streprs import unstr from pygeodesy.units import Distance, Distance_, Height, Lam_, Lat, Lon, Phi_, \ Radians, Radians_, Radius, Radius_, Scalar, _100km from pygeodesy.utily import acos1, atan2b, degrees2m, degrees90, degrees180, \ m2degrees, sincos2, tan_2, unroll180, unrollPI, \ wrap90, wrap180, wrapPI, wrapPI_2 from math import atan, atan2, cos, degrees, radians, sin, sqrt # pow __all__ = _ALL_LAZY.formy __version__ = '21.02.24' def _non0(x): '''Is C{abs(B{x})} > C{EPS0}? ''' return x > EPS0 or (-x) > EPS0 def antipode(lat, lon): '''Return the antipode, the point diametrically opposite to a given point in C{degrees}. @arg lat: Latitude (C{degrees}). @arg lon: Longitude (C{degrees}). @return: A L{LatLon2Tuple}C{(lat, lon)}. @see: U{Geosphere<https://CRAN.R-Project.org/web/packages/geosphere/geosphere.pdf>}. ''' return LatLon2Tuple(-wrap90(lat), wrap180(lon + _180_0)) def antipode_(phi, lam): '''Return the antipode, the point diametrically opposite to a given point in C{radians}. @arg phi: Latitude (C{radians}). @arg lam: Longitude (C{radians}). @return: A L{PhiLam2Tuple}C{(phi, lam)}. @see: U{Geosphere<https://CRAN.R-Project.org/web/packages/geosphere/geosphere.pdf>}. ''' return PhiLam2Tuple(-wrapPI_2(phi), wrapPI(lam + PI)) def _areaOfS(func_, lat1, lat2, radius, d_lon, unused): '''(INTERNAL) Helper for spherical excess and area. ''' r = func_(Phi_(lat2=lat2), Phi_(lat1=lat1), radians(d_lon)) if radius: r *= _mean_radius(radius, lat1, lat2)**2 return r def bearing(lat1, lon1, lat2, lon2, **options): '''Compute the initial or final bearing (forward or reverse azimuth) between a (spherical) start and end point. @arg lat1: Start latitude (C{degrees}). @arg lon1: Start longitude (C{degrees}). @arg lat2: End latitude (C{degrees}). @arg lon2: End longitude (C{degrees}). @kwarg options: Optional keyword arguments for function L{bearing_}. @return: Initial or final bearing (compass C{degrees360}) or zero if start and end point coincide. ''' return degrees(bearing_(Phi_(lat1=lat1), Lam_(lon1=lon1), Phi_(lat2=lat2), Lam_(lon2=lon2), **options)) def bearing_(phi1, lam1, phi2, lam2, final=False, wrap=False): '''Compute the initial or final bearing (forward or reverse azimuth) between a (spherical) start and end point. @arg phi1: Start latitude (C{radians}). @arg lam1: Start longitude (C{radians}). @arg phi2: End latitude (C{radians}). @arg lam2: End longitude (C{radians}). @kwarg final: Return final bearing if C{True}, initial otherwise (C{bool}). @kwarg wrap: Wrap and L{unrollPI} longitudes (C{bool}). @return: Initial or final bearing (compass C{radiansPI2}) or zero if start and end point coincide. ''' if final: phi1, lam1, phi2, lam2 = phi2, lam2, phi1, lam1 r = PI3 else: r = PI2 db, _ = unrollPI(lam1, lam2, wrap=wrap) sa1, ca1, sa2, ca2, sdb, cdb = sincos2(phi1, phi2, db) # see <https://MathForum.org/library/drmath/view/55417.html> x = ca1 * sa2 - sa1 * ca2 * cdb y = sdb * ca2 return (atan2(y, x) + r) % PI2 # wrapPI2 def _bearingTo2(p1, p2, wrap=False): # for points.ispolar, sphericalTrigonometry.areaOf '''(INTERNAL) Compute initial and final bearing. ''' try: # for LatLon_ and ellipsoidalNvector.LatLon return p1.bearingTo2(p2, wrap=wrap) except AttributeError: pass # XXX spherical version, OK for ellipsoidal ispolar? a1, b1 = p1.philam a2, b2 = p2.philam return Bearing2Tuple(degrees(bearing_(a1, b1, a2, b2, final=False, wrap=wrap)), degrees(bearing_(a1, b1, a2, b2, final=True, wrap=wrap))) def compassAngle(lat1, lon1, lat2, lon2, adjust=True, wrap=False): '''Return the angle from North for the direction vector M{(lon2 - lon1, lat2 - lat1)} between two points. Suitable only for short, not near-polar vectors up to a few hundred Km or Miles. Use function L{bearing} for longer vectors. @arg lat1: From latitude (C{degrees}). @arg lon1: From longitude (C{degrees}). @arg lat2: To latitude (C{degrees}). @arg lon2: To longitude (C{degrees}). @kwarg adjust: Adjust the longitudinal delta by the cosine of the mean latitude (C{bool}). @kwarg wrap: Wrap and L{unroll180} longitudes (C{bool}). @return: Compass angle from North (C{degrees360}). @note: Courtesy Martin Schultz. @see: U{Local, flat earth approximation <https://www.EdWilliams.org/avform.htm#flat>}. ''' d_lon, _ = unroll180(lon1, lon2, wrap=wrap) if adjust: # scale delta lon d_lon *= _scale_deg(lat1, lat2) return atan2b(d_lon, lat2 - lat1) def cosineAndoyerLambert(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False): '''Compute the distance between two (ellipsoidal) points using the U{Andoyer-Lambert correction<https://navlib.net/wp-content/uploads/ 2013/10/admiralty-manual-of-navigation-vol-1-1964-english501c.pdf>} of the U{Law of Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} fromula. @arg lat1: Start latitude (C{degrees}). @arg lon1: Start longitude (C{degrees}). @arg lat2: End latitude (C{degrees}). @arg lon2: End longitude (C{degrees}). @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}). @kwarg wrap: Wrap and L{unroll180} longitudes (C{bool}). @return: Distance (C{meter}, same units as the B{C{datum}}'s ellipsoid axes or C{radians} if B{C{datum}} is C{None}). @raise TypeError: Invalid B{C{datum}}. @see: Functions L{cosineAndoyerLambert_}, L{cosineForsytheAndoyerLambert}, L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar}, L{haversine}, L{thomas} and L{vincentys} and method L{Ellipsoid.distance2}. ''' return _distanceToE(cosineAndoyerLambert_, lat1, lat2, datum, *unroll180(lon1, lon2, wrap=wrap)) def cosineAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84): '''Compute the I{angular} distance between two (ellipsoidal) points using the U{Andoyer-Lambert correction<https://navlib.net/wp-content/uploads/2013/10/ admiralty-manual-of-navigation-vol-1-1964-english501c.pdf>} of the U{Law of Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} fromula. @arg phi2: End latitude (C{radians}). @arg phi1: Start latitude (C{radians}). @arg lam21: Longitudinal delta, M{end-start} (C{radians}). @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}). @return: Angular distance (C{radians}). @raise TypeError: Invalid B{C{datum}}. @see: Functions L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_}, L{haversine_}, L{thomas_} and L{vincentys_} and U{Geodesy-PHP <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/ Distance/AndoyerLambert.php>}. ''' s2, c2, s1, c1, r, c21 = _sincosa6(phi2, phi1, lam21) if _non0(c1) and _non0(c2): E = _ellipsoid(datum, cosineAndoyerLambert_) if E.f: # ellipsoidal r2 = atan2(E.b_a * s2, c2) r1 = atan2(E.b_a * s1, c1) s2, c2, s1, c1 = sincos2(r2, r1) r = acos1(s1 * s2 + c1 * c2 * c21) if r: sr, _, sr_2, cr_2 = sincos2(r, r * _0_5) if _non0(sr_2) and _non0(cr_2): s = (sr + r) * ((s1 - s2) / sr_2)**2 c = (sr - r) * ((s1 + s2) / cr_2)**2 r += (c - s) * E.f * _0_125 return r def cosineForsytheAndoyerLambert(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False): '''Compute the distance between two (ellipsoidal) points using the U{Forsythe-Andoyer-Lambert correction<https://www2.UNB.CA/gge/Pubs/TR77.pdf>} of the U{Law of Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula. @arg lat1: Start latitude (C{degrees}). @arg lon1: Start longitude (C{degrees}). @arg lat2: End latitude (C{degrees}). @arg lon2: End longitude (C{degrees}). @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}). @kwarg wrap: Wrap and L{unroll180} longitudes (C{bool}). @return: Distance (C{meter}, same units as the B{C{datum}}'s ellipsoid axes or C{radians} if B{C{datum}} is C{None}). @raise TypeError: Invalid B{C{datum}}. @see: Functions L{cosineForsytheAndoyerLambert_}, L{cosineAndoyerLambert}, L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar}, L{haversine}, L{thomas} and L{vincentys} and method L{Ellipsoid.distance2}. ''' return _distanceToE(cosineForsytheAndoyerLambert_, lat1, lat2, datum, *unroll180(lon1, lon2, wrap=wrap)) def cosineForsytheAndoyerLambert_(phi2, phi1, lam21, datum=_WGS84): '''Compute the I{angular} distance between two (ellipsoidal) points using the U{Forsythe-Andoyer-Lambert correction<https://www2.UNB.CA/gge/Pubs/TR77.pdf>} of the U{Law of Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula. @arg phi2: End latitude (C{radians}). @arg phi1: Start latitude (C{radians}). @arg lam21: Longitudinal delta, M{end-start} (C{radians}). @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}). @return: Angular distance (C{radians}). @raise TypeError: Invalid B{C{datum}}. @see: Functions L{cosineForsytheAndoyerLambert}, L{cosineAndoyerLambert_}, L{cosineLaw_}, L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_}, L{haversine_}, L{thomas_} and L{vincentys_} and U{Geodesy-PHP <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/ Distance/ForsytheCorrection.php>}. ''' s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21) if r and _non0(c1) and _non0(c2): E = _ellipsoid(datum, cosineForsytheAndoyerLambert_) if E.f: # ellipsoidal sr, cr, s2r, _ = sincos2(r, r * _2_0) if _non0(sr) and abs(cr) < EPS1: s = (s1 + s2)**2 / (1 + cr) t = (s1 - s2)**2 / (1 - cr) x = s + t y = s - t s = 8 * r**2 / sr a = 64 * r + _2_0 * s * cr # 16 * r**2 / tan(r) d = 48 * sr + s # 8 * r**2 / tan(r) b = -2 * d e = 30 * s2r c = fsum_(30 * r, e * _0_5, s * cr) # 8 * r**2 / tan(r) t = fsum_( a * x, b * y, -c * x**2, d * x * y, e * y**2) r += fsum_(-r * x, 3 * y * sr, t * E.f / _32_0) * E.f * _0_25 return r def cosineLaw(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): '''Compute the distance between two points using the U{spherical Law of Cosines <https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula. @arg lat1: Start latitude (C{degrees}). @arg lon1: Start longitude (C{degrees}). @arg lat2: End latitude (C{degrees}). @arg lon2: End longitude (C{degrees}). @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter}, L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}). @kwarg wrap: Wrap and L{unroll180} longitudes (C{bool}). @return: Distance (C{meter}, same units as B{C{radius}} or the ellipsoid or datum axes). @raise TypeError: Invalid B{C{radius}}. @see: Functions L{cosineLaw_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar}, L{haversine}, L{thomas} and L{vincentys} and method L{Ellipsoid.distance2}. @note: See note at function L{vincentys_}. ''' return _distanceToS(cosineLaw_, lat1, lat2, radius, *unroll180(lon1, lon2, wrap=wrap)) def cosineLaw_(phi2, phi1, lam21): '''Compute the I{angular} distance between two points using the U{spherical Law of Cosines <https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} formula. @arg phi2: End latitude (C{radians}). @arg phi1: Start latitude (C{radians}). @arg lam21: Longitudinal delta, M{end-start} (C{radians}). @return: Angular distance (C{radians}). @see: Functions L{cosineLaw}, L{cosineAndoyerLambert_}, L{cosineForsytheAndoyerLambert_}, L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_}, L{haversine_}, L{thomas_} and L{vincentys_}. @note: See note at function L{vincentys_}. ''' return _sincosa6(phi2, phi1, lam21)[4] def _distanceToE(func_, lat1, lat2, earth, d_lon, unused): '''(INTERNAL) Helper for ellipsoidal distances. ''' E = _ellipsoid(earth, func_) r = func_(Phi_(lat2=lat2), Phi_(lat1=lat1), radians(d_lon), datum=E) return r * E.a def _distanceToS(func_, lat1, lat2, earth, d_lon, unused, **adjust): '''(INTERNAL) Helper for spherical distances. ''' r = func_(Phi_(lat2=lat2), Phi_(lat1=lat1), radians(d_lon), **adjust) return r * _mean_radius(earth, lat1, lat2) def _ellipsoid(earth, where): '''(INTERNAL) Helper for distances. ''' return earth if isinstance(earth, Ellipsoid) else ( earth if isinstance(earth, Datum) else _ellipsoidal_datum(earth, name=where.__name__)).ellipsoid # PYCHOK indent def equirectangular(lat1, lon1, lat2, lon2, radius=R_M, **options): '''Compute the distance between two points using the U{Equirectangular Approximation / Projection <https://www.Movable-Type.co.UK/scripts/latlong.html#equirectangular>}. @arg lat1: Start latitude (C{degrees}). @arg lon1: Start longitude (C{degrees}). @arg lat2: End latitude (C{degrees}). @arg lon2: End longitude (C{degrees}). @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter}, L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}). @kwarg options: Optional keyword arguments for function L{equirectangular_}. @return: Distance (C{meter}, same units as B{C{radius}} or the ellipsoid or datum axes). @raise TypeError: Invalid B{C{radius}}. @see: Function L{equirectangular_} for more details, the available B{C{options}}, errors, restrictions and other, approximate or accurate distance functions. ''' d = sqrt(equirectangular_(Lat(lat1=lat1), Lon(lon1=lon1), Lat(lat2=lat2), Lon(lon2=lon2), **options).distance2) # PYCHOK 4 vs 2-3 return degrees2m(d, radius=_mean_radius(radius, lat1, lat2)) def equirectangular_(lat1, lon1, lat2, lon2, adjust=True, limit=45, wrap=False): '''Compute the distance between two points using the U{Equirectangular Approximation / Projection <https://www.Movable-Type.co.UK/scripts/latlong.html#equirectangular>}. This approximation is valid for short distance of several hundred Km or Miles, see the B{C{limit}} keyword argument and the L{LimitError}. @arg lat1: Start latitude (C{degrees}). @arg lon1: Start longitude (C{degrees}). @arg lat2: End latitude (C{degrees}). @arg lon2: End longitude (C{degrees}). @kwarg adjust: Adjust the wrapped, unrolled longitudinal delta by the cosine of the mean latitude (C{bool}). @kwarg limit: Optional limit for lat- and longitudinal deltas (C{degrees}) or C{None} or C{0} for unlimited. @kwarg wrap: Wrap and L{unroll180} longitudes (C{bool}). @return: A L{Distance4Tuple}C{(distance2, delta_lat, delta_lon, unroll_lon2)}. @raise LimitError: If the lat- and/or longitudinal delta exceeds the B{C{-limit..+limit}} range and L{limiterrors} set to C{True}. @see: U{Local, flat earth approximation <https://www.EdWilliams.org/avform.htm#flat>}, functions L{equirectangular}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, L{cosineLaw}, L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar}, L{haversine}, L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2}, C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}. ''' d_lat = lat2 - lat1 d_lon, ulon2 = unroll180(lon1, lon2, wrap=wrap) if limit and _limiterrors \ and max(abs(d_lat), abs(d_lon)) > limit > 0: t = unstr(equirectangular_.__name__, lat1, lon1, lat2, lon2, limit=limit) raise LimitError('delta exceeds limit', txt=t) if adjust: # scale delta lon d_lon *= _scale_deg(lat1, lat2) d2 = hypot2(d_lat, d_lon) # degrees squared! return Distance4Tuple(d2, d_lat, d_lon, ulon2 - lon2) def euclidean(lat1, lon1, lat2, lon2, radius=R_M, adjust=True, wrap=False): '''Approximate the C{Euclidean} distance between two (spherical) points. @arg lat1: Start latitude (C{degrees}). @arg lon1: Start longitude (C{degrees}). @arg lat2: End latitude (C{degrees}). @arg lon2: End longitude (C{degrees}). @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter}, L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}). @kwarg adjust: Adjust the longitudinal delta by the cosine of the mean latitude (C{bool}). @kwarg wrap: Wrap and L{unroll180} longitudes (C{bool}). @return: Distance (C{meter}, same units as B{C{radius}} or the ellipsoid or datum axes). @raise TypeError: Invalid B{C{radius}}. @see: U{Distance between two (spherical) points <https://www.EdWilliams.org/avform.htm#Dist>}, functions L{euclid}, L{euclidean_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, L{cosineLaw}, L{equirectangular}, L{flatLocal}/L{hubeny}, L{flatPolar}, L{haversine}, L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2}, C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}. ''' return _distanceToS(euclidean_, lat1, lat2, radius, *unroll180(lon1, lon2, wrap=wrap), adjust=adjust) def euclidean_(phi2, phi1, lam21, adjust=True): '''Approximate the I{angular} C{Euclidean} distance between two (spherical) points. @arg phi2: End latitude (C{radians}). @arg phi1: Start latitude (C{radians}). @arg lam21: Longitudinal delta, M{end-start} (C{radians}). @kwarg adjust: Adjust the longitudinal delta by the cosine of the mean latitude (C{bool}). @return: Angular distance (C{radians}). @see: Functions L{euclid}, L{euclidean}, L{cosineAndoyerLambert_}, L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{equirectangular_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_}, L{haversine_}, L{thomas_} and L{vincentys_}. ''' if adjust: lam21 *= _scale_rad(phi2, phi1) return euclid(phi2 - phi1, lam21) def excessAbc(A, b, c): '''Compute the I{spherical excess} C{E} of a (spherical) triangle from two sides and the included angle. @arg A: An interior triangle angle (C{radians}). @arg b: Frist adjacent triangle side (C{radians}). @arg c: Second adjacent triangle side (C{radians}). @return: Spherical excess ({radians}). @raise UnitError: Invalid B{C{A}}, B{C{b}} or B{C{c}}. @see: Function L{excessGirard}, L{excessLHuilier}, U{Spherical trigonometry<https://WikiPedia.org/wiki/Spherical_trigonometry>}. ''' sA, cA, sb, cb, sc, cc = sincos2(Radians_(A=A), Radians_(b=b) * _0_5, Radians_(c=c) * _0_5) return atan2(sA * sb * sc, cb * cc + cA * sb * sc) * _2_0 def excessGirard(A, B, C): '''Compute the I{spherical excess} C{E} of a (spherical) triangle using U{Girard's<https://MathWorld.Wolfram.com/GirardsSphericalExcessFormula.html>} formula. @arg A: First interior triangle angle (C{radians}). @arg B: Second interior triangle angle (C{radians}). @arg C: Third interior triangle angle (C{radians}). @return: Spherical excess ({radians}). @raise UnitError: Invalid B{C{A}}, B{C{B}} or B{C{C}}. @see: Function L{excessLHuilier}, U{Spherical trigonometry <https://WikiPedia.org/wiki/Spherical_trigonometry>}. ''' return Radians(Girard=fsum_(Radians_(A=A), Radians_(B=B), Radians_(C=C), -PI)) def excessLHuilier(a, b, c): '''Compute the I{spherical excess} C{E} of a (spherical) triangle using U{L'Huilier's<https://MathWorld.Wolfram.com/LHuiliersTheorem.html>} Theorem. @arg a: First triangle side (C{radians}). @arg b: Second triangle side (C{radians}). @arg c: Third triangle side (C{radians}). @return: Spherical excess ({radians}). @raise UnitError: Invalid B{C{a}}, B{C{b}} or B{C{c}}. @see: Function L{excessGirard}, U{Spherical trigonometry <https://WikiPedia.org/wiki/Spherical_trigonometry>}. ''' a = Radians_(a=a) b = Radians_(b=b) c = Radians_(c=c) s = fsum_(a, b, c) * _0_5 r = tan_2(s) * tan_2(s - a) * tan_2(s - b) * tan_2(s - c) return Radians(LHuilier=atan(sqrt(r)) * _4_0) def excessKarney(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): '''Compute the surface area of a (spherical) quadrilateral bounded by a segment of a great circle, two meridians and the equator using U{Karney's <http://OSGeo-org.1560.x6.Nabble.com/Area-of-a-spherical-polygon-td3841625.html>}. @arg lat1: Start latitude (C{degrees}). @arg lon1: Start longitude (C{degrees}). @arg lat2: End latitude (C{degrees}). @arg lon2: End longitude (C{degrees}). @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter}, L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}) or C{None}. @kwarg wrap: Wrap and L{unroll180} longitudes (C{bool}). @return: Surface area, I{signed} (I{square} C{meter}, or units of B{C{radius}} I{squared}) or C{radians} if B{C{radius}} is C{None} or C{0}. @raise TypeError: Invalid B{C{radius}}. @raise UnitError: Invalid B{C{lat2}} or B{C{lat1}}. @raise ValueError: Semi-circular longitudinal delta. @see: Function L{excessKarney_} and L{excessQuad}. ''' return _areaOfS(excessKarney_, lat1, lat2, radius, *unroll180(lon1, lon2, wrap=wrap)) def excessKarney_(phi2, phi1, lam21): '''Compute the I{spherical excess} C{E} of a (spherical) quadrilateral bounded by a segment of a great circle, two meridians and the equator using U{Karney's <http://OSGeo-org.1560.x6.Nabble.com/Area-of-a-spherical-polygon-td3841625.html>} method. @arg phi2: End latitude (C{radians}). @arg phi1: Start latitude (C{radians}). @arg lam21: Longitudinal delta, M{end-start} (C{radians}). @return: Spherical excess, I{signed} (C{radians}). @raise ValueError: Semi-circular longitudinal delta B{C{lam21}}. @see: Function L{excessKarney}, U{Area of a spherical polygon <http://OSGeo-org.1560.x6.Nabble.com/Area-of-a-spherical-polygon-td3841625.html>}. ''' # from: Veness <https://www.Movable-Type.co.UK/scripts/latlong.html> # Area method due to Karney: for each edge of the polygon, # # tan(Δλ/2) · (tan(φ1/2) + tan(φ2/2)) # tan(E/2) = ------------------------------------ # 1 + tan(φ1/2) · tan(φ2/2) # # where E is the spherical excess of the trapezium obtained by # extending the edge to the equator-circle vector for each edge. t2 = tan_2(phi2) t1 = tan_2(phi1) t = tan_2(lam21, lam21=None) return Radians(Karney=atan2(t * (t1 + t2), _1_0 + (t1 * t2)) * _2_0) def excessQuad(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): '''Compute the surface area of a (spherical) quadrilateral bounded by a segment of a great circle, two meridians and the equator. @arg lat1: Start latitude (C{degrees}). @arg lon1: Start longitude (C{degrees}). @arg lat2: End latitude (C{degrees}). @arg lon2: End longitude (C{degrees}). @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter}, L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}) or C{None}. @kwarg wrap: Wrap and L{unroll180} longitudes (C{bool}). @return: Surface area, I{signed} (I{square} C{meter}, or units of B{C{radius}} I{squared}) or C{radians} if B{C{radius}} is C{None} or C{0}. @raise TypeError: Invalid B{C{radius}}. @raise UnitError: Invalid B{C{lat2}} or B{C{lat1}}. @see: Function L{excessQuad_} and L{excessKarney}. ''' return _areaOfS(excessQuad_, lat1, lat2, radius, *unroll180(lon1, lon2, wrap=wrap)) def excessQuad_(phi2, phi1, lam21): '''Compute the I{spherical excess} C{E} of a (spherical) quadrilateral bounded by a segment of a great circle, two meridians and the equator. @arg phi2: End latitude (C{radians}). @arg phi1: Start latitude (C{radians}). @arg lam21: Longitudinal delta, M{end-start} (C{radians}). @return: Spherical excess, I{signed} (C{radians}). @see: Function L{excessQuad}, U{Spherical trigonometry <https://WikiPedia.org/wiki/Spherical_trigonometry>}. ''' s = sin((phi2 + phi1) * _0_5) c = cos((phi2 - phi1) * _0_5) return Radians(Quad=atan2(tan_2(lam21) * s, c) * _2_0) def flatLocal(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False): '''Compute the distance between two (ellipsoidal) points using the U{ellipsoidal Earth to plane projection<https://WikiPedia.org/ wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>} aka U{Hubeny<https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula. @arg lat1: Start latitude (C{degrees}). @arg lon1: Start longitude (C{degrees}). @arg lat2: End latitude (C{degrees}). @arg lon2: End longitude (C{degrees}). @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}). @kwarg wrap: Wrap and L{unroll180} longitudes (C{bool}). @return: Distance (C{meter}, same units as the B{C{datum}}'s ellipsoid axes). @raise TypeError: Invalid B{C{datum}}. @note: The meridional and prime_vertical radii of curvature are taken and scaled at the mean of both latitude. @see: Functions L{flatLocal_}/L{hubeny_}, L{cosineLaw}, L{flatPolar}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, L{equirectangular}, L{euclidean}, L{haversine}, L{thomas}, L{vincentys}, method L{Ellipsoid.distance2} and U{local, flat earth approximation <https://www.EdWilliams.org/avform.htm#flat>}. ''' d, _ = unroll180(lon1, lon2, wrap=wrap) return flatLocal_(Phi_(lat2=lat2), Phi_(lat1=lat1), radians(d), datum=datum) hubeny = flatLocal # for Karl Hubeny def flatLocal_(phi2, phi1, lam21, datum=_WGS84): '''Compute the I{angular} distance between two (ellipsoidal) points using the U{ellipsoidal Earth to plane projection<https://WikiPedia.org/ wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>} aka U{Hubeny<https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula. @arg phi2: End latitude (C{radians}). @arg phi1: Start latitude (C{radians}). @arg lam21: Longitudinal delta, M{end-start} (C{radians}). @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}). @return: Angular distance (C{radians}). @raise TypeError: Invalid B{C{datum}}. @note: The meridional and prime_vertical radii of curvature are taken and scaled I{at the mean of both latitude}. @see: Functions L{flatLocal}/L{hubeny}, L{cosineAndoyerLambert_}, L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{flatPolar_}, L{equirectangular_}, L{euclidean_}, L{haversine_}, L{thomas_} and L{vincentys_} and U{local, flat earth approximation <https://www.EdWilliams.org/avform.htm#flat>}. ''' E = _ellipsoid(datum, flatLocal_) m, n = E.roc2_((phi2 + phi1) * _0_5, scaled=True) return hypot(m * (phi2 - phi1), n * lam21) hubeny_ = flatLocal_ # for Karl Hubeny def flatPolar(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): '''Compute the distance between two (spherical) points using the U{polar coordinate flat-Earth <https://WikiPedia.org/wiki/ Geographical_distance#Polar_coordinate_flat-Earth_formula>} formula. @arg lat1: Start latitude (C{degrees}). @arg lon1: Start longitude (C{degrees}). @arg lat2: End latitude (C{degrees}). @arg lon2: End longitude (C{degrees}). @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter}, L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}). @kwarg wrap: Wrap and L{unroll180} longitudes (C{bool}). @return: Distance (C{meter}, same units as B{C{radius}} or the ellipsoid or datum axes). @raise TypeError: Invalid B{C{radius}}. @see: Functions L{flatPolar_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},L{cosineLaw}, L{flatLocal}/L{hubeny}, L{equirectangular}, L{euclidean}, L{haversine}, L{thomas} and L{vincentys}. ''' return _distanceToS(flatPolar_, lat1, lat2, radius, *unroll180(lon1, lon2, wrap=wrap)) def flatPolar_(phi2, phi1, lam21): '''Compute the I{angular} distance between two (spherical) points using the U{polar coordinate flat-Earth<https://WikiPedia.org/wiki/ Geographical_distance#Polar_coordinate_flat-Earth_formula>} formula. @arg phi2: End latitude (C{radians}). @arg phi1: Start latitude (C{radians}). @arg lam21: Longitudinal delta, M{end-start} (C{radians}). @return: Angular distance (C{radians}). @see: Functions L{flatPolar}, L{cosineAndoyerLambert_}, L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{haversine_}, L{thomas_} and L{vincentys_}. ''' a1 = PI_2 - phi1 # co-latitude a2 = PI_2 - phi2 # co-latitude ab = _2_0 * a1 * a2 * cos(lam21) r2 = fsum_(a1**2, a2**2, -abs(ab)) return sqrt0(r2) def haversine(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): '''Compute the distance between two (spherical) points using the U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>} formula. @arg lat1: Start latitude (C{degrees}). @arg lon1: Start longitude (C{degrees}). @arg lat2: End latitude (C{degrees}). @arg lon2: End longitude (C{degrees}). @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter}, L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}). @kwarg wrap: Wrap and L{unroll180} longitudes (C{bool}). @return: Distance (C{meter}, same units as B{C{radius}}). @raise TypeError: Invalid B{C{radius}}. @see: U{Distance between two (spherical) points <https://www.EdWilliams.org/avform.htm#Dist>}, functions L{cosineLaw}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar}, L{thomas} and L{vincentys} and methods L{Ellipsoid.distance2}, C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}. @note: See note at function L{vincentys_}. ''' return _distanceToS(haversine_, lat1, lat2, radius, *unroll180(lon1, lon2, wrap=wrap)) def haversine_(phi2, phi1, lam21): '''Compute the I{angular} distance between two (spherical) points using the U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>} formula. @arg phi2: End latitude (C{radians}). @arg phi1: Start latitude (C{radians}). @arg lam21: Longitudinal delta, M{end-start} (C{radians}). @return: Angular distance (C{radians}). @see: Functions L{haversine}, L{cosineAndoyerLambert_}, L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_}, L{thomas_} and L{vincentys_}. @note: See note at function L{vincentys_}. ''' def _hsin(rad): return sin(rad * _0_5)**2 h = _hsin(phi2 - phi1) + cos(phi1) * cos(phi2) * _hsin(lam21) # haversine return atan2(sqrt0(h), sqrt0(_1_0 - h)) * _2_0 # == asin(sqrt(h)) * 2 def heightOf(angle, distance, radius=R_M): '''Determine the height above the (spherical) earth after traveling along a straight line at a given tilt. @arg angle: Tilt angle above horizontal (C{degrees}). @arg distance: Distance along the line (C{meter} or same units as B{C{radius}}). @kwarg radius: Optional mean earth radius (C{meter}). @return: Height (C{meter}, same units as B{C{distance}} and B{C{radius}}). @raise ValueError: Invalid B{C{angle}}, B{C{distance}} or B{C{radius}}. @see: U{MultiDop geog_lib.GeogBeamHt<https://GitHub.com/NASA/MultiDop>} (U{Shapiro et al. 2009, JTECH <https://Journals.AMetSoc.org/doi/abs/10.1175/2009JTECHA1256.1>} and U{Potvin et al. 2012, JTECH <https://Journals.AMetSoc.org/doi/abs/10.1175/JTECH-D-11-00019.1>}). ''' r = h = Radius(radius) d = abs(Distance(distance)) if d > h: d, h = h, d if d > EPS0: d = d / h # /= h chokes PyChecker s = sin(Phi_(angle=angle, clip=_180_0)) s = fsum_(_1_0, _2_0 * s * d, d**2) if s > 0: return h * sqrt(s) - r raise _ValueError(angle=angle, distance=distance, radius=radius) def horizon(height, radius=R_M, refraction=False): '''Determine the distance to the horizon from a given altitude above the (spherical) earth. @arg height: Altitude (C{meter} or same units as B{C{radius}}). @kwarg radius: Optional mean earth radius (C{meter}). @kwarg refraction: Consider atmospheric refraction (C{bool}). @return: Distance (C{meter}, same units as B{C{height}} and B{C{radius}}). @raise ValueError: Invalid B{C{height}} or B{C{radius}}. @see: U{Distance to horizon<https://www.EdWilliams.org/avform.htm#Horizon>}. ''' h, r = Height(height), Radius(radius) if min(h, r) < 0: raise _ValueError(height=height, radius=radius) if refraction: d2 = 2.415750694528 * h * r # 2.0 / 0.8279 else: d2 = h * fsum_(r, r, h) return sqrt0(d2) def intersections2(lat1, lon1, radius1, lat2, lon2, radius2, datum=None, wrap=True): '''Conveniently compute the intersections of two circles each defined by a (geodetic) center point and a radius, using either ... 1) L{vector3d.intersections2} for small distances (below 100 KM or about 0.9 degrees) or if no B{C{datum}} is specified, or ... 2) L{sphericalTrigonometry.intersections2} for a spherical B{C{datum}} or if B{C{datum}} is a C{scalar} representing the earth radius, or ... 3) L{ellipsoidalKarney.intersections2} for an ellipsoidal B{C{datum}} and if I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib/>} is installed, or ... 4) L{ellipsoidalVincenty.intersections2} otherwise provided B{C{datum}} is ellipsoidal. @arg lat1: Latitude of the first circle center (C{degrees}). @arg lon1: Longitude of the first circle center (C{degrees}). @arg radius1: Radius of the first circle (C{meter}, conventionally). @arg lat2: Latitude of the second circle center (C{degrees}). @arg lon2: Longitude of the second circle center (C{degrees}). @arg radius2: Radius of the second circle (C{meter}, same units as B{C{radius1}}). @kwarg datum: Optional ellipsoidal or spherical datum (L{Datum}, L{Ellipsoid}, L{Ellipsoid2}, L{a_f2Tuple} or C{scalar} earth radius in C{meter}, same units as B{C{radius1}} and B{C{radius2}}) or C{None}. @kwarg wrap: Wrap and unroll longitudes (C{bool}). @return: 2-Tuple of the intersection points, each a L{LatLon2Tuple}C{(lat, lon)}. For abutting circles, the intersection points are the same instance. @raise IntersectionError: Concentric, antipodal, invalid or non-intersecting circles or no convergence. @raise TypeError: Invalid B{C{datum}}. @raise UnitError: Invalid B{C{lat1}}, B{C{lon1}}, B{C{radius1}} B{C{lat2}}, B{C{lon2}} or B{C{radius2}}. ''' if datum is None or euclidean(lat1, lon1, lat1, lon2, radius=R_M, adjust=True, wrap=wrap) < _100km: import pygeodesy.vector3d as m def _V2T(x, y, _, **unused): # _ == z unused return LatLon2Tuple(y, x, name=intersections2.__name__) r1 = m2degrees(Radius_(radius1=radius1), radius=R_M, lat=lat1) r2 = m2degrees(Radius_(radius2=radius2), radius=R_M, lat=lat2) _, lon2 = unroll180(lon1, lon2, wrap=wrap) t = m.intersections2(m.Vector3d(lon1, lat1, 0), r1, m.Vector3d(lon2, lat2, 0), r2, sphere=False, Vector=_V2T) else: def _LL2T(lat, lon, **unused): return LatLon2Tuple(lat, lon, name=intersections2.__name__) d = _spherical_datum(datum, name=intersections2.__name__) if d.isSpherical: import pygeodesy.sphericalTrigonometry as m elif d.isEllipsoidal: try: if d.ellipsoid.geodesic: pass import pygeodesy.ellipsoidalKarney as m except ImportError: import pygeodesy.ellipsoidalVincenty as m else: raise _AssertionError(datum=d) t = m.intersections2(m.LatLon(lat1, lon1, datum=d), radius1, m.LatLon(lat2, lon2, datum=d), radius2, LatLon=_LL2T, height=0, wrap=wrap) return t def isantipode(lat1, lon1, lat2, lon2, eps=EPS): '''Check whether two points are antipodal, on diametrically opposite sides of the earth. @arg lat1: Latitude of one point (C{degrees}). @arg lon1: Longitude of one point (C{degrees}). @arg lat2: Latitude of the other point (C{degrees}). @arg lon2: Longitude of the other point (C{degrees}). @kwarg eps: Tolerance for near-equality (C{degrees}). @return: C{True} if points are antipodal within the B{C{eps}} tolerance, C{False} otherwise. @see: U{Geosphere<https://CRAN.R-Project.org/web/packages/geosphere/geosphere.pdf>}. ''' return abs(wrap90(lat1) + wrap90(lat2)) < eps and \ abs(abs(wrap180(lon1) - wrap180(lon2)) % _360_0 - _180_0) < eps def isantipode_(phi1, lam1, phi2, lam2, eps=EPS): '''Check whether two points are antipodal, on diametrically opposite sides of the earth. @arg phi1: Latitude of one point (C{radians}). @arg lam1: Longitude of one point (C{radians}). @arg phi2: Latitude of the other point (C{radians}). @arg lam2: Longitude of the other point (C{radians}). @kwarg eps: Tolerance for near-equality (C{radians}). @return: C{True} if points are antipodal within the B{C{eps}} tolerance, C{False} otherwise. @see: U{Geosphere<https://CRAN.R-Project.org/web/packages/geosphere/geosphere.pdf>}. ''' return abs(wrapPI_2(phi1) + wrapPI_2(phi2)) < eps and abs( abs(wrapPI(lam1) - wrapPI(lam2)) % PI2 - PI) < eps def latlon2n_xyz(lat, lon, name=NN): '''Convert lat-, longitude to C{n-vector} (normal to the earth's surface) X, Y and Z components. @arg lat: Latitude (C{degrees}). @arg lon: Longitude (C{degrees}). @kwarg name: Optional name (C{str}). @return: A L{Vector3Tuple}C{(x, y, z)}. @see: Function L{philam2n_xyz}. @note: These are C{n-vector} x, y and z components, I{NOT} geocentric ECEF x, y and z coordinates! ''' return philam2n_xyz(radians(lat), radians(lon), name=name) def n_xyz2latlon(x, y, z, name=NN): '''Convert C{n-vector} components to lat- and longitude in C{degrees}. @arg x: X component (C{scalar}). @arg y: Y component (C{scalar}). @arg z: Z component (C{scalar}). @kwarg name: Optional name (C{str}). @return: A L{LatLon2Tuple}C{(lat, lon)}. @see: Function L{n_xyz2philam}. ''' a, b = n_xyz2philam(x, y, z) # PYCHOK PhiLam2Tuple return LatLon2Tuple(degrees90(a), degrees180(b), name=name) def n_xyz2philam(x, y, z, name=NN): '''Convert C{n-vector} components to lat- and longitude in C{radians}. @arg x: X component (C{scalar}). @arg y: Y component (C{scalar}). @arg z: Z component (C{scalar}). @kwarg name: Optional name (C{str}). @return: A L{PhiLam2Tuple}C{(phi, lam)}. @see: Function L{n_xyz2latlon}. ''' return PhiLam2Tuple(atan2(z, hypot(x, y)), atan2(y, x), name=name) def philam2n_xyz(phi, lam, name=NN): '''Convert lat-, longitude to C{n-vector} (normal to the earth's surface) X, Y and Z components. @arg phi: Latitude (C{radians}). @arg lam: Longitude (C{radians}). @kwarg name: Optional name (C{str}). @return: A L{Vector3Tuple}C{(x, y, z)}. @see: Function L{latlon2n_xyz}. @note: These are C{n-vector} x, y and z components, I{NOT} geocentric ECEF x, y and z coordinates! ''' # Kenneth Gade eqn 3, but using right-handed # vector x -> 0°E,0°N, y -> 90°E,0°N, z -> 90°N sa, ca, sb, cb = sincos2(phi, lam) return Vector3Tuple(ca * cb, ca * sb, sa, name=name) def _radical2(d, r1, r2): # in .ellipsoidalBase, .sphericalTrigonometry, .vector3d # (INTERNAL) See C{radical2} below # assert d > EPS0 r = fsum_(_1_0, (r1 / d)**2, -(r2 / d)**2) * _0_5 return Radical2Tuple(max(_0_0, min(_1_0, r)), r * d) def radical2(distance, radius1, radius2): '''Compute the I{radical ratio} and I{radical line} of two U{intersecting circles<https://MathWorld.Wolfram.com/ Circle-CircleIntersection.html>}. The I{radical line} is perpendicular to the axis thru the centers of the circles at C{(0, 0)} and C{(B{distance}, 0)}. @arg distance: Distance between the circle centers (C{scalar}). @arg radius1: Radius of the first circle (C{scalar}). @arg radius2: Radius of the second circle (C{scalar}). @return: A L{Radical2Tuple}C{(ratio, xline)} where C{0.0 <= ratio <= 1.0} and C{xline} is along the B{C{distance}}. @raise IntersectionError: The B{C{distance}} exceeds the sum of B{C{radius1}} and B{C{radius2}}. @raise UnitError: Invalid B{C{distance}}, B{C{radius1}} or B{C{radius2}}. ''' d = Distance_(distance, low=_0_0) r1 = Radius_(radius1=radius1) r2 = Radius_(radius2=radius2) if d > (r1 + r2): raise IntersectionError(distance=d, radius1=r1, radius2=r2, txt=_too_(_distant_)) return _radical2(d, r1, r2) if d > EPS0 else \ Radical2Tuple(_0_5, _0_0) class Radical2Tuple(_NamedTuple): '''2-Tuple C{(ratio, xline)} of the I{radical} C{ratio} and I{radical} C{xline}, both C{scalar} and C{0.0 <= ratio <= 1.0} ''' _Names_ = ('ratio', 'xline') _Units_ = ( Scalar, Scalar) def _scale_deg(lat1, lat2): # degrees # scale factor cos(mean of lats) for delta lon m = abs(lat1 + lat2) * _0_5 return cos(radians(m)) if m < _90_0 else _0_0 def _scale_rad(phi1, phi2): # radians, by .frechet, .hausdorff, .heights # scale factor cos(mean of phis) for delta lam m = abs(phi1 + phi2) * _0_5 return cos(m) if m < PI_2 else _0_0 def _sincosa6(phi2, phi1, lam21): '''(INTERNAL) C{sin}es, C{cos}ines and C{acos}ine. ''' s2, c2, s1, c1, _, c21 = sincos2(phi2, phi1, lam21) return s2, c2, s1, c1, acos1(s1 * s2 + c1 * c2 * c21), c21 def thomas(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False): '''Compute the distance between two (ellipsoidal) points using U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>} formula. @arg lat1: Start latitude (C{degrees}). @arg lon1: Start longitude (C{degrees}). @arg lat2: End latitude (C{degrees}). @arg lon2: End longitude (C{degrees}). @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}). @kwarg wrap: Wrap and L{unroll180} longitudes (C{bool}). @return: Distance (C{meter}, same units as the B{C{datum}}'s ellipsoid axes). @raise TypeError: Invalid B{C{datum}}. @see: Functions L{thomas_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar}, L{haversine}, L{vincentys} and method L{Ellipsoid.distance2}. ''' return _distanceToE(thomas_, lat1, lat2, datum, *unroll180(lon1, lon2, wrap=wrap)) def thomas_(phi2, phi1, lam21, datum=_WGS84): '''Compute the I{angular} distance between two (ellipsoidal) points using U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>} formula. @arg phi2: End latitude (C{radians}). @arg phi1: Start latitude (C{radians}). @arg lam21: Longitudinal delta, M{end-start} (C{radians}). @kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}). @return: Angular distance (C{radians}). @raise TypeError: Invalid B{C{datum}}. @see: Functions L{thomas}, L{cosineAndoyerLambert_}, L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_}, L{haversine_} and L{vincentys_} and U{Geodesy-PHP <https://GitHub.com/jtejido/geodesy-php/blob/master/src/Geodesy/ Distance/ThomasFormula.php>}. ''' s2, c2, s1, c1, r, _ = _sincosa6(phi2, phi1, lam21) if r and _non0(c1) and _non0(c2): E = _ellipsoid(datum, thomas_) if E.f: r1 = atan2(E.b_a * s1, c1) r2 = atan2(E.b_a * s2, c2) j = (r2 + r1) * _0_5 k = (r2 - r1) * _0_5 sj, cj, sk, ck, h, _ = sincos2(j, k, lam21 * _0_5) h = fsum_(sk**2, (ck * h)**2, -(sj * h)**2) u = _1_0 - h if _non0(u) and _non0(h): r = atan(sqrt0(h / u)) * _2_0 # == acos(1 - 2 * h) sr, cr = sincos2(r) if _non0(sr): u = 2 * (sj * ck)**2 / u h = 2 * (sk * cj)**2 / h x = u + h y = u - h s = r / sr e = 4 * s**2 d = 2 * cr a = e * d b = 2 * r c = s - (a - d) * _0_5 f = E.f * _0_25 t = fsum_(a * x, -b * y, c * x**2, -d * y**2, e * x * y) r -= fsum_(s * x, -y, -t * f * _0_25) * f * sr return r def vincentys(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): '''Compute the distance between two (spherical) points using U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>} spherical formula. @arg lat1: Start latitude (C{degrees}). @arg lon1: Start longitude (C{degrees}). @arg lat2: End latitude (C{degrees}). @arg lon2: End longitude (C{degrees}). @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter}, L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}). @kwarg wrap: Wrap and L{unroll180} longitudes (C{bool}). @return: Distance (C{meter}, same units as B{C{radius}}). @raise UnitError: Invalid B{C{radius}}. @see: Functions L{vincentys_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},L{cosineLaw}, L{equirectangular}, L{euclidean}, L{flatLocal}/L{hubeny}, L{flatPolar}, L{haversine} and L{thomas} and methods L{Ellipsoid.distance2}, C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}. @note: See note at function L{vincentys_}. ''' return _distanceToS(vincentys_, lat1, lat2, radius, *unroll180(lon1, lon2, wrap=wrap)) def vincentys_(phi2, phi1, lam21): '''Compute the I{angular} distance between two (spherical) points using U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>} spherical formula. @arg phi2: End latitude (C{radians}). @arg phi1: Start latitude (C{radians}). @arg lam21: Longitudinal delta, M{end-start} (C{radians}). @return: Angular distance (C{radians}). @see: Functions L{vincentys}, L{cosineAndoyerLambert_}, L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, L{equirectangular_}, L{euclidean_}, L{flatLocal_}/L{hubeny_}, L{flatPolar_}, L{haversine_} and L{thomas_}. @note: Functions L{vincentys_}, L{haversine_} and L{cosineLaw_} produce equivalent results, but L{vincentys_} is suitable for antipodal points and slightly more expensive (M{3 cos, 3 sin, 1 hypot, 1 atan2, 6 mul, 2 add}) than L{haversine_} (M{2 cos, 2 sin, 2 sqrt, 1 atan2, 5 mul, 1 add}) and L{cosineLaw_} (M{3 cos, 3 sin, 1 acos, 3 mul, 1 add}). ''' s1, c1, s2, c2, s21, c21 = sincos2(phi1, phi2, lam21) c = c2 * c21 x = s1 * s2 + c1 * c y = c1 * s2 - s1 * c return atan2(hypot(c2 * s21, y), x) # **) MIT License # # Copyright (C) 2016-2021 -- mrJean1 at Gmail -- All Rights Reserved. # # Permission is hereby granted, free of charge, to any person obtaining a # copy of this software and associated documentation files (the "Software"), # to deal in the Software without restriction, including without limitation # the rights to use, copy, modify, merge, publish, distribute, sublicense, # and/or sell copies of the Software, and to permit persons to whom the # Software is furnished to do so, subject to the following conditions: # # The above copyright notice and this permission notice shall be included # in all copies or substantial portions of the Software. # # THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS # OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, # FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL # THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR # OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, # ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR # OTHER DEALINGS IN THE SOFTWARE.