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-*- coding: utf-8 -*- u'''Formulary of basic geodesy functions and approximations. @newfield example: Example, Examples ''' from pygeodesy.basics import len2 from pygeodesy.datums import Datums, _ellipsoidal_datum, _spherical_datum from pygeodesy.errors import _AssertionError, IntersectionError, LimitError, \ _limiterrors, PointsError, _ValueError from pygeodesy.fmath import euclid, fsum_, hypot, hypot2 from pygeodesy.interns import EPS, EPS1, PI, PI2, PI_2, R_M, _item_sq, \ _too_distant_, _too_few_, _0_0, _0_5, _1_0, \ _4_0, _8_0, _16_0, _32_0, _90_0, _180_0, _360_0 from pygeodesy.lazily import _ALL_LAZY, _ALL_OTHER from pygeodesy.named import _NamedTuple, _xnamed from pygeodesy.namedTuples import Distance4Tuple, LatLon2Tuple, \ PhiLam2Tuple, Points2Tuple, \ Vector3Tuple from pygeodesy.streprs import unstr from pygeodesy.units import Distance, Distance_, Height, Lam_, Lat, Lon, \ Phi_, Radius, Radius_, Scalar from pygeodesy.utily import atan2b, degrees2m, degrees90, degrees180, \ isNumpy2, isTuple2, m2degrees, sincos2, unroll180, \ unrollPI, wrap90, wrap180, wrapPI, wrapPI_2 from math import acos, atan, atan2, cos, degrees, radians, sin, sqrt # pow __all__ = _ALL_LAZY.formy __version__ = '20.10.03' _D_I2_ = 1e5 # meter, 100 Km, about 0.9 degrees 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 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 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 = PI2 + PI 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 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, non-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=Datums.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: Ellipsoidal datum 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{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}. ''' d, _ = unroll180(lon1, lon2, wrap=wrap) r = cosineAndoyerLambert_(Phi_(lat2=lat2), Phi_(lat1=lat1), radians(d), datum=datum) return r * datum.ellipsoid.a def cosineAndoyerLambert_(phi2, phi1, lam21, datum=Datums.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: Ellipsoidal datum 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, _, c21 = sincos2(phi2, phi1, lam21) if c2 and c1: E = _ellipsoidal_datum(datum, name=cosineAndoyerLambert_.__name__).ellipsoid if E.f and abs(c1) > EPS and abs(c2) > EPS: r2 = atan(E.b_a * s2 / c2) r1 = atan(E.b_a * s1 / c1) s2, c2, s1, c1 = sincos2(r2, r1) r = acos(s1 * s2 + c1 * c2 * c21) sr, _, sr_2, cr_2 = sincos2(r, r * _0_5) if abs(sr_2) > EPS and abs(cr_2) > EPS: c = (sr - r) * ((s1 + s2) / cr_2)**2 s = (sr + r) * ((s1 - s2) / sr_2)**2 r += E.f * (c - s) / _8_0 return r # fall back to cosineLaw_ return acos(s1 * s2 + c1 * c2 * c21) def cosineForsytheAndoyerLambert(lat1, lon1, lat2, lon2, datum=Datums.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: Ellipsoidal datum 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{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}. ''' d, _ = unroll180(lon1, lon2, wrap=wrap) r = cosineForsytheAndoyerLambert_(Phi_(lat2=lat2), Phi_(lat1=lat1), radians(d), datum=datum) return r * datum.ellipsoid.a def cosineForsytheAndoyerLambert_(phi2, phi1, lam21, datum=Datums.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: Ellipsoidal datum 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, _, c21 = sincos2(phi2, phi1, lam21) r = acos(s1 * s2 + c1 * c2 * c21) E = _ellipsoidal_datum(datum, name=cosineForsytheAndoyerLambert_.__name__).ellipsoid if E.f: sr, cr, s2r, _ = sincos2(r, r * 2) if abs(sr) > EPS: r2 = r**2 p = (s1 + s2)**2 / (1 + cr) q = (s1 - s2)**2 / (1 - cr) x = p + q y = p - q s = 8 * r2 / sr a = 64 * r + 2 * s * cr # 16 * r2 / tan(r) d = 48 * sr + s # 8 * r2 / tan(r) b = -2 * d e = 30 * s2r c = fsum_(30 * r, e / 2, s * cr) # 8 * r2 / tan(r) d = fsum_(a * x, b * y, -c * x**2, d * x * y, e * y**2) * E.f / _32_0 d = fsum_(d, -x * r, 3 * y * sr) * E.f / _4_0 r += d 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 (C{meter}). @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: 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_}. ''' r = Radius(radius) if r: d, _ = unroll180(lon1, lon2, wrap=wrap) r *= cosineLaw_(Phi_(lat2=lat2), Phi_(lat1=lat1), radians(d)) return r 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_}. ''' s2, c2, s1, c1, _, c21 = sincos2(phi2, phi1, lam21) return acos(s1 * s2 + c1 * c2 * c21) 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 (C{meter}). @kwarg options: Optional keyword arguments for function L{equirectangular_}. @return: Distance (C{meter}, same units as B{C{radius}}). @see: Function L{equirectangular_} for more details, the available B{C{options}}, errors, restrictions and other, approximate or accurate distance functions. ''' _, dy, dx, _ = equirectangular_(Lat(lat1=lat1), Lon(lon1=lon1), Lat(lat2=lat2), Lon(lon2=lon2), **options) # PYCHOK Distance4Tuple return degrees2m(hypot(dx, dy), radius=radius) 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 (C{meter}). @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}}). @raise TypeError: Invalid B{C{radius}}. @see: U{Distance between two (spherical) points <https://www.EdWilliams.org/avform.htm#Dist>}, functions 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}. ''' r = Radius(radius) if r: d, _ = unroll180(lon1, lon2, wrap=wrap) r *= euclidean_(Phi_(lat2=lat2), Phi_(lat1=lat1), radians(d), adjust=adjust) return r 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{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 flatLocal(lat1, lon1, lat2, lon2, datum=Datums.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: Ellipsoidal datum 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=Datums.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: Ellipsoidal datum 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 = _ellipsoidal_datum(datum, name=flatLocal_.__name__).ellipsoid 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 (C{meter}). @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: Functions L{flatPolar_}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert},L{cosineLaw}, L{flatLocal}/L{hubeny}, L{equirectangular}, L{euclidean}, L{haversine}, L{thomas} and L{vincentys}. ''' r = Radius(radius) if r: d, _ = unroll180(lon1, lon2, wrap=wrap) r *= flatPolar_(Phi_(lat2=lat2), Phi_(lat1=lat1), radians(d)) return r 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 = abs(PI_2 - phi1) # co-latitude a2 = abs(PI_2 - phi2) # co-latitude ab = abs(2 * a1 * a2 * cos(lam21)) a = max(a1, a2, ab) if a > EPS: s = fsum_((a1 / a)**2, (a2 / a)**2, -ab / a**2) a *= sqrt(s) if s > 0 else _0_0 return a 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 (C{meter}). @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_}. ''' r = Radius(radius) if r: d, _ = unroll180(lon1, lon2, wrap=wrap) r *= haversine_(Phi_(lat2=lat2), Phi_(lat1=lat1), radians(d)) return r 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 try: r = atan2(sqrt(h), sqrt(1 - h)) * 2 # == asin(sqrt(h)) * 2 except ValueError: r = _0_0 if h < _0_5 else PI return r 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 > EPS: d = d / h # PyChecker chokes on ... /= ... s = sin(Phi_(angle=angle, clip=_180_0)) s = fsum_(_1_0, 2 * 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 sqrt(d2) def intersections2(lat1, lon1, radius1, lat2, lon2, radius2, datum=None, wrap=True): '''Conveniently compute the intersections of two circles each defined by (geodetic/-centric) center point and a radius, using either ... 1) L{vector3d.intersections2} for small distances 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} if B{C{datum}} is ellipsoidal otherwise. @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}). @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}). @kwarg datum: Optional ellipsoidal or spherical datum (L{Datum}, L{Ellipsoid}, L{Ellipsoid2}, L{a_f2Tuple} or C{scalar} earth radius) 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) < _D_I2_: import pygeodesy.vector3d as m def _V2T(x, y, _, **unused): # _ == z unused return _xnamed(LatLon2Tuple(y, x), 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 _xnamed(LatLon2Tuple(lat, lon), 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, wrap=wrap, LatLon=_LL2T, height=0) 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): '''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}). @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)) def n_xyz2latlon(x, y, z): '''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}). @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)) def n_xyz2philam(x, y, z): '''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}). @return: A L{PhiLam2Tuple}C{(phi, lam)}. @see: Function L{n_xyz2latlon}. ''' return PhiLam2Tuple(atan2(z, hypot(x, y)), atan2(y, x)) def philam2n_xyz(phi, lam): '''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}). @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) def points2(points, closed=True, base=None, Error=PointsError): '''Check a path or polygon represented by points. @arg points: The path or polygon points (C{LatLon}[]) @kwarg closed: Optionally, consider the polygon closed, ignoring any duplicate or closing final B{C{points}} (C{bool}). @kwarg base: Optionally, check all B{C{points}} against this base class, if C{None} don't check. @kwarg Error: Exception to raise (C{ValueError}). @return: A L{Points2Tuple}C{(number, points)} with the number of points and the points C{list} or C{tuple}. @raise PointsError: Insufficient number of B{C{points}}. @raise TypeError: Some B{C{points}} are not B{C{base}} compatible. ''' n, points = len2(points) if closed: # remove duplicate or closing final points while n > 1 and points[n-1] in (points[0], points[n-2]): n -= 1 # XXX following line is unneeded if points # are always indexed as ... i in range(n) points = points[:n] # XXX numpy.array slice is a view! if n < (3 if closed else 1): raise Error(points=n, txt=_too_few_) if base and not (isNumpy2(points) or isTuple2(points)): for i in range(n): base.others(points[i], name=_item_sq(points=i)) return Points2Tuple(n, points) def _radical2(d, r1, r2): # in .ellipsoidalBase, .sphericalTrigonometry, .vector3d # (INTERNAL) See C{radical2} below 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) 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) 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 thomas(lat1, lon1, lat2, lon2, datum=Datums.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: Ellipsoidal datum 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}. ''' d, _ = unroll180(lon1, lon2, wrap=wrap) r = thomas_(Phi_(lat2=lat2), Phi_(lat1=lat1), radians(d), datum=datum) return r * datum.ellipsoid.a def thomas_(phi2, phi1, lam21, datum=Datums.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: Ellipsoidal datum 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, _, c21 = sincos2(phi2, phi1, lam21) E = _ellipsoidal_datum(datum, name=thomas_.__name__).ellipsoid if E.f and abs(c1) > EPS and abs(c2) > EPS: r1 = atan(E.b_a * s1 / c1) r2 = atan(E.b_a * s2 / c2) j = (r2 + r1) * _0_5 k = (r2 - r1) * _0_5 sj, cj, sk, ck, sl_2, _ = sincos2(j, k, lam21 * _0_5) h = fsum_(sk**2, (ck * sl_2)**2, -(sj * sl_2)**2) if EPS < abs(h) < EPS1: u = _1_0 / (_1_0 - h) d = atan(sqrt(h * u)) * 2 # == acos(1 - 2 * h) sd, cd = sincos2(d) if abs(sd) > EPS: u = 2 * (sj * ck)**2 * u v = 2 * (sk * cj)**2 / h x = u + v y = u - v t = d / sd s = 4 * t**2 e = 2 * cd a = s * e b = 2 * d c = t - (a - e) * _0_5 s = fsum_(a * x, c * x**2, -b * y, -e * y**2, s * x * y) * E.f / _16_0 s = fsum_(t * x, -y, -s) * E.f / _4_0 return d - s * sd # fall back to cosineLaw_ return acos(s1 * s2 + c1 * c2 * c21) 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 (C{meter}). @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: 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_}. ''' r = Radius(radius) if r: d, _ = unroll180(lon1, lon2, wrap=wrap) r *= vincentys_(Phi_(lat2=lat2), Phi_(lat1=lat1), radians(d)) return r 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}). ''' sa1, ca1, sa2, ca2, sb21, cb21 = sincos2(phi1, phi2, lam21) c = ca2 * cb21 x = sa1 * sa2 + ca1 * c y = ca1 * sa2 - sa1 * c return atan2(hypot(ca2 * sb21, y), x) __all__ += _ALL_OTHER(Radical2Tuple) # **) MIT License # # Copyright (C) 2016-2020 -- 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.