# Coverage for pygeodesy/formy.py : 98%

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functions and approximations. ''' # make sure int/int division yields float quotient, see .basics from __future__ import division as _; del _ # PYCHOK semicolon from pygeodesy.constants import EPS, EPS0, EPS1, PI, PI2, PI3, PI_2, R_M, \ _umod_360, float0, isnon0, remainder, _0_0, \ _0_125, _0_25, _0_5, _1_0, _2_0, _4_0, \ _32_0, _90_0, _180_0, _360_0 from pygeodesy.datums import Datum, Ellipsoid, _ellipsoidal_datum, \ _mean_radius, _spherical_datum, _WGS84 # from pygeodesy.ellipsoids import Ellipsoid # from .datums from pygeodesy.errors import _AssertionError, IntersectionError, \ LimitError, limiterrors, _ValueError from pygeodesy.fmath import euclid, fdot, hypot, hypot2, sqrt0 from pygeodesy.fsums import fsum_, unstr from pygeodesy.interns import NN, _distant_, _inside_, _near_, _null_, \ _opposite_, _outside_, _too_ from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS from pygeodesy.named import _NamedTuple, _xnamed from pygeodesy.namedTuples import Bearing2Tuple, Distance4Tuple, \ LatLon2Tuple, PhiLam2Tuple, Vector3Tuple # from pygeodesy.streprs import unstr # from .fsums from pygeodesy.units import Degrees_, Distance, Distance_, Height, Lam_, Lat, \ Lon, Phi_, Radians, Radians_, Radius, Radius_, \ Scalar, _100km from pygeodesy.utily import acos1, atan2b, atan2d, degrees2m, m2degrees, tan_2, \ sincos2, sincos2_, sincos2d_, unroll180, unrollPI from math import atan, atan2, cos, degrees, fabs, radians, sin, sqrt # pow __all__ = _ALL_LAZY.formy __version__ = '22.09.21' _ratio_ = 'ratio' _xline_ = 'xline' def _anti2(a, b, n_2, n, n2): '''(INTERNAL) Helper for C{antipode} and C{antipode_}. ''' r = remainder(a, n) if fabs(a) > n_2 else a if r == a: r = -r b += n if fabs(b) > n: b = remainder(b, n2) return float0(r, b) def antipode(lat, lon, name=NN): '''Return the antipode, the point diametrically opposite to a given point in C{degrees}. @arg lat: Latitude (C{degrees}). @arg lon: Longitude (C{degrees}). @kwarg name: Optional name (C{str}). @return: A L{LatLon2Tuple}C{(lat, lon)}. @see: Functions L{antipode_} and L{normal} and U{Geosphere <https://CRAN.R-Project.org/web/packages/geosphere/geosphere.pdf>}. ''' return LatLon2Tuple(*_anti2(lat, lon, _90_0, _180_0, _360_0), name=name) def antipode_(phi, lam, name=NN): '''Return the antipode, the point diametrically opposite to a given point in C{radians}. @arg phi: Latitude (C{radians}). @arg lam: Longitude (C{radians}). @kwarg name: Optional name (C{str}). @return: A L{PhiLam2Tuple}C{(phi, lam)}. @see: Functions L{antipode} and L{normal_} and U{Geosphere <https://CRAN.R-Project.org/web/packages/geosphere/geosphere.pdf>}. ''' return PhiLam2Tuple(*_anti2(phi, lam, PI_2, PI, PI2), name=name) def _area_or_(func_, lat1, lat2, radius, d_lon, unused): '''(INTERNAL) Helper for area and spherical excess. ''' 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{pygeodesy.bearing_}. @return: Initial or final bearing (compass C{degrees360}) or zero if start and end point coincide. ''' r = bearing_(Phi_(lat1=lat1), Lam_(lon1=lon1), Phi_(lat2=lat2), Lam_(lon2=lon2), **options) return degrees(r) 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{pygeodesy.unrollPI} longitudes (C{bool}). @return: Initial or final bearing (compass C{radiansPI2}) or zero if start and end point coincide. @see: U{Bearing<https://www.Movable-Type.co.UK/scripts/latlong.html>}, U{Course between two points<https://www.EdWilliams.org/avform147.htm#Crs>} and U{Bearing Between Two Points<https://web.Archive.org/web/20020630205931/ https://MathForum.org/library/drmath/view/55417.html>}. ''' if final: # swap plus PI 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) 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 ellipsoidal 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)), name=_bearingTo2.__name__) 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{pygeodesy.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{pygeodesy.unroll180} longitudes (C{bool}). @return: Compass angle from North (C{degrees360}). @note: Courtesy of 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{pygeodesy.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 isnon0(c1) and isnon0(c2): E = _ellipsoidal(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 isnon0(sr_2) and isnon0(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{pygeodesy.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 isnon0(c1) and isnon0(c2): E = _ellipsoidal(datum, cosineForsytheAndoyerLambert_) if E.f: # ellipsoidal sr, cr, s2r, _ = sincos2_(r, r * _2_0) if isnon0(sr) and fabs(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{pygeodesy.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 = _ellipsoidal(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 _ellipsoidal(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{pygeodesy.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{pygeodesy.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 limit > 0 and limiterrors() and (fabs(d_lat) > limit or fabs(d_lon) > limit): 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{pygeodesy.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 (C{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 (C{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 (C{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 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>} method. @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{pygeodesy.unroll180} longitudes (C{bool}). @return: Surface area, I{signed} (I{square} C{meter}, or units of B{C{radius}} I{squared}) or I{spherical excess} (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 _area_or_(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 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>} 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 <https://MathOverflow.net/questions/97711/the-area-of-spherical-polygons>}. ''' # 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 (see also ***). 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) # ***) Original post no longer available, following is a copy of the main part # <http://OSGeo-org.1560.x6.Nabble.com/Area-of-a-spherical-polygon-td3841625.html> # # The area of a polygon on a (unit) sphere is given by the spherical excess # # A = 2 * pi - sum(exterior angles) # # However this is badly conditioned if the polygon is small. In this case, use # # A = sum S12{i, i+1} over the edges of the polygon # # where S12 is the area of the quadrilateral bounded by an edge of the polygon, # two meridians and the equator, i.e. with vertices (phi1, lambda1), (phi2, # lambda2), (0, lambda1) and (0, lambda2). S12 is given by # # tan(S12 / 2) = tan(lambda21 / 2) * (tan(phi1 / 2) + tan(phi2 / 2)) / # (tan(phi1 / 2) * tan(phi2 / 2) + 1) # # = tan(lambda21 / 2) * tanh((lam(phi1) + lam(phi2)) / 2) # # where lambda21 = lambda2 - lambda1 and lam(x) is the Lambertian (or inverse # Gudermannian) function # # lam(x) = asinh(tan(x)) = atanh(sin(x)) = 2 * atanh(tan(x / 2)) # # Notes: The formula for S12 is exact, except that... # - it is indeterminate if an edge is a semi-circle # - the formula for A applies only if the polygon does not include a pole # (if it does, then add +/- 2 * pi to the result) # - in the limit of small phi and lambda, S12 reduces to the trapezoidal # formula, S12 = (lambda2 - lambda1) * (phi1 + phi2) / 2 # - I derived this result from the equation for the area of a spherical # triangle in terms of two edges and the included angle given by, e.g. # U{Todhunter, I. - Spherical Trigonometry (1871), Sec. 103, Eq. (2) # <http://Books.Google.com/books?id=3uBHAAAAIAAJ&pg=PA71>} # - I would be interested to know if this formula for S12 is already known # - Charles Karney 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{pygeodesy.unroll180} longitudes (C{bool}). @return: Surface area, I{signed} (I{square} C{meter}, or units of B{C{radius}} I{squared}) or I{spherical excess} (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 _area_or_(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{pygeodesy.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_} or 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 # PYCHOK 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} or 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, flatLocal_) m, n = E.roc2_((phi2 + phi1) * _0_5, scaled=True) return hypot(m * (phi2 - phi1), n * lam21) hubeny_ = flatLocal_ # PYCHOK 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{pygeodesy.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_}. ''' a = abs(PI_2 - phi1) # co-latitude b = abs(PI_2 - phi2) # co-latitude if a < b: a, b = b, a if a < EPS0: a = _0_0 elif b > 0: b = b / a # /= chokes PyChecker c = b * cos(lam21) * _2_0 c = fsum_(_1_0, b**2, -fabs(c)) a *= sqrt0(c) return a def hartzell(pov, los=None, earth=_WGS84, **LatLon_and_kwds): '''Compute the intersection of a Line-Of-Sight from a Point-Of-View in space with the surface of the earth. @arg pov: Point-Of-View outside the earth (C{Cartesian}, L{Ecef9Tuple} or L{Vector3d}). @kwarg los: Line-Of-Sight, I{direction} to earth (L{Vector3d}) or C{None} to point to the earth' center. @kwarg earth: The earth model (L{Datum}, L{Ellipsoid}, L{Ellipsoid2}, L{a_f2Tuple} or C{scalar} radius in C{meter}). @kwarg LatLon_and_kwds: Optional C{LatLon} class for the intersection point plus C{LatLon} keyword arguments, include B{C{datum}} if different from B{C{earth}}. @return: The earth intersection (L{Vector3d}, C{Cartesian type} of B{C{pov}} or B{C{LatLon}}). @raise IntersectionError: Null B{C{pov}} or B{C{los}} vector, B{C{pov}} is inside the earth or B{C{los}} points outside the earth or points in an opposite direction. @raise TypeError: Invalid B{C{pov}}, B{C{los}} or B{C{earth}}. @see: U{Stephen Hartzell<https://StephenHartzell.medium.com/ satellite-line-of-sight-intersection-with-earth-d786b4a6a9b6>} and function L{pygeodesy.tyr3d} for B{C{los}}. ''' D = earth if isinstance(earth, Datum) else \ _spherical_datum(earth, name=hartzell.__name__) E = D.ellipsoid a2 = b2 = E.a2 # earth x, y, ... c2 = E.b2 # ... z half-axis squared q2 = E.b2_a2 # == c2 / a2 bc = E.a * E.b # == b * c p3 = _MODS.vector3d._otherV3d(pov=pov) u3 = _MODS.vector3d._otherV3d(los=los) if los else p3.negate() u3 = u3.unit() # unit vector, opposing signs x2, y2, z2 = p3.x2y2z2 # p3.times_(p3).xyz ux, vy, wz = u3.times_(p3).xyz u2, v2, w2 = u3.x2y2z2 # u3.times_(u3).xyz t = c2, c2, b2 # a2 factored out m = fdot(t, u2, v2, w2) if m < EPS0: # zero or near-null LOS vector raise IntersectionError(pov=pov, los=los, earth=earth, txt=_near_(_null_)) # a2 and b2 factored out, b2 == a2 and b2 / a2 == 1 r = fsum_(b2 * w2, c2 * v2, -v2 * z2, vy * wz * 2, c2 * u2, -u2 * z2, -w2 * x2, ux * wz * 2, -w2 * y2, -u2 * y2 * q2, -v2 * x2 * q2, ux * vy * 2 * q2, floats=True) if r > 0: r = bc * sqrt(r) elif r < 0: # LOS pointing away from or missing the earth t = _opposite_ if max(ux, vy, wz) > 0 else _outside_ raise IntersectionError(pov=pov, los=los, earth=earth, txt=t) n = fdot(t, ux, vy, wz) d = (n + r) / m # (n - r) / m for antipode if d > 0: # POV inside or LOS missing the earth t = _inside_ if min(x2 - a2, y2 - b2, z2 - c2) < EPS else _outside_ raise IntersectionError(pov=pov, los=los, earth=earth, txt=t) if fsum_(x2, y2, z2) < d**2: # d beyond earth center raise IntersectionError(pov=pov, los=los, earth=earth, txt=_too_(_distant_)) r = _xnamed(p3.minus(u3.times(d)), hartzell.__name__) if LatLon_and_kwds: c = _MODS.cartesianBase.CartesianBase(r, datum=D, name=r.name) r = c.toLatLon(**LatLon_and_kwds) return r 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{pygeodesy.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' surface 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, conventionally in C{meter} 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, otherwise ... 4) L{ellipsoidalExact.intersections2}, 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 points are the same instance, aka I{radical center}. @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, lat2, lon2, radius=R_M, adjust=True, wrap=wrap) < _100km: 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) m = _MODS.vector3d 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: m = _MODS.sphericalTrigonometry elif d.isEllipsoidal: try: if d.ellipsoid.geodesic: pass m = _MODS.ellipsoidalKarney except ImportError: m = _MODS.ellipsoidalExact 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: Functions L{isantipode_} and L{antipode}. ''' return True if (fabs(lat1 + lat2) <= eps and fabs(lon1 + lon2) <= eps) else \ _MODS.latlonBase._isequalTo(antipode(lat1, lon1), normal(lat2, lon2), eps=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: Functions L{isantipode} and L{antipode_}. ''' return True if (fabs(phi1 + phi2) <= eps and fabs(lam1 + lam2) <= eps) else \ _MODS.latlonBase._isequalTo_(antipode_(phi1, lam1), normal_(phi2, lam2), eps=eps) def isnormal(lat, lon, eps=0): '''Check whether B{C{lat}} I{and} B{C{lon}} are within the I{normal} range in C{degrees}. @arg lat: Latitude (C{degrees}). @arg lon: Longitude (C{degrees}). @kwarg eps: Optional tolerance below C{90} and C{180 degrees}. @return: C{True} if C{(abs(B{lat}) + B{eps}) <= 90} and C{(abs(B{lon}) + B{eps}) <= 180}, C{False} othwerwise. @see: Functions L{isnormal_} and L{normal}. ''' return (_90_0 - fabs(lat)) >= eps and (_180_0 - fabs(lon)) >= eps def isnormal_(phi, lam, eps=0): '''Check whether B{C{phi}} I{and} B{C{lam}} are within the I{normal} range in C{radians}. @arg phi: Latitude (C{radians}). @arg lam: Longitude (C{radians}). @kwarg eps: Optional tolerance below C{PI/2} and C{PI radians}. @return: C{True} if C{(abs(B{phi}) + B{eps}) <= PI/2} and C{(abs(B{lam}) + B{eps}) <= PI}, C{False} othwerwise. @see: Functions L{isnormal} and L{normal_}. ''' return (PI_2 - fabs(phi)) >= eps and (PI - fabs(lam)) >= eps def latlon2n_xyz(lat, lon, name=NN): '''Convert lat-, longitude to C{n-vector} (I{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 _2n_xyz(name, *sincos2d_(lat, lon)) def _normal2(a, b, n_2, n, n2): '''(INTERNAL) Helper for C{normal} and C{normal_}. ''' if fabs(b) > n: b = remainder(b, n2) r = remainder(a, n) if fabs(a) > n_2 else a if r != a: r = -r b -= n if b > 0 else -n return float0(r, b) def normal(lat, lon, name=NN): '''Normalize a lat- I{and} longitude pair in C{degrees}. @arg lat: Latitude (C{degrees}). @arg lon: Longitude (C{degrees}). @kwarg name: Optional name (C{str}). @return: L{LatLon2Tuple}C{(lat, lon)} with C{abs(lat) <= 90} and C{abs(lon) <= 180}. @see: Functions L{normal_} and L{isnormal}. ''' return LatLon2Tuple(*_normal2(lat, lon, _90_0, _180_0, _360_0), name=name) def normal_(phi, lam, name=NN): '''Normalize a lat- I{and} longitude pair in C{radians}. @arg phi: Latitude (C{radians}). @arg lam: Longitude (C{radians}). @kwarg name: Optional name (C{str}). @return: L{PhiLam2Tuple}C{(phi, lam)} with C{abs(phi) <= PI/2} and C{abs(lam) <= PI}. @see: Functions L{normal} and L{isnormal_}. ''' return PhiLam2Tuple(*_normal2(phi, lam, PI_2, PI, PI2), name=name) def _2n_xyz(name, sa, ca, sb, cb): '''(INTERNAL) Helper for C{latlon2n_xyz} and C{philam2n_xyz}. ''' # Kenneth Gade eqn 3, but using right-handed # vector x -> 0°E,0°N, y -> 90°E,0°N, z -> 90°N return Vector3Tuple(ca * cb, ca * sb, sa, 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}. ''' return LatLon2Tuple(atan2d(z, hypot(x, y)), atan2d(y, x), 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 opposing(bearing1, bearing2, margin=None): '''Compare the direction of two bearings given in C{degrees}. @arg bearing1: First bearing (compass C{degrees}). @arg bearing2: Second bearing (compass C{degrees}). @kwarg margin: Optional, interior angle bracket (C{degrees}), default C{90}. @return: C{True} if both bearings point in opposite, C{False} if in similar or C{None} if in perpendicular directions. @see: Function L{opposing_}. ''' m = Degrees_(margin=margin, low=EPS0, high=_90_0) if margin else _90_0 d = _umod_360(bearing2 - bearing1) # -20 % 360 == 340 return False if d < m or d > (_360_0 - m) else ( True if (_180_0 - m) < d < (_180_0 + m) else None) def opposing_(radians1, radians2, margin=None): '''Compare the direction of two bearings given in C{radians}. @arg radians1: First bearing (C{radians}). @arg radians2: Second bearing (C{radians}). @kwarg margin: Optional, interior angle bracket (C{radians}), default C{PI_2}. @return: C{True} if both bearings point in opposite, C{False} if in similar or C{None} if in perpendicular directions. @see: Function L{opposing}. ''' m = Radians_(margin=margin, low=EPS0, high=PI_2) if margin else PI_2 r = (radians2 - radians1) % PI2 # note -1 % PI2 == PI2 - 1 return False if r < m or r > (PI2 - m) else ( True if (PI - m) < r < (PI + m) else None) def philam2n_xyz(phi, lam, name=NN): '''Convert lat-, longitude to C{n-vector} (I{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! ''' return _2n_xyz(name, *sincos2_(phi, lam)) def _radical2(d, r1, r2): # in .ellipsoidalBaseDI, .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}}. @see: U{Circle-Circle Intersection <https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>}. ''' 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 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{pygeodesy.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 isnon0(c1) and isnon0(c2): E = _ellipsoidal(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 isnon0(u) and isnon0(h): r = atan(sqrt0(h / u)) * _2_0 # == acos(1 - 2 * h) sr, cr = sincos2(r) if isnon0(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{pygeodesy.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) 2016-2022 -- 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.