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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.