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# -*- coding: utf-8 -*- 

 

u'''Formulary of basic geodesy functions and approximations. 

''' 

# make sure int/int division yields float quotient, see .basics 

from __future__ import division 

 

from pygeodesy.basics import isnon0 as _non0 

from pygeodesy.datums import Datum, _ellipsoidal_datum, _mean_radius, \ 

_spherical_datum, _WGS84 

from pygeodesy.ellipsoids import Ellipsoid 

from pygeodesy.errors import _AssertionError, IntersectionError, \ 

LimitError, _limiterrors, _ValueError 

from pygeodesy.fmath import euclid, fdot, fsum_, hypot, hypot2, sqrt0, unstr 

from pygeodesy.interns import EPS, EPS0, EPS1, NN, PI, PI2, PI3, PI_2, R_M, \ 

_distant_, _inside_, _near_, _null_, _outside_, _too_, \ 

_0_0, _0_125, _0_25, _0_5, _1_0, _2_0, _4_0, \ 

_32_0, _90_0, _180_0, _360_0 

from pygeodesy.lazily import _ALL_LAZY 

from pygeodesy.named import _NamedTuple, _xnamed 

from pygeodesy.namedTuples import Bearing2Tuple, Distance4Tuple, \ 

LatLon2Tuple, PhiLam2Tuple, Vector3Tuple 

# from pygeodesy.streprs import unstr # from .fmath 

from pygeodesy.units import Degrees_, Distance, Distance_, Height, Lam_, Lat, \ 

Lon, Phi_, Radians, Radians_, Radius, Radius_, \ 

Scalar, _100km 

from pygeodesy.utily import acos1, atan2b, degrees2m, degrees90, degrees180, \ 

m2degrees, sincos2, sincos2_, tan_2, unroll180, \ 

unrollPI, wrap90, wrap180, wrapPI, wrapPI_2 

 

from math import atan, atan2, cos, degrees, radians, sin, sqrt # pow 

 

__all__ = _ALL_LAZY.formy 

__version__ = '21.10.24' 

 

_opposite_ = 'opposite' 

_ratio_ = 'ratio' 

_xline_ = 'xline' 

 

 

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 _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. 

''' 

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) 

 

# see <https://MathForum.org/library/drmath/view/55417.html> 

x = ca1 * sa2 - sa1 * ca2 * cdb 

y = sdb * ca2 

return (atan2(y, x) + r) % PI2 # wrapPI2 

 

 

def _bearingTo2(p1, p2, wrap=False): # for points.ispolar, sphericalTrigonometry.areaOf 

'''(INTERNAL) Compute initial and final bearing. 

''' 

try: # for LatLon_ and 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 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 _non0(c1) and _non0(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 _non0(sr_2) and _non0(cr_2): 

s = (sr + r) * ((s1 - s2) / sr_2)**2 

c = (sr - r) * ((s1 + s2) / cr_2)**2 

r += (c - s) * E.f * _0_125 

return r 

 

 

def cosineForsytheAndoyerLambert(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False): 

'''Compute the distance between two (ellipsoidal) points using the 

U{Forsythe-Andoyer-Lambert correction<https://www2.UNB.CA/gge/Pubs/TR77.pdf>} of 

the U{Law of Cosines<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} 

formula. 

 

@arg lat1: Start latitude (C{degrees}). 

@arg lon1: Start longitude (C{degrees}). 

@arg lat2: End latitude (C{degrees}). 

@arg lon2: End longitude (C{degrees}). 

@kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid}, 

L{Ellipsoid2} or L{a_f2Tuple}). 

@kwarg wrap: Wrap and L{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 _non0(c1) and _non0(c2): 

E = _ellipsoidal(datum, cosineForsytheAndoyerLambert_) 

if E.f: # ellipsoidal 

sr, cr, s2r, _ = sincos2_(r, r * _2_0) 

if _non0(sr) and abs(cr) < EPS1: 

s = (s1 + s2)**2 / (1 + cr) 

t = (s1 - s2)**2 / (1 - cr) 

x = s + t 

y = s - t 

 

s = 8 * r**2 / sr 

a = 64 * r + _2_0 * s * cr # 16 * r**2 / tan(r) 

d = 48 * sr + s # 8 * r**2 / tan(r) 

b = -2 * d 

e = 30 * s2r 

c = fsum_(30 * r, e * _0_5, s * cr) # 8 * r**2 / tan(r) 

 

t = fsum_( a * x, b * y, -c * x**2, d * x * y, e * y**2) 

r += fsum_(-r * x, 3 * y * sr, t * E.f / _32_0) * E.f * _0_25 

return r 

 

 

def cosineLaw(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

'''Compute the distance between two points using the 

U{spherical Law of Cosines 

<https://www.Movable-Type.co.UK/scripts/latlong.html#cosine-law>} 

formula. 

 

@arg lat1: Start latitude (C{degrees}). 

@arg lon1: Start longitude (C{degrees}). 

@arg lat2: End latitude (C{degrees}). 

@arg lon2: End longitude (C{degrees}). 

@kwarg radius: Mean earth radius, ellipsoid or datum (C{meter}, 

L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}). 

@kwarg wrap: Wrap and L{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 _limiterrors \ 

and max(abs(d_lat), abs(d_lon)) > limit > 0: 

t = unstr(equirectangular_.__name__, 

lat1, lon1, lat2, lon2, limit=limit) 

raise LimitError('delta exceeds limit', txt=t) 

 

if adjust: # scale delta lon 

d_lon *= _scale_deg(lat1, lat2) 

 

d2 = hypot2(d_lat, d_lon) # degrees squared! 

return Distance4Tuple(d2, d_lat, d_lon, ulon2 - lon2) 

 

 

def euclidean(lat1, lon1, lat2, lon2, radius=R_M, adjust=True, wrap=False): 

'''Approximate the C{Euclidean} distance between two (spherical) points. 

 

@arg lat1: Start latitude (C{degrees}). 

@arg lon1: Start longitude (C{degrees}). 

@arg lat2: End latitude (C{degrees}). 

@arg lon2: End longitude (C{degrees}). 

@kwarg radius: Mean earth radius, ellipsoid or datum (C{meter}, 

L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}). 

@kwarg adjust: Adjust the longitudinal delta by the cosine of the 

mean latitude (C{bool}). 

@kwarg wrap: Wrap and L{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 ({radians}). 

 

@raise UnitError: Invalid B{C{A}}, B{C{b}} or B{C{c}}. 

 

@see: Function L{excessGirard}, L{excessLHuilier}, U{Spherical 

trigonometry<https://WikiPedia.org/wiki/Spherical_trigonometry>}. 

''' 

sA, cA, sb, cb, sc, cc = sincos2_(Radians_(A=A), Radians_(b=b) * _0_5, 

Radians_(c=c) * _0_5) 

return atan2(sA * sb * sc, cb * cc + cA * sb * sc) * _2_0 

 

 

def excessGirard(A, B, C): 

'''Compute the I{spherical excess} C{E} of a (spherical) triangle using 

U{Girard's<https://MathWorld.Wolfram.com/GirardsSphericalExcessFormula.html>} 

formula. 

 

@arg A: First interior triangle angle (C{radians}). 

@arg B: Second interior triangle angle (C{radians}). 

@arg C: Third interior triangle angle (C{radians}). 

 

@return: Spherical excess ({radians}). 

 

@raise UnitError: Invalid B{C{A}}, B{C{B}} or B{C{C}}. 

 

@see: Function L{excessLHuilier}, U{Spherical trigonometry 

<https://WikiPedia.org/wiki/Spherical_trigonometry>}. 

''' 

return Radians(Girard=fsum_(Radians_(A=A), 

Radians_(B=B), 

Radians_(C=C), -PI)) 

 

 

def excessLHuilier(a, b, c): 

'''Compute the I{spherical excess} C{E} of a (spherical) triangle using 

U{L'Huilier's<https://MathWorld.Wolfram.com/LHuiliersTheorem.html>} 

Theorem. 

 

@arg a: First triangle side (C{radians}). 

@arg b: Second triangle side (C{radians}). 

@arg c: Third triangle side (C{radians}). 

 

@return: Spherical excess ({radians}). 

 

@raise UnitError: Invalid B{C{a}}, B{C{b}} or B{C{c}}. 

 

@see: Function L{excessGirard}, U{Spherical trigonometry 

<https://WikiPedia.org/wiki/Spherical_trigonometry>}. 

''' 

a = Radians_(a=a) 

b = Radians_(b=b) 

c = Radians_(c=c) 

 

s = fsum_(a, b, c) * _0_5 

r = tan_2(s) * tan_2(s - a) * tan_2(s - b) * tan_2(s - c) 

return Radians(LHuilier=atan(sqrt(r)) * _4_0) 

 

 

def excessKarney(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

'''Compute the surface area of a (spherical) quadrilateral bounded by 

a segment of a great circle, two meridians and the equator using U{Karney's 

<http://OSGeo-org.1560.x6.Nabble.com/Area-of-a-spherical-polygon-td3841625.html>} 

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 

<http://OSGeo-org.1560.x6.Nabble.com/Area-of-a-spherical-polygon-td3841625.html>} 

method. 

 

@arg phi2: End latitude (C{radians}). 

@arg phi1: Start latitude (C{radians}). 

@arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

 

@return: Spherical excess, I{signed} (C{radians}). 

 

@raise ValueError: Semi-circular longitudinal delta B{C{lam21}}. 

 

@see: Function L{excessKarney}, U{Area of a spherical polygon 

<http://OSGeo-org.1560.x6.Nabble.com/Area-of-a-spherical-polygon-td3841625.html>}. 

''' 

# from: Veness <https://www.Movable-Type.co.UK/scripts/latlong.html> 

# Area method due to Karney: for each edge of the polygon, 

# 

# tan(Δλ/2) · (tan(φ1/2) + tan(φ2/2)) 

# tan(E/2) = ------------------------------------ 

# 1 + tan(φ1/2) · tan(φ2/2) 

# 

# where E is the spherical excess of the trapezium obtained by 

# extending the edge to the equator-circle vector for each edge. 

t2 = tan_2(phi2) 

t1 = tan_2(phi1) 

t = tan_2(lam21, lam21=None) 

return Radians(Karney=atan2(t * (t1 + t2), 

_1_0 + (t1 * t2)) * _2_0) 

 

 

def excessQuad(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

'''Compute the surface area of a (spherical) quadrilateral bounded by a segment 

of a great circle, two meridians and the equator. 

 

@arg lat1: Start latitude (C{degrees}). 

@arg lon1: Start longitude (C{degrees}). 

@arg lat2: End latitude (C{degrees}). 

@arg lon2: End longitude (C{degrees}). 

@kwarg radius: Mean earth radius, ellipsoid or datum (C{meter}, 

L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}) or C{None}. 

@kwarg wrap: Wrap and L{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_}/L{hubeny_}, L{cosineLaw}, 

L{flatPolar}, L{cosineAndoyerLambert}, L{cosineForsytheAndoyerLambert}, 

L{equirectangular}, L{euclidean}, L{haversine}, L{thomas}, L{vincentys}, 

method L{Ellipsoid.distance2} and U{local, flat earth approximation 

<https://www.EdWilliams.org/avform.htm#flat>}. 

''' 

d, _ = unroll180(lon1, lon2, wrap=wrap) 

return flatLocal_(Phi_(lat2=lat2), 

Phi_(lat1=lat1), radians(d), datum=datum) 

 

 

hubeny = flatLocal # for Karl Hubeny 

 

 

def flatLocal_(phi2, phi1, lam21, datum=_WGS84): 

'''Compute the I{angular} distance between two (ellipsoidal) points using 

the U{ellipsoidal Earth to plane projection<https://WikiPedia.org/ 

wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane>} 

aka U{Hubeny<https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula. 

 

@arg phi2: End latitude (C{radians}). 

@arg phi1: Start latitude (C{radians}). 

@arg lam21: Longitudinal delta, M{end-start} (C{radians}). 

@kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid}, 

L{Ellipsoid2} or L{a_f2Tuple}). 

 

@return: Angular distance (C{radians}). 

 

@raise TypeError: Invalid B{C{datum}}. 

 

@note: The meridional and prime_vertical radii of curvature 

are taken and scaled I{at the mean of both latitude}. 

 

@see: Functions L{flatLocal}/L{hubeny}, L{cosineAndoyerLambert_}, 

L{cosineForsytheAndoyerLambert_}, L{cosineLaw_}, 

L{flatPolar_}, L{equirectangular_}, L{euclidean_}, 

L{haversine_}, L{thomas_} and L{vincentys_} and U{local, flat 

earth approximation <https://www.EdWilliams.org/avform.htm#flat>}. 

''' 

E = _ellipsoidal(datum, flatLocal_) 

m, n = E.roc2_((phi2 + phi1) * _0_5, scaled=True) 

return hypot(m * (phi2 - phi1), n * lam21) 

 

 

hubeny_ = flatLocal_ # for Karl Hubeny 

 

 

def flatPolar(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

'''Compute the distance between two (spherical) points using 

the U{polar coordinate flat-Earth <https://WikiPedia.org/wiki/ 

Geographical_distance#Polar_coordinate_flat-Earth_formula>} 

formula. 

 

@arg lat1: Start latitude (C{degrees}). 

@arg lon1: Start longitude (C{degrees}). 

@arg lat2: End latitude (C{degrees}). 

@arg lon2: End longitude (C{degrees}). 

@kwarg radius: Mean earth radius, ellipsoid or datum (C{meter}, 

L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}). 

@kwarg wrap: Wrap and L{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_}. 

''' 

a1 = PI_2 - phi1 # co-latitude 

a2 = PI_2 - phi2 # co-latitude 

ab = _2_0 * a1 * a2 * cos(lam21) 

r2 = fsum_(a1**2, a2**2, -abs(ab)) 

return sqrt0(r2) 

 

 

def hartzell(pov, los=None, earth=_WGS84, LatLon=None, **LatLon_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: Class to convert insection point (C{LatLon}, L{LatLon_}). 

@kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword 

arguments, ignored if C{B{LatLon} is None}. 

 

@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}}. 

''' 

from pygeodesy.vector3d import _otherV3d 

 

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 = _otherV3d(pov=pov) 

u3 = _otherV3d(los=los) if los else p3.negate() 

u3 = u3.unit() # unit vector, opposing signs 

 

x2, y2, z2 = p3.times_(p3).xyz # == p3.x2y2z2 

ux, vy, wz = u3.times_(p3).xyz 

u2, v2, w2 = u3.times_(u3).xyz # == u3.x2y2z2 

 

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) 

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 is not None: 

from pygeodesy.cartesianBase import CartesianBase as _CB 

# earth datum is overidden in LatLon if datum is specified in LatLon_kwds 

r = _CB(r, datum=D, name=r.name).toLatLon(LatLon=LatLon, **LatLon_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, 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{ellipsoidalExact.intersections2} otherwise provided B{C{datum}} 

is ellipsoidal. 

 

@arg lat1: Latitude of the first circle center (C{degrees}). 

@arg lon1: Longitude of the first circle center (C{degrees}). 

@arg radius1: Radius of the first circle (C{meter}, conventionally). 

@arg lat2: Latitude of the second circle center (C{degrees}). 

@arg lon2: Longitude of the second circle center (C{degrees}). 

@arg radius2: Radius of the second circle (C{meter}, same units as B{C{radius1}}). 

@kwarg datum: Optional ellipsoidal or spherical datum (L{Datum}, 

L{Ellipsoid}, L{Ellipsoid2}, L{a_f2Tuple} or 

C{scalar} earth radius in C{meter}, same units as 

B{C{radius1}} and B{C{radius2}}) or C{None}. 

@kwarg wrap: Wrap and unroll longitudes (C{bool}). 

 

@return: 2-Tuple of the intersection points, each a 

L{LatLon2Tuple}C{(lat, lon)}. For abutting circles, 

the 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, lat1, lon2, radius=R_M, 

adjust=True, wrap=wrap) < _100km: 

import pygeodesy.vector3d as m 

 

def _V2T(x, y, _, **unused): # _ == z unused 

return LatLon2Tuple(y, x, name=intersections2.__name__) 

 

r1 = m2degrees(Radius_(radius1=radius1), radius=R_M, lat=lat1) 

r2 = m2degrees(Radius_(radius2=radius2), radius=R_M, lat=lat2) 

 

_, lon2 = unroll180(lon1, lon2, wrap=wrap) 

t = m.intersections2(m.Vector3d(lon1, lat1, 0), r1, 

m.Vector3d(lon2, lat2, 0), r2, sphere=False, 

Vector=_V2T) 

 

else: 

def _LL2T(lat, lon, **unused): 

return LatLon2Tuple(lat, lon, name=intersections2.__name__) 

 

d = _spherical_datum(datum, name=intersections2.__name__) 

if d.isSpherical: 

import pygeodesy.sphericalTrigonometry as m 

elif d.isEllipsoidal: 

try: 

if d.ellipsoid.geodesic: 

pass 

import pygeodesy.ellipsoidalKarney as m 

except ImportError: 

import pygeodesy.ellipsoidalExact as m 

else: 

raise _AssertionError(datum=d) 

 

t = m.intersections2(m.LatLon(lat1, lon1, datum=d), radius1, 

m.LatLon(lat2, lon2, datum=d), radius2, 

LatLon=_LL2T, height=0, wrap=wrap) 

return t 

 

 

def isantipode(lat1, lon1, lat2, lon2, eps=EPS): 

'''Check whether two points are antipodal, on diametrically 

opposite sides of the earth. 

 

@arg lat1: Latitude of one point (C{degrees}). 

@arg lon1: Longitude of one point (C{degrees}). 

@arg lat2: Latitude of the other point (C{degrees}). 

@arg lon2: Longitude of the other point (C{degrees}). 

@kwarg eps: Tolerance for near-equality (C{degrees}). 

 

@return: C{True} if points are antipodal within the 

B{C{eps}} tolerance, C{False} otherwise. 

 

@see: U{Geosphere<https://CRAN.R-Project.org/web/packages/geosphere/geosphere.pdf>}. 

''' 

return abs(wrap90(lat1) + wrap90(lat2)) < eps and \ 

abs(abs(wrap180(lon1) - wrap180(lon2)) % _360_0 - _180_0) < eps 

 

 

def isantipode_(phi1, lam1, phi2, lam2, eps=EPS): 

'''Check whether two points are antipodal, on diametrically 

opposite sides of the earth. 

 

@arg phi1: Latitude of one point (C{radians}). 

@arg lam1: Longitude of one point (C{radians}). 

@arg phi2: Latitude of the other point (C{radians}). 

@arg lam2: Longitude of the other point (C{radians}). 

@kwarg eps: Tolerance for near-equality (C{radians}). 

 

@return: C{True} if points are antipodal within the 

B{C{eps}} tolerance, C{False} otherwise. 

 

@see: U{Geosphere<https://CRAN.R-Project.org/web/packages/geosphere/geosphere.pdf>}. 

''' 

return abs(wrapPI_2(phi1) + wrapPI_2(phi2)) < eps and abs( 

abs(wrapPI(lam1) - wrapPI(lam2)) % PI2 - PI) < eps 

 

 

def latlon2n_xyz(lat, lon, name=NN): 

'''Convert lat-, longitude to C{n-vector} (normal to the 

earth's surface) X, Y and Z components. 

 

@arg lat: Latitude (C{degrees}). 

@arg lon: Longitude (C{degrees}). 

@kwarg name: Optional name (C{str}). 

 

@return: A L{Vector3Tuple}C{(x, y, z)}. 

 

@see: Function L{philam2n_xyz}. 

 

@note: These are C{n-vector} x, y and z components, 

I{NOT} geocentric ECEF x, y and z coordinates! 

''' 

return philam2n_xyz(radians(lat), radians(lon), name=name) 

 

 

def n_xyz2latlon(x, y, z, name=NN): 

'''Convert C{n-vector} components to lat- and longitude in C{degrees}. 

 

@arg x: X component (C{scalar}). 

@arg y: Y component (C{scalar}). 

@arg z: Z component (C{scalar}). 

@kwarg name: Optional name (C{str}). 

 

@return: A L{LatLon2Tuple}C{(lat, lon)}. 

 

@see: Function L{n_xyz2philam}. 

''' 

a, b = n_xyz2philam(x, y, z) # PYCHOK PhiLam2Tuple 

return LatLon2Tuple(degrees90(a), degrees180(b), name=name) 

 

 

def n_xyz2philam(x, y, z, name=NN): 

'''Convert C{n-vector} components to lat- and longitude in C{radians}. 

 

@arg x: X component (C{scalar}). 

@arg y: Y component (C{scalar}). 

@arg z: Z component (C{scalar}). 

@kwarg name: Optional name (C{str}). 

 

@return: A L{PhiLam2Tuple}C{(phi, lam)}. 

 

@see: Function L{n_xyz2latlon}. 

''' 

return PhiLam2Tuple(atan2(z, hypot(x, y)), atan2(y, x), name=name) 

 

 

def 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 = (bearing2 - bearing1) % _360_0 # note -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} (normal to the 

earth's surface) X, Y and Z components. 

 

@arg phi: Latitude (C{radians}). 

@arg lam: Longitude (C{radians}). 

@kwarg name: Optional name (C{str}). 

 

@return: A L{Vector3Tuple}C{(x, y, z)}. 

 

@see: Function L{latlon2n_xyz}. 

 

@note: These are C{n-vector} x, y and z components, 

I{NOT} geocentric ECEF x, y and z coordinates! 

''' 

# Kenneth Gade eqn 3, but using right-handed 

# vector x -> 0°E,0°N, y -> 90°E,0°N, z -> 90°N 

sa, ca, sb, cb = sincos2_(phi, lam) 

return Vector3Tuple(ca * cb, ca * sb, sa, name=name) 

 

 

def _radical2(d, r1, r2): # in .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_0 else _0_0 

 

 

def _scale_rad(phi1, phi2): # radians, by .frechet, .hausdorff, .heights 

# scale factor cos(mean of phis) for delta lam 

m = abs(phi1 + phi2) * _0_5 

return cos(m) if m < PI_2 else _0_0 

 

 

def _sincosa6(phi2, phi1, lam21): 

'''(INTERNAL) C{sin}es, C{cos}ines and C{acos}ine. 

''' 

s2, c2, s1, c1, _, c21 = sincos2_(phi2, phi1, lam21) 

return s2, c2, s1, c1, acos1(s1 * s2 + c1 * c2 * c21), c21 

 

 

def thomas(lat1, lon1, lat2, lon2, datum=_WGS84, wrap=False): 

'''Compute the distance between two (ellipsoidal) points using 

U{Thomas'<https://apps.DTIC.mil/dtic/tr/fulltext/u2/703541.pdf>} 

formula. 

 

@arg lat1: Start latitude (C{degrees}). 

@arg lon1: Start longitude (C{degrees}). 

@arg lat2: End latitude (C{degrees}). 

@arg lon2: End longitude (C{degrees}). 

@kwarg datum: Datum or ellipsoid to use (L{Datum}, L{Ellipsoid}, 

L{Ellipsoid2} or L{a_f2Tuple}). 

@kwarg wrap: Wrap and L{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 _non0(c1) and _non0(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 _non0(u) and _non0(h): 

r = atan(sqrt0(h / u)) * _2_0 # == acos(1 - 2 * h) 

sr, cr = sincos2(r) 

if _non0(sr): 

u = 2 * (sj * ck)**2 / u 

h = 2 * (sk * cj)**2 / h 

x = u + h 

y = u - h 

 

s = r / sr 

e = 4 * s**2 

d = 2 * cr 

a = e * d 

b = 2 * r 

c = s - (a - d) * _0_5 

f = E.f * _0_25 

 

t = fsum_(a * x, -b * y, c * x**2, -d * y**2, e * x * y) 

r -= fsum_(s * x, -y, -t * f * _0_25) * f * sr 

return r 

 

 

def vincentys(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

'''Compute the distance between two (spherical) points using 

U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>} 

spherical formula. 

 

@arg lat1: Start latitude (C{degrees}). 

@arg lon1: Start longitude (C{degrees}). 

@arg lat2: End latitude (C{degrees}). 

@arg lon2: End longitude (C{degrees}). 

@kwarg radius: Mean earth radius, ellipsoid or datum (C{meter}, 

L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}). 

@kwarg wrap: Wrap and L{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) 

 

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