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# -*- coding: utf-8 -*- 

 

u'''(INTERNAL) Base classes for elliposiodal and spherical C{Nvector}s. 

 

Classes L{LatLonNvectorBase} for C{n-vectorial} ellipsoidal and spherical 

C{LatLon}s, class L{NvectorBase} for C{Cartesian}s and function L{sumOf}. 

 

Pure Python implementation of C{n-vector}-based geodesy tools for 

ellipsoidal earth models, transcoded from JavaScript originals by 

I{(C) Chris Veness 2005-2016} and published under the same MIT Licence**, 

see U{Vector-based geodesy 

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

''' 

 

from pygeodesy.basics import len2, map1 

from pygeodesy.datums import _spherical_datum 

from pygeodesy.errors import IntersectionError, _ValueError, \ 

_xkwds, _xkwds_pop 

from pygeodesy.fmath import fidw, fsum, fsum_, hypot_ 

from pygeodesy.formy import n_xyz2latlon, n_xyz2philam 

from pygeodesy.interns import EPS, EPS1, EPS_2, MISSING, NN, R_M, \ 

_bearing_, _coincident_, _COMMASPACE_, \ 

_distance_, _intersection_, _no_, _NorthPole_, \ 

_points_, _pole_, _SPACE_, _SouthPole_, \ 

_1_, _2_, _3_, _2_0 

from pygeodesy.latlonBase import LatLonBase 

from pygeodesy.lazily import _ALL_DOCS 

from pygeodesy.named import notImplemented, _xother3 

from pygeodesy.namedTuples import Trilaterate5Tuple, Vector3Tuple, \ 

Vector4Tuple 

from pygeodesy.props import deprecated_method, property_doc_, Property_RO 

from pygeodesy.streprs import Fmt, hstr, unstr, _xattrs 

from pygeodesy.units import Bearing, Height, Radius_, Scalar 

from pygeodesy.utily import sincos2d 

from pygeodesy.vector3d import Vector3d, VectorError, \ 

sumOf as _sumOf, _xyzhdn6 

 

from math import fabs, sqrt # atan2, cos, sin 

 

__all__ = (_NorthPole_, _SouthPole_) # constants 

__version__ = '21.06.25' 

 

 

class NvectorBase(Vector3d): # XXX kept private 

'''Base class for ellipsoidal and spherical C{Nvector}s. 

''' 

_datum = None # L{Datum}, overriden 

_h = 0 # height (C{meter}) 

_H = NN # height prefix (C{str}), '↑' in JS version 

 

def __init__(self, x, y=None, z=None, h=0, ll=None, datum=None, name=NN): 

'''New n-vector normal to the earth's surface. 

 

@arg x: An C{Nvector}, L{Vector3Tuple}, L{Vector4Tuple} or 

the C{X} coordinate (C{scalar}). 

@arg y: The C{Y} coordinate (C{scalar}) if B{C{x}} C{scalar}. 

@arg z: The C{Z} coordinate (C{scalar}) if B{C{x}} C{scalar}. 

@kwarg h: Optional height above surface (C{meter}). 

@kwarg ll: Optional, original latlon (C{LatLon}). 

@kwarg datum: Optional, I{pass-thru} datum (L{Datum}). 

@kwarg name: Optional name (C{str}). 

 

@raise TypeError: Non-scalar B{C{x}}, B{C{y}} or B{C{z}} 

coordinate or B{C{x}} not an C{Nvector}, 

L{Vector3Tuple} or L{Vector4Tuple} or 

invalid B{C{datum}}. 

 

@example: 

 

>>> from pygeodesy.sphericalNvector import Nvector 

>>> v = Nvector(0.5, 0.5, 0.7071, 1) 

>>> v.toLatLon() # 45.0°N, 045.0°E, +1.00m 

''' 

x, y, z, h, d, n = _xyzhdn6(x, y, z, h, datum, ll) 

Vector3d.__init__(self, x, y, z, ll=ll, name=name or n) 

if h: 

self.h = h 

if d: 

self._datum = _spherical_datum(d, name=self.name) # pass-thru 

 

@Property_RO 

def datum(self): 

'''Get the I{pass-thru} datum (C{Datum}) or C{None}. 

''' 

return self._datum 

 

@Property_RO 

def Ecef(self): 

'''Get the ECEF I{class} (L{EcefKarney}), I{lazily}. 

''' 

from pygeodesy.ecef import EcefKarney 

return EcefKarney # default 

 

@property_doc_(''' the height above surface (C{meter}).''') 

def h(self): 

'''Get the height above surface (C{meter}). 

''' 

return self._h 

 

@h.setter # PYCHOK setter! 

def h(self, h): 

'''Set the height above surface. 

 

@arg h: New height (C{meter}). 

 

@raise TypeError: If B{C{h}} invalid. 

 

@raise VectorError: If B{C{h}} invalid. 

''' 

h = Height(h=h, Error=VectorError) 

self._update(h != self._h) 

self._h = h 

 

@property_doc_(''' the height prefix (C{str}).''') 

def H(self): 

'''Get the height prefix (C{str}). 

''' 

return self._H 

 

@H.setter # PYCHOK setter! 

def H(self, H): 

'''Set the height prefix. 

 

@arg H: New height prefix (C{str}). 

''' 

self._H = str(H) if H else NN 

 

def hStr(self, prec=-2, m=NN): 

'''Return a string for the height B{C{h}}. 

 

@kwarg prec: Optional number of decimals, unstripped (C{int}). 

@kwarg m: Optional unit of the height (C{str}). 

 

@see: Function L{hstr}. 

''' 

return NN(self.H, hstr(self.h, prec=prec, m=m)) 

 

@Property_RO 

def isEllipsoidal(self): 

'''Check whether this n-vector is ellipsoidal (C{bool} or C{None} if unknown). 

''' 

return self.datum.isEllipsoidal if self.datum else None 

 

@Property_RO 

def isSpherical(self): 

'''Check whether this n-vector is spherical (C{bool} or C{None} if unknown). 

''' 

return self.datum.isSpherical if self.datum else None 

 

@Property_RO 

def lam(self): 

'''Get the (geodetic) longitude in C{radians} (C{float}). 

''' 

return self.philam.lam 

 

@Property_RO 

def lat(self): 

'''Get the (geodetic) latitude in C{degrees} (C{float}). 

''' 

return self.latlon.lat 

 

@Property_RO 

def latlon(self): 

'''Get the (geodetic) lat-, longitude in C{degrees} (L{LatLon2Tuple}C{(lat, lon)}). 

''' 

return n_xyz2latlon(self.x, self.y, self.z, name=self.name) 

 

@Property_RO 

def latlonheight(self): 

'''Get the (geodetic) lat-, longitude in C{degrees} and height (L{LatLon3Tuple}C{(lat, lon, height)}). 

''' 

return self.latlon.to3Tuple(self.h) 

 

@Property_RO 

def latlonheightdatum(self): 

'''Get the lat-, longitude in C{degrees} with height and datum (L{LatLon4Tuple}C{(lat, lon, height, datum)}). 

''' 

return self.latlonheight.to4Tuple(self.datum) 

 

@Property_RO 

def lon(self): 

'''Get the (geodetic) longitude in C{degrees} (C{float}). 

''' 

return self.latlon.lon 

 

@Property_RO 

def phi(self): 

'''Get the (geodetic) latitude in C{radians} (C{float}). 

''' 

return self.philam.phi 

 

@Property_RO 

def philam(self): 

'''Get the (geodetic) lat-, longitude in C{radians} (L{PhiLam2Tuple}C{(phi, lam)}). 

''' 

return n_xyz2philam(self.x, self.y, self.z, name=self.name) 

 

@Property_RO 

def philamheight(self): 

'''Get the (geodetic) lat-, longitude in C{radians} and height (L{PhiLam3Tuple}C{(phi, lam, height)}). 

''' 

return self.philam.to3Tuple(self.h) 

 

@Property_RO 

def philamheightdatum(self): 

'''Get the lat-, longitude in C{radians} with height and datum (L{PhiLam4Tuple}C{(phi, lam, height, datum)}). 

''' 

return self.philamheight.to4Tuple(self.datum) 

 

@deprecated_method 

def to2ab(self): # PYCHOK no cover 

'''DEPRECATED, use property L{philam}. 

 

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

''' 

return self.philam 

 

@deprecated_method 

def to3abh(self, height=None): # PYCHOK no cover 

'''DEPRECATED, use method L{philamheight} or C{philam.to3Tuple(B{height})}. 

 

@kwarg height: Optional height, overriding this 

n-vector's height (C{meter}). 

 

@return: A L{PhiLam3Tuple}C{(phi, lam, height)}. 

 

@raise ValueError: Invalid B{C{height}}. 

''' 

return self.philamheight if height in (None, self.h) else \ 

self.philam.to3Tuple(height) 

 

def toCartesian(self, h=None, Cartesian=None, datum=None, **Cartesian_kwds): 

'''Convert this n-vector to C{Nvector}-based cartesian (ECEF) coordinates. 

 

@kwarg h: Optional height, overriding this n-vector's height (C{meter}). 

@kwarg Cartesian: Optional class to return the (ECEF) coordinates 

(C{Cartesian}). 

@kwarg datum: Optional datum (C{Datum}), overriding this datum. 

@kwarg Cartesian_kwds: Optional, additional B{C{Cartesian}} 

keyword arguments, ignored if 

C{B{Cartesian}=None}. 

 

@return: The cartesian (ECEF) coordinates (B{C{Cartesian}}) or 

if C{B{Cartesian}=None}, an L{Ecef9Tuple}C{(x, y, z, 

lat, lon, height, C, M, datum)} with C{C} and C{M} if 

available. 

 

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

 

@raise ValueError: Invalid B{C{h}}. 

 

@example: 

 

>>> v = Nvector(0.5, 0.5, 0.7071) 

>>> c = v.toCartesian() # [3194434, 3194434, 4487327] 

>>> p = c.toLatLon() # 45.0°N, 45.0°E 

''' 

d = _spherical_datum(datum or self.datum, name=self.name) 

E = d.ellipsoid 

h = self.h if h is None else Height(h=h) 

 

x, y, z = self.x, self.y, self.z 

# Kenneth Gade eqn (22) 

n = E.b / hypot_(x * E.a_b, y * E.a_b, z) 

r = h + n * E.a2_b2 

 

x *= r 

y *= r 

z *= n + h 

 

if Cartesian is None: 

r = self.Ecef(d).reverse(x, y, z, M=True) 

else: 

kwds = _xkwds(Cartesian_kwds, datum=d) # h=0 

r = Cartesian(x, y, z, **kwds) 

return self._xnamed(r) 

 

@deprecated_method 

def to2ll(self): # PYCHOK no cover 

'''DEPRECATED, use property L{latlon}. 

 

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

''' 

return self.latlon 

 

@deprecated_method 

def to3llh(self, height=None): # PYCHOK no cover 

'''DEPRECATED, use property C{latlonheight} or C{latlon.to3Tuple}C{)}B{C{height}}C{)}. 

 

@kwarg height: Optional height, overriding this 

n-vector's height (C{meter}). 

 

@return: A L{LatLon3Tuple}C{(lat, lon, height)}. 

 

@raise ValueError: Invalid B{C{height}}. 

''' 

return self.latlonheight if height in (None, self.h) else \ 

self.latlon.to3Tuple(height) 

 

def toLatLon(self, height=None, LatLon=None, datum=None, **LatLon_kwds): 

'''Convert this n-vector to an C{Nvector}-based geodetic point. 

 

@kwarg height: Optional height, overriding this n-vector's 

height (C{meter}). 

@kwarg LatLon: Optional class to return the geodetic point 

(C{LatLon}) or C{None}. 

@kwarg datum: Optional, spherical datum (C{Datum}). 

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

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

 

@return: The geodetic point (C{LatLon}) or if C{B{LatLon}=None}, 

an L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, 

datum)} with C{C} and C{M} if available. 

 

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

 

@raise ValueError: Invalid B{C{height}}. 

 

@example: 

 

>>> v = Nvector(0.5, 0.5, 0.7071) 

>>> p = v.toLatLon() # 45.0°N, 45.0°E 

''' 

d = _spherical_datum(datum or self.datum, name=self.name) 

h = self.h if height is None else Height(height) 

# use self.Cartesian(Cartesian=None) for better accuracy of the height 

# than self.Ecef(d).forward(self.lat, self.lon, height=h, M=True) 

if LatLon is None: 

r = self.toCartesian(h=h, Cartesian=None, datum=d) 

else: 

kwds = _xkwds(LatLon_kwds, height=h, datum=d) 

r = self._xnamed(LatLon(self.lat, self.lon, **kwds)) 

return r 

 

def toStr(self, prec=5, fmt=Fmt.PAREN, sep=_COMMASPACE_): # PYCHOK expected 

'''Return a string representation of this n-vector. 

 

Height component is only included if non-zero. 

 

@kwarg prec: Optional number of decimals, unstripped (C{int}). 

@kwarg fmt: Optional enclosing backets format (C{str}). 

@kwarg sep: Optional separator between components (C{str}). 

 

@return: Comma-separated C{"(x, y, z [, h])"} enclosed in 

B{C{fmt}} brackets (C{str}). 

 

@example: 

 

>>> Nvector(0.5, 0.5, 0.7071).toStr() # (0.5, 0.5, 0.7071) 

>>> Nvector(0.5, 0.5, 0.7071, 1).toStr(-3) # (0.500, 0.500, 0.707, +1.00) 

''' 

t = Vector3d.toStr(self, prec=prec, fmt=NN, sep=sep) 

if self.h: 

t = sep.join((t, self.hStr())) 

return (fmt % (t,)) if fmt else t 

 

def toVector3d(self, norm=True): 

'''Convert this n-vector to a 3-D vector, I{ignoring 

the height}. 

 

@kwarg norm: Normalize the 3-D vector (C{bool}). 

 

@return: The (normalized) vector (L{Vector3d}). 

''' 

v = Vector3d.unit(self) if norm else self 

return Vector3d(v.x, v.y, v.z, name=self.name) 

 

@deprecated_method 

def to4xyzh(self, h=None): # PYCHOK no cover 

'''DEPRECATED, use property L{xyzh} or C{xyz.to4Tuple(B{h})}. 

''' 

return self.xyzh if h in (None, self.h) else Vector4Tuple( 

self.x, self.y, self.z, h, name=self.name) 

 

def unit(self, ll=None): 

'''Normalize this n-vector to unit length. 

 

@kwarg ll: Optional, original latlon (C{LatLon}). 

 

@return: Normalized vector (C{Nvector}). 

''' 

return _xattrs(Vector3d.unit(self, ll=ll), '_h') 

 

@Property_RO 

def xyzh(self): 

'''Get this n-vector's components (L{Vector4Tuple}C{(x, y, z, h)}) 

''' 

return self.xyz.to4Tuple(self.h) 

 

 

NorthPole = NvectorBase(0, 0, +1, name=_NorthPole_) # North pole (C{Nvector}) 

SouthPole = NvectorBase(0, 0, -1, name=_SouthPole_) # South pole (C{Nvector}) 

 

 

class _N_vector_(NvectorBase): 

'''(INTERNAL) Minimal, low-overhead C{n-vector}. 

''' 

def __init__(self, x, y, z, h=0, name=NN): 

self._x, self._y, self._z = x, y, z 

if h: 

self._h = h 

if name: 

self.name = name 

 

 

class LatLonNvectorBase(LatLonBase): 

'''(INTERNAL) Base class for n-vector-based ellipsoidal 

and spherical C{LatLon} classes. 

''' 

 

def _update(self, updated, *attrs, **kwds): # PYCHOK _Nv=None 

'''(INTERNAL) Zap cached attributes if updated. 

 

@see: C{ellipsoidalNvector.LatLon} and C{sphericalNvector.LatLon} 

for the special case of B{C{_Nv}}. 

''' 

if updated: 

_Nv = _xkwds_pop(kwds, _Nv=None) 

if _Nv is not None: 

if _Nv._fromll is not None: 

_Nv._fromll = None 

self._Nv = None 

LatLonBase._update(self, updated, *attrs) 

 

# def distanceTo(self, other, **kwds): # PYCHOK no cover 

# '''(INTERNAL) I{Must be overloaded}, see function C{notOverloaded}. 

# ''' 

# from pygeodesy.named import notOverloaded 

# notOverloaded(self, other, **kwds) 

 

def intersections2(self, radius1, other, radius2, **kwds): # PYCHOK expected 

'''B{Not implemented}, throws a C{NotImplementedError} always. 

''' 

notImplemented(self, radius1, other, radius2, **kwds) 

 

def others(self, *other, **name_other_up): 

'''Refined class comparison. 

 

@arg other: The other instance (C{LatLonNvectorBase}). 

@kwarg name_other_up: Overriding C{name=other} and C{up=1} 

keyword arguments. 

 

@return: The B{C{other}} if compatible. 

 

@raise TypeError: Incompatible B{C{other}} C{type}. 

''' 

if other: 

other0 = other[0] 

if isinstance(other0, (self.__class__, LatLonNvectorBase)): # XXX NvectorBase? 

return other0 

 

other, name, up = _xother3(self, other, **name_other_up) 

if not isinstance(other, (self.__class__, LatLonNvectorBase)): # XXX NvectorBase? 

LatLonBase.others(self, other, name=name, up=up + 1) 

return other 

 

def toNvector(self, Nvector=NvectorBase, **Nvector_kwds): # PYCHOK signature 

'''Convert this point to C{Nvector} components, I{including 

height}. 

 

@kwarg Nvector_kwds: Optional, additional B{C{Nvector}} keyword 

arguments, ignored if C{B{Nvector}=None}. 

 

@return: An B{C{Nvector}} or a L{Vector4Tuple}C{(x, y, z, h)} if 

B{C{Nvector}} is C{None}. 

 

@raise TypeError: Invalid B{C{Nvector}} or B{C{Nvector_kwds}}. 

''' 

return LatLonBase.toNvector(self, Nvector=Nvector, **Nvector_kwds) 

 

def triangulate(self, bearing1, other, bearing2, height=None): 

'''Locate a point given this and an other point and a bearing 

at this and the other point. 

 

@arg bearing1: Bearing at this point (compass C{degrees360}). 

@arg other: The other point (C{LatLon}). 

@arg bearing2: Bearing at the other point (compass C{degrees360}). 

@kwarg height: Optional height at the triangulated point, 

overriding the mean height (C{meter}). 

 

@return: Triangulated point (C{LatLon}). 

 

@raise TypeError: Invalid B{C{other}} point. 

 

@raise Valuerror: Points coincide. 

 

@example: 

 

>>> p = LatLon("47°18.228'N","002°34.326'W") # Basse Castouillet 

>>> q = LatLon("47°18.664'N","002°31.717'W") # Basse Hergo 

>>> t = p.triangulate(7, q, 295) # 47.323667°N, 002.568501°W' 

''' 

return _triangulate(self, bearing1, self.others(other), bearing2, 

height=height, LatLon=self.classof) 

 

def trilaterate(self, distance1, point2, distance2, point3, distance3, 

radius=R_M, height=None, useZ=False): 

'''Locate a point at given distances from this and two other points. 

 

@arg distance1: Distance to this point (C{meter}, same units 

as B{C{radius}}). 

@arg point2: Second reference point (C{LatLon}). 

@arg distance2: Distance to point2 (C{meter}, same units as 

B{C{radius}}). 

@arg point3: Third reference point (C{LatLon}). 

@arg distance3: Distance to point3 (C{meter}, same units as 

B{C{radius}}). 

@kwarg radius: Mean earth radius (C{meter}). 

@kwarg height: Optional height at trilaterated point, overriding 

the mean height (C{meter}, same units as B{C{radius}}). 

@kwarg useZ: Include Z component iff non-NaN, non-zero (C{bool}). 

 

@return: Trilaterated point (C{LatLon}). 

 

@raise IntersectionError: No intersection, trilateration failed. 

 

@raise TypeError: Invalid B{C{point2}} or B{C{point3}}. 

 

@raise ValueError: Some B{C{points}} coincide or invalid B{C{distance1}}, 

B{C{distance2}}, B{C{distance3}} or B{C{radius}}. 

 

@see: U{Trilateration<https://WikiPedia.org/wiki/Trilateration>}, 

Veness' JavaScript U{Trilateration<https://www.Movable-Type.co.UK/ 

scripts/latlong-vectors.html>} and method C{LatLon.trilaterate2} 

of other, non-C{Nvector LatLon} classes. 

''' 

return _trilaterate(self, distance1, 

self.others(point2=point2), distance2, 

self.others(point3=point3), distance3, 

radius=radius, height=height, useZ=useZ, 

LatLon=self.classof) 

 

def trilaterate5(self, distance1, point2, distance2, point3, distance3, # PYCHOK signature 

area=False, eps=EPS1, radius=R_M, wrap=False): 

'''B{Not implemented} for C{B{area}=True} or C{B{wrap}=True} 

and falls back to method C{trilaterate} otherwise. 

 

@return: A L{Trilaterate5Tuple}C{(min, minPoint, max, maxPoint, n)} 

with a single trilaterated intersection C{minPoint I{is} 

maxPoint}, C{min I{is} max} the nearest intersection 

margin and count C{n = 1}. 

 

@raise IntersectionError: No intersection, trilateration failed. 

 

@raise NotImplementedError: Keyword argument C{B{area}=True} or 

C{B{wrap}=True} not (yet) supported. 

 

@raise TypeError: Invalid B{C{point2}} or B{C{point3}}. 

 

@raise ValueError: Coincident B{C{points}} or invalid B{C{distance1}}, 

B{C{distance2}}, B{C{distance3}} or B{C{radius}}. 

''' 

if area or wrap: 

notImplemented(self, area=area, wrap=wrap) 

 

t = _trilaterate(self, distance1, self.others(point2=point2), distance2, 

self.others(point3=point3), distance3, 

radius=radius, height=None, useZ=True, 

LatLon=self.classof) 

# ... and handle B{C{eps}} and C{IntersectionError} 

# like function C{.latlonBase._trilaterate5} 

d = self.distanceTo(t, radius=radius, wrap=wrap) # PYCHOK distanceTo 

d = min(fabs(distance1 - d), fabs(distance2 - d), fabs(distance3 - d)) 

if d < eps: # min is max, minPoint is maxPoint 

return Trilaterate5Tuple(d, t, d, t, 1) # n = 1 

t = _SPACE_(_no_(_intersection_), Fmt.PAREN(min.__name__, Fmt.f(d, prec=3))) 

raise IntersectionError(area=area, eps=eps, wrap=wrap, txt=t) 

 

 

def sumOf(nvectors, Vector=None, h=None, **Vector_kwds): 

'''Return the vectorial sum of two or more n-vectors. 

 

@arg nvectors: Vectors to be added (C{Nvector}[]). 

@kwarg Vector: Optional class for the vectorial sum (C{Nvector}) 

or C{None}. 

@kwarg h: Optional height, overriding the mean height (C{meter}). 

@kwarg Vector_kwds: Optional, additional B{C{Vector}} keyword 

arguments, ignored if C{B{Vector}=None}. 

 

@return: Vectorial sum (B{C{Vector}}) or a L{Vector4Tuple}C{(x, y, 

z, h)} if B{C{Vector}} is C{None}. 

 

@raise VectorError: No B{C{nvectors}}. 

''' 

n, nvectors = len2(nvectors) 

if n < 1: 

raise VectorError(nvectors=n, txt=MISSING) 

 

if h is None: 

h = fsum(v.h for v in nvectors) / float(n) 

 

if Vector is None: 

r = _sumOf(nvectors, Vector=Vector3Tuple).to4Tuple(h) 

else: 

r = _sumOf(nvectors, Vector=Vector, h=h, **Vector_kwds) 

return r 

 

 

def _triangulate(point1, bearing1, point2, bearing2, height=None, 

**LatLon_LatLon_kwds): 

# (INTERNAL)Locate a point given two known points and initial 

# bearings from those points, see LatLon.triangulate above 

 

def _gc(p, b, _i_): 

n = p.toNvector() 

de = NorthPole.cross(n, raiser=_pole_).unit() # east vector @ n 

dn = n.cross(de) # north vector @ n 

s, c = sincos2d(Bearing(b, name=_bearing_ + _i_)) 

dest = de.times(s) 

dnct = dn.times(c) 

d = dnct.plus(dest) # direction vector @ n 

return n.cross(d) # great circle point + bearing 

 

if point1.isequalTo(point2, EPS): 

raise _ValueError(points=point2, txt=_coincident_) 

 

gc1 = _gc(point1, bearing1, _1_) # great circle p1 + b1 

gc2 = _gc(point2, bearing2, _2_) # great circle p2 + b2 

 

n = gc1.cross(gc2, raiser=_points_) # n-vector of intersection point 

 

h = point1._havg(point2) if height is None else Height(height) 

kwds = _xkwds(LatLon_LatLon_kwds, height=h) 

return n.toLatLon(**kwds) # Nvector(n.x, n.y, n.z).toLatLon(...) 

 

 

def _trilaterate(point1, distance1, point2, distance2, point3, distance3, 

radius=R_M, height=None, useZ=False, 

**LatLon_LatLon_kwds): 

# (INTERNAL) Locate a point at given distances from 

# three other points, see LatLon.triangulate above 

 

def _nd2(p, d, r, _i_, *qs): # .toNvector and angular distance squared 

for q in qs: 

if p.isequalTo(q, EPS): 

raise _ValueError(points=p, txt=_coincident_) 

return p.toNvector(), (Scalar(d, name=_distance_ + _i_) / r)**2 

 

r = Radius_(radius) 

 

n1, r12 = _nd2(point1, distance1, r, _1_) 

n2, r22 = _nd2(point2, distance2, r, _2_, point1) 

n3, r32 = _nd2(point3, distance3, r, _3_, point1, point2) 

 

# the following uses x,y coordinate system with origin at n1, x axis n1->n2 

y = n3.minus(n1) 

x = n2.minus(n1) 

z = None 

 

d = x.length # distance n1->n2 

if d > EPS_2: # and y.length > EPS_2: 

X = x.unit() # unit vector in x direction n1->n2 

i = X.dot(y) # signed magnitude of x component of n1->n3 

Y = y.minus(X.times(i)).unit() # unit vector in y direction 

j = Y.dot(y) # signed magnitude of y component of n1->n3 

if abs(j) > EPS_2: 

# courtesy Carlos Freitas <https://GitHub.com/mrJean1/PyGeodesy/issues/33> 

x = fsum_(r12, -r22, d**2) / (_2_0 * d) # n1->intersection x- and ... 

y = fsum_(r12, -r32, i**2, j**2, -2 * x * i) / (_2_0 * j) # ... y-component 

# courtesy AleixDev <https://GitHub.com/mrJean1/PyGeodesy/issues/43> 

z = fsum_(max(r12, r22, r32), -(x**2), -(y**2)) # XXX not just r12! 

if z > EPS: 

n = n1.plus(X.times(x)).plus(Y.times(y)) 

if useZ: # include Z component 

Z = X.cross(Y) # unit vector perpendicular to plane 

n = n.plus(Z.times(sqrt(z))) 

if height is None: 

h = fidw((point1.height, point2.height, point3.height), 

map1(fabs, distance1, distance2, distance3)) 

else: 

h = Height(height) 

kwds = _xkwds(LatLon_LatLon_kwds, height=h) 

return n.toLatLon(**kwds) # Nvector(n.x, n.y, n.z).toLatLon(...) 

 

# no intersection, d < EPS_2 or abs(j) < EPS_2 or z < EPS 

t = _SPACE_(_no_, _intersection_, NN) 

raise IntersectionError(point1=point1, distance1=distance1, 

point2=point2, distance2=distance2, 

point3=point3, distance3=distance3, 

txt=unstr(t, z=z, useZ=useZ)) 

 

 

__all__ += _ALL_DOCS(LatLonNvectorBase, NvectorBase, sumOf) # classes 

 

# **) MIT License 

# 

# Copyright (C) 2016-2021 -- mrJean1 at Gmail -- All Rights Reserved. 

# 

# Permission is hereby granted, free of charge, to any person obtaining a 

# copy of this software and associated documentation files (the "Software"), 

# to deal in the Software without restriction, including without limitation 

# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

# and/or sell copies of the Software, and to permit persons to whom the 

# Software is furnished to do so, subject to the following conditions: 

# 

# The above copyright notice and this permission notice shall be included 

# in all copies or substantial portions of the Software. 

# 

# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

# OTHER DEALINGS IN THE SOFTWARE.