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# -*- coding: utf-8 -*- 

 

u'''(INTERNAL) Ellipsoidal base classes C{CartesianEllipsoidalBase} and 

C{LatLonEllipsoidalBase}. 

 

Pure Python implementation of geodesy tools for ellipsoidal earth models, 

transcribed in part from JavaScript originals by I{(C) Chris Veness 2005-2016} 

and published under the same MIT Licence**, see for example U{latlon-ellipsoidal 

<https://www.Movable-Type.co.UK/scripts/geodesy/docs/latlon-ellipsoidal.js.html>}. 

 

@newfield example: Example, Examples 

''' 

 

from pygeodesy.basics import issubclassof, property_doc_, property_RO, \ 

_xinstanceof 

from pygeodesy.cartesianBase import CartesianBase 

from pygeodesy.datums import Datum, Datums, _ellipsoidal_datum 

from pygeodesy.ecef import EcefVeness 

from pygeodesy.errors import _AssertionError, _incompatible, IntersectionError, \ 

_IsnotError, _ValueError, _xellipsoidal 

from pygeodesy.fmath import euclid, favg, fsum_ 

from pygeodesy.formy import _radical2 

from pygeodesy.interns import _ellipsoidal_ # PYCHOK used! 

from pygeodesy.interns import EPS, EPS1, NN, PI, _COMMA_, _datum_, _exceed_PI_radians_, \ 

_Missing, _N_, _near_concentric_, _no_convergence_fmt_, \ 

_no_conversion_, _too_distant_fmt_, _3_0 

from pygeodesy.latlonBase import LatLonBase, _trilaterate5 

from pygeodesy.lazily import _ALL_DOCS 

from pygeodesy.named import _xnamed 

from pygeodesy.namedTuples import _LatLon4Tuple, Vector3Tuple 

from pygeodesy.trf import RefFrame, TRFError, _reframeTransforms 

from pygeodesy.units import Epoch, Radius_ 

from pygeodesy.utily import m2degrees, unroll180 

 

__all__ = () 

__version__ = '20.10.12' 

 

_TOL_M = 1e-3 # 1 millimeter, in .ellipsoidKarney, -Vincenty 

_TRIPS = 16 # _intersects2, _nearestOn interations, 6 is sufficient 

 

 

class CartesianEllipsoidalBase(CartesianBase): 

'''(INTERNAL) Base class for ellipsoidal C{Cartesian}s. 

''' 

_datum = Datums.WGS84 # L{Datum} 

_Ecef = EcefVeness # preferred C{Ecef...} class, backward compatible 

 

def convertRefFrame(self, reframe2, reframe, epoch=None): 

'''Convert this cartesian point from one to an other reference frame. 

 

@arg reframe2: Reference frame to convert I{to} (L{RefFrame}). 

@arg reframe: Reference frame to convert I{from} (L{RefFrame}). 

@kwarg epoch: Optional epoch to observe for B{C{reframe}}, a 

fractional calendar year (C{scalar}). 

 

@return: The converted point (C{Cartesian}) or this point if 

conversion is C{nil}. 

 

@raise TRFError: No conversion available from B{C{reframe}} 

to B{C{reframe2}}. 

 

@raise TypeError: B{C{reframe2}} or B{C{reframe}} not a 

L{RefFrame} or B{C{epoch}} not C{scalar}. 

''' 

_xinstanceof(RefFrame, reframe2=reframe2, reframe=reframe) 

 

c, d = self, self.datum 

for t in _reframeTransforms(reframe2, reframe, reframe.epoch if 

epoch is None else Epoch(epoch)): 

c = c._applyHelmert(t, False, datum=d) 

return c 

 

 

class LatLonEllipsoidalBase(LatLonBase): 

'''(INTERNAL) Base class for ellipsoidal C{LatLon}s. 

''' 

_convergence = None # UTM/UPS meridian convergence (C{degrees}) 

_datum = Datums.WGS84 # L{Datum} 

_elevation2 = () # cached C{elevation2} result 

_epoch = None # overriding .reframe.epoch (C{float}) 

_etm = None # cached toEtm (L{Etm}) 

_geoidHeight2 = () # cached C{geoidHeight2} result 

_iteration = None # iteration number (C{int} or C{None}) 

_lcc = None # cached toLcc (C{Lcc}) 

_osgr = None # cached toOsgr (C{Osgr}) 

_reframe = None # reference frame (L{RefFrame}) 

_scale = None # UTM/UPS scale factor (C{float}) 

_ups = None # cached toUps (L{Ups}) 

_utm = None # cached toUtm (L{Utm}) 

_wm = None # cached toWm (webmercator.Wm instance) 

 

def __init__(self, lat, lon, height=0, datum=None, reframe=None, 

epoch=None, name=NN): 

'''Create an ellipsoidal C{LatLon} point frome the given 

lat-, longitude and height on the given datum and with 

the given reference frame and epoch. 

 

@arg lat: Latitude (C{degrees} or DMS C{[N|S]}). 

@arg lon: Longitude (C{degrees} or DMS C{str[E|W]}). 

@kwarg height: Optional elevation (C{meter}, the same units 

as the datum's half-axes). 

@kwarg datum: Optional, ellipsoidal datum to use (L{Datum}, 

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

@kwarg reframe: Optional reference frame (L{RefFrame}). 

@kwarg epoch: Optional epoch to observe for B{C{reframe}} 

(C{scalar}), a non-zero, fractional calendar year. 

@kwarg name: Optional name (string). 

 

@raise TypeError: B{C{datum}} is not a L{datum}, B{C{reframe}} 

is not a L{RefFrame} or B{C{epoch}} is not 

C{scalar} non-zero. 

 

@example: 

 

>>> p = LatLon(51.4778, -0.0016) # height=0, datum=Datums.WGS84 

''' 

LatLonBase.__init__(self, lat, lon, height=height, name=name) 

if datum not in (None, self._datum): 

self.datum = _ellipsoidal_datum(datum, name=name) 

if reframe: 

self.reframe = reframe 

self.epoch = epoch 

 

def _Radjust2(self, adjust, datum, meter_text2): 

'''(INTERNAL) Adjust elevation or geoidHeight with difference 

in Gaussian radii of curvature of given datum and NAD83. 

 

@note: This is an arbitrary, possibly incorrect adjustment. 

''' 

if adjust: # Elevation2Tuple or GeoidHeight2Tuple 

m, t = meter_text2 

if isinstance(m, float): 

n = Datums.NAD83.ellipsoid.rocGauss(self.lat) 

if min(abs(m), n) > EPS: 

# use ratio, datum and NAD83 units may differ 

e = self.ellipsoid(datum).rocGauss(self.lat) 

if min(abs(e - n), e) > EPS: 

m *= e / n 

meter_text2 = meter_text2.classof(m, t) 

return self._xnamed(meter_text2) 

 

def _update(self, updated, *attrs): 

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

''' 

if updated: 

LatLonBase._update(self, updated, '_etm', '_lcc', '_osgr', 

'_ups', '_utm', '_wm', *attrs) 

if self._elevation2: 

self._elevation2 = () 

if self._geoidHeight2: 

self._geoidHeight2 = () 

 

def antipode(self, height=None): 

'''Return the antipode, the point diametrically opposite 

to this point. 

 

@kwarg height: Optional height of the antipode, height 

of this point otherwise (C{meter}). 

 

@return: The antipodal point (C{LatLon}). 

''' 

lla = LatLonBase.antipode(self, height=height) 

if lla.datum != self.datum: 

lla.datum = self.datum 

return lla 

 

@property_RO 

def convergence(self): 

'''Get this point's UTM or UPS meridian convergence (C{degrees}) 

or C{None} if not converted from L{Utm} or L{Ups}. 

''' 

return self._convergence 

 

def convertDatum(self, datum2): 

'''Convert this point to an other datum. 

 

@arg datum2: Datum to convert I{to} (L{Datum}). 

 

@return: The converted point (ellipsoidal C{LatLon}). 

 

@raise TypeError: The B{C{datum2}} invalid. 

 

@example: 

 

>>> p = LatLon(51.4778, -0.0016) # default Datums.WGS84 

>>> p.convertDatum(Datums.OSGB36) # 51.477284°N, 000.00002°E 

''' 

d2 = _ellipsoidal_datum(datum2, name=self.name) 

if self.datum == d2: 

return self.copy() 

 

c = self.toCartesian().convertDatum(d2) 

return c.toLatLon(datum=d2, LatLon=self.classof) 

 

def convertRefFrame(self, reframe2): 

'''Convert this point to an other reference frame. 

 

@arg reframe2: Reference frame to convert I{to} (L{RefFrame}). 

 

@return: The converted point (ellipsoidal C{LatLon}) or 

this point if conversion is C{nil}. 

 

@raise TRFError: No B{C{.reframe}} or no conversion 

available from B{C{.reframe}} to 

B{C{reframe2}}. 

 

@raise TypeError: The B{C{reframe2}} is not a L{RefFrame}. 

 

@example: 

 

>>> p = LatLon(51.4778, -0.0016, reframe=RefFrames.ETRF2000) # default Datums.WGS84 

>>> p.convertRefFrame(RefFrames.ITRF2014) # 51.477803°N, 000.001597°W, +0.01m 

''' 

_xinstanceof(RefFrame, reframe2=reframe2) 

 

if not self.reframe: 

raise TRFError(_no_conversion_, txt='%r.reframe %s' % (self, _Missing)) 

 

ts = _reframeTransforms(reframe2, self.reframe, self.epoch) 

if ts: 

c = self.toCartesian() 

for t in ts: 

c = c._applyHelmert(t, False) 

ll = c.toLatLon(datum=self.datum, LatLon=self.classof, 

epoch=self.epoch, reframe=reframe2) 

# ll.reframe, ll.epoch = reframe2, self.epoch 

else: 

ll = self 

return ll 

 

@property_doc_(''' this points's datum (L{Datum}).''') 

def datum(self): 

'''Get this point's datum (L{Datum}). 

''' 

return self._datum 

 

@datum.setter # PYCHOK setter! 

def datum(self, datum): 

'''Set this point's datum I{without conversion}. 

 

@arg datum: New datum (L{Datum}). 

 

@raise TypeError: The B{C{datum}} is not a L{Datum} 

or not ellipsoidal. 

''' 

_xinstanceof(Datum, datum=datum) 

if not datum.isEllipsoidal: 

raise _IsnotError(_ellipsoidal_, datum=datum) 

self._update(datum != self._datum) 

self._datum = datum 

 

def distanceTo2(self, other): 

'''I{Approximate} the distance and (initial) bearing between this 

and an other (ellipsoidal) point based on the radii of curvature. 

 

I{Suitable only for short distances up to a few hundred Km 

or Miles and only between points not near-polar}. 

 

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

 

@return: An L{Distance2Tuple}C{(distance, initial)}. 

 

@raise TypeError: The B{C{other}} point is not C{LatLon}. 

 

@raise ValueError: Incompatible datum ellipsoids. 

 

@see: Method L{Ellipsoid.distance2} and U{Local, flat earth 

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

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

formula. 

''' 

return self.ellipsoids(other).distance2(self.lat, self.lon, 

other.lat, other.lon) 

 

def elevation2(self, adjust=True, datum=Datums.WGS84, timeout=2): 

'''Return elevation of this point for its or the given datum. 

 

@kwarg adjust: Adjust the elevation for a B{C{datum}} other 

than C{NAD83} (C{bool}). 

@kwarg datum: Optional datum (L{Datum}). 

@kwarg timeout: Optional query timeout (C{seconds}). 

 

@return: An L{Elevation2Tuple}C{(elevation, data_source)} 

or C{(None, error)} in case of errors. 

 

@note: The adjustment applied is the difference in geocentric 

earth radius between the B{C{datum}} and C{NAV83} 

upon which the L{elevations.elevation2} is based. 

 

@note: NED elevation is only available for locations within 

the U{Conterminous US (CONUS) 

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

 

@see: Function L{elevations.elevation2} and method 

L{Ellipsoid.Rgeocentric} for further details and 

possible C{error}s. 

''' 

if not self._elevation2: # get elevation and data source 

from pygeodesy.elevations import elevation2 

self._elevation2 = elevation2(self.lat, self.lon, 

timeout=timeout) 

return self._Radjust2(adjust, datum, self._elevation2) 

 

def ellipsoid(self, datum=Datums.WGS84): 

'''Return the ellipsoid of this point's datum or the given datum. 

 

@kwarg datum: Default datum (L{Datum}). 

 

@return: The ellipsoid (L{Ellipsoid} or L{Ellipsoid2}). 

''' 

return getattr(self, _datum_, datum).ellipsoid 

 

def ellipsoids(self, other): 

'''Check the type and ellipsoid of this and an other point's datum. 

 

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

 

@return: This point's datum ellipsoid (L{Ellipsoid} or L{Ellipsoid2}). 

 

@raise TypeError: The B{C{other}} point is not C{LatLon}. 

 

@raise ValueError: Incompatible datum ellipsoids. 

''' 

self.others(other, up=2) # ellipsoids' caller 

 

E = self.ellipsoid() 

try: # other may be Sphere, etc. 

e = other.ellipsoid() 

except AttributeError: # PYCHOK no cover 

try: # no ellipsoid method, try datum 

e = other.datum.ellipsoid 

except AttributeError: 

e = E # no datum, XXX assume equivalent? 

if e != E: 

raise _ValueError(e.named2, txt=_incompatible(E.named2)) 

return E 

 

@property_doc_(''' this point's observed or C{reframe} epoch (C{float}).''') 

def epoch(self): 

'''Get this point's observed or C{reframe} epoch (C{float}) or C{None}. 

''' 

return self._epoch or (self.reframe.epoch if self.reframe else None) 

 

@epoch.setter # PYCHOK setter! 

def epoch(self, epoch): 

'''Set or clear this point's observed epoch. 

 

@arg epoch: Observed epoch, a fractional calendar year 

(L{Epoch}, C{scalar}) or C{None}. 

 

@raise TRFError: Invalid B{C{epoch}}. 

''' 

self._epoch = None if epoch is None else Epoch(epoch) 

 

def geoidHeight2(self, adjust=False, datum=Datums.WGS84, timeout=2): 

'''Return geoid height of this point for its or the given datum. 

 

@kwarg adjust: Adjust the geoid height for a B{C{datum}} 

other than C{NAD83/NADV88} (C{bool}). 

@kwarg datum: Optional datum (L{Datum}). 

@kwarg timeout: Optional query timeout (C{seconds}). 

 

@return: An L{GeoidHeight2Tuple}C{(height, model_name)} or 

C{(None, error)} in case of errors. 

 

@note: The adjustment applied is the difference in geocentric 

earth radius between the B{C{datum}} and C{NAV83/NADV88} 

upon which the L{elevations.geoidHeight2} is based. 

 

@note: The geoid height is only available for locations within 

the U{Conterminous US (CONUS) 

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

 

@see: Function L{elevations.geoidHeight2} and method 

L{Ellipsoid.Rgeocentric} for further details and 

possible C{error}s. 

''' 

if not self._geoidHeight2: # get elevation and data source 

from pygeodesy.elevations import geoidHeight2 

self._geoidHeight2 = geoidHeight2(self.lat, self.lon, 

model=0, timeout=timeout) 

return self._Radjust2(adjust, datum, self._geoidHeight2) 

 

def intersections2(self, radius1, other, radius2, height=None, wrap=True, 

equidistant=None, tol=_TOL_M): 

'''Compute the intersection points of two circles each defined 

by a center point and a radius. 

 

@arg radius1: Radius of the this circle (C{meter}). 

@arg other: Center of the other circle (C{LatLon}). 

@arg radius2: Radius of the other circle (C{meter}). 

@kwarg height: Optional height for the intersection points, 

overriding the "radical height" at the "radical 

line" between both centers (C{meter}) or C{None}. 

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

@kwarg equidistant: An azimuthal equidistant projection class 

(L{Equidistant} or L{EquidistantKarney}), 

function L{azimuthal.equidistant} will be 

invoked if left unspecified. 

@kwarg tol: Convergence tolerance (C{meter}). 

 

@return: 2-Tuple of the intersection points, each a C{LatLon} 

instance. For abutting circles, both intersection 

points are the same instance. 

 

@raise IntersectionError: Concentric, antipodal, invalid or 

non-intersecting circles or no 

convergence for B{C{tol}}. 

 

@raise ImportError: Package U{geographiclib 

<https://PyPI.org/project/geographiclib>} 

not installed or not found. 

 

@raise TypeError: Invalid B{C{other}} or B{C{equidistant}}. 

 

@raise ValueError: Invalid B{C{radius1}}, B{C{radius2}} or B{C{height}}. 

''' 

self.others(other) 

return _intersects2(self, radius1, other, radius2, height=height, wrap=wrap, 

equidistant=equidistant, tol=tol, 

LatLon=self.classof, datum=self.datum) 

 

def nearestOn(self, point1, point2, within=True, height=None, wrap=True, 

equidistant=None, tol=_TOL_M): 

'''Locate the closest point between two other points. 

 

@arg point1: Start point (C{LatLon}). 

@arg point2: End point (C{LatLon}). 

@kwarg within: If C{True} return the closest point I{between} 

B{C{point1}} and B{C{point2}}, otherwise the 

closest point elsewhere on the arc (C{bool}). 

@kwarg height: Optional height for the closest point (C{meter}) 

or C{None} to interpolate the height. 

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

@kwarg equidistant: An azimuthal equidistant projection class 

(L{Equidistant} or L{EquidistantKarney}), 

function L{azimuthal.equidistant} will be 

invoked if left unspecified. 

@kwarg tol: Convergence tolerance (C{meter}). 

 

@return: Closest point (C{LatLon}). 

 

@raise ImportError: Package U{geographiclib 

<https://PyPI.org/project/geographiclib>} 

not installed or not found. 

 

@raise TypeError: Invalid B{C{point1}}, B{C{point2}} or B{C{equidistant}}. 

''' 

p1 = self.others(point1=point1) 

p2 = self.others(point2=point2) 

return _nearestOn(self, p1, p2, within=within, height=height, wrap=wrap, 

equidistant=equidistant, tol=tol, 

LatLon=self.classof, datum=self.datum) 

 

@property_RO 

def iteration(self): 

'''Get the iteration number (C{int} or C{None} if not available/applicable) 

of the most recent C{intersections2} or C{nearestOn} invokation. 

''' 

return self._iteration 

 

def parse(self, strllh, height=0, datum=None, sep=_COMMA_, name=NN): 

'''Parse a string representing a similar, ellipsoidal C{LatLon} 

point, consisting of C{"lat, lon[, height]"}. 

 

@arg strllh: Lat, lon and optional height (C{str}), 

see function L{parse3llh}. 

@kwarg height: Optional, default height (C{meter} or 

C{None}). 

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

datum I{without conversion}. 

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

@kwarg name: Optional instance name (C{str}), overriding 

this name. 

 

@return: The similar point (ellipsoidal C{LatLon}). 

 

@raise ParseError: Invalid B{C{strllh}}. 

''' 

from pygeodesy.dms import parse3llh 

a, b, h = parse3llh(strllh, height=height, sep=sep) 

r = self.classof(a, b, height=h, datum=self.datum) 

if datum not in (None, self.datum): 

r.datum = datum 

return _xnamed(r, name or self.name, force=True) 

 

@property_doc_(''' this point's reference frame (L{RefFrame}).''') 

def reframe(self): 

'''Get this point's reference frame (L{RefFrame}) or C{None}. 

''' 

return self._reframe 

 

@reframe.setter # PYCHOK setter! 

def reframe(self, reframe): 

'''Set or clear this point's reference frame. 

 

@arg reframe: Reference frame (L{RefFrame}) or C{None}. 

 

@raise TypeError: The B{C{reframe}} is not a L{RefFrame}. 

''' 

if reframe is not None: 

_xinstanceof(RefFrame, reframe=reframe) 

self._reframe = reframe 

elif self.reframe is not None: 

self._reframe = None 

 

@property_RO 

def scale(self): 

'''Get this point's UTM grid or UPS point scale factor (C{float}) 

or C{None} if not converted from L{Utm} or L{Ups}. 

''' 

return self._scale 

 

def to3xyz(self): # PYCHOK no cover 

'''DEPRECATED, use method C{toEcef}. 

 

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

 

@note: Overloads C{LatLonBase.to3xyz} 

''' 

r = self.toEcef() 

return self._xnamed(Vector3Tuple(r.x, r.y, r.z)) 

 

def toEtm(self): 

'''Convert this C{LatLon} point to an ETM coordinate. 

 

@return: The ETM coordinate (L{Etm}). 

 

@see: Function L{toEtm8}. 

''' 

if self._etm is None: 

from pygeodesy.etm import toEtm8, Etm # PYCHOK recursive import 

self._etm = toEtm8(self, datum=self.datum, Etm=Etm) 

return self._etm 

 

def toLcc(self): 

'''Convert this C{LatLon} point to a Lambert location. 

 

@see: Function L{toLcc} in module L{lcc}. 

 

@return: The Lambert location (L{Lcc}). 

''' 

if self._lcc is None: 

from pygeodesy.lcc import Lcc, toLcc # PYCHOK recursive import 

self._lcc = toLcc(self, height=self.height, Lcc=Lcc, 

name=self.name) 

return self._lcc 

 

def toOsgr(self): 

'''Convert this C{LatLon} point to an OSGR coordinate. 

 

@see: Function L{toOsgr} in module L{osgr}. 

 

@return: The OSGR coordinate (L{Osgr}). 

''' 

if self._osgr is None: 

from pygeodesy.osgr import Osgr, toOsgr # PYCHOK recursive import 

self._osgr = toOsgr(self, datum=self.datum, Osgr=Osgr, 

name=self.name) 

return self._osgr 

 

def toUps(self, pole=_N_, falsed=True): 

'''Convert this C{LatLon} point to a UPS coordinate. 

 

@kwarg pole: Optional top/center of (stereographic) 

projection (C{str}, 'N[orth]' or 'S[outh]'). 

@kwarg falsed: False easting and northing (C{bool}). 

 

@return: The UPS coordinate (L{Ups}). 

 

@see: Function L{toUps8}. 

''' 

if self._ups is None: 

from pygeodesy.ups import toUps8, Ups # PYCHOK recursive import 

self._ups = toUps8(self, datum=self.datum, Ups=Ups, 

pole=pole, falsed=falsed) 

return self._ups 

 

def toUtm(self): 

'''Convert this C{LatLon} point to a UTM coordinate. 

 

@return: The UTM coordinate (L{Utm}). 

 

@see: Function L{toUtm8}. 

''' 

if self._utm is None: 

from pygeodesy.utm import toUtm8, Utm # PYCHOK recursive import 

self._utm = toUtm8(self, datum=self.datum, Utm=Utm) 

return self._utm 

 

def toUtmUps(self, pole=NN): 

'''Convert this C{LatLon} point to a UTM or UPS coordinate. 

 

@kwarg pole: Optional top/center of UPS (stereographic) 

projection (C{str}, 'N[orth]' or 'S[outh]'). 

 

@return: The UTM or UPS coordinate (L{Utm} or L{Ups}). 

 

@raise TypeError: Result in L{Utm} or L{Ups}. 

 

@see: Function L{toUtmUps}. 

''' 

if self._utm: 

u = self._utm 

elif self._ups and (self._ups.pole == pole or not pole): 

u = self._ups 

else: 

from pygeodesy.utmups import toUtmUps8, Utm, Ups # PYCHOK recursive import 

u = toUtmUps8(self, datum=self.datum, Utm=Utm, Ups=Ups, pole=pole) 

if isinstance(u, Utm): 

self._utm = u 

elif isinstance(u, Ups): 

self._ups = u 

else: 

_xinstanceof(Utm, Ups, toUtmUps8=u) 

return u 

 

def toWm(self): 

'''Convert this C{LatLon} point to a WM coordinate. 

 

@see: Function L{toWm} in module L{webmercator}. 

 

@return: The WM coordinate (L{Wm}). 

''' 

if self._wm is None: 

from pygeodesy.webmercator import toWm # PYCHOK recursive import 

self._wm = toWm(self) 

return self._wm 

 

def trilaterate5(self, distance1, point2, distance2, point3, distance3, 

area=True, eps=EPS1, wrap=False): 

'''Trilaterate three points by area overlap or perimeter intersection 

three corresponding circles. 

 

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

as B{C{eps}}). 

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

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

B{C{eps}}). 

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

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

B{C{eps}}). 

@kwarg area: If C{True} compute the area overlap, otherwise the 

perimeter intersection of the circles (C{bool}). 

@kwarg eps: The required I{minimal overlap} for C{B{area}=True} 

or the I{intersection margin} for C{B{area}=False} 

(C{meter}, conventionally). 

@kwarg wrap: Wrap/unroll angular distances (C{bool}). 

 

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

with C{min} and C{max} in C{meter}, same units as B{C{eps}}, 

the corresponding trilaterated points C{minPoint} and 

C{maxPoint} as I{ellipsoidal} C{LatLon} and C{n}, the number 

of trilatered points found for the given B{C{eps}}. 

 

If only a single trilaterated point is found, C{min I{is} 

max}, C{minPoint I{is} maxPoint} and C{n = 1}. 

 

For C{B{area}=True}, C{min} and C{max} are the smallest 

respectively largest I{radial} overlap found. 

 

For C{B{area}=False}, C{min} and C{max} represent the 

nearest respectively farthest intersection margin. 

 

If C{B{area}=True} and all 3 circles are concentric, C{n = 

0} and C{minPoint} and C{maxPoint} are both the B{C{point#}} 

with the smallest B{C{distance#}} C{min} and C{max} the 

largest B{C{distance#}}. 

 

@raise IntersectionError: Trilateration failed for the given B{C{eps}}, 

insufficient overlap for C{B{area}=True} or 

no intersection or all (near-)concentric for 

C{B{area}=False}. 

 

@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}} or B{C{distance3}}. 

 

@note: Ellipsoidal trilateration invokes methods C{LatLon.intersections2} 

and C{LatLon.nearestOn}. Install Karney's Python package 

U{geographiclib<https://PyPI.org/project/geographiclib>} to obtain 

the most accurate results for both C{ellipsoidalVincenty.-} and 

C{ellipsoidalKarney.LatLon} points. 

''' 

return _trilaterate5(self, distance1, 

self.others(point2=point2), distance2, 

self.others(point3=point3), distance3, 

area=area, eps=eps, wrap=wrap) 

 

 

def _intersections2(center1, radius1, center2, radius2, height=None, wrap=True, 

equidistant=None, tol=_TOL_M, LatLon=None, **LatLon_kwds): 

# (INTERNAL) Iteratively compute the intersection points of two circles 

# each defined by an (ellipsoidal) center point and a radius, imported 

# by .ellipsoidalKarney and -Vincenty 

 

c1 = _xellipsoidal(center1=center1) 

c2 = c1.others(center2=center2) 

 

r1 = Radius_(radius1=radius1) 

r2 = Radius_(radius2=radius2) 

 

try: 

return _intersects2(c1, r1, c2, r2, height=height, wrap=wrap, 

equidistant=equidistant, tol=tol, 

LatLon=LatLon, **LatLon_kwds) 

except (TypeError, ValueError) as x: 

raise IntersectionError(center1=center1, radius1=radius1, 

center2=center2, radius2=radius2, txt=str(x)) 

 

 

def _intersects2(c1, r1, c2, r2, height=None, wrap=True, # MCCABE 17 

equidistant=None, tol=_TOL_M, LatLon=None, **LatLon_kwds): 

# (INTERNAL) Intersect two spherical circles, see L{_intersections2} 

# above, separated to allow callers to embellish any exceptions 

 

from pygeodesy.sphericalTrigonometry import _intersects2 as _si2, LatLon as _LLS 

from pygeodesy.vector3d import _intersects2 as _vi2 

 

def _latlon4(t, h, n): 

r = _LatLon4Tuple(t.lat, t.lon, h, t.datum, LatLon, LatLon_kwds) 

r._iteration = t.iteration # ._iteration for tests 

return _xnamed(r, n) 

 

if r1 < r2: 

c1, c2 = c2, c1 

r1, r2 = r2, r1 

 

E = c1.ellipsoids(c2) 

if r1 > (min(E.b, E.a) * PI): 

raise ValueError(_exceed_PI_radians_) 

 

if wrap: # unroll180 == .karney._unroll2 

c2 = _unrollon(c1, c2) 

 

# distance between centers and radii are 

# measured along the ellipsoid's surface 

m = c1.distanceTo(c2, wrap=False) # meter 

if m < max(r1 - r2, EPS): 

raise ValueError(_near_concentric_) 

if fsum_(r1, r2, -m) < 0: 

raise ValueError(_too_distant_fmt_ % (m,)) 

 

f = _radical2(m, r1, r2).ratio # "radical fraction" 

r = E.rocMean(favg(c1.lat, c2.lat, f=f)) 

e = max(m2degrees(tol, radius=r), EPS) 

 

# get the azimuthal equidistant projection 

A = _Equidistant(equidistant, datum=c1.datum) 

 

# gu-/estimate initial intersections, spherically ... 

t1, t2 = _si2(_LLS(c1.lat, c1.lon, height=c1.height), r1, 

_LLS(c2.lat, c2.lon, height=c2.height), r2, 

radius=r, height=height, wrap=False, too_d=m) 

h, n = t1.height, t1.name 

 

# ... and then iterate like Karney suggests to find 

# tri-points of median lines, @see: references above 

ts, ta = [], None 

for t in ((t1,) if t1 is t2 else (t1, t2)): 

p = None # force first d == p to False 

for i in range(_TRIPS): 

A.reset(t.lat, t.lon) # gu-/estimate as origin 

# convert centers to projection space 

t1 = A.forward(c1.lat, c1.lon) 

t2 = A.forward(c2.lat, c2.lon) 

# compute intersections in projection space 

v1, v2 = _vi2(t1, r1, # XXX * t1.scale?, 

t2, r2, # XXX * t2.scale?, 

sphere=False, too_d=m) 

# convert intersections back to geodetic 

t1 = A.reverse(v1.x, v1.y) 

d1 = euclid(t1.lat - t.lat, t1.lon - t.lon) 

if v1 is v2: # abutting 

t, d = t1, d1 

else: 

t2 = A.reverse(v2.x, v2.y) 

d2 = euclid(t2.lat - t.lat, t2.lon - t.lon) 

# consider only the closer intersection 

t, d = (t1, d1) if d1 < d2 else (t2, d2) 

# break if below tolerance or if unchanged 

if d < e or d == p: 

t._iteration = i + 1 # _NamedTuple._iteration 

ts.append(t) 

if v1 is v2: # abutting 

ta = t 

break 

p = d 

else: 

raise ValueError(_no_convergence_fmt_ % (tol,)) 

 

if ta: # abutting circles 

r = _latlon4(ta, h, n) 

elif len(ts) == 2: 

return _latlon4(ts[0], h, n), _latlon4(ts[1], h, n) 

elif len(ts) == 1: # XXX assume abutting 

r = _latlon4(ts[0], h, n) 

else: 

raise _AssertionError(ts=ts) 

return r, r 

 

 

def _Equidistant(equidistant, datum): 

# get an C{azimuthal.Equidistant} or {-.Karney} instance 

import pygeodesy.azimuthal as _az 

 

if equidistant is None: 

equidistant = _az.equidistant 

elif not (issubclassof(equidistant, _az.Equidistant) or 

issubclassof(equidistant, _az.EquidistantKarney)): 

raise _IsnotError(_az.Equidistant.__name__, 

_az.EquidistantKarney.__name__, 

equidistant=equidistant) 

return equidistant(0, 0, datum) 

 

 

def _unrollon(p1, p2): # unroll180 == .karney._unroll2 

# wrap, unroll and replace longitude if different 

_, lon = unroll180(p1.lon, p2.lon, wrap=True) 

if abs(lon - p2.lon) > EPS: 

p2 = p2.classof(p2.lat, lon, p2.height, datum=p2.datum) 

return p2 

 

 

def _nearestOn(p, p1, p2, within=True, height=None, wrap=True, 

equidistant=None, tol=_TOL_M, LatLon=None, **LatLon_kwds): 

# (INTERNAL) Get closet point, like L{_intersects2} above, 

# separated to allow callers to embellish any exceptions 

 

from pygeodesy.sphericalNvector import LatLon as _LLS 

from pygeodesy.vector3d import _nearestOn as _vnOn, Vector3d 

 

def _v(t, h): 

return Vector3d(t.x, t.y, h) 

 

_ = p.ellipsoids(p1) 

E = p.ellipsoids(p2) 

 

if wrap: 

p1 = _unrollon(p, p1) 

p2 = _unrollon(p, p2) 

p2 = _unrollon(p1, p2) 

 

r = E.rocMean(fsum_(p.lat, p1.lat, p2.lat) / _3_0) 

e = max(m2degrees(tol, radius=r), EPS) 

 

# get the azimuthal equidistant projection 

A = _Equidistant(equidistant, datum=p.datum) 

 

# gu-/estimate initial nearestOn, spherically ... wrap=False 

t = _LLS(p.lat, p.lon, height=p.height).nearestOn( 

_LLS(p1.lat, p1.lon, height=p1.height), 

_LLS(p2.lat, p2.lon, height=p2.height), within=within, height=height) 

n = t.name 

 

h = h1 = h2 = 0 

if height is False: # use height as Z component 

h = t.height 

h1 = p1.height 

h2 = p2.height 

 

# ... and then iterate like Karney suggests to find 

# tri-points of median lines, @see: references above 

c = None # force first d == c to False 

# closest to origin, .z to interpolate height 

p = Vector3d(0, 0, h) 

for i in range(_TRIPS): 

A.reset(t.lat, t.lon) # gu-/estimate as origin 

# convert points to projection space 

t1 = A.forward(p1.lat, p1.lon) 

t2 = A.forward(p2.lat, p2.lon) 

# compute nearestOn in projection space 

v = _vnOn(p, _v(t1, h1), _v(t2, h2), within=within) 

# convert nearestOn back to geodetic 

r = A.reverse(v.x, v.y) 

d = euclid(r.lat - t.lat, r.lon - t.lon) 

# break if below tolerance or if unchanged 

t = r 

if d < e or d == c: 

t._iteration = i + 1 # _NamedTuple._iteration 

if height is False: 

h = v.z # nearest interpolated 

break 

c = d 

else: 

raise ValueError(_no_convergence_fmt_ % (tol,)) 

 

r = _LatLon4Tuple(t.lat, t.lon, h, t.datum, LatLon, LatLon_kwds) 

r._iteration = t.iteration # ._iteration for tests 

return _xnamed(r, n) 

 

 

__all__ += _ALL_DOCS(CartesianEllipsoidalBase, LatLonEllipsoidalBase) 

 

# **) 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.