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# -*- coding: utf-8 -*- 

 

u'''Ellipsoidal, I{Vincenty}-based geodesy. 

 

I{Thaddeus Vincenty}'s geodetic (lat-/longitude) L{LatLon}, geocentric 

(ECEF) L{Cartesian} and L{VincentyError} classes and functions L{areaOf}, 

L{intersections2}, L{nearestOn} and L{perimeterOf}. 

 

Pure Python implementation of 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{Vincenty geodesics 

<https://www.Movable-Type.co.UK/scripts/LatLongVincenty.html>}. More at 

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

U{GeoPy<https://PyPI.org/project/geopy>}. 

 

Calculate geodesic distance between two points using the U{Vincenty 

<https://WikiPedia.org/wiki/Vincenty's_formulae>} formulae and one of 

several ellipsoidal earth models. The default model is WGS-84, the 

most accurate and widely used globally-applicable model for the earth 

ellipsoid. 

 

Other ellipsoids offering a better fit to the local geoid include Airy 

(1830) in the UK, Clarke (1880) in Africa, International 1924 in much of 

Europe, and GRS-67 in South America. North America (NAD83) and Australia 

(GDA) use GRS-80, which is equivalent to the WGS-84 model. 

 

Great-circle distance uses a spherical model of the earth with the mean 

earth radius defined by the International Union of Geodesy and Geophysics 

(IUGG) as M{(2 * a + b) / 3 = 6371008.7714150598} meter or approx. 6371009 

meter (for WGS-84, resulting in an error of up to about 0.5%). 

 

Here's an example usage of C{ellipsoidalVincenty}: 

 

>>> from pygeodesy.ellipsoidalVincenty import LatLon 

>>> Newport_RI = LatLon(41.49008, -71.312796) 

>>> Cleveland_OH = LatLon(41.499498, -81.695391) 

>>> Newport_RI.distanceTo(Cleveland_OH) 

866,455.4329158525 # meter 

 

You can change the ellipsoid model used by the Vincenty formulae 

as follows: 

 

>>> from pygeodesy import Datums 

>>> from pygeodesy.ellipsoidalVincenty import LatLon 

>>> p = LatLon(0, 0, datum=Datums.OSGB36) 

 

or by converting to anothor datum: 

 

>>> p = p.toDatum(Datums.OSGB36) 

''' 

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

from __future__ import division as _; del _ # PYCHOK semicolon 

 

# from pygeodesy.basics import isnear0 # from .ellipsoidalBaseDI 

# from pygeodesy.datums import _WGS84 # from .ellipsoidalBase 

from pygeodesy.ellipsoidalBase import CartesianEllipsoidalBase, _nearestOn, \ 

_WGS84 

from pygeodesy.ellipsoidalBaseDI import fsum_, isnear0, LatLonEllipsoidalBaseDI, \ 

_TOL_M, _intersection3, _intersections2 

from pygeodesy.errors import _and, _ValueError, _xkwds 

from pygeodesy.fmath import fpolynomial, hypot, hypot1 

# from pygeodesy.fsums import fsum_ # from .ellipsoidalBaseDI 

from pygeodesy.interns import EPS, NN, \ 

_ambiguous_, _antipodal_, _COLONSPACE_, \ 

_convergence_, _no_, _SPACE_, _to_, \ 

_0_0, _1_0, _2_0, _3_0, _4_0, _6_0, _16_0 

from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY, _ALL_MODS as _MODS, \ 

_ALL_OTHER 

from pygeodesy.namedTuples import Destination2Tuple, Destination3Tuple, \ 

Distance3Tuple 

from pygeodesy.points import Fmt, ispolar # PYCHOK exported 

from pygeodesy.props import deprecated_function, deprecated_method, \ 

Property_RO, property_doc_ 

# from pygeodesy.streprs import Fmt # from .points 

from pygeodesy.units import Number_, Scalar_ 

from pygeodesy.utily import atan2b, atan2d, sincos2, unroll180 

 

from math import atan2, cos, degrees, radians, tan 

 

__all__ = _ALL_LAZY.ellipsoidalVincenty 

__version__ = '22.04.27' 

 

_antipodal__ = _antipodal_ + _SPACE_ 

_limit_ = 'limit' # PYCHOK used! 

 

 

class VincentyError(_ValueError): 

'''Error raised from I{Vincenty}'s C{direct} and C{inverse} methods 

for coincident points or lack of convergence. 

''' 

pass 

 

 

class Cartesian(CartesianEllipsoidalBase): 

'''Extended to convert geocentric, L{Cartesian} points to 

Vincenty-based, ellipsoidal, geodetic L{LatLon}. 

''' 

@Property_RO 

def Ecef(self): 

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

''' 

return _MODS.ecef.EcefVeness 

 

def toLatLon(self, **LatLon_and_kwds): # PYCHOK LatLon=LatLon, datum=None 

'''Convert this cartesian point to a C{Vincenty}-based geodetic point. 

 

@kwarg LatLon_and_kwds: Optional L{LatLon} and L{LatLon} keyword 

arguments as C{datum}. Use C{B{LatLon}=..., 

B{datum}=...} to override this L{LatLon} 

class or specify C{B{LatLon}=None}. 

 

@return: The geodetic point (L{LatLon}) or if B{C{LatLon}} is C{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_and_kwds}} argument. 

''' 

kwds = _xkwds(LatLon_and_kwds, LatLon=LatLon, datum=self.datum) 

return CartesianEllipsoidalBase.toLatLon(self, **kwds) 

 

 

class LatLon(LatLonEllipsoidalBaseDI): 

'''Using the formulae devised by U{I{Thaddeus Vincenty (1975)} 

<https://WikiPedia.org/wiki/Vincenty's_formulae>} for an (oblate) 

ellipsoidal model of the earth to compute the geodesic distance 

and bearings between two given points or the destination point 

given an start point and (initial) bearing. 

 

Set the earth model to be used with the keyword argument 

datum. The default is Datums.WGS84, which is the most globally 

accurate. For other models, see the Datums in module datum. 

 

Note: This implementation of I{Vincenty} methods may not converge 

for some valid points, raising a L{VincentyError}. In that case, 

a result may be obtained by increasing the tolerance C{epsilon} 

and/or iteration C{limit}, see properties L{LatLon.epsilon} and 

L{LatLon.iterations}. 

''' 

_epsilon = 1e-12 # radians, about 6 um 

# _iteration = None # iteration number from .named._NamedBase 

_iterations = 101 # default max, 100 vs Veness' 500 

 

@deprecated_method 

def bearingTo(self, other, wrap=False): # PYCHOK no cover 

'''DEPRECATED, use method L{initialBearingTo} or L{bearingTo2}. 

''' 

return self.initialBearingTo(other, wrap=wrap) 

 

@Property_RO 

def Ecef(self): 

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

''' 

return _MODS.ecef.EcefVeness 

 

@property_doc_(''' the convergence epsilon (C{radians}).''') 

def epsilon(self): 

'''Get the convergence epsilon (C{radians}). 

''' 

return self._epsilon 

 

@epsilon.setter # PYCHOK setter! 

def epsilon(self, epsilon): 

'''Set the convergence epsilon. 

 

@arg epsilon: New epsilon (C{radians}). 

 

@raise TypeError: Non-scalar B{C{epsilon}}. 

 

@raise ValueError: Out of bounds B{C{epsilon}}. 

''' 

self._epsilon = Scalar_(epsilon=epsilon) 

 

@property_doc_(''' the iteration limit (C{int}).''') 

def iterations(self): 

'''Get the iteration limit (C{int}). 

''' 

return self._iterations - 1 

 

@iterations.setter # PYCHOK setter! 

def iterations(self, limit): 

'''Set the iteration limit. 

 

@arg limit: New iteration limit (C{int}). 

 

@raise TypeError: Non-scalar B{C{limit}}. 

 

@raise ValueError: Out-of-bounds B{C{limit}}. 

''' 

self._iterations = Number_(limit, name=_limit_, low=4, high=500) + 1 

 

def toCartesian(self, **Cartesian_datum_kwds): # PYCHOK Cartesian=Cartesian, datum=None 

'''Convert this point to C{Vincenty}-based cartesian (ECEF) 

coordinates. 

 

@kwarg Cartesian_datum_kwds: Optional L{Cartesian}, B{C{datum}} 

and other keyword arguments, ignored 

if C{B{Cartesian} is None}. Use 

C{B{Cartesian}=...} to override this 

L{Cartesian} class or specify 

C{B{Cartesian} is None}. 

 

@return: The cartesian point (L{Cartesian}) or if B{C{Cartesian}} 

is C{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}}, B{C{datum}} or other 

B{C{Cartesian_datum_kwds}}. 

''' 

kwds = _xkwds(Cartesian_datum_kwds, Cartesian=Cartesian, 

datum=self.datum) 

return LatLonEllipsoidalBaseDI.toCartesian(self, **kwds) 

 

def _Direct(self, distance, bearing, llr, height): 

'''(INTERNAL) Direct Vincenty method. 

 

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

 

@raise ValueError: If this and the B{C{other}} point's L{Datum} 

ellipsoids are not compatible. 

 

@raise VincentyError: Vincenty fails to converge for the current 

L{LatLon.epsilon} and L{LatLon.iterations} 

limit. 

''' 

E = self.ellipsoid() 

 

c1, s1, t1 = _r3(self.lat, E.f) 

 

i = radians(bearing) # initial bearing (forward azimuth) 

si, ci = sincos2(i) 

s12 = atan2(t1, ci) * _2_0 

 

sa = c1 * si 

c2a = _1_0 - sa**2 

if c2a < EPS: 

c2a = _0_0 

A, B = _1_0, _0_0 

else: # e22 == (a / b)**2 - 1 

A, B = _p2(c2a * E.e22) 

 

s = d = distance / (E.b * A) 

for self._iteration in range(1, self._iterations): # 1-origin 

ss, cs = sincos2(s) 

c2sm = cos(s12 + s) 

s_, s = s, d + _ds(B, cs, ss, c2sm) 

if abs(s - s_) < self._epsilon: 

break 

else: 

raise VincentyError(self._no_convergence(), txt=repr(self)) # self.toRepr() 

 

t = s1 * ss - c1 * cs * ci 

# final bearing (reverse azimuth +/- 180) 

r = atan2b(sa, -t) 

 

if llr: 

a = atan2d(s1 * cs + c1 * ss * ci, E.b_a * hypot(sa, t)) 

b = atan2d(ss * si, c1 * cs - s1 * ss * ci) 

d = degrees(_dl(E.f, c2a, sa, s, cs, ss, c2sm)) 

t = Destination3Tuple(a, fsum_(b, self.lon, -d), r) 

r = self._Direct2Tuple(self.classof, height, t) 

else: 

r = Destination2Tuple(None, r, name=self.name) 

return r 

 

def _Inverse(self, other, wrap, azis=True): # PYCHOK signature 

'''(INTERNAL) Inverse Vincenty method. 

 

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

 

@raise ValueError: If this and the B{C{other}} point's L{Datum} 

ellipsoids are not compatible. 

 

@raise VincentyError: Vincenty fails to converge for the current 

L{LatLon.epsilon} and L{LatLon.iterations} 

limit and/or if this and the B{C{other}} 

point are coincident or near-antipodal. 

''' 

E = self.ellipsoids(other) 

 

c1, s1, _ = _r3(self.lat, E.f) 

c2, s2, _ = _r3(other.lat, E.f) 

 

c1c2, s1c2 = c1 * c2, s1 * c2 

c1s2, s1s2 = c1 * s2, s1 * s2 

 

dl, _ = unroll180(self.lon, other.lon, wrap=wrap) 

ll = dl = radians(dl) 

for self._iteration in range(1, self._iterations): # 1-origin 

ll_ = ll 

sll, cll = sincos2(ll) 

 

ss = hypot(c2 * sll, c1s2 - s1c2 * cll) 

if ss < EPS: # coincident or antipodal, ... 

if self.isantipodeTo(other, eps=self.epsilon): 

t = self._is_to(other, True) 

raise VincentyError(_ambiguous_, txt=t) 

# return zeros like Karney, but unlike Veness 

return Distance3Tuple(_0_0, 0, 0) 

 

cs = s1s2 + c1c2 * cll 

s = atan2(ss, cs) 

 

sa = c1c2 * sll / ss 

c2a = _1_0 - sa**2 

if isnear0(c2a): 

c2a = _0_0 # equatorial line 

ll = dl + E.f * sa * s 

else: 

c2sm = cs - 2 * s1s2 / c2a 

ll = dl + _dl(E.f, c2a, sa, s, cs, ss, c2sm) 

 

if abs(ll - ll_) < self._epsilon: 

break 

# # omitted and applied only after failure to converge below, see footnote 

# # under Inverse at <https://WikiPedia.org/wiki/Vincenty's_formulae> 

# # <https://GitHub.com/ChrisVeness/geodesy/blob/master/latlon-vincenty.js> 

# elif abs(ll) > PI and self.isantipodeTo(other, eps=self._epsilon): 

# raise VincentyError(_ambiguous_, self._is_to(other, True)) 

else: 

raise VincentyError(self._no_convergence(), txt=self._is_to(other)) 

 

if c2a: # e22 == (a / b)**2 - 1 

A, B = _p2(c2a * E.e22) 

s = A * (s - _ds(B, cs, ss, c2sm)) 

 

b = E.b 

# if self.height or other.height: 

# b += self._havg(other) 

d = b * s 

 

if azis: # forward and reverse azimuth 

sll, cll = sincos2(ll) 

f = atan2b(c2 * sll, c1s2 - s1c2 * cll) 

r = atan2b(c1 * sll, -s1c2 + c1s2 * cll) 

else: 

f = r = _0_0 

return Distance3Tuple(d, f, r, name=self.name) 

 

def _is_to(self, other, *anti): 

'''(INTERNAL) Return I{'<self> [antipodal] to <other>'} text (C{str}). 

''' 

t = _antipodal__ if anti or self.isantipodeTo(other, eps=self.epsilon) else NN 

return _SPACE_(repr(self), NN(t, _to_), repr(other)) 

 

def _no_convergence(self): 

'''(INTERNAL) Return I{'no convergence: ...'} text (C{str}). 

''' 

t = (Fmt.PARENSPACED(*t) for t in ((LatLon.epsilon.name, self.epsilon), 

(LatLon.iterations.name, self.iterations))) 

return _COLONSPACE_(_no_(_convergence_), _and(*t)) 

 

 

def _c2sm2(c2sm): 

'''(INTERNAL) 2 * c2sm**2 - 1. 

''' 

return _2_0 * c2sm**2 - _1_0 

 

 

def _dl(f, c2a, sa, s, cs, ss, c2sm): 

'''(INTERNAL) Dl. 

''' 

C = f / _16_0 * c2a * (_4_0 + f * (_4_0 - _3_0 * c2a)) 

return (_1_0 - C) * f * sa * (s + C * ss * (c2sm + 

C * cs * _c2sm2(c2sm))) 

 

 

def _ds(B, cs, ss, c2sm): 

'''(INTERNAL) Ds. 

''' 

if B: 

c2sm2 = _c2sm2(c2sm) 

ss2 = (_4_0 * ss**2 - _3_0) * (_2_0 * c2sm2 - _1_0) 

B *= ss * (c2sm + B / _4_0 * (c2sm2 * cs - 

B / _6_0 * c2sm * ss2)) 

return B 

 

 

def _p2(u2): # e'2 WGS84 = 0.00673949674227643 

'''(INTERNAL) Compute A, B polynomials. 

''' 

A = fpolynomial(u2, 16384, 4096, -768, 320, -175) / 16384 

B = fpolynomial(u2, 0, 256, -128, 74, -47) / 1024 

return A, B 

 

 

def _r3(a, f): 

'''(INTERNAL) Reduced cos, sin, tan. 

''' 

t = (_1_0 - f) * tan(radians(a)) 

c = _1_0 / hypot1(t) 

s = t * c 

return c, s, t 

 

 

@deprecated_function 

def areaOf(points, datum=_WGS84, wrap=True): # PYCHOK no cover 

'''DEPRECATED, use function L{ellipsoidalExact.areaOf} or L{ellipsoidalKarney.areaOf}. 

''' 

return _MODS.ellipsoidalKarney.areaOf(points, datum=datum, wrap=wrap) 

 

 

def intersection3(start1, end1, start2, end2, height=None, wrap=True, 

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

'''Interatively compute the intersection point of two paths, each defined 

by two (ellipsoidal) points or by an (ellipsoidal) start point and a 

bearing from North. 

 

@arg start1: Start point of the first path (L{LatLon}). 

@arg end1: End point of the first path (L{LatLon}) or the initial bearing 

at the first point (compass C{degrees360}). 

@arg start2: Start point of the second path (L{LatLon}). 

@arg end2: End point of the second path (L{LatLon}) or the initial bearing 

at the second point (compass C{degrees360}). 

@kwarg height: Optional height at the intersection (C{meter}, conventionally) 

or C{None} for the mean height. 

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

@kwarg equidistant: An azimuthal equidistant projection (I{class} or function 

L{pygeodesy.equidistant}) or C{None} for the preferred 

C{B{start1}.Equidistant}. 

@kwarg tol: Tolerance for convergence and for skew line distance and length 

(C{meter}, conventionally). 

@kwarg LatLon: Optional class to return the intersection points (L{LatLon}) 

or C{None}. 

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

ignored if C{B{LatLon} is None}. 

 

@return: An L{Intersection3Tuple}C{(point, outside1, outside2)} with C{point} 

a B{C{LatLon}} or if C{B{LatLon} is None}, a L{LatLon4Tuple}C{(lat, 

lon, height, datum)}. 

 

@raise IntersectionError: Skew, colinear, parallel or otherwise 

non-intersecting paths or no convergence 

for the given B{C{tol}}. 

 

@raise TypeError: Invalid or non-ellipsoidal B{C{start1}}, B{C{end1}}, 

B{C{start2}} or B{C{end2}} or invalid B{C{equidistant}}. 

 

@note: For each path specified with an initial bearing, a pseudo-end point 

is computed as the C{destination} along that bearing at about 1.5 

times the distance from the start point to an initial gu-/estimate 

of the intersection point (and between 1/8 and 3/8 of the authalic 

earth perimeter). 

 

@see: U{The B{ellipsoidal} case<https://GIS.StackExchange.com/questions/48937/ 

calculating-intersection-of-two-circles>} and U{Karney's paper 

<https://ArXiv.org/pdf/1102.1215.pdf>}, pp 20-21, section B{14. MARITIME 

BOUNDARIES} for more details about the iteration algorithm. 

''' 

return _intersection3(start1, end1, start2, end2, height=height, wrap=wrap, 

equidistant=equidistant, tol=tol, LatLon=LatLon, **LatLon_kwds) 

 

 

def intersections2(center1, radius1, center2, radius2, height=None, wrap=True, 

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

'''Iteratively compute the intersection points of two circles, each defined 

by an (ellipsoidal) center point and a radius. 

 

@arg center1: Center of the first circle (L{LatLon}). 

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

@arg center2: Center of the second circle (L{LatLon}). 

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

B{C{radius1}}). 

@kwarg height: Optional height for the intersection points (C{meter}, 

conventionally) or C{None} for the I{"radical height"} 

at the I{radical line} between both centers. 

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

@kwarg equidistant: An azimuthal equidistant projection (I{class} or 

function L{pygeodesy.equidistant}) or C{None} for 

the preferred C{B{center1}.Equidistant}. 

@kwarg tol: Convergence tolerance (C{meter}, same units as B{C{radius1}} 

and B{C{radius2}}). 

@kwarg LatLon: Optional class to return the intersection points (L{LatLon}) 

or C{None}. 

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

ignored if C{B{LatLon} is None}. 

 

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

or L{LatLon4Tuple}C{(lat, lon, height, datum)} if C{B{LatLon} is 

None}. For abutting circles, both points are the same instance, 

aka the I{radical center}. 

 

@raise IntersectionError: Concentric, antipodal, invalid or non-intersecting 

circles or no convergence for the B{C{tol}}. 

 

@raise TypeError: Invalid or non-ellipsoidal B{C{center1}} or B{C{center2}} 

or invalid B{C{equidistant}}. 

 

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

 

@see: U{The B{ellipsoidal} case<https://GIS.StackExchange.com/questions/48937/ 

calculating-intersection-of-two-circles>}, U{Karney's paper 

<https://ArXiv.org/pdf/1102.1215.pdf>}, pp 20-21, section B{14. MARITIME BOUNDARIES}, 

U{circle-circle<https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>} and 

U{sphere-sphere<https://MathWorld.Wolfram.com/Sphere-SphereIntersection.html>} 

intersections. 

''' 

return _intersections2(center1, radius1, center2, radius2, height=height, wrap=wrap, 

equidistant=equidistant, tol=tol, LatLon=LatLon, **LatLon_kwds) 

 

 

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

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

'''Iteratively locate the closest point on the geodesic between 

two other (ellipsoidal) points. 

 

@arg point: Reference point (C{LatLon}). 

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

@arg point2: End point of the geodesic (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 geodesic (C{bool}). 

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

conventionally) or C{None} or C{False} for the 

interpolated height. If C{False}, the closest 

takes the heights of the points into account. 

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

@kwarg equidistant: An azimuthal equidistant projection (I{class} 

or function L{pygeodesy.equidistant}) or C{None} 

for the preferred C{B{point}.Equidistant}. 

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

@kwarg LatLon: Optional class to return the closest point 

(L{LatLon}) or C{None}. 

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

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

 

@return: Closest point, a B{C{LatLon}} instance or if C{B{LatLon} 

is None}, a L{LatLon4Tuple}C{(lat, lon, height, datum)}. 

 

@raise ImportError: Package U{geographiclib 

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

not installed or not found, but only if 

C{B{equidistant}=}L{EquidistantKarney}. 

 

@raise TypeError: Invalid or non-ellipsoidal B{C{point}}, B{C{point1}} 

or B{C{point2}} or invalid B{C{equidistant}}. 

 

@raise ValueError: No convergence for the B{C{tol}}. 

 

@see: U{The B{ellipsoidal} case<https://GIS.StackExchange.com/questions/48937/ 

calculating-intersection-of-two-circles>} and U{Karney's paper 

<https://ArXiv.org/pdf/1102.1215.pdf>}, pp 20-21, section B{14. MARITIME 

BOUNDARIES} for more details about the iteration algorithm. 

''' 

return _nearestOn(point, point1, point2, within=within, height=height, wrap=wrap, 

equidistant=equidistant, tol=tol, LatLon=LatLon, **LatLon_kwds) 

 

 

@deprecated_function 

def perimeterOf(points, closed=False, datum=_WGS84, wrap=True): # PYCHOK no cover 

'''DEPRECATED, use function L{ellipsoidalExact.perimeterOf} or L{ellipsoidalKarney.perimeterOf}. 

''' 

return _MODS.ellipsoidalKarney.perimeterOf(points, closed=closed, datum=datum, wrap=wrap) 

 

 

__all__ += _ALL_OTHER(Cartesian, LatLon, 

intersection3, intersections2, ispolar, # from .points 

nearestOn) + _ALL_DOCS(areaOf, perimeterOf) # deprecated 

 

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