# Coverage for pygeodesy/ellipsoidalVincenty.py : 99%

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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 from pygeodesy.basics import isnear0 from pygeodesy.datums import _WGS84 from pygeodesy.ellipsoidalBase import CartesianEllipsoidalBase, _nearestOn from pygeodesy.ellipsoidalBaseDI import LatLonEllipsoidalBaseDI, _TOL_M, \ _intersection3, _intersections2 from pygeodesy.errors import _ValueError, _xkwds from pygeodesy.fmath import fpolynomial, fsum_, hypot, hypot1 from pygeodesy.interns import EPS, NN, _ambiguous_, _antipodal_, \ _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_OTHER from pygeodesy.namedTuples import Destination2Tuple, Destination3Tuple, \ Distance3Tuple from pygeodesy.points import ispolar # PYCHOK exported from pygeodesy.props import deprecated_function, deprecated_method, \ Property_RO, property_doc_ 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__ = '21.09.28' _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}. ''' from pygeodesy.ecef import EcefVeness return 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 _iterations = 100 # 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}. ''' from pygeodesy.ecef import EcefVeness return 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 @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) 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): 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(_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): 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 = '%r %s%s %r' % (self, _antipodal__, _to_, other) 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('%s, %r %sto %r' % ('ambiguous', self, # _antipodal__, other)) else: t = _antipodal__ if self.isantipodeTo(other, eps=self._epsilon) else NN t = '%r %s%s %r' % (self, t, _to_, other) raise VincentyError(_no_(_convergence_), txt=t) 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 _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}. ''' from pygeodesy.ellipsoidalKarney import areaOf return 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{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{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 arc (C{LatLon}). @arg point2: End point of the arc (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}, 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{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}. ''' from pygeodesy.ellipsoidalKarney import perimeterOf return perimeterOf(points, closed=closed, datum=datum, wrap=wrap) __all__ += _ALL_OTHER(Cartesian, LatLon, intersection3, intersections2, ispolar, # from .points nearestOn) + _ALL_DOCS(areaOf, perimeterOf) # deprecated # **) 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.