Coverage for pygeodesy/ellipsoidalVincenty.py: 99%
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2# -*- coding: utf-8 -*-
4u'''Ellipsoidal, I{Vincenty}-based geodesy.
6I{Thaddeus Vincenty}'s geodetic (lat-/longitude) L{LatLon}, geocentric
7(ECEF) L{Cartesian} and L{VincentyError} classes and functions L{areaOf},
8L{intersections2}, L{nearestOn} and L{perimeterOf}.
10Pure Python implementation of geodesy tools for ellipsoidal earth models,
11transcoded from JavaScript originals by I{(C) Chris Veness 2005-2016}
12and published under the same MIT Licence**, see U{Vincenty geodesics
13<https://www.Movable-Type.co.UK/scripts/LatLongVincenty.html>}. More
14at U{geographiclib<https://PyPI.org/project/geographiclib>} and
15U{GeoPy<https://PyPI.org/project/geopy>}.
17Calculate geodesic distance between two points using the U{Vincenty
18<https://WikiPedia.org/wiki/Vincenty's_formulae>} formulae and one of
19several ellipsoidal earth models. The default model is WGS-84, the
20most widely used globally-applicable model for the earth ellipsoid.
22Other ellipsoids offering a better fit to the local geoid include Airy
23(1830) in the UK, Clarke (1880) in Africa, International 1924 in much
24of Europe, and GRS-67 in South America. North America (NAD83) and
25Australia (GDA) use GRS-80, which is equivalent to the WGS-84 model.
27Great-circle distance uses a I{spherical} model of the earth with the
28mean earth radius defined by the International Union of Geodesy and
29Geophysics (IUGG) as M{(2 * a + b) / 3 = 6371008.7714150598} or about
306,371,009 meter (for WGS-84, resulting in an error of up to about 0.5%).
32Here's an example usage of C{ellipsoidalVincenty}:
34 >>> from pygeodesy.ellipsoidalVincenty import LatLon
35 >>> Newport_RI = LatLon(41.49008, -71.312796)
36 >>> Cleveland_OH = LatLon(41.499498, -81.695391)
37 >>> Newport_RI.distanceTo(Cleveland_OH)
38 866,455.4329158525 # meter
40To change the ellipsoid model used by the Vincenty formulae use:
42 >>> from pygeodesy import Datums
43 >>> from pygeodesy.ellipsoidalVincenty import LatLon
44 >>> p = LatLon(0, 0, datum=Datums.OSGB36)
46or by converting to anothor datum:
48 >>> p = p.toDatum(Datums.OSGB36)
49'''
50# make sure int/int division yields float quotient, see .basics
51from __future__ import division as _; del _ # PYCHOK semicolon
53from pygeodesy.constants import EPS, EPS0, _0_0, _1_0, _2_0, _3_0, _4_0, _6_0
54from pygeodesy.ellipsoidalBase import CartesianEllipsoidalBase, _nearestOn
55from pygeodesy.ellipsoidalBaseDI import _intersection3, _intersections2, \
56 LatLonEllipsoidalBaseDI, _TOL_M
57from pygeodesy.errors import _and, _ValueError, _xkwds
58from pygeodesy.fmath import Fpolynomial, hypot, hypot1
59from pygeodesy.interns import _ambiguous_, _antipodal_, _COLONSPACE_, \
60 _to_, _SPACE_, _limit_ # PYCHOK used!
61from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY, _ALL_MODS as _MODS, _ALL_OTHER
62from pygeodesy.namedTuples import Destination2Tuple, Destination3Tuple, \
63 Distance3Tuple
64from pygeodesy.points import Fmt, ispolar # PYCHOK exported
65from pygeodesy.props import deprecated_function, deprecated_method, \
66 Property_RO, property_doc_
67# from pygeodesy.streprs import Fmt # from .points
68from pygeodesy.units import Number_, Scalar_
69from pygeodesy.utily import atan2b, atan2d, sincos2, sincos2d, unroll180, wrap180
71from math import atan2, cos, degrees, fabs, radians, tan
73__all__ = _ALL_LAZY.ellipsoidalVincenty
74__version__ = '23.05.30'
76_antipodal_to_ = _SPACE_(_antipodal_, _to_)
79class VincentyError(_ValueError):
80 '''Error raised from I{Vincenty}'s C{direct} and C{inverse} methods
81 for coincident points or lack of convergence.
82 '''
83 pass
86class Cartesian(CartesianEllipsoidalBase):
87 '''Extended to convert geocentric, L{Cartesian} points to
88 Vincenty-based, ellipsoidal, geodetic L{LatLon}.
89 '''
90 @Property_RO
91 def Ecef(self):
92 '''Get the ECEF I{class} (L{EcefVeness}), I{lazily}.
93 '''
94 return _MODS.ecef.EcefVeness
96 def toLatLon(self, **LatLon_and_kwds): # PYCHOK LatLon=LatLon, datum=None
97 '''Convert this cartesian point to a C{Vincenty}-based geodetic point.
99 @kwarg LatLon_and_kwds: Optional L{LatLon} and L{LatLon} keyword
100 arguments as C{datum}. Use C{B{LatLon}=...,
101 B{datum}=...} to override this L{LatLon}
102 class or specify C{B{LatLon}=None}.
104 @return: The geodetic point (L{LatLon}) or if B{C{LatLon}} is C{None},
105 an L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)}
106 with C{C} and C{M} if available.
108 @raise TypeError: Invalid B{C{LatLon_and_kwds}} argument.
109 '''
110 kwds = _xkwds(LatLon_and_kwds, LatLon=LatLon, datum=self.datum)
111 return CartesianEllipsoidalBase.toLatLon(self, **kwds)
114class LatLon(LatLonEllipsoidalBaseDI):
115 '''Using the formulae devised by U{I{Thaddeus Vincenty (1975)}
116 <https://WikiPedia.org/wiki/Vincenty's_formulae>} for an (oblate)
117 ellipsoidal model of the earth to compute the geodesic distance
118 and bearings between two given points or the destination point
119 given an start point and (initial) bearing.
121 Set the earth model to be used with the keyword argument
122 datum. The default is Datums.WGS84, which is the most globally
123 accurate. For other models, see the Datums in module datum.
125 Note: This implementation of I{Vincenty} methods may not converge
126 for some valid points, raising a L{VincentyError}. In that case,
127 a result may be obtained by increasing the tolerance C{epsilon}
128 and/or iteration C{limit}, see properties L{LatLon.epsilon} and
129 L{LatLon.iterations}.
130 '''
131 _epsilon = 1e-12 # radians, about 6 um
132# _iteration = None # iteration number from .named._NamedBase
133 _iterations = 201 # 5, default max, 200 vs Veness' 1,000
135 @deprecated_method
136 def bearingTo(self, other, wrap=False): # PYCHOK no cover
137 '''DEPRECATED, use method L{initialBearingTo} or L{bearingTo2}.
138 '''
139 return self.initialBearingTo(other, wrap=wrap)
141 @Property_RO
142 def Ecef(self):
143 '''Get the ECEF I{class} (L{EcefVeness}), I{lazily}.
144 '''
145 return _MODS.ecef.EcefVeness
147 @property_doc_(''' the convergence epsilon (C{radians}).''')
148 def epsilon(self):
149 '''Get the convergence epsilon (C{radians}).
150 '''
151 return self._epsilon
153 @epsilon.setter # PYCHOK setter!
154 def epsilon(self, epsilon):
155 '''Set the convergence epsilon (C{radians}).
157 @raise TypeError: Non-scalar B{C{epsilon}}.
159 @raise ValueError: Out of bounds B{C{epsilon}}.
160 '''
161 self._epsilon = Scalar_(epsilon=epsilon)
163 @property_doc_(''' the iteration limit (C{int}).''')
164 def iterations(self):
165 '''Get the iteration limit (C{int}).
166 '''
167 return self._iterations - 1
169 @iterations.setter # PYCHOK setter!
170 def iterations(self, limit):
171 '''Set the iteration limit (C{int}).
173 @raise TypeError: Non-scalar B{C{limit}}.
175 @raise ValueError: Out-of-bounds B{C{limit}}.
176 '''
177 self._iterations = Number_(limit, name=_limit_, low=4, high=1000) + 1
179 def toCartesian(self, **Cartesian_datum_kwds): # PYCHOK Cartesian=Cartesian, datum=None
180 '''Convert this point to C{Vincenty}-based cartesian (ECEF)
181 coordinates.
183 @kwarg Cartesian_datum_kwds: Optional L{Cartesian}, B{C{datum}} and other
184 keyword arguments, ignored if C{B{Cartesian}=None}. Use
185 C{B{Cartesian}=...} to override this L{Cartesian} class
186 or specify C{B{Cartesian}=None}.
188 @return: The cartesian point (L{Cartesian}) or if B{C{Cartesian}}
189 is C{None}, an L{Ecef9Tuple}C{(x, y, z, lat, lon, height,
190 C, M, datum)} with C{C} and C{M} if available.
192 @raise TypeError: Invalid B{C{Cartesian}}, B{C{datum}} or other
193 B{C{Cartesian_datum_kwds}}.
194 '''
195 kwds = _xkwds(Cartesian_datum_kwds, Cartesian=Cartesian,
196 datum=self.datum)
197 return LatLonEllipsoidalBaseDI.toCartesian(self, **kwds)
199 def _Direct(self, distance, bearing, llr, height):
200 '''(INTERNAL) Direct Vincenty method.
202 @raise TypeError: The B{C{other}} point is not L{LatLon}.
204 @raise ValueError: If this and the B{C{other}} point's L{Datum}
205 ellipsoids are not compatible.
207 @raise VincentyError: Vincenty fails to converge for the current
208 L{LatLon.epsilon} and L{LatLon.iterations}
209 limits.
210 '''
211 E = self.ellipsoid()
212 f = E.f
214 sb, cb = sincos2d(bearing)
215 s1, c1, t1 = _sincostan3r(self.phi, f)
217 eps = self.epsilon
218 s12 = atan2(t1, cb) * _2_0
219 sa, ca2 = _sincos22(c1 * sb)
220 A, B = _AB2(ca2 * E.e22) # e22 == (a / b)**2 - 1
221 s = d = distance / (A * E.b)
222 for i in range(1, self._iterations): # 1-origin
223 ss, cs = sincos2(s)
224 c2sm, e = cos(s12 + s), s
225 s = _Ds(B, cs, ss, c2sm, d)
226 e = fabs(s - e)
227 if e < eps:
228 self._iteration = i
229 break
230 else:
231 t = self._no_convergence(e)
232 raise VincentyError(t, txt=repr(self)) # self.toRepr()
234 t = s1 * ss - c1 * cs * cb
235 # final bearing (reverse azimuth +/- 180)
236 d = atan2b(sa, -t)
237 if llr:
238 b = cb * ss
239 a = atan2d(s1 * cs + c1 * b, hypot(sa, t) * E.b_a)
240 b = atan2d(sb * ss, -s1 * b + c1 * cs) + self.lon \
241 - degrees(_Dl(f, ca2, sa, s, cs, ss, c2sm))
242 t = Destination3Tuple(a, wrap180(b), d)
243 r = self._Direct2Tuple(self.classof, height, t)
244 else:
245 r = Destination2Tuple(None, d, name=self.name)
246 r._iteration = i
247 return r
249 def _Inverse(self, other, wrap, azis=True): # PYCHOK signature
250 '''(INTERNAL) Inverse Vincenty method.
252 @raise TypeError: The B{C{other}} point is not L{LatLon}.
254 @raise ValueError: If this and the B{C{other}} point's L{Datum}
255 ellipsoids are not compatible.
257 @raise VincentyError: Vincenty fails to converge for the current
258 L{LatLon.epsilon} and L{LatLon.iterations}
259 limits and/or if this and the B{C{other}}
260 point are coincident or near-antipodal.
261 '''
262 E = self.ellipsoids(other)
263 f = E.f
265 s1, c1, _ = _sincostan3r( self.phi, f)
266 s2, c2, _ = _sincostan3r(other.phi, f)
268 c1c2, s1c2 = c1 * c2, s1 * c2
269 c1s2, s1s2 = c1 * s2, s1 * s2
271 eps = self.epsilon
272 d, _ = unroll180(self.lon, other.lon, wrap=wrap)
273 dl = ll = radians(d)
274 for i in range(1, self._iterations): # 1-origin
275 sll, cll = sincos2(ll)
277 ss = hypot(c2 * sll, c1s2 - s1c2 * cll)
278 if ss < EPS: # coincident or antipodal, ...
279 if self.isantipodeTo(other, eps=eps):
280 t = self._is_to(other, True)
281 raise VincentyError(_ambiguous_, txt=t)
282 self._iteration = i
283 # return zeros like Karney, unlike Veness
284 return Distance3Tuple(_0_0, 0, 0, iteration=i)
286 cs = s1s2 + c1c2 * cll
287 s, e = atan2(ss, cs), ll
288 sa, ca2 = _sincos22(c1c2 * sll / ss)
289 if ca2:
290 c2sm = cs - _2_0 * s1s2 / ca2
291 ll = _Dl(f, ca2, sa, s, cs, ss, c2sm, dl)
292 else: # equatorial line
293 ll = dl + f * sa * s
294 e = fabs(ll - e)
295 if e < eps:
296 self._iteration = i
297 break
298# elif abs(ll) > PI and self.isantipodeTo(other, eps=eps):
299# # omitted and applied *after* failure to converge below,
300# # see footnote under Inverse <https://WikiPedia.org/wiki/
301# # Vincenty's_formulae> and <https://GitHub.com/chrisveness/
302# # geodesy/blob/master/latlon-ellipsoidal-vincenty.js>
303# raise VincentyError(_ambiguous_, self._is_to(other, True))
304 else:
305 t = self._is_to(other, self.isantipodeTo(other, eps=eps))
306 raise VincentyError(self._no_convergence(e), txt=t)
308 if ca2: # e22 == (a / b)**2 - 1
309 A, B = _AB2(ca2 * E.e22)
310 s = -A * _Ds(B, cs, ss, c2sm, -s)
312 b = E.b
313# if self.height or other.height:
314# b += self._havg(other)
315 d = b * s
317 if azis: # forward and reverse azimuth
318 s, c = sincos2(ll)
319 f = atan2b(c2 * s, c1s2 - s1c2 * c)
320 r = atan2b(c1 * s, -s1c2 + c1s2 * c)
321 else:
322 f = r = _0_0
323 return Distance3Tuple(d, f, r, name=self.name, iteration=i)
325 def _is_to(self, other, anti):
326 '''(INTERNAL) Return I{'<self> [antipodal] to <other>'} text (C{str}).
327 '''
328 t = _antipodal_to_ if anti else _to_
329 return _SPACE_(repr(self), t, repr(other))
331 def _no_convergence(self, e):
332 '''(INTERNAL) Return I{'no convergence (..): ...'} text (C{str}).
333 '''
334 t = (Fmt.PARENSPACED(*t) for t in ((LatLon.epsilon.name, self.epsilon),
335 (LatLon.iterations.name, self.iterations)))
336 return _COLONSPACE_(Fmt.no_convergence(e), _and(*t))
339def _AB2(u2): # WGS84 e22 = 0.00673949674227643
340 # 2-Tuple C{(A, B)} polynomials
341 if u2:
342 A = Fpolynomial(u2, 16384, 4096, -768, 320, -175).fover(16384)
343 B = Fpolynomial(u2, 0, 256, -128, 74, -47).fover( 1024)
344 return A, B
345 return _1_0, _0_0
348def _c2sm2(c2sm):
349 # C{2 * c2sm**2 - 1}
350 return c2sm**2 * _2_0 - _1_0
353def _Dl(f, ca2, sa, s, cs, ss, c2sm, dl=_0_0):
354 # C{Dl}
355 if f and sa:
356 C = f * ca2 / _4_0
357 C *= f - C * _3_0 + _1_0
358 if C and ss:
359 s += C * ss * (c2sm +
360 C * cs * _c2sm2(c2sm))
361 dl += (_1_0 - C) * f * sa * s
362 return dl
365def _Ds(B, cs, ss, c2sm, d):
366 # C{Ds - d}
367 if B and ss:
368 c2sm2 = _c2sm2(c2sm)
369 ss2 = (ss**2 * _4_0 - _3_0) * (c2sm2 * _2_0 - _1_0)
370 B *= ss * (c2sm + B / _4_0 * (c2sm2 * cs -
371 B / _6_0 * c2sm * ss2))
372 d += B
373 return d
376def _sincos22(sa):
377 # 2-Tuple C{(sin(a), cos(a)**2)}
378 ca2 = _1_0 - sa**2
379 return sa, (_0_0 if ca2 < EPS0 else ca2) # XXX EPS?
382def _sincostan3r(a, f):
383 # I{Reduced} C{(sin(B{a}), cos(B{a}), tan(B{a}))}
384 if a: # see L{sincostan3}
385 t = (_1_0 - f) * tan(a)
386 if t:
387 c = _1_0 / hypot1(t)
388 s = t * c
389 return s, c, t
390 return _0_0, _1_0, _0_0
393@deprecated_function
394def areaOf(points, **datum_wrap):
395 '''DEPRECATED, use function L{ellipsoidalExact.areaOf} or L{ellipsoidalKarney.areaOf}.
396 '''
397 try:
398 return _MODS.ellipsoidalKarney.areaOf(points, **datum_wrap)
399 except ImportError:
400 return _MODS.ellipsoidalExact.areaOf(points, **datum_wrap)
403def intersection3(start1, end1, start2, end2, height=None, wrap=False, # was=True
404 equidistant=None, tol=_TOL_M, LatLon=LatLon, **LatLon_kwds):
405 '''I{Iteratively} compute the intersection point of two lines, each defined
406 by two (ellipsoidal) points or by an (ellipsoidal) start point and an
407 (initial) bearing from North.
409 @arg start1: Start point of the first line (L{LatLon}).
410 @arg end1: End point of the first line (L{LatLon}) or the initial bearing
411 at the first point (compass C{degrees360}).
412 @arg start2: Start point of the second line (L{LatLon}).
413 @arg end2: End point of the second line (L{LatLon}) or the initial bearing
414 at the second point (compass C{degrees360}).
415 @kwarg height: Optional height at the intersection (C{meter}, conventionally)
416 or C{None} for the mean height.
417 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the B{C{start2}}
418 and B{C{end*}} points (C{bool}).
419 @kwarg equidistant: An azimuthal equidistant projection (I{class} or function
420 L{pygeodesy.equidistant}) or C{None} for the preferred
421 C{B{start1}.Equidistant}.
422 @kwarg tol: Tolerance for convergence and for skew line distance and length
423 (C{meter}, conventionally).
424 @kwarg LatLon: Optional class to return the intersection points (L{LatLon})
425 or C{None}.
426 @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword arguments,
427 ignored if C{B{LatLon} is None}.
429 @return: An L{Intersection3Tuple}C{(point, outside1, outside2)} with C{point}
430 a B{C{LatLon}} or if C{B{LatLon} is None}, a L{LatLon4Tuple}C{(lat,
431 lon, height, datum)}.
433 @raise IntersectionError: Skew, colinear, parallel or otherwise
434 non-intersecting lines or no convergence
435 for the given B{C{tol}}.
437 @raise TypeError: Invalid or non-ellipsoidal B{C{start1}}, B{C{end1}},
438 B{C{start2}} or B{C{end2}} or invalid B{C{equidistant}}.
440 @note: For each line specified with an initial bearing, a pseudo-end point
441 is computed as the C{destination} along that bearing at about 1.5
442 times the distance from the start point to an initial gu-/estimate
443 of the intersection point (and between 1/8 and 3/8 of the authalic
444 earth perimeter).
446 @see: U{The B{ellipsoidal} case<https://GIS.StackExchange.com/questions/48937/
447 calculating-intersection-of-two-circles>} and U{Karney's paper
448 <https://ArXiv.org/pdf/1102.1215.pdf>}, pp 20-21, section B{14. MARITIME
449 BOUNDARIES} for more details about the iteration algorithm.
450 '''
451 return _intersection3(start1, end1, start2, end2, height=height, wrap=wrap,
452 equidistant=equidistant, tol=tol, LatLon=LatLon, **LatLon_kwds)
455def intersections2(center1, radius1, center2, radius2, height=None, wrap=False, # was=True
456 equidistant=None, tol=_TOL_M, LatLon=LatLon, **LatLon_kwds):
457 '''I{Iteratively} compute the intersection points of two circles, each defined
458 by an (ellipsoidal) center point and a radius.
460 @arg center1: Center of the first circle (L{LatLon}).
461 @arg radius1: Radius of the first circle (C{meter}, conventionally).
462 @arg center2: Center of the second circle (L{LatLon}).
463 @arg radius2: Radius of the second circle (C{meter}, same units as
464 B{C{radius1}}).
465 @kwarg height: Optional height for the intersection points (C{meter},
466 conventionally) or C{None} for the I{"radical height"}
467 at the I{radical line} between both centers.
468 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{center2}}
469 (C{bool}).
470 @kwarg equidistant: An azimuthal equidistant projection (I{class} or
471 function L{pygeodesy.equidistant}) or C{None} for
472 the preferred C{B{center1}.Equidistant}.
473 @kwarg tol: Convergence tolerance (C{meter}, same units as B{C{radius1}}
474 and B{C{radius2}}).
475 @kwarg LatLon: Optional class to return the intersection points (L{LatLon})
476 or C{None}.
477 @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword arguments,
478 ignored if C{B{LatLon} is None}.
480 @return: 2-Tuple of the intersection points, each a B{C{LatLon}} instance
481 or L{LatLon4Tuple}C{(lat, lon, height, datum)} if C{B{LatLon} is
482 None}. For abutting circles, both points are the same instance,
483 aka the I{radical center}.
485 @raise IntersectionError: Concentric, antipodal, invalid or non-intersecting
486 circles or no convergence for the B{C{tol}}.
488 @raise TypeError: Invalid or non-ellipsoidal B{C{center1}} or B{C{center2}}
489 or invalid B{C{equidistant}}.
491 @raise UnitError: Invalid B{C{radius1}}, B{C{radius2}} or B{C{height}}.
493 @see: U{The B{ellipsoidal} case<https://GIS.StackExchange.com/questions/48937/
494 calculating-intersection-of-two-circles>}, U{Karney's paper
495 <https://ArXiv.org/pdf/1102.1215.pdf>}, pp 20-21, section B{14. MARITIME BOUNDARIES},
496 U{circle-circle<https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>} and
497 U{sphere-sphere<https://MathWorld.Wolfram.com/Sphere-SphereIntersection.html>}
498 intersections.
499 '''
500 return _intersections2(center1, radius1, center2, radius2, height=height, wrap=wrap,
501 equidistant=equidistant, tol=tol, LatLon=LatLon, **LatLon_kwds)
504def nearestOn(point, point1, point2, within=True, height=None, wrap=False,
505 equidistant=None, tol=_TOL_M, LatLon=LatLon, **LatLon_kwds):
506 '''I{Iteratively} locate the closest point on the geodesic between
507 two other (ellipsoidal) points.
509 @arg point: Reference point (C{LatLon}).
510 @arg point1: Start point of the geodesic (C{LatLon}).
511 @arg point2: End point of the geodesic (C{LatLon}).
512 @kwarg within: If C{True} return the closest point I{between}
513 B{C{point1}} and B{C{point2}}, otherwise the
514 closest point elsewhere on the geodesic (C{bool}).
515 @kwarg height: Optional height for the closest point (C{meter},
516 conventionally) or C{None} or C{False} for the
517 interpolated height. If C{False}, the closest
518 takes the heights of the points into account.
519 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll both
520 B{C{point1}} and B{C{point2}} (C{bool}).
521 @kwarg equidistant: An azimuthal equidistant projection (I{class}
522 or function L{pygeodesy.equidistant}) or C{None}
523 for the preferred C{B{point}.Equidistant}.
524 @kwarg tol: Convergence tolerance (C{meter}).
525 @kwarg LatLon: Optional class to return the closest point
526 (L{LatLon}) or C{None}.
527 @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword
528 arguments, ignored if C{B{LatLon} is None}.
530 @return: Closest point, a B{C{LatLon}} instance or if C{B{LatLon}
531 is None}, a L{LatLon4Tuple}C{(lat, lon, height, datum)}.
533 @raise ImportError: Package U{geographiclib
534 <https://PyPI.org/project/geographiclib>}
535 not installed or not found, but only if
536 C{B{equidistant}=}L{EquidistantKarney}.
538 @raise TypeError: Invalid or non-ellipsoidal B{C{point}}, B{C{point1}}
539 or B{C{point2}} or invalid B{C{equidistant}}.
541 @raise ValueError: No convergence for the B{C{tol}}.
543 @see: U{The B{ellipsoidal} case<https://GIS.StackExchange.com/questions/48937/
544 calculating-intersection-of-two-circles>} and U{Karney's paper
545 <https://ArXiv.org/pdf/1102.1215.pdf>}, pp 20-21, section B{14. MARITIME
546 BOUNDARIES} for more details about the iteration algorithm.
547 '''
548 return _nearestOn(point, point1, point2, within=within, height=height, wrap=wrap,
549 equidistant=equidistant, tol=tol, LatLon=LatLon, **LatLon_kwds)
552@deprecated_function
553def perimeterOf(points, **closed_datum_wrap):
554 '''DEPRECATED, use function L{ellipsoidalExact.perimeterOf} or L{ellipsoidalKarney.perimeterOf}.
555 '''
556 try:
557 return _MODS.ellipsoidalKarney.perimeterOf(points, **closed_datum_wrap)
558 except ImportError:
559 return _MODS.ellipsoidalExact.perimeterOf(points, **closed_datum_wrap)
562__all__ += _ALL_OTHER(Cartesian, LatLon,
563 intersection3, intersections2, ispolar, # from .points
564 nearestOn) + _ALL_DOCS(areaOf, perimeterOf) # deprecated
566# **) MIT License
567#
568# Copyright (C) 2016-2023 -- mrJean1 at Gmail -- All Rights Reserved.
569#
570# Permission is hereby granted, free of charge, to any person obtaining a
571# copy of this software and associated documentation files (the "Software"),
572# to deal in the Software without restriction, including without limitation
573# the rights to use, copy, modify, merge, publish, distribute, sublicense,
574# and/or sell copies of the Software, and to permit persons to whom the
575# Software is furnished to do so, subject to the following conditions:
576#
577# The above copyright notice and this permission notice shall be included
578# in all copies or substantial portions of the Software.
579#
580# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
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