Coverage for pygeodesy/ellipsoidalVincenty.py: 99%

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1 

2# -*- coding: utf-8 -*- 

3 

4u'''Ellipsoidal, I{Vincenty}-based geodesy. 

5 

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}. 

9 

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>}. 

16 

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. 

21 

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. 

26 

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%). 

31 

32Here's an example usage of C{ellipsoidalVincenty}: 

33 

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 

39 

40To change the ellipsoid model used by the Vincenty formulae use: 

41 

42 >>> from pygeodesy import Datums 

43 >>> from pygeodesy.ellipsoidalVincenty import LatLon 

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

45 

46or by converting to anothor datum: 

47 

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 

52 

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 LatLonEllipsoidalBaseDI, \ 

56 _intersection3, _intersections2, \ 

57 _TOL_M, intersecant2 

58from pygeodesy.errors import _and, _ValueError, _xkwds 

59from pygeodesy.fmath import Fpolynomial, hypot, hypot1 

60from pygeodesy.interns import _ambiguous_, _antipodal_, _COLONSPACE_, \ 

61 _to_, _SPACE_, _limit_ # PYCHOK used! 

62from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY, _ALL_MODS as _MODS, _ALL_OTHER 

63from pygeodesy.namedTuples import Destination2Tuple, Destination3Tuple, \ 

64 Distance3Tuple 

65from pygeodesy.points import Fmt, ispolar # PYCHOK exported 

66from pygeodesy.props import deprecated_function, deprecated_method, \ 

67 property_doc_, property_RO 

68# from pygeodesy.streprs import Fmt # from .points 

69from pygeodesy.units import Number_, Scalar_ 

70from pygeodesy.utily import atan2b, atan2d, sincos2, sincos2d, unroll180, wrap180 

71 

72from math import atan2, cos, degrees, fabs, radians, tan 

73 

74__all__ = _ALL_LAZY.ellipsoidalVincenty 

75__version__ = '23.11.20' 

76 

77_antipodal_to_ = _SPACE_(_antipodal_, _to_) 

78 

79 

80class VincentyError(_ValueError): 

81 '''Error raised by I{Vincenty}'s C{Direct} and C{Inverse} methods 

82 for coincident points or lack of convergence. 

83 ''' 

84 pass 

85 

86 

87class Cartesian(CartesianEllipsoidalBase): 

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

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

90 ''' 

91 @property_RO 

92 def Ecef(self): 

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

94 ''' 

95 return _Ecef() 

96 

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

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

99 

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

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

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

103 class or specify C{B{LatLon}=None}. 

104 

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

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

107 with C{C} and C{M} if available. 

108 

109 @raise TypeError: Invalid B{C{LatLon_and_kwds}} argument. 

110 ''' 

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

112 return CartesianEllipsoidalBase.toLatLon(self, **kwds) 

113 

114 

115class LatLon(LatLonEllipsoidalBaseDI): 

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

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

118 ellipsoidal model of the earth to compute the geodesic distance 

119 and bearings between two given points or the destination point 

120 given an start point and (initial) bearing. 

121 

122 Set the earth model to be used with the keyword argument 

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

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

125 

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

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

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

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

130 L{LatLon.iterations}. 

131 ''' 

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

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

134 _iterations = 201 # 5, default max, 200 vs Veness' 1,000 

135 

136 @deprecated_method 

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

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

139 ''' 

140 return self.initialBearingTo(other, wrap=wrap) 

141 

142 @property_RO 

143 def Ecef(self): 

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

145 ''' 

146 return _Ecef() 

147 

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

149 def epsilon(self): 

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

151 ''' 

152 return self._epsilon 

153 

154 @epsilon.setter # PYCHOK setter! 

155 def epsilon(self, epsilon): 

156 '''Set the convergence epsilon (C{radians}). 

157 

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

159 

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

161 ''' 

162 self._epsilon = Scalar_(epsilon=epsilon) 

163 

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

165 def iterations(self): 

166 '''Get the iteration limit (C{int}). 

167 ''' 

168 return self._iterations - 1 

169 

170 @iterations.setter # PYCHOK setter! 

171 def iterations(self, limit): 

172 '''Set the iteration limit (C{int}). 

173 

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

175 

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

177 ''' 

178 self._iterations = Number_(limit, name=_limit_, low=4, high=1000) + 1 

179 

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

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

182 coordinates. 

183 

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

185 keyword arguments, ignored if C{B{Cartesian}=None}. Use 

186 C{B{Cartesian}=...} to override this L{Cartesian} class 

187 or specify C{B{Cartesian}=None}. 

188 

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

190 is C{None}, an L{Ecef9Tuple}C{(x, y, z, lat, lon, height, 

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

192 

193 @raise TypeError: Invalid B{C{Cartesian}}, B{C{datum}} or other 

194 B{C{Cartesian_datum_kwds}}. 

195 ''' 

196 kwds = _xkwds(Cartesian_datum_kwds, Cartesian=Cartesian, 

197 datum=self.datum) 

198 return LatLonEllipsoidalBaseDI.toCartesian(self, **kwds) 

199 

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

201 '''(INTERNAL) Direct Vincenty method. 

202 

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

204 

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

206 ellipsoids are not compatible. 

207 

208 @raise VincentyError: Vincenty fails to converge for the current 

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

210 limits. 

211 ''' 

212 E = self.ellipsoid() 

213 f = E.f 

214 

215 sb, cb = sincos2d(bearing) 

216 s1, c1, t1 = _sincostan3r(self.phi, f) 

217 

218 eps = self.epsilon 

219 s12 = atan2(t1, cb) * _2_0 

220 sa, ca2 = _sincos22(c1 * sb) 

221 A, B = _AB2(ca2 * E.e22) # e22 == (a / b)**2 - 1 

222 s = d = distance / (A * E.b) 

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

224 ss, cs = sincos2(s) 

225 c2sm, e = cos(s12 + s), s 

226 s = _Ds(B, cs, ss, c2sm, d) 

227 e = fabs(s - e) 

228 if e < eps: 

229 self._iteration = i 

230 break 

231 else: 

232 t = self._no_convergence(e) 

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

234 

235 t = s1 * ss - c1 * cs * cb 

236 # final bearing (reverse azimuth +/- 180) 

237 d = atan2b(sa, -t) 

238 if llr: 

239 b = cb * ss 

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

241 b = atan2d(sb * ss, -s1 * b + c1 * cs) + self.lon \ 

242 - degrees(_Dl(f, ca2, sa, s, cs, ss, c2sm)) 

243 t = Destination3Tuple(a, wrap180(b), d) 

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

245 else: 

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

247 r._iteration = i 

248 return r 

249 

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

251 '''(INTERNAL) Inverse Vincenty method. 

252 

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

254 

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

256 ellipsoids are not compatible. 

257 

258 @raise VincentyError: Vincenty fails to converge for the current 

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

260 limits and/or if this and the B{C{other}} 

261 point are coincident or near-antipodal. 

262 ''' 

263 E = self.ellipsoids(other) 

264 f = E.f 

265 

266 s1, c1, _ = _sincostan3r( self.phi, f) 

267 s2, c2, _ = _sincostan3r(other.phi, f) 

268 

269 c1c2, s1c2 = c1 * c2, s1 * c2 

270 c1s2, s1s2 = c1 * s2, s1 * s2 

271 

272 eps = self.epsilon 

273 d, _ = unroll180(self.lon, other.lon, wrap=wrap) 

274 dl = ll = radians(d) 

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

276 sll, cll = sincos2(ll) 

277 

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

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

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

281 t = self._is_to(other, True) 

282 raise VincentyError(_ambiguous_, txt=t) 

283 self._iteration = i 

284 # return zeros like Karney, unlike Veness 

285 return Distance3Tuple(_0_0, 0, 0, iteration=i) 

286 

287 cs = s1s2 + c1c2 * cll 

288 s, e = atan2(ss, cs), ll 

289 sa, ca2 = _sincos22(c1c2 * sll / ss) 

290 if ca2: 

291 c2sm = cs - _2_0 * s1s2 / ca2 

292 ll = _Dl(f, ca2, sa, s, cs, ss, c2sm, dl) 

293 else: # equatorial line 

294 ll = dl + f * sa * s 

295 e = fabs(ll - e) 

296 if e < eps: 

297 self._iteration = i 

298 break 

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

300# # omitted and applied *after* failure to converge below, 

301# # see footnote under Inverse <https://WikiPedia.org/wiki/ 

302# # Vincenty's_formulae> and <https://GitHub.com/chrisveness/ 

303# # geodesy/blob/master/latlon-ellipsoidal-vincenty.js> 

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

305 else: 

306 t = self._is_to(other, self.isantipodeTo(other, eps=eps)) 

307 raise VincentyError(self._no_convergence(e), txt=t) 

308 

309 if ca2: # e22 == (a / b)**2 - 1 

310 A, B = _AB2(ca2 * E.e22) 

311 s = -A * _Ds(B, cs, ss, c2sm, -s) 

312 

313 b = E.b 

314# if self.height or other.height: 

315# b += self._havg(other) 

316 d = b * s 

317 

318 if azis: # forward and reverse azimuth 

319 s, c = sincos2(ll) 

320 f = atan2b(c2 * s, c1s2 - s1c2 * c) 

321 r = atan2b(c1 * s, -s1c2 + c1s2 * c) 

322 else: 

323 f = r = _0_0 # NAN 

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

325 

326 def _is_to(self, other, anti): 

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

328 ''' 

329 t = _antipodal_to_ if anti else _to_ 

330 return _SPACE_(repr(self), t, repr(other)) 

331 

332 def _no_convergence(self, e): 

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

334 ''' 

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

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

337 return _COLONSPACE_(Fmt.no_convergence(e), _and(*t)) 

338 

339 

340def _AB2(u2): # WGS84 e22 = 0.00673949674227643 

341 # 2-Tuple C{(A, B)} polynomials 

342 if u2: 

343 A = Fpolynomial(u2, 16384, 4096, -768, 320, -175).fover(16384) 

344 B = Fpolynomial(u2, 0, 256, -128, 74, -47).fover( 1024) 

345 return A, B 

346 return _1_0, _0_0 

347 

348 

349def _c2sm2(c2sm): 

350 # C{2 * c2sm**2 - 1} 

351 return c2sm**2 * _2_0 - _1_0 

352 

353 

354def _Dl(f, ca2, sa, s, cs, ss, c2sm, dl=_0_0): 

355 # C{Dl} 

356 if f and sa: 

357 C = f * ca2 / _4_0 

358 C *= f - C * _3_0 + _1_0 

359 if C and ss: 

360 s += C * ss * (c2sm + 

361 C * cs * _c2sm2(c2sm)) 

362 dl += (_1_0 - C) * f * sa * s 

363 return dl 

364 

365 

366def _Ds(B, cs, ss, c2sm, d): 

367 # C{Ds - d} 

368 if B and ss: 

369 c2sm2 = _c2sm2(c2sm) 

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

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

372 B / _6_0 * c2sm * ss2)) 

373 d += B 

374 return d 

375 

376 

377def _Ecef(): 

378 # get the Ecef class and overwrite property_RO 

379 Cartesian.Ecef = LatLon.Ecef = E = _MODS.ecef.EcefVeness 

380 return E 

381 

382 

383def _sincos22(sa): 

384 # 2-Tuple C{(sin(a), cos(a)**2)} 

385 ca2 = _1_0 - sa**2 

386 return sa, (_0_0 if ca2 < EPS0 else ca2) # XXX EPS? 

387 

388 

389def _sincostan3r(a, f): 

390 # I{Reduced} 3-tuple C{(sin(B{a}), cos(B{a}), tan(B{a}))} 

391 if a: # see L{sincostan3} 

392 t = (_1_0 - f) * tan(a) 

393 if t: 

394 c = _1_0 / hypot1(t) 

395 s = c * t 

396 return s, c, t 

397 return _0_0, _1_0, _0_0 

398 

399 

400@deprecated_function 

401def areaOf(points, **datum_wrap): 

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

403 ''' 

404 try: 

405 return _MODS.ellipsoidalKarney.areaOf(points, **datum_wrap) 

406 except ImportError: 

407 return _MODS.ellipsoidalExact.areaOf(points, **datum_wrap) 

408 

409 

410def intersection3(start1, end1, start2, end2, height=None, wrap=False, # was=True 

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

412 '''I{Iteratively} compute the intersection point of two lines, each defined 

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

414 (initial) bearing from North. 

415 

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

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

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

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

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

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

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

423 or C{None} for the mean height. 

424 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the B{C{start2}} 

425 and B{C{end*}} points (C{bool}). 

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

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

428 C{B{start1}.Equidistant}. 

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

430 (C{meter}, conventionally). 

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

432 or C{None}. 

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

434 ignored if C{B{LatLon} is None}. 

435 

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

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

438 lon, height, datum)}. 

439 

440 @raise IntersectionError: Skew, colinear, parallel or otherwise 

441 non-intersecting lines or no convergence 

442 for the given B{C{tol}}. 

443 

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

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

446 

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

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

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

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

451 earth perimeter). 

452 

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

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

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

456 BOUNDARIES} for more details about the iteration algorithm. 

457 ''' 

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

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

460 

461 

462def intersections2(center1, radius1, center2, radius2, height=None, wrap=False, # was=True 

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

464 '''I{Iteratively} compute the intersection points of two circles, each defined 

465 by an (ellipsoidal) center point and a radius. 

466 

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

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

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

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

471 B{C{radius1}}). 

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

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

474 at the I{radical line} between both centers. 

475 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{center2}} 

476 (C{bool}). 

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

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

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

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

481 and B{C{radius2}}). 

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

483 or C{None}. 

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

485 ignored if C{B{LatLon} is None}. 

486 

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

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

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

490 aka the I{radical center}. 

491 

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

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

494 

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

496 or invalid B{C{equidistant}}. 

497 

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

499 

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

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

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

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

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

505 intersections. 

506 ''' 

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

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

509 

510 

511def nearestOn(point, point1, point2, within=True, height=None, wrap=False, 

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

513 '''I{Iteratively} locate the closest point on the geodesic between 

514 two other (ellipsoidal) points. 

515 

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

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

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

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

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

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

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

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

524 interpolated height. If C{False}, the closest 

525 takes the heights of the points into account. 

526 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll both 

527 B{C{point1}} and B{C{point2}} (C{bool}). 

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

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

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

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

532 @kwarg LatLon: Optional class to return the closest point 

533 (L{LatLon}) or C{None}. 

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

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

536 

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

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

539 

540 @raise ImportError: Package U{geographiclib 

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

542 not installed or not found, but only if 

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

544 

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

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

547 

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

549 

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

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

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

553 BOUNDARIES} for more details about the iteration algorithm. 

554 ''' 

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

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

557 

558 

559@deprecated_function 

560def perimeterOf(points, **closed_datum_wrap): 

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

562 ''' 

563 try: 

564 return _MODS.ellipsoidalKarney.perimeterOf(points, **closed_datum_wrap) 

565 except ImportError: 

566 return _MODS.ellipsoidalExact.perimeterOf(points, **closed_datum_wrap) 

567 

568 

569__all__ += _ALL_OTHER(Cartesian, LatLon, intersecant2, # from .ellipsoidalBaseDI 

570 intersection3, intersections2, ispolar, # from .points 

571 nearestOn) + _ALL_DOCS(areaOf, perimeterOf) # deprecated 

572 

573# **) MIT License 

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