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__ = '23.12.18' 

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{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 '''New point on an (oblate) ellipsoidal earth model, using the formulae devised 

117 by U{I{Thaddeus Vincenty}<https://WikiPedia.org/wiki/Vincenty's_formulae>} 

118 (1975) to compute geodesic distances, bearings (azimuths), etc. 

119 

120 Set the earth model to be used with the keyword argument datum. The default 

121 is C{Datums.WGS84}, which is the most globally accurate. For other models, 

122 see the L{Datums<pygeodesy.datums>}. 

123 

124 @note: This implementation of I{Vincenty} methods may not converge for some 

125 valid points, raising a L{VincentyError}. In that case, a result may 

126 be obtained by increasing the tolerance C{epsilon} and/or iteration 

127 C{limit}, see properties L{LatLon.epsilon} and L{LatLon.iterations}. 

128 ''' 

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

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

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

132 

133 @deprecated_method 

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

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

136 ''' 

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

138 

139 @property_RO 

140 def Ecef(self): 

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

142 ''' 

143 return _Ecef() 

144 

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

146 def epsilon(self): 

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

148 ''' 

149 return self._epsilon 

150 

151 @epsilon.setter # PYCHOK setter! 

152 def epsilon(self, epsilon): 

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

154 

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

156 

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

158 ''' 

159 self._epsilon = Scalar_(epsilon=epsilon) 

160 

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

162 def iterations(self): 

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

164 ''' 

165 return self._iterations - 1 

166 

167 @iterations.setter # PYCHOK setter! 

168 def iterations(self, limit): 

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

170 

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

172 

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

174 ''' 

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

176 

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

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

179 coordinates. 

180 

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

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

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

184 or specify C{B{Cartesian}=None}. 

185 

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

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

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

189 

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

191 B{C{Cartesian_datum_kwds}}. 

192 ''' 

193 kwds = _xkwds(Cartesian_datum_kwds, Cartesian=Cartesian, 

194 datum=self.datum) 

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

196 

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

198 '''(INTERNAL) Direct Vincenty method. 

199 

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

201 

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

203 ellipsoids are not compatible. 

204 

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

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

207 limits. 

208 ''' 

209 E = self.ellipsoid() 

210 f = E.f 

211 

212 sb, cb = sincos2d(bearing) 

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

214 

215 eps = self.epsilon 

216 s12 = atan2(t1, cb) * _2_0 

217 sa, ca2 = _sincos22(c1 * sb) 

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

219 s = d = distance / (A * E.b) 

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

221 ss, cs = sincos2(s) 

222 c2sm, e = cos(s12 + s), s 

223 s = _Ds(B, cs, ss, c2sm, d) 

224 e = fabs(s - e) 

225 if e < eps: 

226 self._iteration = i 

227 break 

228 else: 

229 t = self._no_convergence(e) 

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

231 

232 t = s1 * ss - c1 * cs * cb 

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

234 d = atan2b(sa, -t) 

235 if llr: 

236 b = cb * ss 

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

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

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

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

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

242 else: 

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

244 r._iteration = i 

245 return r 

246 

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

248 '''(INTERNAL) Inverse Vincenty method. 

249 

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

251 

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

253 ellipsoids are not compatible. 

254 

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

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

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

258 point are coincident or near-antipodal. 

259 ''' 

260 E = self.ellipsoids(other) 

261 f = E.f 

262 

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

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

265 

266 c1c2, s1c2 = c1 * c2, s1 * c2 

267 c1s2, s1s2 = c1 * s2, s1 * s2 

268 

269 eps = self.epsilon 

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

271 dl = ll = radians(d) 

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

273 sll, cll = sincos2(ll) 

274 

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

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

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

278 t = self._is_to(other, True) 

279 raise VincentyError(_ambiguous_, txt=t) 

280 self._iteration = i 

281 # return zeros like Karney, unlike Veness 

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

283 

284 cs = s1s2 + c1c2 * cll 

285 s, e = atan2(ss, cs), ll 

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

287 if ca2: 

288 c2sm = cs - _2_0 * s1s2 / ca2 

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

290 else: # equatorial line 

291 ll = dl + f * sa * s 

292 e = fabs(ll - e) 

293 if e < eps: 

294 self._iteration = i 

295 break 

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

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

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

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

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

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

302 else: 

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

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

305 

306 if ca2: # e22 == (a / b)**2 - 1 

307 A, B = _AB2(ca2 * E.e22) 

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

309 

310 b = E.b 

311# if self.height or other.height: 

312# b += self._havg(other) 

313 d = b * s 

314 

315 if azis: # forward and reverse azimuth 

316 s, c = sincos2(ll) 

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

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

319 else: 

320 f = r = _0_0 # NAN 

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

322 

323 def _is_to(self, other, anti): 

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

325 ''' 

326 t = _antipodal_to_ if anti else _to_ 

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

328 

329 def _no_convergence(self, e): 

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

331 ''' 

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

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

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

335 

336 

337def _AB2(u2): # WGS84 e22 = 0.00673949674227643 

338 # 2-Tuple C{(A, B)} polynomials 

339 if u2: 

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

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

342 return A, B 

343 return _1_0, _0_0 

344 

345 

346def _c2sm2(c2sm): 

347 # C{2 * c2sm**2 - 1} 

348 return c2sm**2 * _2_0 - _1_0 

349 

350 

351def _Dl(f, ca2, sa, s, cs, ss, c2sm, dl=_0_0): 

352 # C{Dl} 

353 if f and sa: 

354 C = f * ca2 / _4_0 

355 C *= f - C * _3_0 + _1_0 

356 if C and ss: 

357 s += C * ss * (c2sm + 

358 C * cs * _c2sm2(c2sm)) 

359 dl += (_1_0 - C) * f * sa * s 

360 return dl 

361 

362 

363def _Ds(B, cs, ss, c2sm, d): 

364 # C{Ds - d} 

365 if B and ss: 

366 c2sm2 = _c2sm2(c2sm) 

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

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

369 B / _6_0 * c2sm * ss2)) 

370 d += B 

371 return d 

372 

373 

374def _Ecef(): 

375 # get the Ecef class and overwrite property_RO 

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

377 return E 

378 

379 

380def _sincos22(sa): 

381 # 2-Tuple C{(sin(a), cos(a)**2)} 

382 ca2 = _1_0 - sa**2 

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

384 

385 

386def _sincostan3r(a, f): 

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

388 if a: # see L{sincostan3} 

389 t = (_1_0 - f) * tan(a) 

390 if t: 

391 c = _1_0 / hypot1(t) 

392 s = c * t 

393 return s, c, t 

394 return _0_0, _1_0, _0_0 

395 

396 

397@deprecated_function 

398def areaOf(points, **datum_wrap): 

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

400 ''' 

401 try: 

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

403 except ImportError: 

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

405 

406 

407def intersection3(start1, end1, start2, end2, height=None, wrap=False, # was=True 

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

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

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

411 (initial) bearing from North. 

412 

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

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

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

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

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

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

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

420 or C{None} for the mean height. 

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

422 and B{C{end*}} points (C{bool}). 

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

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

425 C{B{start1}.Equidistant}. 

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

427 (C{meter}, conventionally). 

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

429 or C{None}. 

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

431 ignored if C{B{LatLon} is None}. 

432 

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

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

435 lon, height, datum)}. 

436 

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

438 non-intersecting lines or no convergence 

439 for the given B{C{tol}}. 

440 

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

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

443 

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

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

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

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

448 earth perimeter). 

449 

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

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

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

453 BOUNDARIES} for more details about the iteration algorithm. 

454 ''' 

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

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

457 

458 

459def intersections2(center1, radius1, center2, radius2, height=None, wrap=False, # was=True 

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

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

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

463 

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

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

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

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

468 B{C{radius1}}). 

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

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

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

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

473 (C{bool}). 

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

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

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

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

478 and B{C{radius2}}). 

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

480 or C{None}. 

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

482 ignored if C{B{LatLon} is None}. 

483 

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

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

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

487 aka the I{radical center}. 

488 

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

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

491 

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

493 or invalid B{C{equidistant}}. 

494 

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

496 

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

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

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

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

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

502 intersections. 

503 ''' 

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

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

506 

507 

508def nearestOn(point, point1, point2, within=True, height=None, wrap=False, 

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

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

511 two other (ellipsoidal) points. 

512 

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

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

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

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

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

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

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

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

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

522 takes the heights of the points into account. 

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

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

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

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

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

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

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

530 (L{LatLon}) or C{None}. 

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

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

533 

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

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

536 

537 @raise ImportError: Package U{geographiclib 

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

539 not installed or not found, but only if 

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

541 

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

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

544 

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

546 

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

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

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

550 BOUNDARIES} for more details about the iteration algorithm. 

551 ''' 

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

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

554 

555 

556@deprecated_function 

557def perimeterOf(points, **closed_datum_wrap): 

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

559 ''' 

560 try: 

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

562 except ImportError: 

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

564 

565 

566__all__ += _ALL_OTHER(Cartesian, LatLon, intersecant2, # from .ellipsoidalBaseDI 

567 intersection3, intersections2, ispolar, # from .points 

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

569 

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