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 2005-2024} 

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 

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 atan2, atan2b, atan2d, sincos2, sincos2d, \ 

71 unroll180, wrap180 

72 

73from math import cos, degrees, fabs, radians, tan as _tan 

74 

75__all__ = _ALL_LAZY.ellipsoidalVincenty 

76__version__ = '24.11.26' 

77 

78_antipodal_to_ = _SPACE_(_antipodal_, _to_) 

79 

80 

81class VincentyError(_ValueError): 

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

83 for coincident points or lack of convergence. 

84 ''' 

85 pass 

86 

87 

88class Cartesian(CartesianEllipsoidalBase): 

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

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

91 ''' 

92 @property_RO 

93 def Ecef(self): 

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

95 ''' 

96 return _Ecef() 

97 

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

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

100 

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

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

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

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

105 

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

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

108 with C{C} and C{M} if available. 

109 

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

111 ''' 

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

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

114 

115 

116class LatLon(LatLonEllipsoidalBaseDI): 

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

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

119 (1975) to compute geodesic distances, bearings (azimuths), etc. 

120 

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

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

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

124 

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

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

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

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

129 ''' 

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

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

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

133 

134 @deprecated_method 

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

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

137 ''' 

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

139 

140 @property_RO 

141 def Ecef(self): 

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

143 ''' 

144 return _Ecef() 

145 

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

147 def epsilon(self): 

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

149 ''' 

150 return self._epsilon 

151 

152 @epsilon.setter # PYCHOK setter! 

153 def epsilon(self, epsilon): 

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

155 

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

157 

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

159 ''' 

160 self._epsilon = Scalar_(epsilon=epsilon) 

161 

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

163 def iterations(self): 

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

165 ''' 

166 return self._iterations - 1 

167 

168 @iterations.setter # PYCHOK setter! 

169 def iterations(self, limit): 

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

171 

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

173 

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

175 ''' 

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

177 

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

179 '''Convert this point to C{Vincenty}-based cartesian (ECEF) 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 C{B{Cartesian} is None}, 

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

188 C{C} and C{M} if available. 

189 

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

191 ''' 

192 kwds = _xkwds(Cartesian_datum_kwds, Cartesian=Cartesian, 

193 datum=self.datum) 

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

195 

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

197 '''(INTERNAL) Direct Vincenty method. 

198 

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

200 

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

202 not compatible. 

203 

204 @raise VincentyError: Vincenty fails to converge for the current limits, see 

205 L{epsilon<LatLon.epsilon>} and L{iterations<LatLon.iterations>}. 

206 ''' 

207 E = self.ellipsoid() 

208 f = E.f 

209 

210 sb, cb = sincos2d(bearing) 

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

212 

213 eps = self.epsilon 

214 s12 = atan2(t1, cb) * _2_0 

215 sa, ca2 = _sincos22(c1 * sb) 

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

217 s = d = distance / (A * E.b) 

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

219 ss, cs = sincos2(s) 

220 c2sm, e = cos(s12 + s), s 

221 s = _Ds(B, cs, ss, c2sm, d) 

222 e = fabs(s - e) 

223 if e < eps: 

224 self._iteration = i 

225 break 

226 else: 

227 t = self._no_convergence(e) 

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

229 

230 t = s1 * ss - c1 * cs * cb 

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

232 d = atan2b(sa, -t) 

233 if llr: 

234 b = cb * ss 

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

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

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

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

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

240 else: 

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

242 r._iteration = i 

243 return r 

244 

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

246 '''(INTERNAL) Inverse Vincenty method. 

247 

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

249 

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

251 ellipsoids are not compatible. 

252 

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

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

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

256 point are coincident or near-antipodal. 

257 ''' 

258 E = self.ellipsoids(other) 

259 f = E.f 

260 

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

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

263 

264 c1c2, s1c2 = c1 * c2, s1 * c2 

265 c1s2, s1s2 = c1 * s2, s1 * s2 

266 

267 eps = self.epsilon 

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

269 dl = ll = radians(d) 

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

271 sll, cll = sincos2(ll) 

272 

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

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

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

276 t = self._is_to(other, True) 

277 raise VincentyError(_ambiguous_, txt=t) 

278 self._iteration = i 

279 # return zeros like Karney, unlike Veness 

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

281 

282 cs = s1s2 + c1c2 * cll 

283 s, e = atan2(ss, cs), ll 

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

285 if ca2: 

286 c2sm = cs - _2_0 * s1s2 / ca2 

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

288 else: # equatorial line 

289 ll = dl + f * sa * s 

290 e = fabs(ll - e) 

291 if e < eps: 

292 self._iteration = i 

293 break 

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

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

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

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

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

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

300 else: 

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

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

303 

304 if ca2: # e22 == (a / b)**2 - 1 

305 A, B = _AB2(ca2 * E.e22) 

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

307 

308 b = E.b 

309# if self.height or other.height: 

310# b += self._havg(other) 

311 d = b * s 

312 

313 if azis: # forward and reverse azimuth 

314 s, c = sincos2(ll) 

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

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

317 else: 

318 f = r = _0_0 # NAN 

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

320 

321 def _is_to(self, other, anti): 

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

323 ''' 

324 t = _antipodal_to_ if anti else _to_ 

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

326 

327 def _no_convergence(self, e): 

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

329 ''' 

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

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

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

333 

334 

335def _AB2(u2): # WGS84 e22 = 0.00673949674227643 

336 # 2-Tuple C{(A, B)} polynomials 

337 if u2: 

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

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

340 return A, B 

341 return _1_0, _0_0 

342 

343 

344def _c2sm2(c2sm): 

345 # C{2 * c2sm**2 - 1} 

346 return c2sm**2 * _2_0 - _1_0 

347 

348 

349def _Dl(f, ca2, sa, s, cs, ss, c2sm, dl=_0_0): 

350 # C{Dl} 

351 if f and sa: 

352 C = f * ca2 / _4_0 

353 C *= f - C * _3_0 + _1_0 

354 if C and ss: 

355 s += C * ss * (c2sm + 

356 C * cs * _c2sm2(c2sm)) 

357 dl += (_1_0 - C) * f * sa * s 

358 return dl 

359 

360 

361def _Ds(B, cs, ss, c2sm, d): 

362 # C{Ds - d} 

363 if B and ss: 

364 c2sm2 = _c2sm2(c2sm) 

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

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

367 B / _6_0 * c2sm * ss2)) 

368 d += B 

369 return d 

370 

371 

372def _Ecef(): 

373 # get the Ecef class and overwrite property_RO 

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

375 return E 

376 

377 

378def _sincos22(sa): 

379 # 2-Tuple C{(sin(a), cos(a)**2)} 

380 ca2 = _1_0 - sa**2 

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

382 

383 

384def _sincostan3r(a, f): 

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

386 if a: # see L{sincostan3} 

387 t = (_1_0 - f) * _tan(a) 

388 if t: 

389 c = _1_0 / hypot1(t) 

390 s = c * t 

391 return s, c, t 

392 return _0_0, _1_0, _0_0 

393 

394 

395@deprecated_function 

396def areaOf(points, **datum_wrap): 

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

398 ''' 

399 try: 

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

401 except ImportError: 

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

403 

404 

405def intersection3(start1, end1, start2, end2, height=None, wrap=False, # was=True 

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

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

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

409 (initial) bearing from North. 

410 

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

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

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

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

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

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

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

418 or C{None} for the mean height. 

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

420 and B{C{end*}} points (C{bool}). 

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

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

423 C{B{start1}.Equidistant}. 

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

425 (C{meter}, conventionally). 

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

427 or C{None}. 

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

429 ignored if C{B{LatLon} is None}. 

430 

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

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

433 lon, height, datum)}. 

434 

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

436 non-intersecting lines or no convergence 

437 for the given B{C{tol}}. 

438 

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

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

441 

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

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

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

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

446 earth perimeter). 

447 

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

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

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

451 BOUNDARIES} for more details about the iteration algorithm. 

452 ''' 

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

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

455 

456 

457def intersections2(center1, radius1, center2, radius2, height=None, wrap=False, # was=True 

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

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

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

461 

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

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

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

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

466 B{C{radius1}}). 

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

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

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

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

471 (C{bool}). 

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

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

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

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

476 and B{C{radius2}}). 

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

478 or C{None}. 

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

480 ignored if C{B{LatLon} is None}. 

481 

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

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

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

485 aka the I{radical center}. 

486 

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

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

489 

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

491 or invalid B{C{equidistant}}. 

492 

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

494 

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

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

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

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

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

500 intersections. 

501 ''' 

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

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

504 

505 

506def nearestOn(point, point1, point2, within=True, height=None, wrap=False, 

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

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

509 two other (ellipsoidal) points. 

510 

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

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

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

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

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

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

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

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

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

520 takes the heights of the points into account. 

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

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

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

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

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

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

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

528 (L{LatLon}) or C{None}. 

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

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

531 

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

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

534 

535 @raise ImportError: Package U{geographiclib 

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

537 not installed or not found, but only if 

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

539 

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

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

542 

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

544 

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

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

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

548 BOUNDARIES} for more details about the iteration algorithm. 

549 ''' 

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

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

552 

553 

554@deprecated_function 

555def perimeterOf(points, **closed_datum_wrap): 

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

557 ''' 

558 try: 

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

560 except ImportError: 

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

562 

563 

564__all__ += _ALL_DOCS(Cartesian, LatLon, intersecant2, # from .ellipsoidalBaseDI 

565 intersection3, intersections2, ispolar, # from .points 

566 nearestOn, 

567 areaOf, perimeterOf) # DEPRECATED 

568 

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