Coverage for pygeodesy/sphericalTrigonometry.py: 94%

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1 

2# -*- coding: utf-8 -*- 

3 

4u'''Spherical, C{trigonometry}-based geodesy. 

5 

6Trigonometric classes geodetic (lat-/longitude) L{LatLon} and 

7geocentric (ECEF) L{Cartesian} and functions L{areaOf}, L{intersection}, 

8L{intersections2}, L{isPoleEnclosedBy}, L{meanOf}, L{nearestOn3} and 

9L{perimeterOf}, I{all spherical}. 

10 

11Pure Python implementation of geodetic (lat-/longitude) methods using 

12spherical trigonometry, transcoded from JavaScript originals by 

13I{(C) Chris Veness 2011-2016} published under the same MIT Licence**, see 

14U{Latitude/Longitude<https://www.Movable-Type.co.UK/scripts/latlong.html>}. 

15''' 

16# make sure int/int division yields float quotient, see .basics 

17from __future__ import division as _; del _ # PYCHOK semicolon 

18 

19from pygeodesy.basics import copysign0, isscalar, map1, signOf 

20from pygeodesy.constants import EPS, EPS1, EPS4, PI, PI2, PI_2, PI_4, R_M, \ 

21 isnear0, isnear1, isnon0, _0_0, _0_5, \ 

22 _1_0, _2_0, _90_0 

23from pygeodesy.datums import _ellipsoidal_datum, _mean_radius 

24from pygeodesy.errors import _AssertionError, CrossError, crosserrors, \ 

25 _ValueError, IntersectionError, _xError, \ 

26 _xkwds, _xkwds_get, _xkwds_pop 

27from pygeodesy.fmath import favg, fdot, fmean, hypot 

28from pygeodesy.fsums import Fsum, fsum, fsumf_ 

29from pygeodesy.formy import antipode_, bearing_, _bearingTo2, excessAbc_, \ 

30 excessGirard_, excessLHuilier_, opposing_, _radical2, \ 

31 vincentys_ 

32from pygeodesy.interns import _1_, _2_, _coincident_, _composite_, _colinear_, \ 

33 _concentric_, _convex_, _end_, _infinite_, \ 

34 _invalid_, _line_, _near_, _not_, _null_, \ 

35 _point_, _SPACE_, _too_ 

36from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS, _ALL_OTHER 

37# from pygeodesy.named import notImplemented # from .points 

38# from pygeodesy.nvectorBase import NvectorBase, sumOf # _MODE 

39from pygeodesy.namedTuples import LatLon2Tuple, LatLon3Tuple, \ 

40 NearestOn3Tuple, Triangle7Tuple, \ 

41 Triangle8Tuple 

42from pygeodesy.points import ispolar, nearestOn5 as _nearestOn5, \ 

43 notImplemented, Fmt as _Fmt # XXX shadowed 

44from pygeodesy.props import deprecated_function, deprecated_method 

45from pygeodesy.sphericalBase import _angular, CartesianSphericalBase, _intersecant2, \ 

46 LatLonSphericalBase, _rads3, _trilaterate5 

47# from pygeodesy.streprs import Fmt as _Fmt # from .points XXX shadowed 

48from pygeodesy.units import Bearing_, Height, Lam_, Phi_, Radius, \ 

49 Radius_, Scalar 

50from pygeodesy.utily import acos1, asin1, degrees90, degrees180, degrees2m, \ 

51 m2radians, radiansPI2, sincos2_, tan_2, _unrollon, \ 

52 unrollPI, _unrollon3, _Wrap, wrap180, wrapPI 

53from pygeodesy.vector3d import sumOf, Vector3d 

54 

55from math import asin, atan2, cos, degrees, fabs, radians, sin 

56 

57__all__ = _ALL_LAZY.sphericalTrigonometry 

58__version__ = '23.05.25' 

59 

60_parallel_ = 'parallel' 

61 

62_PI_EPS4 = PI - EPS4 

63if _PI_EPS4 >= PI: 

64 raise _AssertionError(EPS4=EPS4, PI=PI, PI_EPS4=_PI_EPS4) 

65 

66 

67class Cartesian(CartesianSphericalBase): 

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

69 spherical, geodetic L{LatLon}. 

70 ''' 

71 

72 def toLatLon(self, **LatLon_and_kwds): # PYCHOK LatLon=LatLon 

73 '''Convert this cartesian point to a C{spherical} geodetic point. 

74 

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

76 arguments. Use C{B{LatLon}=...} to override 

77 this L{LatLon} class or specify C{B{LatLon}=None}. 

78 

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

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

81 with C{C} and C{M} if available. 

82 

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

84 ''' 

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

86 return CartesianSphericalBase.toLatLon(self, **kwds) 

87 

88 

89class LatLon(LatLonSphericalBase): 

90 '''New point on spherical model earth model. 

91 

92 @example: 

93 

94 >>> p = LatLon(52.205, 0.119) # height=0 

95 ''' 

96 

97 def _ab1_ab2_db5(self, other, wrap): 

98 '''(INTERNAL) Helper for several methods. 

99 ''' 

100 a1, b1 = self.philam 

101 a2, b2 = self.others(other, up=2).philam 

102 if wrap: 

103 a2, b2 = _Wrap.philam(a2, b2) 

104 db, b2 = unrollPI(b1, b2, wrap=wrap) 

105 else: # unrollPI shortcut 

106 db = b2 - b1 

107 return a1, b1, a2, b2, db 

108 

109 def alongTrackDistanceTo(self, start, end, radius=R_M, wrap=False): 

110 '''Compute the (angular) distance (signed) from the start to 

111 the closest point on the great circle line defined by a 

112 start and an end point. 

113 

114 That is, if a perpendicular is drawn from this point to the 

115 great circle line, the along-track distance is the distance 

116 from the start point to the point where the perpendicular 

117 crosses the line. 

118 

119 @arg start: Start point of the great circle line (L{LatLon}). 

120 @arg end: End point of the great circle line (L{LatLon}). 

121 @kwarg radius: Mean earth radius (C{meter}) or C{None}. 

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

123 the B{C{start}} and B{C{end}} point (C{bool}). 

124 

125 @return: Distance along the great circle line (C{meter}, 

126 same units as B{C{radius}}) or C{radians} if 

127 C{B{radius} is None}, positive if I{after} the 

128 B{C{start}} toward the B{C{end}} point of the 

129 line, I{negative} if before or C{0} if at the 

130 B{C{start}} point. 

131 

132 @raise TypeError: Invalid B{C{start}} or B{C{end}} point. 

133 

134 @raise ValueError: Invalid B{C{radius}}. 

135 

136 @example: 

137 

138 >>> p = LatLon(53.2611, -0.7972) 

139 

140 >>> s = LatLon(53.3206, -1.7297) 

141 >>> e = LatLon(53.1887, 0.1334) 

142 >>> d = p.alongTrackDistanceTo(s, e) # 62331.58 

143 ''' 

144 r, x, b = self._a_x_b3(start, end, radius, wrap) 

145 cx = cos(x) 

146 return _0_0 if isnear0(cx) else \ 

147 _r2m(copysign0(acos1(cos(r) / cx), cos(b)), radius) 

148 

149 def _a_x_b3(self, start, end, radius, wrap): 

150 '''(INTERNAL) Helper for .along-/crossTrackDistanceTo. 

151 ''' 

152 s = self.others(start=start) 

153 e = self.others(end=end) 

154 s, e, w = _unrollon3(self, s, e, wrap) 

155 

156 r = Radius_(radius) 

157 r = s.distanceTo(self, r, wrap=w) / r 

158 

159 b = radians(s.initialBearingTo(self, wrap=w) 

160 - s.initialBearingTo(e, wrap=w)) 

161 x = asin(sin(r) * sin(b)) 

162 return r, x, -b 

163 

164 @deprecated_method 

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

166 '''DEPRECATED, use method L{initialBearingTo}. 

167 ''' 

168 return self.initialBearingTo(other, wrap=wrap, raiser=raiser) 

169 

170 def crossingParallels(self, other, lat, wrap=False): 

171 '''Return the pair of meridians at which a great circle defined 

172 by this and an other point crosses the given latitude. 

173 

174 @arg other: The other point defining great circle (L{LatLon}). 

175 @arg lat: Latitude at the crossing (C{degrees}). 

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

177 B{C{other}} point (C{bool}). 

178 

179 @return: 2-Tuple C{(lon1, lon2)}, both in C{degrees180} or 

180 C{None} if the great circle doesn't reach B{C{lat}}. 

181 ''' 

182 a1, b1, a2, b2, db = self._ab1_ab2_db5(other, wrap) 

183 sa, ca, sa1, ca1, \ 

184 sa2, ca2, sdb, cdb = sincos2_(radians(lat), a1, a2, db) 

185 sa1 *= ca2 * ca 

186 

187 x = sa1 * sdb 

188 y = sa1 * cdb - ca1 * sa2 * ca 

189 z = ca1 * sdb * ca2 * sa 

190 

191 h = hypot(x, y) 

192 if h < EPS or fabs(z) > h: # PYCHOK no cover 

193 return None # great circle doesn't reach latitude 

194 

195 m = atan2(-y, x) + b1 # longitude at max latitude 

196 d = acos1(z / h) # delta longitude to intersections 

197 return degrees180(m - d), degrees180(m + d) 

198 

199 def crossTrackDistanceTo(self, start, end, radius=R_M, wrap=False): 

200 '''Compute the (signed, angular) distance from this point to 

201 the great circle defined by a start and an end point. 

202 

203 @arg start: Start point of the great circle line (L{LatLon}). 

204 @arg end: End point of the great circle line (L{LatLon}). 

205 @kwarg radius: Mean earth radius (C{meter}) or C{None}. 

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

207 the B{C{start}} and B{C{end}} point (C{bool}). 

208 

209 @return: Distance to the great circle (I{negative} if to 

210 the left or I{positive} if to the right of the 

211 line) (C{meter}, same units as B{C{radius}} or 

212 C{radians} if B{C{radius}} is C{None}). 

213 

214 @raise TypeError: If B{C{start}} or B{C{end}} is not L{LatLon}. 

215 

216 @raise ValueError: Invalid B{C{radius}}. 

217 

218 @example: 

219 

220 >>> p = LatLon(53.2611, -0.7972) 

221 

222 >>> s = LatLon(53.3206, -1.7297) 

223 >>> e = LatLon(53.1887, 0.1334) 

224 >>> d = p.crossTrackDistanceTo(s, e) # -307.5 

225 ''' 

226 _, x, _ = self._a_x_b3(start, end, radius, wrap) 

227 return _r2m(x, radius) 

228 

229 def destination(self, distance, bearing, radius=R_M, height=None): 

230 '''Locate the destination from this point after having 

231 travelled the given distance on the given initial bearing. 

232 

233 @arg distance: Distance travelled (C{meter}, same units as 

234 B{C{radius}}). 

235 @arg bearing: Bearing from this point (compass C{degrees360}). 

236 @kwarg radius: Mean earth radius (C{meter}). 

237 @kwarg height: Optional height at destination (C{meter}, same 

238 units a B{C{radius}}). 

239 

240 @return: Destination point (L{LatLon}). 

241 

242 @raise ValueError: Invalid B{C{distance}}, B{C{bearing}}, 

243 B{C{radius}} or B{C{height}}. 

244 

245 @example: 

246 

247 >>> p1 = LatLon(51.4778, -0.0015) 

248 >>> p2 = p1.destination(7794, 300.7) 

249 >>> p2.toStr() # '51.5135°N, 000.0983°W' 

250 ''' 

251 a, b = self.philam 

252 r, t = _angular(distance, radius), Bearing_(bearing) 

253 

254 a, b = _destination2(a, b, r, t) 

255 h = self._heigHt(height) 

256 return self.classof(degrees90(a), degrees180(b), height=h) 

257 

258 def distanceTo(self, other, radius=R_M, wrap=False): 

259 '''Compute the (angular) distance from this to an other point. 

260 

261 @arg other: The other point (L{LatLon}). 

262 @kwarg radius: Mean earth radius (C{meter}) or C{None}. 

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

264 the B{C{other}} point (C{bool}). 

265 

266 @return: Distance between this and the B{C{other}} point 

267 (C{meter}, same units as B{C{radius}} or 

268 C{radians} if B{C{radius}} is C{None}). 

269 

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

271 

272 @raise ValueError: Invalid B{C{radius}}. 

273 

274 @example: 

275 

276 >>> p1 = LatLon(52.205, 0.119) 

277 >>> p2 = LatLon(48.857, 2.351); 

278 >>> d = p1.distanceTo(p2) # 404300 

279 ''' 

280 a1, _, a2, _, db = self._ab1_ab2_db5(other, wrap) 

281 return _r2m(vincentys_(a2, a1, db), radius) 

282 

283# @Property_RO 

284# def Ecef(self): 

285# '''Get the ECEF I{class} (L{EcefVeness}), I{lazily}. 

286# ''' 

287# return _MODS.ecef.EcefKarney 

288 

289 def greatCircle(self, bearing, Vector=Vector3d, **Vector_kwds): 

290 '''Compute the vector normal to great circle obtained by heading 

291 on the given initial bearing from this point. 

292 

293 Direction of vector is such that initial bearing vector 

294 b = c × n, where n is an n-vector representing this point. 

295 

296 @arg bearing: Bearing from this point (compass C{degrees360}). 

297 @kwarg Vector: Vector class to return the great circle, 

298 overriding the default L{Vector3d}. 

299 @kwarg Vector_kwds: Optional, additional keyword argunents 

300 for B{C{Vector}}. 

301 

302 @return: Vector representing great circle (C{Vector}). 

303 

304 @raise ValueError: Invalid B{C{bearing}}. 

305 

306 @example: 

307 

308 >>> p = LatLon(53.3206, -1.7297) 

309 >>> g = p.greatCircle(96.0) 

310 >>> g.toStr() # (-0.794, 0.129, 0.594) 

311 ''' 

312 a, b = self.philam 

313 sa, ca, sb, cb, st, ct = sincos2_(a, b, Bearing_(bearing)) 

314 

315 return Vector(sb * ct - cb * sa * st, 

316 -cb * ct - sb * sa * st, 

317 ca * st, **Vector_kwds) # XXX .unit()? 

318 

319 def initialBearingTo(self, other, wrap=False, raiser=False): 

320 '''Compute the initial bearing (forward azimuth) from this 

321 to an other point. 

322 

323 @arg other: The other point (spherical L{LatLon}). 

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

325 the B{C{other}} point (C{bool}). 

326 @kwarg raiser: Optionally, raise L{CrossError} (C{bool}), 

327 use C{B{raiser}=True} for behavior like 

328 C{sphericalNvector.LatLon.initialBearingTo}. 

329 

330 @return: Initial bearing (compass C{degrees360}). 

331 

332 @raise CrossError: If this and the B{C{other}} point coincide, 

333 provided both B{C{raiser}} is C{True} and 

334 L{pygeodesy.crosserrors} is C{True}. 

335 

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

337 

338 @example: 

339 

340 >>> p1 = LatLon(52.205, 0.119) 

341 >>> p2 = LatLon(48.857, 2.351) 

342 >>> b = p1.initialBearingTo(p2) # 156.2 

343 ''' 

344 a1, b1, a2, b2, db = self._ab1_ab2_db5(other, wrap) 

345 # XXX behavior like sphericalNvector.LatLon.initialBearingTo 

346 if raiser and crosserrors() and max(fabs(a2 - a1), fabs(db)) < EPS: 

347 raise CrossError(_point_, self, other=other, wrap=wrap, txt=_coincident_) 

348 

349 return degrees(bearing_(a1, b1, a2, b2, final=False)) 

350 

351 def intermediateTo(self, other, fraction, height=None, wrap=False): 

352 '''Locate the point at given fraction between (or along) this 

353 and an other point. 

354 

355 @arg other: The other point (L{LatLon}). 

356 @arg fraction: Fraction between both points (C{scalar}, 

357 0.0 at this and 1.0 at the other point). 

358 @kwarg height: Optional height, overriding the intermediate 

359 height (C{meter}). 

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

361 B{C{other}} point (C{bool}). 

362 

363 @return: Intermediate point (L{LatLon}). 

364 

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

366 

367 @raise ValueError: Invalid B{C{fraction}} or B{C{height}}. 

368 

369 @see: Methods C{midpointTo} and C{rhumbMidpointTo}. 

370 

371 @example: 

372 

373 >>> p1 = LatLon(52.205, 0.119) 

374 >>> p2 = LatLon(48.857, 2.351) 

375 >>> p = p1.intermediateTo(p2, 0.25) # 51.3721°N, 000.7073°E 

376 ''' 

377 p = self 

378 f = Scalar(fraction=fraction) 

379 if not isnear0(f): 

380 p = p.others(other) 

381 if wrap: 

382 p = _Wrap.point(p) 

383 if not isnear1(f): # and not near0 

384 a1, b1 = self.philam 

385 a2, b2 = p.philam 

386 db, b2 = unrollPI(b1, b2, wrap=wrap) 

387 r = vincentys_(a2, a1, db) 

388 sr = sin(r) 

389 if isnon0(sr): 

390 sa1, ca1, sa2, ca2, \ 

391 sb1, cb1, sb2, cb2 = sincos2_(a1, a2, b1, b2) 

392 

393 t = f * r 

394 a = sin(r - t) # / sr superflous 

395 b = sin( t) # / sr superflous 

396 

397 x = a * ca1 * cb1 + b * ca2 * cb2 

398 y = a * ca1 * sb1 + b * ca2 * sb2 

399 z = a * sa1 + b * sa2 

400 

401 a = atan2(z, hypot(x, y)) 

402 b = atan2(y, x) 

403 

404 else: # PYCHOK no cover 

405 a = favg(a1, a2, f=f) # coincident 

406 b = favg(b1, b2, f=f) 

407 

408 h = self._havg(other, f=f, h=height) 

409 p = self.classof(degrees90(a), degrees180(b), height=h) 

410 return p 

411 

412 def intersection(self, end1, other, end2, height=None, wrap=False): 

413 '''Compute the intersection point of two lines, each defined by 

414 two points or a start point and bearing from North. 

415 

416 @arg end1: End point of this line (L{LatLon}) or the initial 

417 bearing at this point (compass C{degrees360}). 

418 @arg other: Start point of the other line (L{LatLon}). 

419 @arg end2: End point of the other line (L{LatLon}) or the 

420 initial bearing at the B{C{other}} point (compass 

421 C{degrees360}). 

422 @kwarg height: Optional height for intersection point, 

423 overriding the mean height (C{meter}). 

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

425 B{C{start2}} and both B{C{end*}} points (C{bool}). 

426 

427 @return: The intersection point (L{LatLon}). An alternate 

428 intersection point might be the L{antipode} to 

429 the returned result. 

430 

431 @raise IntersectionError: Ambiguous or infinite intersection 

432 or colinear, parallel or otherwise 

433 non-intersecting lines. 

434 

435 @raise TypeError: If B{C{other}} is not L{LatLon} or B{C{end1}} 

436 or B{C{end2}} not C{scalar} nor L{LatLon}. 

437 

438 @raise ValueError: Invalid B{C{height}} or C{null} line. 

439 

440 @example: 

441 

442 >>> p = LatLon(51.8853, 0.2545) 

443 >>> s = LatLon(49.0034, 2.5735) 

444 >>> i = p.intersection(108.547, s, 32.435) # '50.9078°N, 004.5084°E' 

445 ''' 

446 try: 

447 s2 = self.others(other) 

448 return _intersect(self, end1, s2, end2, height=height, wrap=wrap, 

449 LatLon=self.classof) 

450 except (TypeError, ValueError) as x: 

451 raise _xError(x, start1=self, end1=end1, 

452 other=other, end2=end2, wrap=wrap) 

453 

454 def intersections2(self, rad1, other, rad2, radius=R_M, eps=_0_0, 

455 height=None, wrap=True): 

456 '''Compute the intersection points of two circles, each defined 

457 by a center point and radius. 

458 

459 @arg rad1: Radius of the this circle (C{meter} or C{radians}, 

460 see B{C{radius}}). 

461 @arg other: Center point of the other circle (L{LatLon}). 

462 @arg rad2: Radius of the other circle (C{meter} or C{radians}, 

463 see B{C{radius}}). 

464 @kwarg radius: Mean earth radius (C{meter} or C{None} if B{C{rad1}}, 

465 B{C{rad2}} and B{C{eps}} are given in C{radians}). 

466 @kwarg eps: Required overlap (C{meter} or C{radians}, see 

467 B{C{radius}}). 

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

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

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

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

472 B{C{other}} point (C{bool}). 

473 

474 @return: 2-Tuple of the intersection points, each a L{LatLon} 

475 instance. For abutting circles, both intersection 

476 points are the same instance, aka the I{radical center}. 

477 

478 @raise IntersectionError: Concentric, antipodal, invalid or 

479 non-intersecting circles. 

480 

481 @raise TypeError: If B{C{other}} is not L{LatLon}. 

482 

483 @raise ValueError: Invalid B{C{rad1}}, B{C{rad2}}, B{C{radius}}, 

484 B{C{eps}} or B{C{height}}. 

485 ''' 

486 try: 

487 c2 = self.others(other) 

488 return _intersects2(self, rad1, c2, rad2, radius=radius, eps=eps, 

489 height=height, wrap=wrap, 

490 LatLon=self.classof) 

491 except (TypeError, ValueError) as x: 

492 raise _xError(x, center=self, rad1=rad1, 

493 other=other, rad2=rad2, wrap=wrap) 

494 

495 @deprecated_method 

496 def isEnclosedBy(self, points): # PYCHOK no cover 

497 '''DEPRECATED, use method C{isenclosedBy}.''' 

498 return self.isenclosedBy(points) 

499 

500 def isenclosedBy(self, points, wrap=False): 

501 '''Check whether a (convex) polygon or composite encloses this point. 

502 

503 @arg points: The polygon points or composite (L{LatLon}[], 

504 L{BooleanFHP} or L{BooleanGH}). 

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

506 B{C{points}} (C{bool}). 

507 

508 @return: C{True} if this point is inside the polygon or 

509 composite, C{False} otherwise. 

510 

511 @raise PointsError: Insufficient number of B{C{points}}. 

512 

513 @raise TypeError: Some B{C{points}} are not L{LatLon}. 

514 

515 @raise ValueError: Invalid B{C{points}}, non-convex polygon. 

516 

517 @see: Functions L{pygeodesy.isconvex}, L{pygeodesy.isenclosedBy} 

518 and L{pygeodesy.ispolar} especially if the B{C{points}} may 

519 enclose a pole or wrap around the earth I{longitudinally}. 

520 ''' 

521 if _MODS.booleans.isBoolean(points): 

522 return points._encloses(self.lat, self.lon, wrap=wrap) 

523 

524 Ps = self.PointsIter(points, loop=2, dedup=True, wrap=wrap) 

525 n0 = self._N_vector 

526 

527 v2 = Ps[0]._N_vector 

528 p1 = Ps[1] 

529 v1 = p1._N_vector 

530 # check whether this point on same side of all 

531 # polygon edges (to the left or right depending 

532 # on the anti-/clockwise polygon direction) 

533 gc1 = v2.cross(v1) 

534 t0 = gc1.angleTo(n0) > PI_2 

535 s0 = None 

536 # get great-circle vector for each edge 

537 for i, p2 in Ps.enumerate(closed=True): 

538 if wrap and not Ps.looped: 

539 p2 = _unrollon(p1, p2) 

540 p1 = p2 

541 v2 = p2._N_vector 

542 gc = v1.cross(v2) 

543 t = gc.angleTo(n0) > PI_2 

544 if t != t0: # different sides of edge i 

545 return False # outside 

546 

547 # check for convex polygon: angle between 

548 # gc vectors, signed by direction of n0 

549 # (otherwise the test above is not reliable) 

550 s = signOf(gc1.angleTo(gc, vSign=n0)) 

551 if s != s0: 

552 if s0 is None: 

553 s0 = s 

554 else: 

555 t = _Fmt.SQUARE(points=i) 

556 raise _ValueError(t, p2, wrap=wrap, txt=_not_(_convex_)) 

557 gc1, v1 = gc, v2 

558 

559 return True # inside 

560 

561 def midpointTo(self, other, height=None, fraction=_0_5, wrap=False): 

562 '''Find the midpoint between this and an other point. 

563 

564 @arg other: The other point (L{LatLon}). 

565 @kwarg height: Optional height for midpoint, overriding 

566 the mean height (C{meter}). 

567 @kwarg fraction: Midpoint location from this point (C{scalar}), 

568 may be negative or greater than 1.0. 

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

570 B{C{other}} point (C{bool}). 

571 

572 @return: Midpoint (L{LatLon}). 

573 

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

575 

576 @raise ValueError: Invalid B{C{height}}. 

577 

578 @see: Methods C{intermediateTo} and C{rhumbMidpointTo}. 

579 

580 @example: 

581 

582 >>> p1 = LatLon(52.205, 0.119) 

583 >>> p2 = LatLon(48.857, 2.351) 

584 >>> m = p1.midpointTo(p2) # '50.5363°N, 001.2746°E' 

585 ''' 

586 if fraction is _0_5: 

587 # see <https://MathForum.org/library/drmath/view/51822.html> 

588 a1, b, a2, _, db = self._ab1_ab2_db5(other, wrap) 

589 sa1, ca1, sa2, ca2, sdb, cdb = sincos2_(a1, a2, db) 

590 

591 x = ca2 * cdb + ca1 

592 y = ca2 * sdb 

593 

594 a = atan2(sa1 + sa2, hypot(x, y)) 

595 b += atan2(y, x) 

596 

597 h = self._havg(other, h=height) 

598 r = self.classof(degrees90(a), degrees180(b), height=h) 

599 else: 

600 r = self.intermediateTo(other, fraction, height=height, wrap=wrap) 

601 return r 

602 

603 def nearestOn(self, point1, point2, radius=R_M, **wrap_adjust_limit): 

604 '''Locate the point between two points closest to this point. 

605 

606 Distances are approximated by function L{pygeodesy.equirectangular_}, 

607 subject to the supplied B{C{options}}. 

608 

609 @arg point1: Start point (L{LatLon}). 

610 @arg point2: End point (L{LatLon}). 

611 @kwarg radius: Mean earth radius (C{meter}). 

612 @kwarg wrap_adjust_limit: Optional keyword arguments for functions 

613 L{sphericalTrigonometry.nearestOn3} and 

614 L{pygeodesy.equirectangular_}, 

615 

616 @return: Closest point on the great circle line (L{LatLon}). 

617 

618 @raise LimitError: Lat- and/or longitudinal delta exceeds B{C{limit}}, 

619 see function L{pygeodesy.equirectangular_}. 

620 

621 @raise NotImplementedError: Keyword argument C{B{within}=False} 

622 is not (yet) supported. 

623 

624 @raise TypeError: Invalid B{C{point1}} or B{C{point2}}. 

625 

626 @raise ValueError: Invalid B{C{radius}} or B{C{options}}. 

627 

628 @see: Functions L{pygeodesy.equirectangular_} and L{pygeodesy.nearestOn5} 

629 and method L{sphericalTrigonometry.LatLon.nearestOn3}. 

630 ''' 

631 # remove kwarg B{C{within}} if present 

632 w = _xkwds_pop(wrap_adjust_limit, within=True) 

633 if not w: 

634 notImplemented(self, within=w) 

635 

636# # UNTESTED - handle C{B{within}=False} and C{B{within}=True} 

637# wrap = _xkwds_get(options, wrap=False) 

638# a = self.alongTrackDistanceTo(point1, point2, radius=radius, wrap=wrap) 

639# if fabs(a) < EPS or (within and a < EPS): 

640# return point1 

641# d = point1.distanceTo(point2, radius=radius, wrap=wrap) 

642# if isnear0(d): 

643# return point1 # or point2 

644# elif fabs(d - a) < EPS or (a + EPS) > d: 

645# return point2 

646# f = a / d 

647# if within: 

648# if f > EPS1: 

649# return point2 

650# elif f < EPS: 

651# return point1 

652# return point1.intermediateTo(point2, f, wrap=wrap) 

653 

654 # without kwarg B{C{within}}, use backward compatible .nearestOn3 

655 return self.nearestOn3([point1, point2], closed=False, radius=radius, 

656 **wrap_adjust_limit)[0] 

657 

658 @deprecated_method 

659 def nearestOn2(self, points, closed=False, radius=R_M, **options): # PYCHOK no cover 

660 '''DEPRECATED, use method L{sphericalTrigonometry.LatLon.nearestOn3}. 

661 

662 @return: ... 2-Tuple C{(closest, distance)} of the closest 

663 point (L{LatLon}) on the polygon and the distance 

664 to that point from this point in C{meter}, same 

665 units of B{C{radius}}. 

666 ''' 

667 r = self.nearestOn3(points, closed=closed, radius=radius, **options) 

668 return r.closest, r.distance 

669 

670 def nearestOn3(self, points, closed=False, radius=R_M, **wrap_adjust_limit): 

671 '''Locate the point on a polygon closest to this point. 

672 

673 Distances are approximated by function L{pygeodesy.equirectangular_}, 

674 subject to the supplied B{C{options}}. 

675 

676 @arg points: The polygon points (L{LatLon}[]). 

677 @kwarg closed: Optionally, close the polygon (C{bool}). 

678 @kwarg radius: Mean earth radius (C{meter}). 

679 @kwarg wrap_adjust_limit: Optional keyword arguments for function 

680 L{sphericalTrigonometry.nearestOn3} and 

681 L{pygeodesy.equirectangular_}, 

682 

683 @return: A L{NearestOn3Tuple}C{(closest, distance, angle)} of the 

684 C{closest} point (L{LatLon}), the L{pygeodesy.equirectangular_} 

685 C{distance} between this and the C{closest} point converted to 

686 C{meter}, same units as B{C{radius}}. The C{angle} from this 

687 to the C{closest} point is in compass C{degrees360}, like 

688 function L{pygeodesy.compassAngle}. 

689 

690 @raise LimitError: Lat- and/or longitudinal delta exceeds B{C{limit}}, 

691 see function L{pygeodesy.equirectangular_}. 

692 

693 @raise PointsError: Insufficient number of B{C{points}}. 

694 

695 @raise TypeError: Some B{C{points}} are not C{LatLon}. 

696 

697 @raise ValueError: Invalid B{C{radius}} or B{C{options}}. 

698 

699 @see: Functions L{pygeodesy.compassAngle}, L{pygeodesy.equirectangular_} 

700 and L{pygeodesy.nearestOn5}. 

701 ''' 

702 return nearestOn3(self, points, closed=closed, radius=radius, 

703 LatLon=self.classof, **wrap_adjust_limit) 

704 

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

706 '''Convert this point to C{Karney}-based cartesian (ECEF) 

707 coordinates. 

708 

709 @kwarg Cartesian_datum_kwds: Optional L{Cartesian}, B{C{datum}} 

710 and other keyword arguments, ignored 

711 if C{B{Cartesian} is None}. Use 

712 C{B{Cartesian}=...} to override 

713 this L{Cartesian} class or specify 

714 C{B{Cartesian}=None}. 

715 

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

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

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

719 

720 @raise TypeError: Invalid B{C{Cartesian_datum_kwds}} argument. 

721 ''' 

722 kwds = _xkwds(Cartesian_datum_kwds, Cartesian=Cartesian, datum=self.datum) 

723 return LatLonSphericalBase.toCartesian(self, **kwds) 

724 

725 def triangle7(self, otherB, otherC, radius=R_M, wrap=False): 

726 '''Compute the angles, sides and area of a spherical triangle. 

727 

728 @arg otherB: Second triangle point (C{LatLon}). 

729 @arg otherC: Third triangle point (C{LatLon}). 

730 @kwarg radius: Mean earth radius, ellipsoid or datum 

731 (C{meter}, L{Ellipsoid}, L{Ellipsoid2}, 

732 L{Datum} or L{a_f2Tuple}) or C{None}. 

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

734 B{C{otherB}} and B{C{otherC}} points (C{bool}). 

735 

736 @return: L{Triangle7Tuple}C{(A, a, B, b, C, c, area)} or if 

737 B{C{radius}} is C{None}, a L{Triangle8Tuple}C{(A, 

738 a, B, b, C, c, D, E)}. 

739 

740 @see: Function L{triangle7} and U{Spherical trigonometry 

741 <https://WikiPedia.org/wiki/Spherical_trigonometry>}. 

742 ''' 

743 B = self.others(otherB=otherB) 

744 C = self.others(otherC=otherC) 

745 B, C, _ = _unrollon3(self, B, C, wrap) 

746 

747 r = self.philam + B.philam + C.philam 

748 t = triangle8_(*r, wrap=wrap) 

749 return self._xnamed(_t7Tuple(t, radius)) 

750 

751 def trilaterate5(self, distance1, point2, distance2, point3, distance3, 

752 area=True, eps=EPS1, radius=R_M, wrap=False): 

753 '''Trilaterate three points by area overlap or perimeter intersection 

754 of three corresponding circles. 

755 

756 @arg distance1: Distance to this point (C{meter}, same units 

757 as B{C{radius}}). 

758 @arg point2: Second center point (C{LatLon}). 

759 @arg distance2: Distance to point2 (C{meter}, same units as 

760 B{C{radius}}). 

761 @arg point3: Third center point (C{LatLon}). 

762 @arg distance3: Distance to point3 (C{meter}, same units as 

763 B{C{radius}}). 

764 @kwarg area: If C{True} compute the area overlap, otherwise the 

765 perimeter intersection of the circles (C{bool}). 

766 @kwarg eps: The required I{minimal overlap} for C{B{area}=True} 

767 or the I{intersection margin} for C{B{area}=False} 

768 (C{meter}, same units as B{C{radius}}). 

769 @kwarg radius: Mean earth radius (C{meter}, conventionally). 

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

771 B{C{point2}} and B{C{point3}} (C{bool}). 

772 

773 @return: A L{Trilaterate5Tuple}C{(min, minPoint, max, maxPoint, n)} 

774 with C{min} and C{max} in C{meter}, same units as B{C{eps}}, 

775 the corresponding trilaterated points C{minPoint} and 

776 C{maxPoint} as I{spherical} C{LatLon} and C{n}, the number 

777 of trilatered points found for the given B{C{eps}}. 

778 

779 If only a single trilaterated point is found, C{min I{is} 

780 max}, C{minPoint I{is} maxPoint} and C{n = 1}. 

781 

782 For C{B{area}=True}, C{min} and C{max} are the smallest 

783 respectively largest I{radial} overlap found. 

784 

785 For C{B{area}=False}, C{min} and C{max} represent the 

786 nearest respectively farthest intersection margin. 

787 

788 If C{B{area}=True} and all 3 circles are concentric, C{n = 

789 0} and C{minPoint} and C{maxPoint} are both the B{C{point#}} 

790 with the smallest B{C{distance#}} C{min} and C{max} the 

791 largest B{C{distance#}}. 

792 

793 @raise IntersectionError: Trilateration failed for the given B{C{eps}}, 

794 insufficient overlap for C{B{area}=True} or 

795 no intersection or all (near-)concentric for 

796 C{B{area}=False}. 

797 

798 @raise TypeError: Invalid B{C{point2}} or B{C{point3}}. 

799 

800 @raise ValueError: Coincident B{C{point2}} or B{C{point3}} or invalid 

801 B{C{distance1}}, B{C{distance2}}, B{C{distance3}} 

802 or B{C{radius}}. 

803 ''' 

804 return _trilaterate5(self, distance1, 

805 self.others(point2=point2), distance2, 

806 self.others(point3=point3), distance3, 

807 area=area, radius=radius, eps=eps, wrap=wrap) 

808 

809 

810_T00 = LatLon(0, 0, name='T00') # reference instance (L{LatLon}) 

811 

812 

813def areaOf(points, radius=R_M, wrap=False): # was=True 

814 '''Calculate the area of a (spherical) polygon or composite 

815 (with the pointsjoined by great circle arcs). 

816 

817 @arg points: The polygon points or clips (L{LatLon}[], L{BooleanFHP} 

818 or L{BooleanGH}). 

819 @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter}, 

820 L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}) 

821 or C{None}. 

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

823 (C{bool}). 

824 

825 @return: Polygon area (C{meter} I{quared}, same units as B{C{radius}} 

826 or C{radians} if B{C{radius}} is C{None}). 

827 

828 @raise PointsError: Insufficient number of B{C{points}}. 

829 

830 @raise TypeError: Some B{C{points}} are not L{LatLon}. 

831 

832 @raise ValueError: Invalid B{C{radius}} or semi-circular polygon edge. 

833 

834 @note: The area is based on I{Karney}'s U{'Area of a spherical 

835 polygon'<https://MathOverflow.net/questions/97711/ 

836 the-area-of-spherical-polygons>}, 3rd Answer. 

837 

838 @see: Functions L{pygeodesy.areaOf}, L{sphericalNvector.areaOf}, 

839 L{pygeodesy.excessKarney}, L{ellipsoidalExact.areaOf} and 

840 L{ellipsoidalKarney.areaOf}. 

841 

842 @example: 

843 

844 >>> b = LatLon(45, 1), LatLon(45, 2), LatLon(46, 2), LatLon(46, 1) 

845 >>> areaOf(b) # 8666058750.718977 

846 

847 >>> c = LatLon(0, 0), LatLon(1, 0), LatLon(0, 1) 

848 >>> areaOf(c) # 6.18e9 

849 ''' 

850 if _MODS.booleans.isBoolean(points): 

851 return points._sum2(LatLon, areaOf, radius=radius, wrap=wrap) 

852 

853 _at2, _t_2 = atan2, tan_2 

854 _un, _w180 = unrollPI, wrap180 

855 

856 Ps = _T00.PointsIter(points, loop=1, wrap=wrap) 

857 p1 = p2 = Ps[0] 

858 a1, b1 = p1.philam 

859 ta1, z1 = _t_2(a1), None 

860 

861 A = Fsum() # mean phi 

862 R = Fsum() # see L{pygeodesy.excessKarney_} 

863 # ispolar: Summation of course deltas around pole is 0° rather than normally ±360° 

864 # <https://blog.Element84.com/determining-if-a-spherical-polygon-contains-a-pole.html> 

865 # XXX duplicate of function C{points.ispolar} to avoid copying all iterated points 

866 D = Fsum() 

867 for i, p2 in Ps.enumerate(closed=True): 

868 a2, b2 = p2.philam 

869 db, b2 = _un(b1, b2, wrap=wrap and not Ps.looped) 

870 A += a2 

871 ta2 = _t_2(a2) 

872 tdb = _t_2(db, points=i) 

873 R += _at2(tdb * (ta1 + ta2), 

874 _1_0 + ta1 * ta2) 

875 ta1, b1 = ta2, b2 

876 

877 if not p2.isequalTo(p1, eps=EPS): 

878 z, z2 = _bearingTo2(p1, p2, wrap=wrap) 

879 if z1 is not None: 

880 D += _w180(z - z1) # (z - z1 + 540) ... 

881 D += _w180(z2 - z) # (z2 - z + 540) % 360 - 180 

882 p1, z1 = p2, z2 

883 

884 R = abs(R * _2_0) 

885 if abs(D) < _90_0: # ispolar(points) 

886 R = abs(R - PI2) 

887 if radius: 

888 a = degrees(A.fover(len(A))) # mean lat 

889 R *= _mean_radius(radius, a)**2 

890 return float(R) 

891 

892 

893def _destination2(a, b, r, t): 

894 '''(INTERNAL) Destination lat- and longitude in C{radians}. 

895 

896 @arg a: Latitude (C{radians}). 

897 @arg b: Longitude (C{radians}). 

898 @arg r: Angular distance (C{radians}). 

899 @arg t: Bearing (compass C{radians}). 

900 

901 @return: 2-Tuple (phi, lam) of (C{radians}, C{radiansPI}). 

902 ''' 

903 # see <https://www.EdWilliams.org/avform.htm#LL> 

904 sa, ca, sr, cr, st, ct = sincos2_(a, r, t) 

905 ca *= sr 

906 

907 a = asin1(ct * ca + cr * sa) 

908 d = atan2(st * ca, cr - sa * sin(a)) 

909 # note, in EdWilliams.org/avform.htm W is + and E is - 

910 return a, (b + d) # (mod(b + d + PI, PI2) - PI) 

911 

912 

913def _int3d2(s, end, wrap, _i_, Vector, hs): 

914 # see <https://www.EdWilliams.org/intersect.htm> (5) ff 

915 # and similar logic in .ellipsoidalBaseDI._intersect3 

916 a1, b1 = s.philam 

917 

918 if isscalar(end): # bearing, get pseudo-end point 

919 a2, b2 = _destination2(a1, b1, PI_4, radians(end)) 

920 else: # must be a point 

921 s.others(end, name=_end_ + _i_) 

922 hs.append(end.height) 

923 a2, b2 = end.philam 

924 if wrap: 

925 a2, b2 = _Wrap.philam(a2, b2) 

926 

927 db, b2 = unrollPI(b1, b2, wrap=wrap) 

928 if max(fabs(db), fabs(a2 - a1)) < EPS: 

929 raise _ValueError(_SPACE_(_line_ + _i_, _null_)) 

930 # note, in EdWilliams.org/avform.htm W is + and E is - 

931 sb21, cb21, sb12, cb12 = sincos2_(db * _0_5, 

932 -(b1 + b2) * _0_5) 

933 cb21 *= sin(a1 - a2) # sa21 

934 sb21 *= sin(a1 + a2) # sa12 

935 x = Vector(sb12 * cb21 - cb12 * sb21, 

936 cb12 * cb21 + sb12 * sb21, 

937 cos(a1) * cos(a2) * sin(db)) # ll=start 

938 return x.unit(), (db, (a2 - a1)) # negated d 

939 

940 

941def _intdot(ds, a1, b1, a, b, wrap): 

942 # compute dot product ds . (-b + b1, a - a1) 

943 db, _ = unrollPI(b1, b, wrap=wrap) 

944 return fdot(ds, db, a - a1) 

945 

946 

947def intersecant2(center, circle, point, bearing, radius=R_M, exact=False, 

948 height=None, wrap=False): # was=True 

949 '''Compute the intersections of a circle and a line. 

950 

951 @arg center: Center of the circle (L{LatLon}). 

952 @arg circle: Radius of the circle (C{meter}, same units as B{C{radius}}) 

953 or a point on the circle (L{LatLon}). 

954 @arg point: A point in- or outside the circle (L{LatLon}). 

955 @arg bearing: Bearing at the B{C{point}} (compass C{degrees360}) or 

956 a second point on the line (L{LatLon}). 

957 @kwarg radius: Mean earth radius (C{meter}, conventionally). 

958 @kwarg exact: If C{True} use the I{exact} rhumb methods for azimuth, 

959 destination and distance, if C{False} use the basic 

960 rhumb methods (C{bool}) or if C{None} use the I{great 

961 circle} methods. 

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

963 conventionally) or C{None}. 

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

965 and the B{C{circle}} and B{C{bearing}} points (C{bool}). 

966 

967 @return: 2-Tuple of the intersection points (representing a chord), 

968 each an instance of this class. For a tangent line, each 

969 point C{is} this very instance. 

970 

971 @raise IntersectionError: The circle and line do not intersect. 

972 

973 @raise TypeError: If B{C{center}} or B{C{point}} not L{LatLon} or 

974 B{C{circle}} or B{C{bearing}} invalid. 

975 

976 @raise ValueError: Invalid B{C{circle}}, B{C{bearing}}, B{C{radius}}, 

977 B{C{exact}} or B{C{height}}. 

978 ''' 

979 c = _T00.others(center=center) 

980 p = _T00.others(point=point) 

981 try: 

982 return _intersecant2(c, circle, p, bearing, radius=radius, exact=exact, 

983 height=height, wrap=wrap) 

984 except (TypeError, ValueError) as x: 

985 raise _xError(x, center=center, circle=circle, point=point, bearing=bearing, exact=exact) 

986 

987 

988def _intersect(start1, end1, start2, end2, height=None, wrap=False, # in.ellipsoidalBaseDI._intersect3 

989 LatLon=None, **LatLon_kwds): 

990 # (INTERNAL) Intersect two (spherical) lines, see L{intersection} 

991 # above, separated to allow callers to embellish any exceptions 

992 

993 s1, s2 = start1, start2 

994 if wrap: 

995 s2 = _Wrap.point(s2) 

996 hs = [s1.height, s2.height] 

997 

998 a1, b1 = s1.philam 

999 a2, b2 = s2.philam 

1000 db, b2 = unrollPI(b1, b2, wrap=wrap) 

1001 r12 = vincentys_(a2, a1, db) 

1002 if fabs(r12) < EPS: # [nearly] coincident points 

1003 a, b = favg(a1, a2), favg(b1, b2) 

1004 

1005 # see <https://www.EdWilliams.org/avform.htm#Intersection> 

1006 elif isscalar(end1) and isscalar(end2): # both bearings 

1007 sa1, ca1, sa2, ca2, sr12, cr12 = sincos2_(a1, a2, r12) 

1008 

1009 x1, x2 = (sr12 * ca1), (sr12 * ca2) 

1010 if isnear0(x1) or isnear0(x2): 

1011 raise IntersectionError(_parallel_) 

1012 # handle domain error for equivalent longitudes, 

1013 # see also functions asin_safe and acos_safe at 

1014 # <https://www.EdWilliams.org/avform.htm#Math> 

1015 t1, t2 = acos1((sa2 - sa1 * cr12) / x1), \ 

1016 acos1((sa1 - sa2 * cr12) / x2) 

1017 if sin(db) > 0: 

1018 t12, t21 = t1, PI2 - t2 

1019 else: 

1020 t12, t21 = PI2 - t1, t2 

1021 t13, t23 = radiansPI2(end1), radiansPI2(end2) 

1022 sx1, cx1, sx2, cx2 = sincos2_(wrapPI(t13 - t12), # angle 2-1-3 

1023 wrapPI(t21 - t23)) # angle 1-2-3) 

1024 if isnear0(sx1) and isnear0(sx2): 

1025 raise IntersectionError(_infinite_) 

1026 sx3 = sx1 * sx2 

1027# XXX if sx3 < 0: 

1028# XXX raise ValueError(_ambiguous_) 

1029 x3 = acos1(cr12 * sx3 - cx2 * cx1) 

1030 r13 = atan2(sr12 * sx3, cx2 + cx1 * cos(x3)) 

1031 

1032 a, b = _destination2(a1, b1, r13, t13) 

1033 # like .ellipsoidalBaseDI,_intersect3, if this intersection 

1034 # is "before" the first point, use the antipodal intersection 

1035 if opposing_(t13, bearing_(a1, b1, a, b, wrap=wrap)): 

1036 a, b = antipode_(a, b) # PYCHOK PhiLam2Tuple 

1037 

1038 else: # end point(s) or bearing(s) 

1039 _N_vector_ = _MODS.nvectorBase._N_vector_ 

1040 

1041 x1, d1 = _int3d2(s1, end1, wrap, _1_, _N_vector_, hs) 

1042 x2, d2 = _int3d2(s2, end2, wrap, _2_, _N_vector_, hs) 

1043 x = x1.cross(x2) 

1044 if x.length < EPS: # [nearly] colinear or parallel lines 

1045 raise IntersectionError(_colinear_) 

1046 a, b = x.philam 

1047 # choose intersection similar to sphericalNvector 

1048 if not (_intdot(d1, a1, b1, a, b, wrap) * 

1049 _intdot(d2, a2, b2, a, b, wrap)) > 0: 

1050 a, b = antipode_(a, b) # PYCHOK PhiLam2Tuple 

1051 

1052 h = fmean(hs) if height is None else Height(height) 

1053 return _LL3Tuple(degrees90(a), degrees180(b), h, 

1054 intersection, LatLon, LatLon_kwds) 

1055 

1056 

1057def intersection(start1, end1, start2, end2, height=None, wrap=False, 

1058 LatLon=LatLon, **LatLon_kwds): 

1059 '''Compute the intersection point of two lines, each defined 

1060 by two points or a start point and bearing from North. 

1061 

1062 @arg start1: Start point of the first line (L{LatLon}). 

1063 @arg end1: End point of the first line (L{LatLon}) or 

1064 the initial bearing at the first start point 

1065 (compass C{degrees360}). 

1066 @arg start2: Start point of the second line (L{LatLon}). 

1067 @arg end2: End point of the second line (L{LatLon}) or 

1068 the initial bearing at the second start point 

1069 (compass C{degrees360}). 

1070 @kwarg height: Optional height for the intersection point, 

1071 overriding the mean height (C{meter}). 

1072 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

1073 B{C{start2}} and both B{C{end*}} points (C{bool}). 

1074 @kwarg LatLon: Optional class to return the intersection 

1075 point (L{LatLon}) or C{None}. 

1076 @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword 

1077 arguments, ignored if C{B{LatLon} is None}. 

1078 

1079 @return: The intersection point as a (B{C{LatLon}}) or if 

1080 C{B{LatLon} is None} a L{LatLon3Tuple}C{(lat, lon, 

1081 height)}. An alternate intersection point might 

1082 be the L{antipode} to the returned result. 

1083 

1084 @raise IntersectionError: Ambiguous or infinite intersection 

1085 or colinear, parallel or otherwise 

1086 non-intersecting lines. 

1087 

1088 @raise TypeError: A B{C{start1}}, B{C{end1}}, B{C{start2}} 

1089 or B{C{end2}} point not L{LatLon}. 

1090 

1091 @raise ValueError: Invalid B{C{height}} or C{null} line. 

1092 

1093 @example: 

1094 

1095 >>> p = LatLon(51.8853, 0.2545) 

1096 >>> s = LatLon(49.0034, 2.5735) 

1097 >>> i = intersection(p, 108.547, s, 32.435) # '50.9078°N, 004.5084°E' 

1098 ''' 

1099 s1 = _T00.others(start1=start1) 

1100 s2 = _T00.others(start2=start2) 

1101 try: 

1102 return _intersect(s1, end1, s2, end2, height=height, wrap=wrap, 

1103 LatLon=LatLon, **LatLon_kwds) 

1104 except (TypeError, ValueError) as x: 

1105 raise _xError(x, start1=start1, end1=end1, start2=start2, end2=end2) 

1106 

1107 

1108def intersections2(center1, rad1, center2, rad2, radius=R_M, eps=_0_0, 

1109 height=None, wrap=False, # was=True 

1110 LatLon=LatLon, **LatLon_kwds): 

1111 '''Compute the intersection points of two circles each defined 

1112 by a center point and a radius. 

1113 

1114 @arg center1: Center of the first circle (L{LatLon}). 

1115 @arg rad1: Radius of the first circle (C{meter} or C{radians}, 

1116 see B{C{radius}}). 

1117 @arg center2: Center of the second circle (L{LatLon}). 

1118 @arg rad2: Radius of the second circle (C{meter} or C{radians}, 

1119 see B{C{radius}}). 

1120 @kwarg radius: Mean earth radius (C{meter} or C{None} if B{C{rad1}}, 

1121 B{C{rad2}} and B{C{eps}} are given in C{radians}). 

1122 @kwarg eps: Required overlap (C{meter} or C{radians}, see 

1123 B{C{radius}}). 

1124 @kwarg height: Optional height for the intersection points (C{meter}, 

1125 conventionally) or C{None} for the I{"radical height"} 

1126 at the I{radical line} between both centers. 

1127 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{center2}} 

1128 (C{bool}). 

1129 @kwarg LatLon: Optional class to return the intersection 

1130 points (L{LatLon}) or C{None}. 

1131 @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword 

1132 arguments, ignored if C{B{LatLon} is None}. 

1133 

1134 @return: 2-Tuple of the intersection points, each a B{C{LatLon}} 

1135 instance or if C{B{LatLon} is None} a L{LatLon3Tuple}C{(lat, 

1136 lon, height)}. For abutting circles, both intersection 

1137 points are the same instance, aka the I{radical center}. 

1138 

1139 @raise IntersectionError: Concentric, antipodal, invalid or 

1140 non-intersecting circles. 

1141 

1142 @raise TypeError: If B{C{center1}} or B{C{center2}} not L{LatLon}. 

1143 

1144 @raise ValueError: Invalid B{C{rad1}}, B{C{rad2}}, B{C{radius}}, 

1145 B{C{eps}} or B{C{height}}. 

1146 

1147 @note: Courtesy of U{Samuel Čavoj<https://GitHub.com/mrJean1/PyGeodesy/issues/41>}. 

1148 

1149 @see: This U{Answer<https://StackOverflow.com/questions/53324667/ 

1150 find-intersection-coordinates-of-two-circles-on-earth/53331953>}. 

1151 ''' 

1152 c1 = _T00.others(center1=center1) 

1153 c2 = _T00.others(center2=center2) 

1154 try: 

1155 return _intersects2(c1, rad1, c2, rad2, radius=radius, eps=eps, 

1156 height=height, wrap=wrap, 

1157 LatLon=LatLon, **LatLon_kwds) 

1158 except (TypeError, ValueError) as x: 

1159 raise _xError(x, center1=center1, rad1=rad1, 

1160 center2=center2, rad2=rad2, wrap=wrap) 

1161 

1162 

1163def _intersects2(c1, rad1, c2, rad2, radius=R_M, eps=_0_0, # in .ellipsoidalBaseDI._intersects2 

1164 height=None, too_d=None, wrap=False, # was=True 

1165 LatLon=LatLon, **LatLon_kwds): 

1166 # (INTERNAL) Intersect two spherical circles, see L{intersections2} 

1167 # above, separated to allow callers to embellish any exceptions 

1168 

1169 def _dest3(bearing, h): 

1170 a, b = _destination2(a1, b1, r1, bearing) 

1171 return _LL3Tuple(degrees90(a), degrees180(b), h, 

1172 intersections2, LatLon, LatLon_kwds) 

1173 

1174 a1, b1 = c1.philam 

1175 a2, b2 = c2.philam 

1176 if wrap: 

1177 a2, b2 = _Wrap.philam(a2, b2) 

1178 

1179 r1, r2, f = _rads3(rad1, rad2, radius) 

1180 if f: # swapped radii, swap centers 

1181 a1, a2 = a2, a1 # PYCHOK swap! 

1182 b1, b2 = b2, b1 # PYCHOK swap! 

1183 

1184 db, b2 = unrollPI(b1, b2, wrap=wrap) 

1185 d = vincentys_(a2, a1, db) # radians 

1186 if d < max(r1 - r2, EPS): 

1187 raise IntersectionError(_near_(_concentric_)) # XXX ConcentricError? 

1188 

1189 r = eps if radius is None else (m2radians( 

1190 eps, radius=radius) if eps else _0_0) 

1191 if r < _0_0: 

1192 raise _ValueError(eps=r) 

1193 

1194 x = fsumf_(r1, r2, -d) # overlap 

1195 if x > max(r, EPS): 

1196 sd, cd, sr1, cr1, _, cr2 = sincos2_(d, r1, r2) 

1197 x = sd * sr1 

1198 if isnear0(x): 

1199 raise _ValueError(_invalid_) 

1200 x = acos1((cr2 - cd * cr1) / x) # 0 <= x <= PI 

1201 

1202 elif x < r: # PYCHOK no cover 

1203 t = (d * radius) if too_d is None else too_d 

1204 raise IntersectionError(_too_(_Fmt.distant(t))) 

1205 

1206 if height is None: # "radical height" 

1207 f = _radical2(d, r1, r2).ratio 

1208 h = Height(favg(c1.height, c2.height, f=f)) 

1209 else: 

1210 h = Height(height) 

1211 

1212 b = bearing_(a1, b1, a2, b2, final=False, wrap=wrap) 

1213 if x < EPS4: # externally ... 

1214 r = _dest3(b, h) 

1215 elif x > _PI_EPS4: # internally ... 

1216 r = _dest3(b + PI, h) 

1217 else: 

1218 return _dest3(b + x, h), _dest3(b - x, h) 

1219 return r, r # ... abutting circles 

1220 

1221 

1222@deprecated_function 

1223def isPoleEnclosedBy(points, wrap=False): # PYCHOK no cover 

1224 '''DEPRECATED, use function L{pygeodesy.ispolar}. 

1225 ''' 

1226 return ispolar(points, wrap=wrap) 

1227 

1228 

1229def _LL3Tuple(lat, lon, height, where, LatLon, LatLon_kwds): 

1230 '''(INTERNAL) Helper for L{intersection}, L{intersections2} and L{meanOf}. 

1231 ''' 

1232 n = where.__name__ 

1233 if LatLon is None: 

1234 r = LatLon3Tuple(lat, lon, height, name=n) 

1235 else: 

1236 kwds = _xkwds(LatLon_kwds, height=height, name=n) 

1237 r = LatLon(lat, lon, **kwds) 

1238 return r 

1239 

1240 

1241def meanOf(points, height=None, wrap=False, LatLon=LatLon, **LatLon_kwds): 

1242 '''Compute the I{geographic} mean of several points. 

1243 

1244 @arg points: Points to be averaged (L{LatLon}[]). 

1245 @kwarg height: Optional height at mean point, overriding the mean 

1246 height (C{meter}). 

1247 @kwarg wrap: If C{True}, wrap or I{normalize} the B{C{points}} 

1248 (C{bool}). 

1249 @kwarg LatLon: Optional class to return the mean point (L{LatLon}) 

1250 or C{None}. 

1251 @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword 

1252 arguments, ignored if C{B{LatLon} is None}. 

1253 

1254 @return: The geographic mean and height (B{C{LatLon}}) or a 

1255 L{LatLon3Tuple}C{(lat, lon, height)} if B{C{LatLon}} 

1256 is C{None}. 

1257 

1258 @raise TypeError: Some B{C{points}} are not L{LatLon}. 

1259 

1260 @raise ValueError: No B{C{points}} or invalid B{C{height}}. 

1261 ''' 

1262 def _N_vs(ps, w): 

1263 Ps = _T00.PointsIter(ps, wrap=w) 

1264 for p in Ps.iterate(closed=False): 

1265 yield p._N_vector 

1266 

1267 m = _MODS.nvectorBase 

1268 # geographic, vectorial mean 

1269 n = m.sumOf(_N_vs(points, wrap), h=height, Vector=m.NvectorBase) 

1270 lat, lon, h = n.latlonheight 

1271 return _LL3Tuple(lat, lon, h, meanOf, LatLon, LatLon_kwds) 

1272 

1273 

1274@deprecated_function 

1275def nearestOn2(point, points, **closed_radius_LatLon_options): # PYCHOK no cover 

1276 '''DEPRECATED, use function L{sphericalTrigonometry.nearestOn3}. 

1277 

1278 @return: ... 2-tuple C{(closest, distance)} of the C{closest} 

1279 point (L{LatLon}) on the polygon and the C{distance} 

1280 between the C{closest} and the given B{C{point}}. The 

1281 C{closest} is a B{C{LatLon}} or a L{LatLon2Tuple}C{(lat, 

1282 lon)} if B{C{LatLon}} is C{None} ... 

1283 ''' 

1284 ll, d, _ = nearestOn3(point, points, **closed_radius_LatLon_options) # PYCHOK 3-tuple 

1285 if _xkwds_get(closed_radius_LatLon_options, LatLon=LatLon) is None: 

1286 ll = LatLon2Tuple(ll.lat, ll.lon) 

1287 return ll, d 

1288 

1289 

1290def nearestOn3(point, points, closed=False, radius=R_M, wrap=False, adjust=True, 

1291 limit=9, **LatLon_and_kwds): 

1292 '''Locate the point on a path or polygon closest to a reference point. 

1293 

1294 Distances are I{approximated} using function L{pygeodesy.equirectangular_}, 

1295 subject to the supplied B{C{options}}. 

1296 

1297 @arg point: The reference point (L{LatLon}). 

1298 @arg points: The path or polygon points (L{LatLon}[]). 

1299 @kwarg closed: Optionally, close the polygon (C{bool}). 

1300 @kwarg radius: Mean earth radius (C{meter}). 

1301 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the 

1302 B{C{points}} (C{bool}). 

1303 @kwarg adjust: See function L{pygeodesy.equirectangular_} (C{bool}). 

1304 @kwarg limit: See function L{pygeodesy.equirectangular_} (C{degrees}), 

1305 default C{9 degrees} is about C{1,000 Kmeter} (for mean 

1306 spherical earth radius L{R_KM}). 

1307 @kwarg LatLon: Optional class to return the closest point (L{LatLon}) 

1308 or C{None}. 

1309 @kwarg options: Optional keyword arguments for function 

1310 L{pygeodesy.equirectangular_}. 

1311 

1312 @return: A L{NearestOn3Tuple}C{(closest, distance, angle)} with the 

1313 C{closest} point as B{C{LatLon}} or L{LatLon3Tuple}C{(lat, 

1314 lon, height)} if B{C{LatLon}} is C{None}. The C{distance} 

1315 is the L{pygeodesy.equirectangular_} distance between the 

1316 C{closest} and the given B{C{point}} converted to C{meter}, 

1317 same units as B{C{radius}}. The C{angle} from the given 

1318 B{C{point}} to the C{closest} is in compass C{degrees360}, 

1319 like function L{pygeodesy.compassAngle}. The C{height} is 

1320 the (interpolated) height at the C{closest} point. 

1321 

1322 @raise LimitError: Lat- and/or longitudinal delta exceeds the B{C{limit}}, 

1323 see function L{pygeodesy.equirectangular_}. 

1324 

1325 @raise PointsError: Insufficient number of B{C{points}}. 

1326 

1327 @raise TypeError: Some B{C{points}} are not C{LatLon}. 

1328 

1329 @raise ValueError: Invalid B{C{radius}}. 

1330 

1331 @see: Functions L{pygeodesy.equirectangular_} and L{pygeodesy.nearestOn5}. 

1332 ''' 

1333 t = _nearestOn5(point, points, closed=closed, wrap=wrap, 

1334 adjust=adjust, limit=limit) 

1335 d = degrees2m(t.distance, radius=radius) 

1336 h = t.height 

1337 n = nearestOn3.__name__ 

1338 

1339 kwds = _xkwds(LatLon_and_kwds, height=h, name=n) 

1340 LL = _xkwds_pop(kwds, LatLon=LatLon) 

1341 r = LatLon3Tuple(t.lat, t.lon, h, name=n) if LL is None else \ 

1342 LL(t.lat, t.lon, **kwds) 

1343 return NearestOn3Tuple(r, d, t.angle, name=n) 

1344 

1345 

1346def perimeterOf(points, closed=False, radius=R_M, wrap=True): 

1347 '''Compute the perimeter of a (spherical) polygon or composite 

1348 (with great circle arcs joining the points). 

1349 

1350 @arg points: The polygon points or clips (L{LatLon}[], L{BooleanFHP} 

1351 or L{BooleanGH}). 

1352 @kwarg closed: Optionally, close the polygon (C{bool}). 

1353 @kwarg radius: Mean earth radius (C{meter}) or C{None}. 

1354 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the 

1355 B{C{points}} (C{bool}). 

1356 

1357 @return: Polygon perimeter (C{meter}, same units as B{C{radius}} 

1358 or C{radians} if B{C{radius}} is C{None}). 

1359 

1360 @raise PointsError: Insufficient number of B{C{points}}. 

1361 

1362 @raise TypeError: Some B{C{points}} are not L{LatLon}. 

1363 

1364 @raise ValueError: Invalid B{C{radius}} or C{B{closed}=False} with 

1365 C{B{points}} a composite. 

1366 

1367 @note: Distances are based on function L{pygeodesy.vincentys_}. 

1368 

1369 @see: Functions L{pygeodesy.perimeterOf}, L{sphericalNvector.perimeterOf} 

1370 and L{ellipsoidalKarney.perimeterOf}. 

1371 ''' 

1372 def _rads(ps, c, w): # angular edge lengths in radians 

1373 Ps = _T00.PointsIter(ps, loop=1, wrap=w) 

1374 a1, b1 = Ps[0].philam 

1375 for p in Ps.iterate(closed=c): 

1376 a2, b2 = p.philam 

1377 db, b2 = unrollPI(b1, b2, wrap=w and not (c and Ps.looped)) 

1378 yield vincentys_(a2, a1, db) 

1379 a1, b1 = a2, b2 

1380 

1381 if _MODS.booleans.isBoolean(points): 

1382 if not closed: 

1383 raise _ValueError(closed=closed, points=_composite_) 

1384 r = points._sum2(LatLon, perimeterOf, closed=True, radius=radius, wrap=wrap) 

1385 else: 

1386 r = fsum(_rads(points, closed, wrap), floats=True) 

1387 return _r2m(r, radius) 

1388 

1389 

1390def _r2m(r, radius): 

1391 '''(INTERNAL) Angular distance in C{radians} to C{meter}. 

1392 ''' 

1393 if radius is not None: # not in (None, _0_0) 

1394 r *= R_M if radius is R_M else Radius(radius) 

1395 return r 

1396 

1397 

1398def triangle7(latA, lonA, latB, lonB, latC, lonC, radius=R_M, 

1399 excess=excessAbc_, 

1400 wrap=False): 

1401 '''Compute the angles, sides, and area of a (spherical) triangle. 

1402 

1403 @arg latA: First corner latitude (C{degrees}). 

1404 @arg lonA: First corner longitude (C{degrees}). 

1405 @arg latB: Second corner latitude (C{degrees}). 

1406 @arg lonB: Second corner longitude (C{degrees}). 

1407 @arg latC: Third corner latitude (C{degrees}). 

1408 @arg lonC: Third corner longitude (C{degrees}). 

1409 @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter}, 

1410 L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}) 

1411 or C{None}. 

1412 @kwarg excess: I{Spherical excess} callable (L{excessAbc_}, 

1413 L{excessGirard_} or L{excessLHuilier_}). 

1414 @kwarg wrap: If C{True}, wrap and L{pygeodesy.unroll180} 

1415 longitudes (C{bool}). 

1416 

1417 @return: A L{Triangle7Tuple}C{(A, a, B, b, C, c, area)} with 

1418 spherical angles C{A}, C{B} and C{C}, angular sides 

1419 C{a}, C{b} and C{c} all in C{degrees} and C{area} 

1420 in I{square} C{meter} or same units as B{C{radius}} 

1421 I{squared} or if C{B{radius}=0} or C{None}, a 

1422 L{Triangle8Tuple}C{(A, a, B, b, C, c, D, E)} all in 

1423 C{radians} with the I{spherical excess} C{E} as the 

1424 C{unit area} in C{radians}. 

1425 ''' 

1426 t = triangle8_(Phi_(latA=latA), Lam_(lonA=lonA), 

1427 Phi_(latB=latB), Lam_(lonB=lonB), 

1428 Phi_(latC=latC), Lam_(lonC=lonC), 

1429 excess=excess, wrap=wrap) 

1430 return _t7Tuple(t, radius) 

1431 

1432 

1433def triangle8_(phiA, lamA, phiB, lamB, phiC, lamC, excess=excessAbc_, 

1434 wrap=False): 

1435 '''Compute the angles, sides, I{spherical deficit} and I{spherical 

1436 excess} of a (spherical) triangle. 

1437 

1438 @arg phiA: First corner latitude (C{radians}). 

1439 @arg lamA: First corner longitude (C{radians}). 

1440 @arg phiB: Second corner latitude (C{radians}). 

1441 @arg lamB: Second corner longitude (C{radians}). 

1442 @arg phiC: Third corner latitude (C{radians}). 

1443 @arg lamC: Third corner longitude (C{radians}). 

1444 @kwarg excess: I{Spherical excess} callable (L{excessAbc_}, 

1445 L{excessGirard_} or L{excessLHuilier_}). 

1446 @kwarg wrap: If C{True}, L{pygeodesy.unrollPI} the 

1447 longitudinal deltas (C{bool}). 

1448 

1449 @return: A L{Triangle8Tuple}C{(A, a, B, b, C, c, D, E)} with 

1450 spherical angles C{A}, C{B} and C{C}, angular sides 

1451 C{a}, C{b} and C{c}, I{spherical deficit} C{D} and 

1452 I{spherical excess} C{E}, all in C{radians}. 

1453 ''' 

1454 def _a_r(w, phiA, lamA, phiB, lamB, phiC, lamC): 

1455 d, _ = unrollPI(lamB, lamC, wrap=w) 

1456 a = vincentys_(phiC, phiB, d) 

1457 return a, (phiB, lamB, phiC, lamC, phiA, lamA) 

1458 

1459 def _A_r(a, sa, ca, sb, cb, sc, cc): 

1460 s = sb * sc 

1461 A = acos1((ca - cb * cc) / s) if isnon0(s) else a 

1462 return A, (sb, cb, sc, cc, sa, ca) # rotate sincos2's 

1463 

1464 # notation: side C{a} is oposite to corner C{A}, etc. 

1465 a, r = _a_r(wrap, phiA, lamA, phiB, lamB, phiC, lamC) 

1466 b, r = _a_r(wrap, *r) 

1467 c, _ = _a_r(wrap, *r) 

1468 

1469 A, r = _A_r(a, *sincos2_(a, b, c)) 

1470 B, r = _A_r(b, *r) 

1471 C, _ = _A_r(c, *r) 

1472 

1473 D = fsumf_(PI2, -a, -b, -c) # deficit aka defect 

1474 E = excessGirard_(A, B, C) if excess in (excessGirard_, True) else ( 

1475 excessLHuilier_(a, b, c) if excess in (excessLHuilier_, False) else 

1476 excessAbc_(*max((A, b, c), (B, c, a), (C, a, b)))) 

1477 

1478 return Triangle8Tuple(A, a, B, b, C, c, D, E) 

1479 

1480 

1481def _t7Tuple(t, radius): 

1482 '''(INTERNAL) Convert a L{Triangle8Tuple} to L{Triangle7Tuple}. 

1483 ''' 

1484 if radius: # not in (None, _0_0) 

1485 r = radius if isscalar(radius) else \ 

1486 _ellipsoidal_datum(radius).ellipsoid.Rmean 

1487 A, B, C = map1(degrees, t.A, t.B, t.C) 

1488 t = Triangle7Tuple(A, (r * t.a), 

1489 B, (r * t.b), 

1490 C, (r * t.c), t.E * r**2) 

1491 return t 

1492 

1493 

1494__all__ += _ALL_OTHER(Cartesian, LatLon, # classes 

1495 areaOf, # functions 

1496 intersecant2, intersection, intersections2, ispolar, 

1497 isPoleEnclosedBy, # DEPRECATED, use ispolar 

1498 meanOf, 

1499 nearestOn2, nearestOn3, 

1500 perimeterOf, 

1501 sumOf, # XXX == vector3d.sumOf 

1502 triangle7, triangle8_) 

1503 

1504# **) MIT License 

1505# 

1506# Copyright (C) 2016-2023 -- mrJean1 at Gmail -- All Rights Reserved. 

1507# 

1508# Permission is hereby granted, free of charge, to any person obtaining a 

1509# copy of this software and associated documentation files (the "Software"), 

1510# to deal in the Software without restriction, including without limitation 

1511# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1512# and/or sell copies of the Software, and to permit persons to whom the 

1513# Software is furnished to do so, subject to the following conditions: 

1514# 

1515# The above copyright notice and this permission notice shall be included 

1516# in all copies or substantial portions of the Software. 

1517# 

1518# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

1519# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1520# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

1521# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

1522# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

1523# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

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