Coverage for pygeodesy/sphericalTrigonometry.py: 93%

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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, 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 _TypeError, _ValueError, IntersectionError, \ 

26 _xError, _xkwds, _xkwds_get, _xkwds_pop2 

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_, _invalid_,\ 

34 _line_, _near_, _not_, _null_, _parallel_, _point_, \ 

35 _SPACE_, _too_ 

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

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

38from pygeodesy.namedTuples import LatLon2Tuple, LatLon3Tuple, NearestOn3Tuple, \ 

39 Triangle7Tuple, Triangle8Tuple 

40from pygeodesy.points import ispolar, nearestOn5 as _nearestOn5, \ 

41 Fmt as _Fmt # XXX shadowed 

42from pygeodesy.props import deprecated_function, deprecated_method 

43from pygeodesy.sphericalBase import _m2radians, CartesianSphericalBase, \ 

44 _intersecant2, LatLonSphericalBase, \ 

45 _rads3, _radians2m, _trilaterate5 

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

47from pygeodesy.units import Bearing_, Height, _isDegrees, _isRadius, Lam_, \ 

48 Phi_, Radius_, Scalar 

49from pygeodesy.utily import acos1, asin1, atan1d, atan2d, degrees90, degrees180, \ 

50 degrees2m, m2radians, radiansPI2, sincos2_, tan_2, \ 

51 unrollPI, _unrollon, _unrollon3, _Wrap, wrap180, wrapPI 

52from pygeodesy.vector3d import sumOf, Vector3d 

53 

54from math import asin, atan2, cos, degrees, fabs, radians, sin 

55 

56__all__ = _ALL_LAZY.sphericalTrigonometry 

57__version__ = '24.04.07' 

58 

59_PI_EPS4 = PI - EPS4 

60if _PI_EPS4 >= PI: 

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

62 

63 

64class Cartesian(CartesianSphericalBase): 

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

66 spherical, geodetic L{LatLon}. 

67 ''' 

68 

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

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

71 

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

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

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

75 

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

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

78 with C{C} and C{M} if available. 

79 

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

81 ''' 

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

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

84 

85 

86class LatLon(LatLonSphericalBase): 

87 '''New point on a spherical earth model, based on trigonometry formulae. 

88 ''' 

89 

90 def _ab1_ab2_db5(self, other, wrap): 

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

92 ''' 

93 a1, b1 = self.philam 

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

95 if wrap: 

96 a2, b2 = _Wrap.philam(a2, b2) 

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

98 else: # unrollPI shortcut 

99 db = b2 - b1 

100 return a1, b1, a2, b2, db 

101 

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

103 '''Compute the (signed) distance from the start to the closest 

104 point on the great circle line defined by a start and an 

105 end point. 

106 

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

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

109 from the start point to the point where the perpendicular 

110 crosses the line. 

111 

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

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

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

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

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

117 

118 @return: Distance along the great circle line (C{radians} 

119 if C{B{radius} is None} or C{meter}, same units 

120 as B{C{radius}}), positive if I{after} the 

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

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

123 B{C{start}} point. 

124 

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

126 

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

128 ''' 

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

130 cx = cos(x) 

131 return _0_0 if isnear0(cx) else \ 

132 _radians2m(copysign0(acos1(cos(r) / cx), cos(b)), radius) 

133 

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

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

136 ''' 

137 s = self.others(start=start) 

138 e = self.others(end=end) 

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

140 

141 r = Radius_(radius) 

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

143 

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

145 - s.initialBearingTo(e, wrap=w)) 

146 x = asin(sin(r) * sin(b)) 

147 return r, x, -b 

148 

149 @deprecated_method 

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

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

152 ''' 

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

154 

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

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

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

158 

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

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

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

162 B{C{other}} point (C{bool}). 

163 

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

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

166 ''' 

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

168 sa, ca, sa1, ca1, \ 

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

170 sa1 *= ca2 * ca 

171 

172 x = sa1 * sdb 

173 y = sa1 * cdb - ca1 * sa2 * ca 

174 z = ca1 * sdb * ca2 * sa 

175 

176 h = hypot(x, y) 

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

178 return None # great circle doesn't reach latitude 

179 

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

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

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

183 

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

185 '''Compute the (signed) distance from this point to 

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

187 

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

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

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

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

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

193 

194 @return: Distance to the great circle (C{radians} if 

195 B{C{radius}} or C{meter}, same units as 

196 B{C{radius}}), I{negative} if to the left or 

197 I{positive} if to the right of the line. 

198 

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

200 

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

202 ''' 

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

204 return _radians2m(x, radius) 

205 

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

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

208 travelled the given distance on the given initial bearing. 

209 

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

211 B{C{radius}}). 

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

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

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

215 units a B{C{radius}}). 

216 

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

218 

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

220 B{C{radius}} or B{C{height}}. 

221 ''' 

222 a, b = self.philam 

223 r, t = _m2radians(distance, radius, low=None), Bearing_(bearing) 

224 

225 a, b = _destination2(a, b, r, t) 

226 h = self._heigHt(height) 

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

228 

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

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

231 

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

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

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

235 the B{C{other}} point (C{bool}). 

236 

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

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

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

240 

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

242 

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

244 ''' 

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

246 return _radians2m(vincentys_(a2, a1, db), radius) 

247 

248# @Property_RO 

249# def Ecef(self): 

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

251# ''' 

252# return _MODS.ecef.EcefKarney 

253 

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

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

256 on the given initial bearing from this point. 

257 

258 Direction of vector is such that initial bearing vector 

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

260 

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

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

263 overriding the default L{Vector3d}. 

264 @kwarg Vector_kwds: Optional, additional keyword argunents 

265 for B{C{Vector}}. 

266 

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

268 

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

270 ''' 

271 a, b = self.philam 

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

273 

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

275 -cb * ct - sb * sa * st, 

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

277 

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

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

280 to an other point. 

281 

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

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

284 the B{C{other}} point (C{bool}). 

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

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

287 C{sphericalNvector.LatLon.initialBearingTo}. 

288 

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

290 

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

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

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

294 

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

296 ''' 

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

298 # XXX behavior like sphericalNvector.LatLon.initialBearingTo 

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

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

301 

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

303 

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

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

306 and an other point. 

307 

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

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

310 0.0 at this and 1.0 at the other point). 

311 @kwarg height: Optional height, overriding the intermediate 

312 height (C{meter}). 

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

314 B{C{other}} point (C{bool}). 

315 

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

317 

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

319 

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

321 

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

323 ''' 

324 p = self 

325 f = Scalar(fraction=fraction) 

326 if not isnear0(f): 

327 p = p.others(other) 

328 if wrap: 

329 p = _Wrap.point(p) 

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

331 a1, b1 = self.philam 

332 a2, b2 = p.philam 

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

334 r = vincentys_(a2, a1, db) 

335 sr = sin(r) 

336 if isnon0(sr): 

337 sa1, ca1, sa2, ca2, \ 

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

339 

340 t = f * r 

341 a = sin(r - t) # / sr superflous 

342 b = sin( t) # / sr superflous 

343 

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

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

346 z = a * sa1 + b * sa2 

347 

348 a = atan1d(z, hypot(x, y)) 

349 b = atan2d(y, x) 

350 

351 else: # PYCHOK no cover 

352 a = degrees90( favg(a1, a2, f=f)) # coincident 

353 b = degrees180(favg(b1, b2, f=f)) 

354 

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

356 p = self.classof(a, b, height=h) 

357 return p 

358 

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

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

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

362 

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

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

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

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

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

368 C{degrees360}). 

369 @kwarg height: Optional height for intersection point, 

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

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

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

373 

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

375 intersection point might be the L{antipode} to 

376 the returned result. 

377 

378 @raise IntersectionError: Ambiguous or infinite intersection 

379 or colinear, parallel or otherwise 

380 non-intersecting lines. 

381 

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

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

384 

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

386 ''' 

387 try: 

388 s2 = self.others(other) 

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

390 LatLon=self.classof) 

391 except (TypeError, ValueError) as x: 

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

393 other=other, end2=end2, wrap=wrap) 

394 

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

396 height=None, wrap=True): 

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

398 by a center point and radius. 

399 

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

401 see B{C{radius}}). 

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

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

404 see B{C{radius}}). 

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

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

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

408 B{C{radius}}). 

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

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

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

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

413 B{C{other}} point (C{bool}). 

414 

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

416 instance. For abutting circles, both intersection 

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

418 

419 @raise IntersectionError: Concentric, antipodal, invalid or 

420 non-intersecting circles. 

421 

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

423 

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

425 B{C{eps}} or B{C{height}}. 

426 ''' 

427 try: 

428 c2 = self.others(other) 

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

430 height=height, wrap=wrap, 

431 LatLon=self.classof) 

432 except (TypeError, ValueError) as x: 

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

434 other=other, rad2=rad2, wrap=wrap) 

435 

436 @deprecated_method 

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

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

439 return self.isenclosedBy(points) 

440 

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

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

443 

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

445 L{BooleanFHP} or L{BooleanGH}). 

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

447 B{C{points}} (C{bool}). 

448 

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

450 composite, C{False} otherwise. 

451 

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

453 

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

455 

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

457 

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

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

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

461 ''' 

462 if _MODS.booleans.isBoolean(points): 

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

464 

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

466 n0 = self._N_vector 

467 

468 v2 = Ps[0]._N_vector 

469 p1 = Ps[1] 

470 v1 = p1._N_vector 

471 # check whether this point on same side of all 

472 # polygon edges (to the left or right depending 

473 # on the anti-/clockwise polygon direction) 

474 gc1 = v2.cross(v1) 

475 t0 = gc1.angleTo(n0) > PI_2 

476 s0 = None 

477 # get great-circle vector for each edge 

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

479 if wrap and not Ps.looped: 

480 p2 = _unrollon(p1, p2) 

481 p1 = p2 

482 v2 = p2._N_vector 

483 gc = v1.cross(v2) 

484 t = gc.angleTo(n0) > PI_2 

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

486 return False # outside 

487 

488 # check for convex polygon: angle between 

489 # gc vectors, signed by direction of n0 

490 # (otherwise the test above is not reliable) 

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

492 if s != s0: 

493 if s0 is None: 

494 s0 = s 

495 else: 

496 t = _Fmt.SQUARE(points=i) 

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

498 gc1, v1 = gc, v2 

499 

500 return True # inside 

501 

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

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

504 

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

506 @kwarg height: Optional height for midpoint, overriding 

507 the mean height (C{meter}). 

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

509 may be negative or greater than 1.0. 

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

511 B{C{other}} point (C{bool}). 

512 

513 @return: Midpoint (L{LatLon}). 

514 

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

516 

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

518 

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

520 ''' 

521 if fraction is _0_5: 

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

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

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

525 

526 x = ca2 * cdb + ca1 

527 y = ca2 * sdb 

528 

529 a = atan1d(sa1 + sa2, hypot(x, y)) 

530 b = degrees180(b + atan2(y, x)) 

531 

532 h = self._havg(other, h=height) 

533 r = self.classof(a, b, height=h) 

534 else: 

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

536 return r 

537 

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

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

540 

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

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

543 

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

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

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

547 @kwarg wrap_adjust_limit: Optional keyword arguments for functions 

548 L{sphericalTrigonometry.nearestOn3} and 

549 L{pygeodesy.equirectangular_}, 

550 

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

552 

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

554 see function L{pygeodesy.equirectangular_}. 

555 

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

557 is not (yet) supported. 

558 

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

560 

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

562 

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

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

565 ''' 

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

567 w, kwds = _xkwds_pop2(wrap_adjust_limit, within=True) 

568 if not w: 

569 self._notImplemented(within=w) 

570 

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

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

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

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

575# return point1 

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

577# if isnear0(d): 

578# return point1 # or point2 

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

580# return point2 

581# f = a / d 

582# if within: 

583# if f > EPS1: 

584# return point2 

585# elif f < EPS: 

586# return point1 

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

588 

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

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

591 **kwds)[0] 

592 

593 @deprecated_method 

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

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

596 

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

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

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

600 units of B{C{radius}}. 

601 ''' 

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

603 return r.closest, r.distance 

604 

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

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

607 

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

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

610 

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

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

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

614 @kwarg wrap_adjust_limit: Optional keyword arguments for function 

615 L{sphericalTrigonometry.nearestOn3} and 

616 L{pygeodesy.equirectangular_}, 

617 

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

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

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

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

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

623 function L{pygeodesy.compassAngle}. 

624 

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

626 see function L{pygeodesy.equirectangular_}. 

627 

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

629 

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

631 

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

633 

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

635 and L{pygeodesy.nearestOn5}. 

636 ''' 

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

638 LatLon=self.classof, **wrap_adjust_limit) 

639 

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

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

642 coordinates. 

643 

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

645 and other keyword arguments, ignored 

646 if C{B{Cartesian} is None}. Use 

647 C{B{Cartesian}=...} to override 

648 this L{Cartesian} class or specify 

649 C{B{Cartesian}=None}. 

650 

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

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

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

654 

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

656 ''' 

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

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

659 

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

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

662 

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

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

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

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

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

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

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

670 

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

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

673 a, B, b, C, c, D, E)}. 

674 

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

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

677 ''' 

678 B = self.others(otherB=otherB) 

679 C = self.others(otherC=otherC) 

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

681 

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

683 t = triangle8_(*r, wrap=wrap) 

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

685 

686 def triangulate(self, bearing1, other, bearing2, **height_wrap): 

687 '''Locate a point given this, an other point and the (initial) bearing 

688 at this and at the other point. 

689 

690 @arg bearing1: Bearing at this point (compass C{degrees360}). 

691 @arg other: The other point (C{LatLon}). 

692 @arg bearing2: Bearing at the other point (compass C{degrees360}). 

693 @kwarg height_wrap_tol: Optional keyword arguments C{B{height}=None}, 

694 C{B{wrap}=False}, see method L{intersection}. 

695 

696 @return: Triangulated point (C{LatLon}). 

697 

698 @see: Method L{intersection} for further details. 

699 ''' 

700 if _isDegrees(bearing1) and _isDegrees(bearing2): 

701 return self.intersection(bearing1, other, bearing2, **height_wrap) 

702 raise _TypeError(bearing1=bearing1, bearing2=bearing2, **height_wrap) 

703 

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

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

706 '''Trilaterate three points by I{area overlap} or I{perimeter 

707 intersection} of three corresponding circles. 

708 

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

710 as B{C{radius}}). 

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

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

713 B{C{radius}}). 

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

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

716 B{C{radius}}). 

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

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

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

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

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

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

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

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

725 

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

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

728 the corresponding trilaterated points C{minPoint} and 

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

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

731 

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

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

734 

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

736 respectively largest I{radial} overlap found. 

737 

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

739 nearest respectively farthest intersection margin. 

740 

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

742 0} and C{minPoint} and C{maxPoint} are both the B{C{point#}} 

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

744 largest B{C{distance#}}. 

745 

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

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

748 no intersection or all (near-)concentric for 

749 C{B{area}=False}. 

750 

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

752 

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

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

755 or B{C{radius}}. 

756 ''' 

757 return _trilaterate5(self, distance1, 

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

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

760 area=area, radius=radius, eps=eps, wrap=wrap) 

761 

762 

763_T00 = LatLon(0, 0, name='T00') # reference instance (L{LatLon}) 

764 

765 

766def areaOf(points, radius=R_M, wrap=False): # was=True 

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

768 (with the pointsjoined by great circle arcs). 

769 

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

771 or L{BooleanGH}). 

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

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

774 or C{None}. 

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

776 (C{bool}). 

777 

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

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

780 

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

782 

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

784 

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

786 

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

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

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

790 

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

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

793 L{ellipsoidalKarney.areaOf}. 

794 ''' 

795 if _MODS.booleans.isBoolean(points): 

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

797 

798 _at2, _t_2 = atan2, tan_2 

799 _un, _w180 = unrollPI, wrap180 

800 

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

802 p1 = p2 = Ps[0] 

803 a1, b1 = p1.philam 

804 ta1, z1 = _t_2(a1), None 

805 

806 A = Fsum() # mean phi 

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

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

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

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

811 D = Fsum() 

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

813 a2, b2 = p2.philam 

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

815 A += a2 

816 ta2 = _t_2(a2) 

817 tdb = _t_2(db, points=i) 

818 R += _at2(tdb * (ta1 + ta2), 

819 _1_0 + ta1 * ta2) 

820 ta1, b1 = ta2, b2 

821 

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

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

824 if z1 is not None: 

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

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

827 p1, z1 = p2, z2 

828 

829 R = abs(R * _2_0) 

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

831 R = abs(R - PI2) 

832 if radius: 

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

834 R *= _mean_radius(radius, a)**2 

835 return float(R) 

836 

837 

838def _destination2(a, b, r, t): 

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

840 

841 @arg a: Latitude (C{radians}). 

842 @arg b: Longitude (C{radians}). 

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

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

845 

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

847 ''' 

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

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

850 ca *= sr 

851 

852 a = asin1(ct * ca + cr * sa) 

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

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

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

856 

857 

858def _int3d2(s, end, wrap, _i_, Vector, hs): 

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

860 # and similar logic in .ellipsoidalBaseDI._intersect3 

861 a1, b1 = s.philam 

862 

863 if _isDegrees(end): # bearing, get pseudo-end point 

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

865 else: # must be a point 

866 s.others(end, name=_end_ + _i_) 

867 hs.append(end.height) 

868 a2, b2 = end.philam 

869 if wrap: 

870 a2, b2 = _Wrap.philam(a2, b2) 

871 

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

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

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

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

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

877 -(b1 + b2) * _0_5) 

878 cb21 *= sin(a1 - a2) # sa21 

879 sb21 *= sin(a1 + a2) # sa12 

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

881 cb12 * cb21 + sb12 * sb21, 

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

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

884 

885 

886def _intdot(ds, a1, b1, a, b, wrap): 

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

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

889 return fdot(ds, db, a - a1) 

890 

891 

892def intersecant2(center, circle, point, other, **radius_exact_height_wrap): 

893 '''Compute the intersections of a circle and a (great circle) line given as 

894 two points or as a point and bearing. 

895 

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

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

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

899 @arg point: A point on the (great circle) line (L{LatLon}). 

900 @arg other: An other point on the (great circle) line (L{LatLon}) or 

901 the bearing at the B{C{point}} (compass C{degrees360}). 

902 @kwarg radius_exact_height_wrap: Optional keyword arguments, see 

903 method L{LatLon.intersecant2} for further details. 

904 

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

906 an instance of the B{C{point}} class. Both points are the same 

907 instance if the (great circle) line is tangent to the circle. 

908 

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

910 

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

912 B{C{circle}} or B{C{other}} invalid. 

913 

914 @raise UnitError: Invalid B{C{circle}}, B{C{other}}, B{C{radius}}, 

915 B{C{exact}}, B{C{height}} or B{C{napieradius}}. 

916 ''' 

917 c = _T00.others(center=center) 

918 p = _T00.others(point=point) 

919 try: 

920 return _intersecant2(c, circle, p, other, **radius_exact_height_wrap) 

921 except (TypeError, ValueError) as x: 

922 raise _xError(x, center=center, circle=circle, point=point, other=other, 

923 **radius_exact_height_wrap) 

924 

925 

926def _intersect(start1, end1, start2, end2, height=None, wrap=False, # in.ellipsoidalBaseDI._intersect3 

927 LatLon=None, **LatLon_kwds): 

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

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

930 

931 s1, s2 = start1, start2 

932 if wrap: 

933 s2 = _Wrap.point(s2) 

934 hs = [s1.height, s2.height] 

935 

936 a1, b1 = s1.philam 

937 a2, b2 = s2.philam 

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

939 r12 = vincentys_(a2, a1, db) 

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

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

942 

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

944 elif _isDegrees(end1) and _isDegrees(end2): # both bearings 

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

946 

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

948 if isnear0(x1) or isnear0(x2): 

949 raise IntersectionError(_parallel_) 

950 # handle domain error for equivalent longitudes, 

951 # see also functions asin_safe and acos_safe at 

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

953 t12, t13 = acos1((sa2 - sa1 * cr12) / x1), radiansPI2(end1) 

954 t21, t23 = acos1((sa1 - sa2 * cr12) / x2), radiansPI2(end2) 

955 if sin(db) > 0: 

956 t21 = PI2 - t21 

957 else: 

958 t12 = PI2 - t12 

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

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

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

962 raise IntersectionError(_infinite_) 

963 sx3 = sx1 * sx2 

964# XXX if sx3 < 0: 

965# XXX raise ValueError(_ambiguous_) 

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

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

968 

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

970 # like .ellipsoidalBaseDI,_intersect3, if this intersection 

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

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

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

974 

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

976 _N_vector_ = _MODS.nvectorBase._N_vector_ 

977 

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

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

980 x = x1.cross(x2) 

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

982 raise IntersectionError(_colinear_) 

983 a, b = x.philam 

984 # choose intersection similar to sphericalNvector 

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

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

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

988 

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

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

991 intersection, LatLon, LatLon_kwds) 

992 

993 

994def intersection(start1, end1, start2, end2, height=None, wrap=False, 

995 LatLon=LatLon, **LatLon_kwds): 

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

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

998 

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

1000 @arg end1: End point of the first line (L{LatLon}) or 

1001 the initial bearing at the first start point 

1002 (compass C{degrees360}). 

1003 @arg start2: Start point of the second line (L{LatLon}). 

1004 @arg end2: End point of the second line (L{LatLon}) or 

1005 the initial bearing at the second start point 

1006 (compass C{degrees360}). 

1007 @kwarg height: Optional height for the intersection point, 

1008 overriding the mean height (C{meter}). 

1009 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

1010 B{C{start2}} and both B{C{end*}} points (C{bool}). 

1011 @kwarg LatLon: Optional class to return the intersection 

1012 point (L{LatLon}) or C{None}. 

1013 @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword 

1014 arguments, ignored if C{B{LatLon} is None}. 

1015 

1016 @return: The intersection point as a (B{C{LatLon}}) or if 

1017 C{B{LatLon} is None} a L{LatLon3Tuple}C{(lat, lon, 

1018 height)}. An alternate intersection point might 

1019 be the L{antipode} to the returned result. 

1020 

1021 @raise IntersectionError: Ambiguous or infinite intersection 

1022 or colinear, parallel or otherwise 

1023 non-intersecting lines. 

1024 

1025 @raise TypeError: A B{C{start1}}, B{C{end1}}, B{C{start2}} 

1026 or B{C{end2}} point not L{LatLon}. 

1027 

1028 @raise ValueError: Invalid B{C{height}} or C{null} line. 

1029 ''' 

1030 s1 = _T00.others(start1=start1) 

1031 s2 = _T00.others(start2=start2) 

1032 try: 

1033 return _intersect(s1, end1, s2, end2, height=height, wrap=wrap, 

1034 LatLon=LatLon, **LatLon_kwds) 

1035 except (TypeError, ValueError) as x: 

1036 raise _xError(x, start1=start1, end1=end1, start2=start2, end2=end2) 

1037 

1038 

1039def intersections2(center1, rad1, center2, rad2, radius=R_M, eps=_0_0, 

1040 height=None, wrap=False, # was=True 

1041 LatLon=LatLon, **LatLon_kwds): 

1042 '''Compute the intersection points of two circles each defined 

1043 by a center point and a radius. 

1044 

1045 @arg center1: Center of the first circle (L{LatLon}). 

1046 @arg rad1: Radius of the first circle (C{meter} or C{radians}, 

1047 see B{C{radius}}). 

1048 @arg center2: Center of the second circle (L{LatLon}). 

1049 @arg rad2: Radius of the second circle (C{meter} or C{radians}, 

1050 see B{C{radius}}). 

1051 @kwarg radius: Mean earth radius (C{meter} or C{None} if B{C{rad1}}, 

1052 B{C{rad2}} and B{C{eps}} are given in C{radians}). 

1053 @kwarg eps: Required overlap (C{meter} or C{radians}, see 

1054 B{C{radius}}). 

1055 @kwarg height: Optional height for the intersection points (C{meter}, 

1056 conventionally) or C{None} for the I{"radical height"} 

1057 at the I{radical line} between both centers. 

1058 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{center2}} 

1059 (C{bool}). 

1060 @kwarg LatLon: Optional class to return the intersection 

1061 points (L{LatLon}) or C{None}. 

1062 @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword 

1063 arguments, ignored if C{B{LatLon} is None}. 

1064 

1065 @return: 2-Tuple of the intersection points, each a B{C{LatLon}} 

1066 instance or if C{B{LatLon} is None} a L{LatLon3Tuple}C{(lat, 

1067 lon, height)}. For abutting circles, both intersection 

1068 points are the same instance, aka the I{radical center}. 

1069 

1070 @raise IntersectionError: Concentric, antipodal, invalid or 

1071 non-intersecting circles. 

1072 

1073 @raise TypeError: If B{C{center1}} or B{C{center2}} not L{LatLon}. 

1074 

1075 @raise ValueError: Invalid B{C{rad1}}, B{C{rad2}}, B{C{radius}}, 

1076 B{C{eps}} or B{C{height}}. 

1077 

1078 @note: Courtesy of U{Samuel Čavoj<https://GitHub.com/mrJean1/PyGeodesy/issues/41>}. 

1079 

1080 @see: This U{Answer<https://StackOverflow.com/questions/53324667/ 

1081 find-intersection-coordinates-of-two-circles-on-earth/53331953>}. 

1082 ''' 

1083 c1 = _T00.others(center1=center1) 

1084 c2 = _T00.others(center2=center2) 

1085 try: 

1086 return _intersects2(c1, rad1, c2, rad2, radius=radius, eps=eps, 

1087 height=height, wrap=wrap, 

1088 LatLon=LatLon, **LatLon_kwds) 

1089 except (TypeError, ValueError) as x: 

1090 raise _xError(x, center1=center1, rad1=rad1, 

1091 center2=center2, rad2=rad2, wrap=wrap) 

1092 

1093 

1094def _intersects2(c1, rad1, c2, rad2, radius=R_M, eps=_0_0, # in .ellipsoidalBaseDI._intersects2 

1095 height=None, too_d=None, wrap=False, # was=True 

1096 LatLon=LatLon, **LatLon_kwds): 

1097 # (INTERNAL) Intersect two spherical circles, see L{intersections2} 

1098 # above, separated to allow callers to embellish any exceptions 

1099 

1100 def _dest3(bearing, h): 

1101 a, b = _destination2(a1, b1, r1, bearing) 

1102 return _LL3Tuple(degrees90(a), degrees180(b), h, 

1103 intersections2, LatLon, LatLon_kwds) 

1104 

1105 a1, b1 = c1.philam 

1106 a2, b2 = c2.philam 

1107 if wrap: 

1108 a2, b2 = _Wrap.philam(a2, b2) 

1109 

1110 r1, r2, f = _rads3(rad1, rad2, radius) 

1111 if f: # swapped radii, swap centers 

1112 a1, a2 = a2, a1 # PYCHOK swap! 

1113 b1, b2 = b2, b1 # PYCHOK swap! 

1114 

1115 db, b2 = unrollPI(b1, b2, wrap=wrap) 

1116 d = vincentys_(a2, a1, db) # radians 

1117 if d < max(r1 - r2, EPS): 

1118 raise IntersectionError(_near_(_concentric_)) # XXX ConcentricError? 

1119 

1120 r = eps if radius is None else (m2radians( 

1121 eps, radius=radius) if eps else _0_0) 

1122 if r < _0_0: 

1123 raise _ValueError(eps=r) 

1124 

1125 x = fsumf_(r1, r2, -d) # overlap 

1126 if x > max(r, EPS): 

1127 sd, cd, sr1, cr1, _, cr2 = sincos2_(d, r1, r2) 

1128 x = sd * sr1 

1129 if isnear0(x): 

1130 raise _ValueError(_invalid_) 

1131 x = acos1((cr2 - cd * cr1) / x) # 0 <= x <= PI 

1132 

1133 elif x < r: # PYCHOK no cover 

1134 t = (d * radius) if too_d is None else too_d 

1135 raise IntersectionError(_too_(_Fmt.distant(t))) 

1136 

1137 if height is None: # "radical height" 

1138 f = _radical2(d, r1, r2).ratio 

1139 h = Height(favg(c1.height, c2.height, f=f)) 

1140 else: 

1141 h = Height(height) 

1142 

1143 b = bearing_(a1, b1, a2, b2, final=False, wrap=wrap) 

1144 if x < EPS4: # externally ... 

1145 r = _dest3(b, h) 

1146 elif x > _PI_EPS4: # internally ... 

1147 r = _dest3(b + PI, h) 

1148 else: 

1149 return _dest3(b + x, h), _dest3(b - x, h) 

1150 return r, r # ... abutting circles 

1151 

1152 

1153@deprecated_function 

1154def isPoleEnclosedBy(points, wrap=False): # PYCHOK no cover 

1155 '''DEPRECATED, use function L{pygeodesy.ispolar}. 

1156 ''' 

1157 return ispolar(points, wrap=wrap) 

1158 

1159 

1160def _LL3Tuple(lat, lon, height, where, LatLon, LatLon_kwds): 

1161 '''(INTERNAL) Helper for L{intersection}, L{intersections2} and L{meanOf}. 

1162 ''' 

1163 n = where.__name__ 

1164 if LatLon is None: 

1165 r = LatLon3Tuple(lat, lon, height, name=n) 

1166 else: 

1167 kwds = _xkwds(LatLon_kwds, height=height, name=n) 

1168 r = LatLon(lat, lon, **kwds) 

1169 return r 

1170 

1171 

1172def meanOf(points, height=None, wrap=False, LatLon=LatLon, **LatLon_kwds): 

1173 '''Compute the I{geographic} mean of several points. 

1174 

1175 @arg points: Points to be averaged (L{LatLon}[]). 

1176 @kwarg height: Optional height at mean point, overriding the mean 

1177 height (C{meter}). 

1178 @kwarg wrap: If C{True}, wrap or I{normalize} the B{C{points}} 

1179 (C{bool}). 

1180 @kwarg LatLon: Optional class to return the mean point (L{LatLon}) 

1181 or C{None}. 

1182 @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword 

1183 arguments, ignored if C{B{LatLon} is None}. 

1184 

1185 @return: The geographic mean and height (B{C{LatLon}}) or a 

1186 L{LatLon3Tuple}C{(lat, lon, height)} if B{C{LatLon}} 

1187 is C{None}. 

1188 

1189 @raise TypeError: Some B{C{points}} are not L{LatLon}. 

1190 

1191 @raise ValueError: No B{C{points}} or invalid B{C{height}}. 

1192 ''' 

1193 def _N_vs(ps, w): 

1194 Ps = _T00.PointsIter(ps, wrap=w) 

1195 for p in Ps.iterate(closed=False): 

1196 yield p._N_vector 

1197 

1198 m = _MODS.nvectorBase 

1199 # geographic, vectorial mean 

1200 n = m.sumOf(_N_vs(points, wrap), h=height, Vector=m.NvectorBase) 

1201 lat, lon, h = n.latlonheight 

1202 return _LL3Tuple(lat, lon, h, meanOf, LatLon, LatLon_kwds) 

1203 

1204 

1205@deprecated_function 

1206def nearestOn2(point, points, **closed_radius_LatLon_options): # PYCHOK no cover 

1207 '''DEPRECATED, use function L{sphericalTrigonometry.nearestOn3}. 

1208 

1209 @return: ... 2-tuple C{(closest, distance)} of the C{closest} 

1210 point (L{LatLon}) on the polygon and the C{distance} 

1211 between the C{closest} and the given B{C{point}}. The 

1212 C{closest} is a B{C{LatLon}} or a L{LatLon2Tuple}C{(lat, 

1213 lon)} if B{C{LatLon}} is C{None} ... 

1214 ''' 

1215 ll, d, _ = nearestOn3(point, points, **closed_radius_LatLon_options) # PYCHOK 3-tuple 

1216 if _xkwds_get(closed_radius_LatLon_options, LatLon=LatLon) is None: 

1217 ll = LatLon2Tuple(ll.lat, ll.lon) 

1218 return ll, d 

1219 

1220 

1221def nearestOn3(point, points, closed=False, radius=R_M, wrap=False, adjust=True, 

1222 limit=9, **LatLon_and_kwds): 

1223 '''Locate the point on a path or polygon closest to a reference point. 

1224 

1225 Distances are I{approximated} using function L{pygeodesy.equirectangular_}, 

1226 subject to the supplied B{C{options}}. 

1227 

1228 @arg point: The reference point (L{LatLon}). 

1229 @arg points: The path or polygon points (L{LatLon}[]). 

1230 @kwarg closed: Optionally, close the polygon (C{bool}). 

1231 @kwarg radius: Mean earth radius (C{meter}). 

1232 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the 

1233 B{C{points}} (C{bool}). 

1234 @kwarg adjust: See function L{pygeodesy.equirectangular_} (C{bool}). 

1235 @kwarg limit: See function L{pygeodesy.equirectangular_} (C{degrees}), 

1236 default C{9 degrees} is about C{1,000 Kmeter} (for mean 

1237 spherical earth radius L{R_KM}). 

1238 @kwarg LatLon: Optional class to return the closest point (L{LatLon}) 

1239 or C{None}. 

1240 @kwarg options: Optional keyword arguments for function 

1241 L{pygeodesy.equirectangular_}. 

1242 

1243 @return: A L{NearestOn3Tuple}C{(closest, distance, angle)} with the 

1244 C{closest} point as B{C{LatLon}} or L{LatLon3Tuple}C{(lat, 

1245 lon, height)} if B{C{LatLon}} is C{None}. The C{distance} 

1246 is the L{pygeodesy.equirectangular_} distance between the 

1247 C{closest} and the given B{C{point}} converted to C{meter}, 

1248 same units as B{C{radius}}. The C{angle} from the given 

1249 B{C{point}} to the C{closest} is in compass C{degrees360}, 

1250 like function L{pygeodesy.compassAngle}. The C{height} is 

1251 the (interpolated) height at the C{closest} point. 

1252 

1253 @raise LimitError: Lat- and/or longitudinal delta exceeds the B{C{limit}}, 

1254 see function L{pygeodesy.equirectangular_}. 

1255 

1256 @raise PointsError: Insufficient number of B{C{points}}. 

1257 

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

1259 

1260 @raise ValueError: Invalid B{C{radius}}. 

1261 

1262 @see: Functions L{pygeodesy.equirectangular_} and L{pygeodesy.nearestOn5}. 

1263 ''' 

1264 t = _nearestOn5(point, points, closed=closed, wrap=wrap, 

1265 adjust=adjust, limit=limit) 

1266 d = degrees2m(t.distance, radius=radius) 

1267 h = t.height 

1268 n = nearestOn3.__name__ 

1269 

1270 LL, kwds = _xkwds_pop2(LatLon_and_kwds, LatLon=LatLon) 

1271 r = LatLon3Tuple(t.lat, t.lon, h, name=n) if LL is None else \ 

1272 LL(t.lat, t.lon, **_xkwds(kwds, height=h, name=n)) 

1273 return NearestOn3Tuple(r, d, t.angle, name=n) 

1274 

1275 

1276def perimeterOf(points, closed=False, radius=R_M, wrap=True): 

1277 '''Compute the perimeter of a (spherical) polygon or composite 

1278 (with great circle arcs joining the points). 

1279 

1280 @arg points: The polygon points or clips (L{LatLon}[], L{BooleanFHP} 

1281 or L{BooleanGH}). 

1282 @kwarg closed: Optionally, close the polygon (C{bool}). 

1283 @kwarg radius: Mean earth radius (C{meter}) or C{None}. 

1284 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the 

1285 B{C{points}} (C{bool}). 

1286 

1287 @return: Polygon perimeter (C{meter}, same units as B{C{radius}} 

1288 or C{radians} if B{C{radius}} is C{None}). 

1289 

1290 @raise PointsError: Insufficient number of B{C{points}}. 

1291 

1292 @raise TypeError: Some B{C{points}} are not L{LatLon}. 

1293 

1294 @raise ValueError: Invalid B{C{radius}} or C{B{closed}=False} with 

1295 C{B{points}} a composite. 

1296 

1297 @note: Distances are based on function L{pygeodesy.vincentys_}. 

1298 

1299 @see: Functions L{perimeterOf<pygeodesy.perimeterOf>}, 

1300 L{sphericalNvector.perimeterOf} and L{ellipsoidalKarney.perimeterOf}. 

1301 ''' 

1302 def _rads(ps, c, w): # angular edge lengths in radians 

1303 Ps = _T00.PointsIter(ps, loop=1, wrap=w) 

1304 a1, b1 = Ps[0].philam 

1305 for p in Ps.iterate(closed=c): 

1306 a2, b2 = p.philam 

1307 db, b2 = unrollPI(b1, b2, wrap=w and not (c and Ps.looped)) 

1308 yield vincentys_(a2, a1, db) 

1309 a1, b1 = a2, b2 

1310 

1311 if _MODS.booleans.isBoolean(points): 

1312 if not closed: 

1313 raise _ValueError(closed=closed, points=_composite_) 

1314 r = points._sum2(LatLon, perimeterOf, closed=True, radius=radius, wrap=wrap) 

1315 else: 

1316 r = fsum(_rads(points, closed, wrap), floats=True) 

1317 return _radians2m(r, radius) 

1318 

1319 

1320def triangle7(latA, lonA, latB, lonB, latC, lonC, radius=R_M, 

1321 excess=excessAbc_, 

1322 wrap=False): 

1323 '''Compute the angles, sides, and area of a (spherical) triangle. 

1324 

1325 @arg latA: First corner latitude (C{degrees}). 

1326 @arg lonA: First corner longitude (C{degrees}). 

1327 @arg latB: Second corner latitude (C{degrees}). 

1328 @arg lonB: Second corner longitude (C{degrees}). 

1329 @arg latC: Third corner latitude (C{degrees}). 

1330 @arg lonC: Third corner longitude (C{degrees}). 

1331 @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter}, 

1332 L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}) 

1333 or C{None}. 

1334 @kwarg excess: I{Spherical excess} callable (L{excessAbc_}, 

1335 L{excessGirard_} or L{excessLHuilier_}). 

1336 @kwarg wrap: If C{True}, wrap and L{pygeodesy.unroll180} 

1337 longitudes (C{bool}). 

1338 

1339 @return: A L{Triangle7Tuple}C{(A, a, B, b, C, c, area)} with 

1340 spherical angles C{A}, C{B} and C{C}, angular sides 

1341 C{a}, C{b} and C{c} all in C{degrees} and C{area} 

1342 in I{square} C{meter} or same units as B{C{radius}} 

1343 I{squared} or if C{B{radius}=0} or C{None}, a 

1344 L{Triangle8Tuple}C{(A, a, B, b, C, c, D, E)} all in 

1345 C{radians} with the I{spherical excess} C{E} as the 

1346 C{unit area} in C{radians}. 

1347 ''' 

1348 t = triangle8_(Phi_(latA=latA), Lam_(lonA=lonA), 

1349 Phi_(latB=latB), Lam_(lonB=lonB), 

1350 Phi_(latC=latC), Lam_(lonC=lonC), 

1351 excess=excess, wrap=wrap) 

1352 return _t7Tuple(t, radius) 

1353 

1354 

1355def triangle8_(phiA, lamA, phiB, lamB, phiC, lamC, excess=excessAbc_, 

1356 wrap=False): 

1357 '''Compute the angles, sides, I{spherical deficit} and I{spherical 

1358 excess} of a (spherical) triangle. 

1359 

1360 @arg phiA: First corner latitude (C{radians}). 

1361 @arg lamA: First corner longitude (C{radians}). 

1362 @arg phiB: Second corner latitude (C{radians}). 

1363 @arg lamB: Second corner longitude (C{radians}). 

1364 @arg phiC: Third corner latitude (C{radians}). 

1365 @arg lamC: Third corner longitude (C{radians}). 

1366 @kwarg excess: I{Spherical excess} callable (L{excessAbc_}, 

1367 L{excessGirard_} or L{excessLHuilier_}). 

1368 @kwarg wrap: If C{True}, L{pygeodesy.unrollPI} the 

1369 longitudinal deltas (C{bool}). 

1370 

1371 @return: A L{Triangle8Tuple}C{(A, a, B, b, C, c, D, E)} with 

1372 spherical angles C{A}, C{B} and C{C}, angular sides 

1373 C{a}, C{b} and C{c}, I{spherical deficit} C{D} and 

1374 I{spherical excess} C{E}, all in C{radians}. 

1375 ''' 

1376 def _a_r(w, phiA, lamA, phiB, lamB, phiC, lamC): 

1377 d, _ = unrollPI(lamB, lamC, wrap=w) 

1378 a = vincentys_(phiC, phiB, d) 

1379 return a, (phiB, lamB, phiC, lamC, phiA, lamA) # rotate A, B, C 

1380 

1381 def _A_r(a, sa, ca, sb, cb, sc, cc): 

1382 s = sb * sc 

1383 A = acos1((ca - cb * cc) / s) if isnon0(s) else a 

1384 return A, (sb, cb, sc, cc, sa, ca) # rotate sincos2_'s 

1385 

1386 # notation: side C{a} is oposite to corner C{A}, etc. 

1387 a, r = _a_r(wrap, phiA, lamA, phiB, lamB, phiC, lamC) 

1388 b, r = _a_r(wrap, *r) 

1389 c, _ = _a_r(wrap, *r) 

1390 

1391 A, r = _A_r(a, *sincos2_(a, b, c)) 

1392 B, r = _A_r(b, *r) 

1393 C, _ = _A_r(c, *r) 

1394 

1395 D = fsumf_(PI2, -a, -b, -c) # deficit aka defect 

1396 E = excessGirard_(A, B, C) if excess in (excessGirard_, True) else ( 

1397 excessLHuilier_(a, b, c) if excess in (excessLHuilier_, False) else 

1398 excessAbc_(*max((A, b, c), (B, c, a), (C, a, b)))) 

1399 

1400 return Triangle8Tuple(A, a, B, b, C, c, D, E) 

1401 

1402 

1403def _t7Tuple(t, radius): 

1404 '''(INTERNAL) Convert a L{Triangle8Tuple} to L{Triangle7Tuple}. 

1405 ''' 

1406 if radius: # not in (None, _0_0) 

1407 r = radius if _isRadius(radius) else \ 

1408 _ellipsoidal_datum(radius).ellipsoid.Rmean 

1409 A, B, C = map1(degrees, t.A, t.B, t.C) 

1410 t = Triangle7Tuple(A, (r * t.a), 

1411 B, (r * t.b), 

1412 C, (r * t.c), t.E * r**2) 

1413 return t 

1414 

1415 

1416__all__ += _ALL_OTHER(Cartesian, LatLon, # classes 

1417 areaOf, # functions 

1418 intersecant2, intersection, intersections2, ispolar, 

1419 isPoleEnclosedBy, # DEPRECATED, use ispolar 

1420 meanOf, 

1421 nearestOn2, nearestOn3, 

1422 perimeterOf, 

1423 sumOf, # XXX == vector3d.sumOf 

1424 triangle7, triangle8_) 

1425 

1426# **) MIT License 

1427# 

1428# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved. 

1429# 

1430# Permission is hereby granted, free of charge, to any person obtaining a 

1431# copy of this software and associated documentation files (the "Software"), 

1432# to deal in the Software without restriction, including without limitation 

1433# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1434# and/or sell copies of the Software, and to permit persons to whom the 

1435# Software is furnished to do so, subject to the following conditions: 

1436# 

1437# The above copyright notice and this permission notice shall be included 

1438# in all copies or substantial portions of the Software. 

1439# 

1440# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

1441# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1442# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

1443# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

1444# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

1445# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

1446# OTHER DEALINGS IN THE SOFTWARE.