1

2# -*- coding: utf-8 -*-

3

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

5

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

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

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

9 L{perimeterOf}, I{all spherical}.

10

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

12 spherical trigonometry, transcoded from JavaScript originals by

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

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

15'''

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

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

18

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

20 from 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

23 from pygeodesy.datums import _ellipsoidal_datum, _mean_radius

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

25 _ValueError, IntersectionError, _xError, \

26 _xkwds, _xkwds_get, _xkwds_pop

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

28 from pygeodesy.fsums import Fsum, fsum, fsumf_

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

30 excessGirard_, excessLHuilier_, opposing_, _radical2, \

31 vincentys_

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

33 _concentric_, _convex_, _end_, _infinite_, \

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

35 _point_, _SPACE_, _too_

36 from 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

39 from pygeodesy.namedTuples import LatLon2Tuple, LatLon3Tuple, \

40 NearestOn3Tuple, Triangle7Tuple, \

41 Triangle8Tuple

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

43 notImplemented, Fmt as _Fmt # XXX shadowed

44 from pygeodesy.props import deprecated_function, deprecated_method

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

46 LatLonSphericalBase, _rads3, _trilaterate5

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

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

49 Radius_, Scalar

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

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

52 unrollPI, _unrollon3, _Wrap, wrap180, wrapPI

53 from pygeodesy.vector3d import sumOf, Vector3d

54

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

56

57__all__ = _ALL_LAZY.sphericalTrigonometry

58__version__ = '23.06.12'

59

60_parallel_ = 'parallel'

61

62_PI_EPS4 = PI - EPS4

63 if _PI_EPS4 >= PI:

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

65

66

67 class 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

89 class 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, low=None), 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

813 def 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

893 def _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

913 def _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

941 def _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

947 def 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, both

969 points are the same instance, the B{C{point}} or wrapped

970 or I{normalized}.

971

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

973

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

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

976

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

978 B{C{exact}} or B{C{height}}.

979 '''

980 c = _T00.others(center=center)

981 p = _T00.others(point=point)

982 try:

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

984 height=height, wrap=wrap)

985 except (TypeError, ValueError) as x:

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

987

988

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

990 LatLon=None, **LatLon_kwds):

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

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

993

994 s1, s2 = start1, start2

995 if wrap:

996 s2 = _Wrap.point(s2)

997 hs = [s1.height, s2.height]

998

999 a1, b1 = s1.philam

1000 a2, b2 = s2.philam

1001 db, b2 = unrollPI(b1, b2, wrap=wrap)

1002 r12 = vincentys_(a2, a1, db)

1003 if fabs(r12) < EPS: # [nearly] coincident points

1004 a, b = favg(a1, a2), favg(b1, b2)

1005

1006 # see <https://www.EdWilliams.org/avform.htm#Intersection>

1007 elif isscalar(end1) and isscalar(end2): # both bearings

1008 sa1, ca1, sa2, ca2, sr12, cr12 = sincos2_(a1, a2, r12)

1009

1010 x1, x2 = (sr12 * ca1), (sr12 * ca2)

1011 if isnear0(x1) or isnear0(x2):

1012 raise IntersectionError(_parallel_)

1013 # handle domain error for equivalent longitudes,

1014 # see also functions asin_safe and acos_safe at

1015 # <https://www.EdWilliams.org/avform.htm#Math>

1016 t1, t2 = acos1((sa2 - sa1 * cr12) / x1), \

1017 acos1((sa1 - sa2 * cr12) / x2)

1018 if sin(db) > 0:

1019 t12, t21 = t1, PI2 - t2

1020 else:

1021 t12, t21 = PI2 - t1, t2

1022 t13, t23 = radiansPI2(end1), radiansPI2(end2)

1023 sx1, cx1, sx2, cx2 = sincos2_(wrapPI(t13 - t12), # angle 2-1-3

1024 wrapPI(t21 - t23)) # angle 1-2-3)

1025 if isnear0(sx1) and isnear0(sx2):

1026 raise IntersectionError(_infinite_)

1027 sx3 = sx1 * sx2

1028# XXX if sx3 < 0:

1029# XXX raise ValueError(_ambiguous_)

1030 x3 = acos1(cr12 * sx3 - cx2 * cx1)

1031 r13 = atan2(sr12 * sx3, cx2 + cx1 * cos(x3))

1032

1033 a, b = _destination2(a1, b1, r13, t13)

1034 # like .ellipsoidalBaseDI,_intersect3, if this intersection

1035 # is "before" the first point, use the antipodal intersection

1036 if opposing_(t13, bearing_(a1, b1, a, b, wrap=wrap)):

1037 a, b = antipode_(a, b) # PYCHOK PhiLam2Tuple

1038

1039 else: # end point(s) or bearing(s)

1040 _N_vector_ = _MODS.nvectorBase._N_vector_

1041

1042 x1, d1 = _int3d2(s1, end1, wrap, _1_, _N_vector_, hs)

1043 x2, d2 = _int3d2(s2, end2, wrap, _2_, _N_vector_, hs)

1044 x = x1.cross(x2)

1045 if x.length < EPS: # [nearly] colinear or parallel lines

1046 raise IntersectionError(_colinear_)

1047 a, b = x.philam

1048 # choose intersection similar to sphericalNvector

1049 if not (_intdot(d1, a1, b1, a, b, wrap) *

1050 _intdot(d2, a2, b2, a, b, wrap)) > 0:

1051 a, b = antipode_(a, b) # PYCHOK PhiLam2Tuple

1052

1053 h = fmean(hs) if height is None else Height(height)

1054 return _LL3Tuple(degrees90(a), degrees180(b), h,

1055 intersection, LatLon, LatLon_kwds)

1056

1057

1058 def intersection(start1, end1, start2, end2, height=None, wrap=False,

1059 LatLon=LatLon, **LatLon_kwds):

1060 '''Compute the intersection point of two lines, each defined

1061 by two points or a start point and bearing from North.

1062

1063 @arg start1: Start point of the first line (L{LatLon}).

1064 @arg end1: End point of the first line (L{LatLon}) or

1065 the initial bearing at the first start point

1066 (compass C{degrees360}).

1067 @arg start2: Start point of the second line (L{LatLon}).

1068 @arg end2: End point of the second line (L{LatLon}) or

1069 the initial bearing at the second start point

1070 (compass C{degrees360}).

1071 @kwarg height: Optional height for the intersection point,

1072 overriding the mean height (C{meter}).

1073 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll

1074 B{C{start2}} and both B{C{end*}} points (C{bool}).

1075 @kwarg LatLon: Optional class to return the intersection

1076 point (L{LatLon}) or C{None}.

1077 @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword

1078 arguments, ignored if C{B{LatLon} is None}.

1079

1080 @return: The intersection point as a (B{C{LatLon}}) or if

1081 C{B{LatLon} is None} a L{LatLon3Tuple}C{(lat, lon,

1082 height)}. An alternate intersection point might

1083 be the L{antipode} to the returned result.

1084

1085 @raise IntersectionError: Ambiguous or infinite intersection

1086 or colinear, parallel or otherwise

1087 non-intersecting lines.

1088

1089 @raise TypeError: A B{C{start1}}, B{C{end1}}, B{C{start2}}

1090 or B{C{end2}} point not L{LatLon}.

1091

1092 @raise ValueError: Invalid B{C{height}} or C{null} line.

1093

1094 @example:

1095

1096 >>> p = LatLon(51.8853, 0.2545)

1097 >>> s = LatLon(49.0034, 2.5735)

1098 >>> i = intersection(p, 108.547, s, 32.435) # '50.9078°N, 004.5084°E'

1099 '''

1100 s1 = _T00.others(start1=start1)

1101 s2 = _T00.others(start2=start2)

1102 try:

1103 return _intersect(s1, end1, s2, end2, height=height, wrap=wrap,

1104 LatLon=LatLon, **LatLon_kwds)

1105 except (TypeError, ValueError) as x:

1106 raise _xError(x, start1=start1, end1=end1, start2=start2, end2=end2)

1107

1108

1109 def intersections2(center1, rad1, center2, rad2, radius=R_M, eps=_0_0,

1110 height=None, wrap=False, # was=True

1111 LatLon=LatLon, **LatLon_kwds):

1112 '''Compute the intersection points of two circles each defined

1113 by a center point and a radius.

1114

1115 @arg center1: Center of the first circle (L{LatLon}).

1116 @arg rad1: Radius of the first circle (C{meter} or C{radians},

1117 see B{C{radius}}).

1118 @arg center2: Center of the second circle (L{LatLon}).

1119 @arg rad2: Radius of the second circle (C{meter} or C{radians},

1120 see B{C{radius}}).

1121 @kwarg radius: Mean earth radius (C{meter} or C{None} if B{C{rad1}},

1122 B{C{rad2}} and B{C{eps}} are given in C{radians}).

1123 @kwarg eps: Required overlap (C{meter} or C{radians}, see

1124 B{C{radius}}).

1125 @kwarg height: Optional height for the intersection points (C{meter},

1126 conventionally) or C{None} for the I{"radical height"}

1127 at the I{radical line} between both centers.

1128 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{center2}}

1129 (C{bool}).

1130 @kwarg LatLon: Optional class to return the intersection

1131 points (L{LatLon}) or C{None}.

1132 @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword

1133 arguments, ignored if C{B{LatLon} is None}.

1134

1135 @return: 2-Tuple of the intersection points, each a B{C{LatLon}}

1136 instance or if C{B{LatLon} is None} a L{LatLon3Tuple}C{(lat,

1137 lon, height)}. For abutting circles, both intersection

1138 points are the same instance, aka the I{radical center}.

1139

1140 @raise IntersectionError: Concentric, antipodal, invalid or

1141 non-intersecting circles.

1142

1143 @raise TypeError: If B{C{center1}} or B{C{center2}} not L{LatLon}.

1144

1145 @raise ValueError: Invalid B{C{rad1}}, B{C{rad2}}, B{C{radius}},

1146 B{C{eps}} or B{C{height}}.

1147

1148 @note: Courtesy of U{Samuel Čavoj<https://GitHub.com/mrJean1/PyGeodesy/issues/41>}.

1149

1150 @see: This U{Answer<https://StackOverflow.com/questions/53324667/

1151 find-intersection-coordinates-of-two-circles-on-earth/53331953>}.

1152 '''

1153 c1 = _T00.others(center1=center1)

1154 c2 = _T00.others(center2=center2)

1155 try:

1156 return _intersects2(c1, rad1, c2, rad2, radius=radius, eps=eps,

1157 height=height, wrap=wrap,

1158 LatLon=LatLon, **LatLon_kwds)

1159 except (TypeError, ValueError) as x:

1160 raise _xError(x, center1=center1, rad1=rad1,

1161 center2=center2, rad2=rad2, wrap=wrap)

1162

1163

1164 def _intersects2(c1, rad1, c2, rad2, radius=R_M, eps=_0_0, # in .ellipsoidalBaseDI._intersects2

1165 height=None, too_d=None, wrap=False, # was=True

1166 LatLon=LatLon, **LatLon_kwds):

1167 # (INTERNAL) Intersect two spherical circles, see L{intersections2}

1168 # above, separated to allow callers to embellish any exceptions

1169

1170 def _dest3(bearing, h):

1171 a, b = _destination2(a1, b1, r1, bearing)

1172 return _LL3Tuple(degrees90(a), degrees180(b), h,

1173 intersections2, LatLon, LatLon_kwds)

1174

1175 a1, b1 = c1.philam

1176 a2, b2 = c2.philam

1177 if wrap:

1178 a2, b2 = _Wrap.philam(a2, b2)

1179

1180 r1, r2, f = _rads3(rad1, rad2, radius)

1181 if f: # swapped radii, swap centers

1182 a1, a2 = a2, a1 # PYCHOK swap!

1183 b1, b2 = b2, b1 # PYCHOK swap!

1184

1185 db, b2 = unrollPI(b1, b2, wrap=wrap)

1186 d = vincentys_(a2, a1, db) # radians

1187 if d < max(r1 - r2, EPS):

1188 raise IntersectionError(_near_(_concentric_)) # XXX ConcentricError?

1189

1190 r = eps if radius is None else (m2radians(

1191 eps, radius=radius) if eps else _0_0)

1192 if r < _0_0:

1193 raise _ValueError(eps=r)

1194

1195 x = fsumf_(r1, r2, -d) # overlap

1196 if x > max(r, EPS):

1197 sd, cd, sr1, cr1, _, cr2 = sincos2_(d, r1, r2)

1198 x = sd * sr1

1199 if isnear0(x):

1200 raise _ValueError(_invalid_)

1201 x = acos1((cr2 - cd * cr1) / x) # 0 <= x <= PI

1202

1203 elif x < r: # PYCHOK no cover

1204 t = (d * radius) if too_d is None else too_d

1205 raise IntersectionError(_too_(_Fmt.distant(t)))

1206

1207 if height is None: # "radical height"

1208 f = _radical2(d, r1, r2).ratio

1209 h = Height(favg(c1.height, c2.height, f=f))

1210 else:

1211 h = Height(height)

1212

1213 b = bearing_(a1, b1, a2, b2, final=False, wrap=wrap)

1214 if x < EPS4: # externally ...

1215 r = _dest3(b, h)

1216 elif x > _PI_EPS4: # internally ...

1217 r = _dest3(b + PI, h)

1218 else:

1219 return _dest3(b + x, h), _dest3(b - x, h)

1220 return r, r # ... abutting circles

1221

1222

1223@deprecated_function

1224 def isPoleEnclosedBy(points, wrap=False): # PYCHOK no cover

1225 '''DEPRECATED, use function L{pygeodesy.ispolar}.

1226 '''

1227 return ispolar(points, wrap=wrap)

1228

1229

1230 def _LL3Tuple(lat, lon, height, where, LatLon, LatLon_kwds):

1231 '''(INTERNAL) Helper for L{intersection}, L{intersections2} and L{meanOf}.

1232 '''

1233 n = where.__name__

1234 if LatLon is None:

1235 r = LatLon3Tuple(lat, lon, height, name=n)

1236 else:

1237 kwds = _xkwds(LatLon_kwds, height=height, name=n)

1238 r = LatLon(lat, lon, **kwds)

1239 return r

1240

1241

1242 def meanOf(points, height=None, wrap=False, LatLon=LatLon, **LatLon_kwds):

1243 '''Compute the I{geographic} mean of several points.

1244

1245 @arg points: Points to be averaged (L{LatLon}[]).

1246 @kwarg height: Optional height at mean point, overriding the mean

1247 height (C{meter}).

1248 @kwarg wrap: If C{True}, wrap or I{normalize} the B{C{points}}

1249 (C{bool}).

1250 @kwarg LatLon: Optional class to return the mean point (L{LatLon})

1251 or C{None}.

1252 @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword

1253 arguments, ignored if C{B{LatLon} is None}.

1254

1255 @return: The geographic mean and height (B{C{LatLon}}) or a

1256 L{LatLon3Tuple}C{(lat, lon, height)} if B{C{LatLon}}

1257 is C{None}.

1258

1259 @raise TypeError: Some B{C{points}} are not L{LatLon}.

1260

1261 @raise ValueError: No B{C{points}} or invalid B{C{height}}.

1262 '''

1263 def _N_vs(ps, w):

1264 Ps = _T00.PointsIter(ps, wrap=w)

1265 for p in Ps.iterate(closed=False):

1266 yield p._N_vector

1267

1268 m = _MODS.nvectorBase

1269 # geographic, vectorial mean

1270 n = m.sumOf(_N_vs(points, wrap), h=height, Vector=m.NvectorBase)

1271 lat, lon, h = n.latlonheight

1272 return _LL3Tuple(lat, lon, h, meanOf, LatLon, LatLon_kwds)

1273

1274

1275@deprecated_function

1276 def nearestOn2(point, points, **closed_radius_LatLon_options): # PYCHOK no cover

1277 '''DEPRECATED, use function L{sphericalTrigonometry.nearestOn3}.

1278

1279 @return: ... 2-tuple C{(closest, distance)} of the C{closest}

1280 point (L{LatLon}) on the polygon and the C{distance}

1281 between the C{closest} and the given B{C{point}}. The

1282 C{closest} is a B{C{LatLon}} or a L{LatLon2Tuple}C{(lat,

1283 lon)} if B{C{LatLon}} is C{None} ...

1284 '''

1285 ll, d, _ = nearestOn3(point, points, **closed_radius_LatLon_options) # PYCHOK 3-tuple

1286 if _xkwds_get(closed_radius_LatLon_options, LatLon=LatLon) is None:

1287 ll = LatLon2Tuple(ll.lat, ll.lon)

1288 return ll, d

1289

1290

1291 def nearestOn3(point, points, closed=False, radius=R_M, wrap=False, adjust=True,

1292 limit=9, **LatLon_and_kwds):

1293 '''Locate the point on a path or polygon closest to a reference point.

1294

1295 Distances are I{approximated} using function L{pygeodesy.equirectangular_},

1296 subject to the supplied B{C{options}}.

1297

1298 @arg point: The reference point (L{LatLon}).

1299 @arg points: The path or polygon points (L{LatLon}[]).

1300 @kwarg closed: Optionally, close the polygon (C{bool}).

1301 @kwarg radius: Mean earth radius (C{meter}).

1302 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the

1303 B{C{points}} (C{bool}).

1304 @kwarg adjust: See function L{pygeodesy.equirectangular_} (C{bool}).

1305 @kwarg limit: See function L{pygeodesy.equirectangular_} (C{degrees}),

1306 default C{9 degrees} is about C{1,000 Kmeter} (for mean

1307 spherical earth radius L{R_KM}).

1308 @kwarg LatLon: Optional class to return the closest point (L{LatLon})

1309 or C{None}.

1310 @kwarg options: Optional keyword arguments for function

1311 L{pygeodesy.equirectangular_}.

1312

1313 @return: A L{NearestOn3Tuple}C{(closest, distance, angle)} with the

1314 C{closest} point as B{C{LatLon}} or L{LatLon3Tuple}C{(lat,

1315 lon, height)} if B{C{LatLon}} is C{None}. The C{distance}

1316 is the L{pygeodesy.equirectangular_} distance between the

1317 C{closest} and the given B{C{point}} converted to C{meter},

1318 same units as B{C{radius}}. The C{angle} from the given

1319 B{C{point}} to the C{closest} is in compass C{degrees360},

1320 like function L{pygeodesy.compassAngle}. The C{height} is

1321 the (interpolated) height at the C{closest} point.

1322

1323 @raise LimitError: Lat- and/or longitudinal delta exceeds the B{C{limit}},

1324 see function L{pygeodesy.equirectangular_}.

1325

1326 @raise PointsError: Insufficient number of B{C{points}}.

1327

1328 @raise TypeError: Some B{C{points}} are not C{LatLon}.

1329

1330 @raise ValueError: Invalid B{C{radius}}.

1331

1332 @see: Functions L{pygeodesy.equirectangular_} and L{pygeodesy.nearestOn5}.

1333 '''

1334 t = _nearestOn5(point, points, closed=closed, wrap=wrap,

1335 adjust=adjust, limit=limit)

1336 d = degrees2m(t.distance, radius=radius)

1337 h = t.height

1338 n = nearestOn3.__name__

1339

1340 kwds = _xkwds(LatLon_and_kwds, height=h, name=n)

1341 LL = _xkwds_pop(kwds, LatLon=LatLon)

1342 r = LatLon3Tuple(t.lat, t.lon, h, name=n) if LL is None else \

1343 LL(t.lat, t.lon, **kwds)

1344 return NearestOn3Tuple(r, d, t.angle, name=n)

1345

1346

1347 def perimeterOf(points, closed=False, radius=R_M, wrap=True):

1348 '''Compute the perimeter of a (spherical) polygon or composite

1349 (with great circle arcs joining the points).

1350

1351 @arg points: The polygon points or clips (L{LatLon}[], L{BooleanFHP}

1352 or L{BooleanGH}).

1353 @kwarg closed: Optionally, close the polygon (C{bool}).

1354 @kwarg radius: Mean earth radius (C{meter}) or C{None}.

1355 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the

1356 B{C{points}} (C{bool}).

1357

1358 @return: Polygon perimeter (C{meter}, same units as B{C{radius}}

1359 or C{radians} if B{C{radius}} is C{None}).

1360

1361 @raise PointsError: Insufficient number of B{C{points}}.

1362

1363 @raise TypeError: Some B{C{points}} are not L{LatLon}.

1364

1365 @raise ValueError: Invalid B{C{radius}} or C{B{closed}=False} with

1366 C{B{points}} a composite.

1367

1368 @note: Distances are based on function L{pygeodesy.vincentys_}.

1369

1370 @see: Functions L{pygeodesy.perimeterOf}, L{sphericalNvector.perimeterOf}

1371 and L{ellipsoidalKarney.perimeterOf}.

1372 '''

1373 def _rads(ps, c, w): # angular edge lengths in radians

1374 Ps = _T00.PointsIter(ps, loop=1, wrap=w)

1375 a1, b1 = Ps[0].philam

1376 for p in Ps.iterate(closed=c):

1377 a2, b2 = p.philam

1378 db, b2 = unrollPI(b1, b2, wrap=w and not (c and Ps.looped))

1379 yield vincentys_(a2, a1, db)

1380 a1, b1 = a2, b2

1381

1382 if _MODS.booleans.isBoolean(points):

1383 if not closed:

1384 raise _ValueError(closed=closed, points=_composite_)

1385 r = points._sum2(LatLon, perimeterOf, closed=True, radius=radius, wrap=wrap)

1386 else:

1387 r = fsum(_rads(points, closed, wrap), floats=True)

1388 return _r2m(r, radius)

1389

1390

1391 def _r2m(r, radius):

1392 '''(INTERNAL) Angular distance in C{radians} to C{meter}.

1393 '''

1394 if radius is not None: # not in (None, _0_0)

1395 r *= R_M if radius is R_M else Radius(radius)

1396 return r

1397

1398

1399 def triangle7(latA, lonA, latB, lonB, latC, lonC, radius=R_M,

1400 excess=excessAbc_,

1401 wrap=False):

1402 '''Compute the angles, sides, and area of a (spherical) triangle.

1403

1404 @arg latA: First corner latitude (C{degrees}).

1405 @arg lonA: First corner longitude (C{degrees}).

1406 @arg latB: Second corner latitude (C{degrees}).

1407 @arg lonB: Second corner longitude (C{degrees}).

1408 @arg latC: Third corner latitude (C{degrees}).

1409 @arg lonC: Third corner longitude (C{degrees}).

1410 @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter},

1411 L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple})

1412 or C{None}.

1413 @kwarg excess: I{Spherical excess} callable (L{excessAbc_},

1414 L{excessGirard_} or L{excessLHuilier_}).

1415 @kwarg wrap: If C{True}, wrap and L{pygeodesy.unroll180}

1416 longitudes (C{bool}).

1417

1418 @return: A L{Triangle7Tuple}C{(A, a, B, b, C, c, area)} with

1419 spherical angles C{A}, C{B} and C{C}, angular sides

1420 C{a}, C{b} and C{c} all in C{degrees} and C{area}

1421 in I{square} C{meter} or same units as B{C{radius}}

1422 I{squared} or if C{B{radius}=0} or C{None}, a

1423 L{Triangle8Tuple}C{(A, a, B, b, C, c, D, E)} all in

1424 C{radians} with the I{spherical excess} C{E} as the

1425 C{unit area} in C{radians}.

1426 '''

1427 t = triangle8_(Phi_(latA=latA), Lam_(lonA=lonA),

1428 Phi_(latB=latB), Lam_(lonB=lonB),

1429 Phi_(latC=latC), Lam_(lonC=lonC),

1430 excess=excess, wrap=wrap)

1431 return _t7Tuple(t, radius)

1432

1433

1434 def triangle8_(phiA, lamA, phiB, lamB, phiC, lamC, excess=excessAbc_,

1435 wrap=False):

1436 '''Compute the angles, sides, I{spherical deficit} and I{spherical

1437 excess} of a (spherical) triangle.

1438

1439 @arg phiA: First corner latitude (C{radians}).

1440 @arg lamA: First corner longitude (C{radians}).

1441 @arg phiB: Second corner latitude (C{radians}).

1442 @arg lamB: Second corner longitude (C{radians}).

1443 @arg phiC: Third corner latitude (C{radians}).

1444 @arg lamC: Third corner longitude (C{radians}).

1445 @kwarg excess: I{Spherical excess} callable (L{excessAbc_},

1446 L{excessGirard_} or L{excessLHuilier_}).

1447 @kwarg wrap: If C{True}, L{pygeodesy.unrollPI} the

1448 longitudinal deltas (C{bool}).

1449

1450 @return: A L{Triangle8Tuple}C{(A, a, B, b, C, c, D, E)} with

1451 spherical angles C{A}, C{B} and C{C}, angular sides

1452 C{a}, C{b} and C{c}, I{spherical deficit} C{D} and

1453 I{spherical excess} C{E}, all in C{radians}.

1454 '''

1455 def _a_r(w, phiA, lamA, phiB, lamB, phiC, lamC):

1456 d, _ = unrollPI(lamB, lamC, wrap=w)

1457 a = vincentys_(phiC, phiB, d)

1458 return a, (phiB, lamB, phiC, lamC, phiA, lamA)

1459

1460 def _A_r(a, sa, ca, sb, cb, sc, cc):

1461 s = sb * sc

1462 A = acos1((ca - cb * cc) / s) if isnon0(s) else a

1463 return A, (sb, cb, sc, cc, sa, ca) # rotate sincos2's

1464

1465 # notation: side C{a} is oposite to corner C{A}, etc.

1466 a, r = _a_r(wrap, phiA, lamA, phiB, lamB, phiC, lamC)

1467 b, r = _a_r(wrap, *r)

1468 c, _ = _a_r(wrap, *r)

1469

1470 A, r = _A_r(a, *sincos2_(a, b, c))

1471 B, r = _A_r(b, *r)

1472 C, _ = _A_r(c, *r)

1473

1474 D = fsumf_(PI2, -a, -b, -c) # deficit aka defect

1475 E = excessGirard_(A, B, C) if excess in (excessGirard_, True) else (

1476 excessLHuilier_(a, b, c) if excess in (excessLHuilier_, False) else

1477 excessAbc_(*max((A, b, c), (B, c, a), (C, a, b))))

1478

1479 return Triangle8Tuple(A, a, B, b, C, c, D, E)

1480

1481

1482 def _t7Tuple(t, radius):

1483 '''(INTERNAL) Convert a L{Triangle8Tuple} to L{Triangle7Tuple}.

1484 '''

1485 if radius: # not in (None, _0_0)

1486 r = radius if isscalar(radius) else \

1487 _ellipsoidal_datum(radius).ellipsoid.Rmean

1488 A, B, C = map1(degrees, t.A, t.B, t.C)

1489 t = Triangle7Tuple(A, (r * t.a),

1490 B, (r * t.b),

1491 C, (r * t.c), t.E * r**2)

1492 return t

1493

1494

1495__all__ += _ALL_OTHER(Cartesian, LatLon, # classes

1496 areaOf, # functions

1497 intersecant2, intersection, intersections2, ispolar,

1498 isPoleEnclosedBy, # DEPRECATED, use ispolar

1499 meanOf,

1500 nearestOn2, nearestOn3,

1501 perimeterOf,

1502 sumOf, # XXX == vector3d.sumOf

1503 triangle7, triangle8_)

1504

1505# **) MIT License

1506#

1507# Copyright (C) 2016-2023 -- mrJean1 at Gmail -- All Rights Reserved.

1508#

1509# Permission is hereby granted, free of charge, to any person obtaining a

1510# copy of this software and associated documentation files (the "Software"),

1511# to deal in the Software without restriction, including without limitation

1512# the rights to use, copy, modify, merge, publish, distribute, sublicense,

1513# and/or sell copies of the Software, and to permit persons to whom the

1514# Software is furnished to do so, subject to the following conditions:

1515#

1516# The above copyright notice and this permission notice shall be included

1517# in all copies or substantial portions of the Software.

1518#

1519# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS

1520# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

1521# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL

1522# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR

1523# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,

1524# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR

1525# OTHER DEALINGS IN THE SOFTWARE.