1

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

3

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

5

6 N-vector-based classes geodetic (lat-/longitude) L{LatLon}, geocentric

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

8 L{meanOf}, L{nearestOn3}, L{perimeterOf}, L{sumOf}, L{triangulate} and

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

10

11 Pure Python implementation of n-vector-based spherical geodetic (lat-/longitude)

12 methods, transcoded from JavaScript originals by I{(C) Chris Veness 2011-2016},

13 published under the same MIT Licence**. See U{Vector-based geodesy

14<https://www.Movable-Type.co.UK/scripts/latlong-vectors.html>} and

15 U{Module latlon-nvector-spherical

16<https://www.Movable-Type.co.UK/scripts/geodesy/docs/module-latlon-nvector-spherical.html>}.

17

18 Tools for working with points and lines on (a spherical model of) the

19 earth’s surface using using n-vectors rather than the more common

20 spherical trigonometry. N-vectors make many calculations much simpler,

21 and easier to follow, compared with the trigonometric equivalents.

22

23 Based on Kenneth Gade’s U{‘Non-singular Horizontal Position Representation’

24<https://www.NavLab.net/Publications/A_Nonsingular_Horizontal_Position_Representation.pdf>},

25 The Journal of Navigation (2010), vol 63, nr 3, pp 395-417.

26

27 Note that the formulations below take x => 0°N,0°E, y => 0°N,90°E and

28 z => 90°N while Gade uses x => 90°N, y => 0°N,90°E, z => 0°N,0°E.

29

30 Also note that on a spherical earth model, an n-vector is equivalent

31 to a normalised version of an (ECEF) cartesian coordinate.

32'''

33# make sure int/int division yields float quosient, see .basics

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

35

36 from pygeodesy.basics import isscalar, _xinstanceof

37 from pygeodesy.constants import EPS, EPS0, PI, PI2, PI_2, R_M, \

38 _0_0, _0_5, _1_0

39# from pygeodesy.datums import Datums # from .sphericalBase

40 from pygeodesy.errors import _ValueError, _xError, _xkwds, _xkwds_pop

41 from pygeodesy.fmath import fmean, fsum

42# from pygeodesy.fsums import fsum # from .fmath

43 from pygeodesy.interns import _composite_, _end_, _Nv00_, _other_, _point_, \

44 _points_, _pole_

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

46# from pygeodesy.named import notImplemented # from .nvectorBase

47# from pygeodesy.namedTuples import NearestOn3Tuple # from .points

48 from pygeodesy.nvectorBase import NvectorBase, NorthPole, LatLonNvectorBase, \

49 sumOf as _sumOf, _triangulate, _trilaterate, \

50 notImplemented

51 from pygeodesy.points import NearestOn3Tuple, ispolar # PYCHOK exported

52 from pygeodesy.props import deprecated_function, deprecated_method

53 from pygeodesy.sphericalBase import _angular, CartesianSphericalBase, \

54 Datums, _intersecant2, LatLonSphericalBase

55 from pygeodesy.units import Bearing, Bearing_, Height, Radius, Scalar

56 from pygeodesy.utily import atan2, degrees360, fabs, sincos2, sincos2_, sincos2d

57

58# from math import atan2, fabs # from utily

59

60__all__ = _ALL_LAZY.sphericalNvector

61__version__ = '23.04.11'

62

63_lines_ = 'lines'

64

65

66 class Cartesian(CartesianSphericalBase):

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

68 L{Nvector} and n-vector-based, spherical L{LatLon}.

69 '''

70

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

72 '''Convert this cartesian to an C{Nvector}-based geodetic point.

73

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

75 arguments, like C{datum}. Use C{B{LatLon}=...}

76 to override this L{LatLon} class or specify

77 C{B{LatLon}=None}.

78

79 @return: The geodetic point (L{LatLon}) or if B{C{LatLon}} is set

80 to C{None}, an L{Ecef9Tuple}C{(x, y, z, lat, lon, height,

81 C, M, datum)} 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 def toNvector(self, **Nvector_and_kwds): # PYCHOK Datums.WGS84

89 '''Convert this cartesian to L{Nvector} components, I{including height}.

90

91 @kwarg Nvector_and_kwds: Optional L{Nvector} and L{Nvector} keyword

92 arguments, like C{datum}. Use C{B{Nvector}=...}

93 to override this L{Nvector} class or specify

94 C{B{Nvector}=None}.

95

96 @return: The C{n-vector} components (L{Nvector}) or if B{C{Nvector}}

97 is set to C{None}, a L{Vector4Tuple}C{(x, y, z, h)}

98

99 @raise TypeError: Invalid B{C{Nvector_and_kwds}} argument.

100 '''

101 # ll = CartesianBase.toLatLon(self, LatLon=LatLon,

102 # datum=datum or self.datum)

103 # kwds = _xkwds(kwds, Nvector=Nvector)

104 # return ll.toNvector(**kwds)

105 kwds = _xkwds(Nvector_and_kwds, Nvector=Nvector, datum=self.datum)

106 return CartesianSphericalBase.toNvector(self, **kwds)

107

108

109 class LatLon(LatLonNvectorBase, LatLonSphericalBase):

110 '''New n-vector based point on a spherical earth model.

111

112 Tools for working with points, lines and paths on (a spherical

113 model of) the earth's surface using vector-based methods.

114

115 @example:

116

117 >>> from sphericalNvector import LatLon

118 >>> p = LatLon(52.205, 0.119)

119 '''

120 _Nv = None # cached_toNvector L{Nvector})

121

122 def _update(self, updated, *attrs, **setters): # PYCHOK args

123 '''(INTERNAL) Zap cached attributes if updated.

124 '''

125 if updated: # reset caches

126 LatLonNvectorBase._update(self, updated, _Nv=self._Nv) # special case

127 LatLonSphericalBase._update(self, updated, *attrs, **setters)

128

129 def alongTrackDistanceTo(self, start, end, radius=R_M):

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

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

132 end point.

133

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

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

136 from the start point to the point where the perpendicular

137 crosses the line.

138

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

140 @arg end: End point of great circle line (L{LatLon}) or

141 initial bearing from start point (compass

142 C{degrees360}).

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

144

145 @return: Distance along the great circle line (positive if

146 after the start toward the end point of the line

147 or negative if before the start point).

148

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

150

151 @raise Valuerror: Some points coincide.

152

153 @example:

154

155 >>> p = LatLon(53.2611, -0.7972)

156

157 >>> s = LatLon(53.3206, -1.7297)

158 >>> e = LatLon(53.1887, 0.1334)

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

160 '''

161 self.others(start=start)

162 gc, _, _ = self._gc3(start, end, _end_)

163

164 p = self.toNvector()

165 a = gc.cross(p).cross(gc) # along-track point gc × p × gc

166 return start.toNvector().angleTo(a, vSign=gc) * radius

167

168 @deprecated_method

169 def bearingTo(self, other, **unused): # PYCHOK no cover

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

171 '''

172 return self.initialBearingTo(other)

173

174 def crossTrackDistanceTo(self, start, end, radius=R_M):

175 '''Compute the (signed) distance from this point to great circle

176 defined by a start and end point.

177

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

179 @arg end: End point of great circle line (L{LatLon}) or

180 initial bearing from start point (compass

181 C{degrees360}).

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

183

184 @return: Distance to great circle (negative if to the

185 left or positive if to the right of the line).

186

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

188

189 @raise Valuerror: Some points coincide.

190

191 @example:

192

193 >>> p = LatLon(53.2611, -0.7972)

194

195 >>> s = LatLon(53.3206, -1.7297)

196 >>> d = p.crossTrackDistanceTo(s, 96) # -305.7

197

198 >>> e = LatLon(53.1887, 0.1334)

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

200 '''

201 self.others(start=start)

202 gc, _, _ = self._gc3(start, end, _end_)

203

204 p = self.toNvector()

205 return (gc.angleTo(p) - PI_2) * radius

206

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

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

209 travelled the given distance on the given bearing.

210

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

212 as B{C{radius}}).

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

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

215 @kwarg height: Optional height at destination, overriding the

216 default height (C{meter}, same units as B{C{radius}}).

217

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

219

220 @raise Valuerror: Polar coincidence or invalid B{C{distance}},

221 B{C{bearing}}, B{C{radius}} or B{C{height}}.

222

223 @example:

224

225 >>> p = LatLon(51.4778, -0.0015)

226 >>> q = p.destination(7794, 300.7)

227 >>> q.toStr() # 51.513546°N, 000.098345°W

228 '''

229 a = _angular(distance, radius)

230 sa, ca, sb, cb = sincos2_(a, Bearing_(bearing))

231

232 p = self.toNvector()

233 e = NorthPole.cross(p, raiser=_pole_).unit() # east vector at p

234 n = p.cross(e) # north vector at p

235 q = n.times(cb).plus(e.times(sb)) # direction vector @ p

236 n = p.times(ca).plus(q.times(sa))

237 return n.toLatLon(height=height, LatLon=self.classof) # Nvector(n.x, n.y, n.z).toLatLon(...)

238

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

240 '''Compute the distance from this to an other point.

241

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

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

244 @kwarg wrap: Wrap/unroll the angular distance (C{bool}).

245

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

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

248 if B{C{radius}} is C{None}).

249

250 @raise TypeError: Invalid B{C{other}} point.

251

252 @example:

253

254 >>> p = LatLon(52.205, 0.119)

255 >>> q = LatLon(48.857, 2.351);

256 >>> d = p.distanceTo(q) # 404.3 km

257 '''

258 self.others(other)

259

260 r = fabs(self.toNvector().angleTo(other.toNvector(), wrap=wrap))

261 return r if radius is None else (Radius(radius) * r)

262

263# @Property_RO

264# def Ecef(self):

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

266# '''

267# return _ALL_MODS.ecef.EcefKarney

268

269 def _gc3(self, start, end, namend, raiser=_points_):

270 '''(INTERNAL) Return great circle, start and end Nvectors.

271 '''

272 s = start.toNvector()

273 if isscalar(end): # bearing

274 gc = s.greatCircle(end)

275 e = None

276 else:

277 self.others(end, name=namend)

278 e = end.toNvector()

279 gc = s.cross(e, raiser=raiser) # XXX .unit()?

280 return gc, s, e

281

282 def greatCircle(self, bearing):

283 '''Compute the vector normal to great circle obtained by

284 heading on the given bearing from this point.

285

286 Direction of vector is such that initial bearing vector

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

288

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

290

291 @return: N-vector representing the great circle (L{Nvector}).

292 '''

293 a, b = self.philam

294 t = Bearing_(bearing)

295

296 sa, ca, sb, cb, st, ct = sincos2_(a, b, t)

297 return Nvector(sb * ct - sa * cb * st,

298 -cb * ct - sa * sb * st,

299 ca * st, name=self.name) # XXX .unit()

300

301 def greatCircleTo(self, other):

302 '''Compute the vector normal to great circle obtained by

303 heading from this to an other point or on a given bearing.

304

305 Direction of vector is such that initial bearing vector

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

307

308 @arg other: The other point (L{LatLon}) or the bearing from

309 this point (compass C{degrees360}).

310

311 @return: N-vector representing the great circle (L{Nvector}).

312

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

314

315 @raise Valuerror: Points coincide.

316

317 @example:

318

319 >>> p = LatLon(53.3206, -1.7297)

320 >>> gc = p.greatCircle(96.0)

321 >>> gc.toStr() # (-0.79408, 0.12856, 0.59406)

322

323 >>> q = LatLon(53.1887, 0.1334)

324 >>> g = p.greatCircleTo(q)

325 >>> g.toStr() # (-0.79408, 0.12859, 0.59406)

326 '''

327 gc, _, _ = self._gc3(self, other, _other_)

328 return gc.unit()

329

330 def initialBearingTo(self, other, **unused):

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

332 to an other point.

333

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

335 @arg unused: Optional keyword argument B{C{wrap}} ignored.

336

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

338

339 @raise Crosserror: This point coincides with the B{C{other}}

340 point or the C{NorthPole}, provided

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

342

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

344

345 @example:

346

347 >>> p1 = LatLon(52.205, 0.119)

348 >>> p2 = LatLon(48.857, 2.351)

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

350 '''

351 self.others(other)

352 # see <https://MathForum.org/library/drmath/view/55417.html>

353 n = self.toNvector()

354# gc1 = self.greatCircleTo(other)

355 gc1 = n.cross(other.toNvector(), raiser=_points_) # .unit()

356# gc2 = self.greatCircleTo(NorthPole)

357 gc2 = n.cross(NorthPole, raiser=_pole_) # .unit()

358 return degrees360(gc1.angleTo(gc2, vSign=n))

359

360 def intermediateChordTo(self, other, fraction, height=None):

361 '''Locate the point projected from the point at given fraction

362 on a straight line (chord) between this and an other point.

363

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

365 @arg fraction: Fraction between both points (float, between

366 0.0 for this and 1.0 for the other point).

367 @kwarg height: Optional height at the intermediate point,

368 overriding the fractional height (C{meter}).

369

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

371

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

373

374 @example:

375

376 >>> p = LatLon(52.205, 0.119)

377 >>> q = LatLon(48.857, 2.351)

378 >>> i = p.intermediateChordTo(q, 0.25) # 51.3723°N, 000.7072°E

379 '''

380 self.others(other)

381

382 f = Scalar(fraction=fraction)

383 i = other.toNvector().times(f).plus(

384 self.toNvector().times(1 - f))

385# i = other.toNvector() * f + \

386# self.toNvector() * (1 - f))

387

388 h = self._havg(other, f=f) if height is None else Height(height)

389 return i.toLatLon(height=h, LatLon=self.classof) # Nvector(i.x, i.y, i.z).toLatLon(...)

390

391 def intermediateTo(self, other, fraction, height=None, **unused): # wrap=False

392 '''Locate the point at a given fraction between this and an

393 other point.

394

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

396 @arg fraction: Fraction between both points (C{float}, between

397 0.0 for this and 1.0 for the other point).

398 @kwarg height: Optional height at the intermediate point,

399 overriding the fractional height (C{meter}).

400

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

402

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

404

405 @raise Valuerror: Points coincide or invalid B{C{height}}.

406

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

408

409 @example:

410

411 >>> p = LatLon(52.205, 0.119)

412 >>> q = LatLon(48.857, 2.351)

413 >>> i = p.intermediateTo(q, 0.25) # 51.3721°N, 000.7074°E

414 '''

415 q = self.others(other).toNvector()

416 p = self.toNvector()

417 f = Scalar(fraction=fraction)

418

419 x = p.cross(q, raiser=_points_)

420 d = x.unit().cross(p) # unit(p × q) × p

421 # angular distance α, tan(α) = |p × q| / p ⋅ q

422 s, c = sincos2(atan2(x.length, p.dot(q)) * f) # interpolated

423 i = p.times(c).plus(d.times(s)) # p * cosα + d * sinα

424

425 h = self._havg(other, f=f) if height is None else Height(height)

426 return i.toLatLon(height=h, LatLon=self.classof) # Nvector(i.x, i.y, i.z).toLatLon(...)

427

428 def intersection(self, end1, start2, end2, height=None):

429 '''Locate the intersection point of two lines each defined

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

431

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

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

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

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

436 initial bearing at the second point (compass

437 C{degrees}).

438 @kwarg height: Optional height at the intersection point,

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

440

441 @return: The intersection point (L{LatLon}) or C{None}

442 if no unique intersection exists.

443

444 @raise TypeError: If B{C{start2}}, B{C{end1}} or B{C{end2}} point

445 is not L{LatLon}.

446

447 @raise ValueError: Intersection is ambiguous or infinite or

448 the lines are parallel, coincident or null.

449

450 @example:

451

452 >>> s = LatLon(51.8853, 0.2545)

453 >>> e = LatLon(49.0034, 2.5735)

454 >>> i = s.intersection(108.55, e, 32.44) # 50.9076°N, 004.5086°E

455 '''

456 return intersection(self, end1, start2, end2,

457 height=height, LatLon=self.classof)

458

459 def isenclosedBy(self, points):

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

461

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

463 L{BooleanFHP} or L{BooleanGH}).

464

465 @return: C{True} if this point is inside the polygon or composite,

466 C{False} otherwise.

467

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

469

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

471

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

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

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

475 '''

476 if _MODS.booleans.isBoolean(points):

477 return points._encloses(self.lat, self.lon)

478

479 # sum subtended angles of each edge (using n0, the

480 # normal vector to this point for sign of α)

481 def _subtangles(Ps, n0):

482 vs1 = n0.minus(Ps[0].toNvector())

483 for p in Ps.iterate(closed=True):

484 vs2 = n0.minus(p.toNvector())

485 yield vs1.angleTo(vs2, vSign=n0) # PYCHOK false

486 vs1 = vs2

487

488 # Note, this method uses angle summation test: on a plane,

489 # angles for an enclosed point will sum to 360°, angles for

490 # an exterior point will sum to 0°. On a sphere, enclosed

491 # point angles will sum to less than 360° (due to spherical

492 # excess), exterior point angles will be small but non-zero.

493 s = fsum(_subtangles(self.PointsIter(points, loop=1),

494 self.toNvector()), floats=True) # normal vector

495 # XXX are winding number optimisations equally applicable to

496 # spherical surface?

497 return fabs(s) > PI

498

499 @deprecated_method

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

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

502 return self.isenclosedBy(points)

503

504 def iswithin(self, point1, point2):

505 '''Check whether this point is between two other points.

506

507 If this point is not on the great circle arc defined by

508 both points, return whether it is within the area bound

509 by perpendiculars to the great circle at each point (in

510 the same hemispere).

511

512 @arg point1: Start point of the arc (L{LatLon}).

513 @arg point2: End point of the arc (L{LatLon}).

514

515 @return: C{True} if this point is within the arc,

516 C{False} otherwise.

517

518 @raise TypeError: If B{C{point1}} or B{C{point2}} is not L{LatLon}.

519 '''

520 n0 = self.toNvector()

521 n1 = self.others(point1=point1).toNvector()

522 n2 = self.others(point2=point2).toNvector()

523

524 # corner case, null arc

525 if n1.isequalTo(n2):

526 return n0.isequalTo(n1) or n0.isequalTo(n2) # PYCHOK returns

527

528 if n0.dot(n1) < 0 or n0.dot(n2) < 0: # different hemisphere

529 return False # PYCHOK returns

530

531 # get vectors representing d0=p0->p1 and d2=p2->p1 and the

532 # dot product d0⋅d2 tells us if p0 is on the p2 side of p1 or

533 # on the other side (similarly for d0=p0->p2 and d1=p1->p2

534 # and dot product d0⋅d1 and p0 on the p1 side of p2 or not)

535 return n0.minus(n1).dot(n2.minus(n1)) >= 0 and \

536 n0.minus(n2).dot(n1.minus(n2)) >= 0

537

538 @deprecated_method

539 def isWithin(self, point1, point2): # PYCHOK no cover

540 '''DEPRECATED, use method C{iswithin}.'''

541 return self.iswithin(point1, point2)

542

543 def midpointTo(self, other, height=None, fraction=_0_5):

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

545

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

547 @kwarg height: Optional height at the midpoint, overriding

548 the mean height (C{meter}).

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

550 may be negative or greater than 1.0.

551

552 @return: Midpoint (L{LatLon}).

553

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

555

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

557

558 @example:

559

560 >>> p1 = LatLon(52.205, 0.119)

561 >>> p2 = LatLon(48.857, 2.351)

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

563 '''

564 if fraction is _0_5:

565 self.others(other)

566

567 m = self.toNvector().plus(other.toNvector())

568 h = self._havg(other) if height is None else height

569 r = m.toLatLon(height=h, LatLon=self.classof)

570 else:

571 r = self.intermediateTo(other, fraction, height=height)

572 return r

573

574 def nearestOn(self, point1, point2, height=None, within=True, wrap=False):

575 '''Locate the point on the great circle arc between two

576 points closest to this point.

577

578 @arg point1: Start point of the arc (L{LatLon}).

579 @arg point2: End point of the arc (L{LatLon}).

580 @kwarg height: Optional height, overriding the mean height

581 for the point within the arc (C{meter}), or

582 C{None} to interpolate the height.

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

584 both given points, otherwise the closest

585 point elsewhere on the arc (C{bool}).

586 @kwarg wrap: Wrap and unroll longitudes (C{bool}).

587

588 @return: Closest point on the arc (L{LatLon}).

589

590 @raise NotImplementedError: Keyword argument C{B{wrap}=True}

591 not supported.

592

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

594

595 @example:

596

597 >>> s1 = LatLon(51.0, 1.0)

598 >>> s2 = LatLon(51.0, 2.0)

599

600 >>> s = LatLon(51.0, 1.9)

601 >>> p = s.nearestOn(s1, s2) # 51.0004°N, 001.9000°E

602

603 >>> d = p.distanceTo(s) # 42.71 m

604

605 >>> s = LatLon(51.0, 2.1)

606 >>> p = s.nearestOn(s1, s2) # 51.0000°N, 002.0000°E

607 '''

608 if wrap: # wrap=True throws C{NotImplementedError} always.

609 notImplemented(self, wrap=wrap)

610

611 if self.iswithin(point1, point2) and not point1.isequalTo(point2, EPS):

612 # closer to arc than to its endpoints,

613 # find the closest point on the arc

614 gc1 = point1.toNvector().cross(point2.toNvector())

615 gc2 = self.toNvector().cross(gc1)

616 n = gc1.cross(gc2)

617

618 elif within: # for backward compatibility

619 return point1 if self.distanceTo(point1) < self.distanceTo(point2) else point2

620

621 else: # handle beyond arc extent by .vector3d.nearestOn

622 n1 = point1.toNvector()

623 n2 = point2.toNvector()

624 n = self.toNvector().nearestOn(n1, n2, within=False)

625 if n is n1:

626 return point1

627 elif n is n2:

628 return point2

629

630 p = n.toLatLon(height=height or 0, LatLon=self.classof)

631 if height in (None, False): # interpolate height within extent

632 d = point1.distanceTo(point2)

633 f = (point1.distanceTo(p) / d) if d > EPS0 else _0_5

634 p.height = point1._havg(point2, f=max(_0_0, min(f, _1_0)))

635 return p

636

637 # @deprecated_method

638 def nearestOn2(self, points, **closed_radius_height): # PYCHOK no cover

639 '''DEPRECATED, use method L{sphericalNvector.LatLon.nearestOn3}.

640

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

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

643 to that point from this point ...

644 '''

645 r = self.nearestOn3(points, **closed_radius_height)

646 return r.closest, r.distance

647

648 def nearestOn3(self, points, closed=False, radius=R_M, height=None):

649 '''Locate the point on a path or polygon (with great circle

650 arcs joining consecutive points) closest to this point.

651

652 The closest point is either on within the extent of any great

653 circle arc or the nearest of the arc's end points.

654

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

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

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

658 @kwarg height: Optional height, overriding the mean height

659 for a point within the arc (C{meter}).

660

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

662 the C{closest} point (L{LatLon}), the C{distance}

663 between this and the C{closest} point in C{meter},

664 same units as B{C{radius}} or in C{radians} if

665 B{C{radius}} is C{None} and the C{angle} from this

666 to the C{closest} point in compass C{degrees360}.

667

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

669

670 @raise ValueError: No B{C{points}}.

671 '''

672 Ps = self.PointsIter(points, loop=1)

673

674 R = self.distanceTo

675 N = self.nearestOn

676

677 c = p1 = Ps[0]

678 r = R(c, radius=None) # radians

679 for p2 in Ps.iterate(closed=closed):

680 p = N(p1, p2, height=height)

681 d = R(p, radius=None) # radians

682 if d < r:

683 c, r = p, d

684 p1 = p2

685 d = r if radius is None else (Radius(radius) * r)

686 return NearestOn3Tuple(c, d, degrees360(r))

687

688 def toCartesian(self, **Cartesian_and_kwds): # PYCHOK Cartesian=Cartesian, datum=None

689 '''Convert this point to C{Nvector}-based cartesian (ECEF) coordinates.

690

691 @kwarg Cartesian_and_kwds: Optional L{Cartesian} and L{Cartesian} keyword

692 arguments, like C{datum}. Use C{B{Cartesian}=...}

693 to override this L{Cartesian} class or specify

694 C{B{Cartesian}=None}.

695

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

697 set to C{None}, an L{Ecef9Tuple}C{(x, y, z, lat, lon, height,

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

699

700 @raise TypeError: Invalid B{C{Cartesian_and_kwds}} argument.

701 '''

702 kwds = _xkwds(Cartesian_and_kwds, Cartesian=Cartesian, datum=self.datum)

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

704

705 def toNvector(self, **Nvector_and_kwds): # PYCHOK signature

706 '''Convert this point to L{Nvector} components, I{including height}.

707

708 @kwarg Nvector_and_kwds: Optional L{Nvector} and L{Nvector} keyword

709 arguments. Use C{B{Nvector}=...} to override

710 this L{Nvector} class or specify

711 C{B{Nvector}=None}.

712

713 @return: The C{n-vector} components (L{Nvector}) or if B{C{Nvector}} is

714 set to C{None}, a L{Vector4Tuple}C{(x, y, z, h)} if B{C{Nvector}}.

715

716 @raise TypeError: Invalid B{C{Nvector_and_kwds}} argument.

717

718 @example:

719

720 >>> p = LatLon(45, 45)

721 >>> n = p.toNvector()

722 >>> n.toStr() # [0.50, 0.50, 0.70710]

723 '''

724 kwds = _xkwds(Nvector_and_kwds, Nvector=Nvector)

725 return LatLonNvectorBase.toNvector(self, **kwds)

726

727

728 class Nvector(NvectorBase):

729 '''An n-vector is a position representation using a (unit) vector

730 normal to the earth's surface. Unlike lat-/longitude points,

731 n-vectors have no singularities or discontinuities.

732

733 For many applications, n-vectors are more convenient to work

734 with than other position representations like lat-/longitude,

735 earth-centred earth-fixed (ECEF) vectors, UTM coordinates, etc.

736

737 On a spherical model earth, an n-vector is equivalent to an

738 earth-centred earth-fixed (ECEF) vector.

739

740 Note commonality with L{ellipsoidalNvector.Nvector}.

741 '''

742 _datum = Datums.Sphere # default datum (L{Datum})

743

744 def toCartesian(self, **Cartesian_and_kwds): # PYCHOK Cartesian=Cartesian

745 '''Convert this n-vector to C{Nvector}-based cartesian

746 (ECEF) coordinates.

747

748 @kwarg Cartesian_and_kwds: Optional L{Cartesian} and L{Cartesian} keyword

749 arguments, like C{h}. Use C{B{Cartesian}=...}

750 to override this L{Cartesian} class or specify

751 C{B{Cartesian}=None}.

752

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

754 set to C{None}, an L{Ecef9Tuple}C{(x, y, z, lat, lon, height,

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

756

757 @raise TypeError: Invalid B{C{Cartesian_and_kwds}} argument.

758 '''

759 kwds = _xkwds(Cartesian_and_kwds, h=self.h, Cartesian=Cartesian)

760 return NvectorBase.toCartesian(self, **kwds) # class or .classof

761

762 def toLatLon(self, **LatLon_and_kwds): # PYCHOK height=None, LatLon=LatLon

763 '''Convert this n-vector to an C{Nvector}-based geodetic point.

764

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

766 arguments, like C{height}. Use C{B{LatLon}=...}

767 to override this L{LatLon} class or specify

768 C{B{LatLon}=None}.

769

770 @return: The geodetic point (L{LatLon}) or if B{C{LatLon}} is set

771 to C{None}, an L{Ecef9Tuple}C{(x, y, z, lat, lon, height,

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

773

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

775

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

777 '''

778 kwds = _xkwds(LatLon_and_kwds, height=self.h, LatLon=LatLon)

779 return NvectorBase.toLatLon(self, **kwds) # class or .classof

780

781 def greatCircle(self, bearing):

782 '''Compute the n-vector normal to great circle obtained by

783 heading on given compass bearing from this point as its

784 n-vector.

785

786 Direction of vector is such that initial bearing vector

787 b = c × p.

788

789 @arg bearing: Initial compass bearing (C{degrees}).

790

791 @return: N-vector representing great circle (L{Nvector}).

792

793 @raise Valuerror: Polar coincidence.

794

795 @example:

796

797 >>> n = LatLon(53.3206, -1.7297).toNvector()

798 >>> gc = n.greatCircle(96.0) # [-0.794, 0.129, 0.594]

799 '''

800 s, c = sincos2d(Bearing(bearing))

801

802 e = NorthPole.cross(self, raiser=_pole_) # easting

803 n = self.cross(e, raiser=_point_) # northing

804

805 e = e.times(c / e.length)

806 n = n.times(s / n.length)

807 return n.minus(e)

808

809

810_Nvll = LatLon(_0_0, _0_0, name=_Nv00_) # reference instance (L{LatLon})

811

812

813 def areaOf(points, radius=R_M):

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

815 (with great circle arcs joining consecutive points).

816

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

818 L{BooleanFHP} or L{BooleanGH}).

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

820

821 @return: Polygon area (C{meter} I{squared} , same units as

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

823

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

825

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

827

828 @see: Functions L{pygeodesy.areaOf}, L{sphericalTrigonometry.areaOf}

829 and L{ellipsoidalKarney.areaOf}.

830

831 @example:

832

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

834 >>> areaOf(b) # 8666058750.718977

835 '''

836 def _interangles(Ps):

837 # use vector to 1st point as plane normal for sign of α

838 n0 = Ps[0].toNvector()

839

840 v2 = Ps[0]._N_vector # XXX v2 == no?

841 v1 = Ps[1]._N_vector

842 gc = v2.cross(v1)

843 for p in Ps.iterate(closed=True):

844 v2 = p._N_vector

845 gc1 = v1.cross(v2)

846 v1 = v2

847 yield gc.angleTo(gc1, vSign=n0)

848 gc = gc1

849

850 if _MODS.booleans.isBoolean(points):

851 r = points._sum2(LatLon, areaOf, radius=None)

852 else:

853 # sum interior angles: depending on whether polygon is cw or ccw,

854 # angle between edges is π−α or π+α, where α is angle between

855 # great-circle vectors; so sum α, then take n·π − |Σα| (cannot

856 # use Σ(π−|α|) as concave polygons would fail)

857 s = fsum(_interangles(_Nvll.PointsIter(points, loop=2)), floats=True)

858 # using Girard’s theorem: A = [Σθᵢ − (n−2)·π]·R²

859 # (PI2 - abs(s) == (n*PI - abs(s)) - (n-2)*PI)

860 r = fabs(PI2 - fabs(s))

861 return r if radius is None else (r * Radius(radius)**2)

862

863

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

865 height=None, wrap=True):

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

867

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

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

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

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

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

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

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

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

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

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

878 circle} methods.

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

880 conventionally) or C{None}.

881 @kwarg wrap: Wrap and unroll longitudes (C{bool}).

882

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

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

885 point C{is} this very instance.

886

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

888

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

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

891

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

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

894 '''

895 c = _Nvll.others(center=center)

896 p = _Nvll.others(point=point)

897 try:

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

899 height=height, wrap=wrap)

900 except (TypeError, ValueError) as x:

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

902

903

904 def intersection(start1, end1, start2, end2,

905 height=None, LatLon=LatLon, **LatLon_kwds):

906 '''Locate the intersection of two lines each defined by two

907 points or by a start point and an (initial) bearing.

908

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

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

911 initial bearing at the first start point

912 (compass C{degrees360}).

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

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

915 initial bearing at the second start point

916 (compass C{degrees360}).

917 @kwarg height: Optional height at the intersection point,

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

919 @kwarg LatLon: Optional class to return the intersection

920 point (L{LatLon}).

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

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

923

924 @return: The intersection point (B{C{LatLon}}) or 3-tuple

925 (C{degrees90}, C{degrees180}, height) if B{C{LatLon}}

926 is C{None} or C{None} if no unique intersection

927 exists.

928

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

930

931 @raise ValueError: Intersection is ambiguous or infinite or

932 the lines are parallel, coincident or null.

933

934 @example:

935

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

937 >>> q = LatLon(49.0034, 2.5735)

938 >>> i = intersection(p, 108.55, q, 32.44) # 50.9076°N, 004.5086°E

939 '''

940 _Nvll.others(start1=start1)

941 _Nvll.others(start2=start2)

942

943 # If gc1 and gc2 are great circles through start and end points

944 # (or defined by start point and bearing), then the candidate

945 # intersections are simply gc1 × gc2 and gc2 × gc1. Most of the

946 # work is deciding the correct intersection point to select! If

947 # bearing is given, that determines the intersection, but if both

948 # lines are defined by start/end points, take closer intersection.

949 gc1, s1, e1 = _Nvll._gc3(start1, end1, 'end1')

950 gc2, s2, e2 = _Nvll._gc3(start2, end2, 'end2')

951

952 hs = start1.height, start2.height

953 # there are two (antipodal) candidate intersection

954 # points ... we have to choose the one to return

955 i1 = gc1.cross(gc2, raiser=_lines_)

956 # postpone computing i2 until needed

957 # i2 = gc2.cross(gc1, raiser=_lines_)

958

959 # selection of intersection point depends on how

960 # lines are defined (by bearings or endpoints)

961 if e1 and e2: # endpoint+endpoint

962 d = sumOf((s1, s2, e1, e2)).dot(i1)

963 hs += end1.height, end2.height

964 elif e1 and not e2: # endpoint+bearing

965 # gc2 x v2 . i1 +ve means v2 bearing points to i1

966 d = gc2.cross(s2).dot(i1)

967 hs += end1.height,

968 elif e2 and not e1: # bearing+endpoint

969 # gc1 x v1 . i1 +ve means v1 bearing points to i1

970 d = gc1.cross(s1).dot(i1)

971 hs += end2.height,

972 else: # bearing+bearing

973 # if gc x v . i1 is +ve, initial bearing is

974 # towards i1, otherwise towards antipodal i2

975 d1 = gc1.cross(s1).dot(i1) # +ve means p1 bearing points to i1

976 d2 = gc2.cross(s2).dot(i1) # +ve means p2 bearing points to i1

977 if d1 > 0 and d2 > 0:

978 d = 1 # both point to i1

979 elif d1 < 0 and d2 < 0:

980 d = -1 # both point to i2

981 else: # d1, d2 opposite signs

982 # intersection is at further-away intersection point,

983 # take opposite intersection from mid- point of v1

984 # and v2 [is this always true?] XXX changed to always

985 # get intersection p1 bearing points to, aka being

986 # located "after" p1 along the bearing at p1, like

987 # function .sphericalTrigonometry._intersect and

988 # .ellipsoidalBaseDI._intersect3

989 d = d1 # neg(s1.plus(s2).dot(i1))

990

991 i = i1 if d > 0 else gc2.cross(gc1, raiser=_lines_)

992

993 h = fmean(hs) if height is None else height

994 kwds = _xkwds(LatLon_kwds, height=h, LatLon=LatLon)

995 _ = _xkwds_pop(kwds, wrap=None) # from .formy.intersection2

996 return i.toLatLon(**kwds) # Nvector(i.x, i.y, i.z).toLatLon(...)

997

998

999 def meanOf(points, height=None, LatLon=LatLon, **LatLon_kwds):

1000 '''Compute the geographic mean of the supplied points.

1001

1002 @arg points: Array of points to be averaged (L{LatLon}[]).

1003 @kwarg height: Optional height, overriding the mean height

1004 (C{meter}).

1005 @kwarg LatLon: Optional class to return the mean point

1006 (L{LatLon}).

1007 @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword

1008 arguments, ignored if C{B{LatLon} is None}.

1009

1010 @return: Point at geographic mean and mean height (B{C{LatLon}}).

1011

1012 @raise PointsError: Insufficient number of B{C{points}}.

1013

1014 @raise TypeError: Some B{C{points}} are not C{LatLon}.

1015 '''

1016 Ps = _Nvll.PointsIter(points)

1017 # geographic mean

1018 m = sumOf(p._N_vector for p in Ps.iterate(closed=False))

1019 kwds = _xkwds(LatLon_kwds, height=height, LatLon=LatLon,

1020 name=meanOf.__name__)

1021 return m.toLatLon(**kwds)

1022

1023

1024@deprecated_function

1025 def nearestOn2(point, points, **closed_radius_height): # PYCHOK no cover

1026 '''DEPRECATED, use method L{sphericalNvector.nearestOn3}.

1027

1028 @return: ... 2-Tuple C{(closest, distance)} of the C{closest}

1029 point (L{LatLon}) on the polygon and the C{distance}

1030 between the C{closest} and the given B{C{point}} ...

1031 '''

1032 r = nearestOn3(point, points, **closed_radius_height)

1033 return r.closest, r.distance

1034

1035

1036 def nearestOn3(point, points, closed=False, radius=R_M, height=None):

1037 '''Locate the point on a polygon (with great circle arcs

1038 joining consecutive points) closest to an other point.

1039

1040 If the given point is within the extent of any great circle

1041 arc, the closest point is on that arc. Otherwise, the

1042 closest is the nearest of the arc's end points.

1043

1044 @arg point: The other, reference point (L{LatLon}).

1045 @arg points: The polygon points (L{LatLon}[]).

1046 @kwarg closed: Optionally, close the polygon (C{bool}).

1047 @kwarg radius: Mean earth radius (C{meter}) or C{None}.

1048 @kwarg height: Optional height, overriding the mean height

1049 for a point within the arc (C{meter}).

1050

1051 @return: A L{NearestOn3Tuple}C{(closest, distance, angle)} of

1052 the C{closest} point (L{LatLon}) on the polygon, the

1053 C{distance} and the C{angle} between the C{closest}

1054 and the given B{C{point}}. The C{distance} is in

1055 C{meter}, same units as B{C{radius}} or in C{radians}

1056 if B{C{radius}} is C{None}, the C{angle} is in compass

1057 C{degrees360}.

1058

1059 @raise PointsError: Insufficient number of B{C{points}}.

1060

1061 @raise TypeError: Some B{C{points}} or B{C{point}} not C{LatLon}.

1062 '''

1063 _xinstanceof(LatLon, point=point)

1064

1065 return point.nearestOn3(points, closed=closed, radius=radius, height=height)

1066

1067

1068 def perimeterOf(points, closed=False, radius=R_M):

1069 '''Compute the perimeter of a (spherical) polygon or composite

1070 (with great circle arcs joining consecutive points).

1071

1072 @arg points: The polygon points (L{LatLon}[]).

1073 @kwarg closed: Optionally, close the polygon (C{bool}).

1074 @kwarg radius: Mean earth radius (C{meter}) or C{None}.

1075

1076 @return: Polygon perimeter (C{meter}, same units as B{C{radius}} or

1077 C{radians} if B{C{radius}} is C{None}).

1078

1079 @raise PointsError: Insufficient number of B{C{points}}.

1080

1081 @raise TypeError: Some B{C{points}} are not L{LatLon}.

1082

1083 @raise ValueError: Invalid B{C{radius}} or C{B{closed}=False} with

1084 C{B{points}} a composite.

1085

1086 @see: Functions L{pygeodesy.perimeterOf}, L{sphericalTrigonometry.perimeterOf}

1087 and L{ellipsoidalKarney.perimeterOf}.

1088 '''

1089 def _rads(Ps, closed): # angular edge lengths in radians

1090 v1 = Ps[0]._N_vector

1091 for p in Ps.iterate(closed=closed):

1092 v2 = p._N_vector

1093 yield v1.angleTo(v2)

1094 v1 = v2

1095

1096 if _MODS.booleans.isBoolean(points):

1097 if not closed:

1098 raise _ValueError(closed=closed, points=_composite_)

1099 r = points._sum2(LatLon, perimeterOf, closed=True, radius=None)

1100 else:

1101 r = fsum(_rads(_Nvll.PointsIter(points, loop=1), closed), floats=True)

1102 return r if radius is None else (Radius(radius) * r)

1103

1104

1105 def sumOf(nvectors, Vector=Nvector, h=None, **Vector_kwds):

1106 '''Return the vectorial sum of two or more n-vectors.

1107

1108 @arg nvectors: Vectors to be added (L{Nvector}[]).

1109 @kwarg Vector: Optional class for the vectorial sum (L{Nvector}).

1110 @kwarg h: Optional height, overriding the mean height (C{meter}).

1111 @kwarg Vector_kwds: Optional, additional B{C{Vector}} keyword arguments.

1112

1113 @return: Vectorial sum (B{C{Vector}}).

1114

1115 @raise VectorError: No B{C{nvectors}}.

1116 '''

1117 return _sumOf(nvectors, Vector=Vector, h=h, **Vector_kwds)

1118

1119

1120 def triangulate(point1, bearing1, point2, bearing2,

1121 height=None, LatLon=LatLon, **LatLon_kwds):

1122 '''Locate a point given two known points and the initial bearings

1123 from those points.

1124

1125 @arg point1: First reference point (L{LatLon}).

1126 @arg bearing1: Bearing at the first point (compass C{degrees360}).

1127 @arg point2: Second reference point (L{LatLon}).

1128 @arg bearing2: Bearing at the second point (compass C{degrees360}).

1129 @kwarg height: Optional height at the triangulated point, overriding

1130 the mean height (C{meter}).

1131 @kwarg LatLon: Optional class to return the triangulated point

1132 (L{LatLon}).

1133 @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword

1134 arguments, ignored if C{B{LatLon} is None}.

1135

1136 @return: Triangulated point (B{C{LatLon}}).

1137

1138 @raise TypeError: If B{C{point1}} or B{C{point2}} is not L{LatLon}.

1139

1140 @raise Valuerror: Points coincide.

1141

1142 @example:

1143

1144 >>> p = LatLon("47°18.228'N","002°34.326'W") # Basse Castouillet

1145 >>> q = LatLon("47°18.664'N","002°31.717'W") # Basse Hergo

1146 >>> t = triangulate(p, 7, q, 295) # 47.323667°N, 002.568501°W'

1147 '''

1148 return _triangulate(_Nvll.others(point1=point1), bearing1,

1149 _Nvll.others(point2=point2), bearing2,

1150 height=height, LatLon=LatLon, **LatLon_kwds)

1151

1152

1153 def trilaterate(point1, distance1, point2, distance2, point3, distance3, # PYCHOK args

1154 radius=R_M, height=None, useZ=False,

1155 LatLon=LatLon, **LatLon_kwds):

1156 '''Locate a point at given distances from three other points.

1157

1158 @arg point1: First point (L{LatLon}).

1159 @arg distance1: Distance to the first point (C{meter}, same units

1160 as B{C{radius}}).

1161 @arg point2: Second point (L{LatLon}).

1162 @arg distance2: Distance to the second point (C{meter}, same units

1163 as B{C{radius}}).

1164 @arg point3: Third point (L{LatLon}).

1165 @arg distance3: Distance to the third point (C{meter}, same units

1166 as B{C{radius}}).

1167 @kwarg radius: Mean earth radius (C{meter}).

1168 @kwarg height: Optional height at the trilaterated point, overriding

1169 the IDW height (C{meter}, same units as B{C{radius}}).

1170 @kwarg useZ: Include Z component iff non-NaN, non-zero (C{bool}).

1171 @kwarg LatLon: Optional class to return the trilaterated

1172 @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword arguments,

1173 ignored if C{B{LatLon} is None}.

1174

1175 @return: Trilaterated point (B{C{LatLon}}).

1176

1177 @raise IntersectionError: No intersection, trilateration failed.

1178

1179 @raise TypeError: Invalid B{C{point1}}, B{C{point2}} or B{C{point3}}.

1180

1181 @raise ValueError: Coincident B{C{points}} or invalid B{C{distance1}},

1182 B{C{distance2}}, B{C{distance3}} or B{C{radius}}.

1183

1184 @see: U{Trilateration<https://WikiPedia.org/wiki/Trilateration>}.

1185 '''

1186 return _trilaterate(_Nvll.others(point1=point1), distance1,

1187 _Nvll.others(point2=point2), distance2,

1188 _Nvll.others(point3=point3), distance3,

1189 radius=radius, height=height, useZ=useZ,

1190 LatLon=LatLon, **LatLon_kwds)

1191

1192

1193__all__ += _ALL_OTHER(Cartesian, LatLon, Nvector, # classes

1194 areaOf, # functions

1195 intersecant2, intersection, ispolar,

1196 meanOf,

1197 nearestOn2, nearestOn3,

1198 perimeterOf,

1199 sumOf,

1200 triangulate, trilaterate)

1201

1202# **) MIT License

1203#

1204# Copyright (C) 2016-2023 -- mrJean1 at Gmail -- All Rights Reserved.

1205#

1206# Permission is hereby granted, free of charge, to any person obtaining a

1207# copy of this software and associated documentation files (the "Software"),

1208# to deal in the Software without restriction, including without limitation

1209# the rights to use, copy, modify, merge, publish, distribute, sublicense,

1210# and/or sell copies of the Software, and to permit persons to whom the

1211# Software is furnished to do so, subject to the following conditions:

1212#

1213# The above copyright notice and this permission notice shall be included

1214# in all copies or substantial portions of the Software.

1215#

1216# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS

1217# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

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1220# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,

1221# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR

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