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, 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 _TypeError, _ValueError, IntersectionError, \

26 _xError, _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_, _invalid_,\

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

35 _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, NearestOn3Tuple, \

40 Triangle7Tuple, Triangle8Tuple

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

42 Fmt as _Fmt, notImplemented # XXX shadowed

43 from pygeodesy.props import deprecated_function, deprecated_method

44 from pygeodesy.sphericalBase import _m2radians, CartesianSphericalBase, \

45 _intersecant2, LatLonSphericalBase, \

46 _rads3, _radians2m, _trilaterate5

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

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

49 Phi_, Radius_, Scalar

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

51 degrees2m, m2radians, radiansPI2, sincos2_, tan_2, \

52 unrollPI, _unrollon, _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__ = '23.12.18'

59

60_PI_EPS4 = PI - EPS4

61 if _PI_EPS4 >= PI:

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

63

64

65 class Cartesian(CartesianSphericalBase):

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

67 spherical, geodetic L{LatLon}.

68 '''

69

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

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

72

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

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

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

76

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

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

79 with C{C} and C{M} if available.

80

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

82 '''

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

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

85

86

87 class LatLon(LatLonSphericalBase):

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

89 '''

90

91 def _ab1_ab2_db5(self, other, wrap):

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

93 '''

94 a1, b1 = self.philam

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

96 if wrap:

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

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

99 else: # unrollPI shortcut

100 db = b2 - b1

101 return a1, b1, a2, b2, db

102

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

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

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

106 end point.

107

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

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

110 from the start point to the point where the perpendicular

111 crosses the line.

112

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

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

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

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

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

118

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

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

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

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

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

124 B{C{start}} point.

125

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

127

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

129 '''

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

131 cx = cos(x)

132 return _0_0 if isnear0(cx) else \

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

134

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

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

137 '''

138 s = self.others(start=start)

139 e = self.others(end=end)

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

141

142 r = Radius_(radius)

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

144

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

146 - s.initialBearingTo(e, wrap=w))

147 x = asin(sin(r) * sin(b))

148 return r, x, -b

149

150 @deprecated_method

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

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

153 '''

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

155

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

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

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

159

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

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

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

163 B{C{other}} point (C{bool}).

164

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

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

167 '''

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

169 sa, ca, sa1, ca1, \

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

171 sa1 *= ca2 * ca

172

173 x = sa1 * sdb

174 y = sa1 * cdb - ca1 * sa2 * ca

175 z = ca1 * sdb * ca2 * sa

176

177 h = hypot(x, y)

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

179 return None # great circle doesn't reach latitude

180

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

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

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

184

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

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

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

188

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

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

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

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

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

194

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

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

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

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

199

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

201

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

203 '''

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

205 return _radians2m(x, 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 initial bearing.

210

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

212 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 (C{meter}, same

216 units a B{C{radius}}).

217

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

219

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

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

222 '''

223 a, b = self.philam

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

225

226 a, b = _destination2(a, b, r, t)

227 h = self._heigHt(height)

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

229

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

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

232

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

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

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

236 the B{C{other}} point (C{bool}).

237

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

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

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

241

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

243

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

245 '''

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

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

248

249# @Property_RO

250# def Ecef(self):

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

252# '''

253# return _MODS.ecef.EcefKarney

254

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

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

257 on the given initial bearing from this point.

258

259 Direction of vector is such that initial bearing vector

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

261

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

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

264 overriding the default L{Vector3d}.

265 @kwarg Vector_kwds: Optional, additional keyword argunents

266 for B{C{Vector}}.

267

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

269

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

271 '''

272 a, b = self.philam

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

274

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

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

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

278

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

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

281 to an other point.

282

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

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

285 the B{C{other}} point (C{bool}).

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

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

288 C{sphericalNvector.LatLon.initialBearingTo}.

289

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

291

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

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

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

295

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

297 '''

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

299 # XXX behavior like sphericalNvector.LatLon.initialBearingTo

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

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

302

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

304

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

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

307 and an other point.

308

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

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

311 0.0 at this and 1.0 at the other point).

312 @kwarg height: Optional height, overriding the intermediate

313 height (C{meter}).

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

315 B{C{other}} point (C{bool}).

316

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

318

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

320

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

322

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

324 '''

325 p = self

326 f = Scalar(fraction=fraction)

327 if not isnear0(f):

328 p = p.others(other)

329 if wrap:

330 p = _Wrap.point(p)

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

332 a1, b1 = self.philam

333 a2, b2 = p.philam

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

335 r = vincentys_(a2, a1, db)

336 sr = sin(r)

337 if isnon0(sr):

338 sa1, ca1, sa2, ca2, \

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

340

341 t = f * r

342 a = sin(r - t) # / sr superflous

343 b = sin( t) # / sr superflous

344

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

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

347 z = a * sa1 + b * sa2

348

349 a = atan1d(z, hypot(x, y))

350 b = atan2d(y, x)

351

352 else: # PYCHOK no cover

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

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

355

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

357 p = self.classof(a, b, height=h)

358 return p

359

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

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

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

363

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

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

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

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

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

369 C{degrees360}).

370 @kwarg height: Optional height for intersection point,

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

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

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

374

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

376 intersection point might be the L{antipode} to

377 the returned result.

378

379 @raise IntersectionError: Ambiguous or infinite intersection

380 or colinear, parallel or otherwise

381 non-intersecting lines.

382

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

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

385

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

387 '''

388 try:

389 s2 = self.others(other)

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

391 LatLon=self.classof)

392 except (TypeError, ValueError) as x:

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

394 other=other, end2=end2, wrap=wrap)

395

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

397 height=None, wrap=True):

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

399 by a center point and radius.

400

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

402 see B{C{radius}}).

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

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

405 see B{C{radius}}).

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

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

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

409 B{C{radius}}).

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

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

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

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

414 B{C{other}} point (C{bool}).

415

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

417 instance. For abutting circles, both intersection

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

419

420 @raise IntersectionError: Concentric, antipodal, invalid or

421 non-intersecting circles.

422

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

424

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

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

427 '''

428 try:

429 c2 = self.others(other)

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

431 height=height, wrap=wrap,

432 LatLon=self.classof)

433 except (TypeError, ValueError) as x:

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

435 other=other, rad2=rad2, wrap=wrap)

436

437 @deprecated_method

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

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

440 return self.isenclosedBy(points)

441

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

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

444

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

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

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

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

449

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

451 composite, C{False} otherwise.

452

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

454

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

456

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

458

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

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

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

462 '''

463 if _MODS.booleans.isBoolean(points):

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

465

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

467 n0 = self._N_vector

468

469 v2 = Ps[0]._N_vector

470 p1 = Ps[1]

471 v1 = p1._N_vector

472 # check whether this point on same side of all

473 # polygon edges (to the left or right depending

474 # on the anti-/clockwise polygon direction)

475 gc1 = v2.cross(v1)

476 t0 = gc1.angleTo(n0) > PI_2

477 s0 = None

478 # get great-circle vector for each edge

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

480 if wrap and not Ps.looped:

481 p2 = _unrollon(p1, p2)

482 p1 = p2

483 v2 = p2._N_vector

484 gc = v1.cross(v2)

485 t = gc.angleTo(n0) > PI_2

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

487 return False # outside

488

489 # check for convex polygon: angle between

490 # gc vectors, signed by direction of n0

491 # (otherwise the test above is not reliable)

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

493 if s != s0:

494 if s0 is None:

495 s0 = s

496 else:

497 t = _Fmt.SQUARE(points=i)

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

499 gc1, v1 = gc, v2

500

501 return True # inside

502

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

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

505

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

507 @kwarg height: Optional height for midpoint, overriding

508 the mean height (C{meter}).

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

510 may be negative or greater than 1.0.

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

512 B{C{other}} point (C{bool}).

513

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

515

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

517

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

519

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

521 '''

522 if fraction is _0_5:

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

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

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

526

527 x = ca2 * cdb + ca1

528 y = ca2 * sdb

529

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

531 b = degrees180(b + atan2(y, x))

532

533 h = self._havg(other, h=height)

534 r = self.classof(a, b, height=h)

535 else:

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

537 return r

538

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

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

541

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

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

544

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

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

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

548 @kwarg wrap_adjust_limit: Optional keyword arguments for functions

549 L{sphericalTrigonometry.nearestOn3} and

550 L{pygeodesy.equirectangular_},

551

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

553

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

555 see function L{pygeodesy.equirectangular_}.

556

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

558 is not (yet) supported.

559

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

561

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

563

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

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

566 '''

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

568 w = _xkwds_pop(wrap_adjust_limit, within=True)

569 if not w:

570 notImplemented(self, within=w)

571

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

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

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

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

576# return point1

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

578# if isnear0(d):

579# return point1 # or point2

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

581# return point2

582# f = a / d

583# if within:

584# if f > EPS1:

585# return point2

586# elif f < EPS:

587# return point1

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

589

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

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

592 **wrap_adjust_limit)[0]

593

594 @deprecated_method

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

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

597

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

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

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

601 units of B{C{radius}}.

602 '''

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

604 return r.closest, r.distance

605

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

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

608

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

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

611

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

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

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

615 @kwarg wrap_adjust_limit: Optional keyword arguments for function

616 L{sphericalTrigonometry.nearestOn3} and

617 L{pygeodesy.equirectangular_},

618

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

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

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

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

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

624 function L{pygeodesy.compassAngle}.

625

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

627 see function L{pygeodesy.equirectangular_}.

628

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

630

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

632

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

634

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

636 and L{pygeodesy.nearestOn5}.

637 '''

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

639 LatLon=self.classof, **wrap_adjust_limit)

640

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

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

643 coordinates.

644

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

646 and other keyword arguments, ignored

647 if C{B{Cartesian} is None}. Use

648 C{B{Cartesian}=...} to override

649 this L{Cartesian} class or specify

650 C{B{Cartesian}=None}.

651

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

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

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

655

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

657 '''

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

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

660

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

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

663

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

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

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

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

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

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

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

671

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

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

674 a, B, b, C, c, D, E)}.

675

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

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

678 '''

679 B = self.others(otherB=otherB)

680 C = self.others(otherC=otherC)

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

682

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

684 t = triangle8_(*r, wrap=wrap)

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

686

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

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

689 at this and at the other point.

690

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

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

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

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

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

696

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

698

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

700 '''

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

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

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

704

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

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

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

708 intersection} of three corresponding circles.

709

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

711 as B{C{radius}}).

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

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

714 B{C{radius}}).

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

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

717 B{C{radius}}).

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

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

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

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

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

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

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

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

726

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

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

729 the corresponding trilaterated points C{minPoint} and

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

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

732

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

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

735

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

737 respectively largest I{radial} overlap found.

738

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

740 nearest respectively farthest intersection margin.

741

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

743 0} and C{minPoint} and C{maxPoint} are both the B{C{point#}}

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

745 largest B{C{distance#}}.

746

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

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

749 no intersection or all (near-)concentric for

750 C{B{area}=False}.

751

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

753

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

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

756 or B{C{radius}}.

757 '''

758 return _trilaterate5(self, distance1,

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

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

761 area=area, radius=radius, eps=eps, wrap=wrap)

762

763

764_T00 = LatLon(0, 0, name='T00') # reference instance (L{LatLon})

765

766

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

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

769 (with the pointsjoined by great circle arcs).

770

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

772 or L{BooleanGH}).

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

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

775 or C{None}.

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

777 (C{bool}).

778

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

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

781

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

783

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

785

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

787

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

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

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

791

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

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

794 L{ellipsoidalKarney.areaOf}.

795 '''

796 if _MODS.booleans.isBoolean(points):

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

798

799 _at2, _t_2 = atan2, tan_2

800 _un, _w180 = unrollPI, wrap180

801

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

803 p1 = p2 = Ps[0]

804 a1, b1 = p1.philam

805 ta1, z1 = _t_2(a1), None

806

807 A = Fsum() # mean phi

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

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

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

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

812 D = Fsum()

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

814 a2, b2 = p2.philam

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

816 A += a2

817 ta2 = _t_2(a2)

818 tdb = _t_2(db, points=i)

819 R += _at2(tdb * (ta1 + ta2),

820 _1_0 + ta1 * ta2)

821 ta1, b1 = ta2, b2

822

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

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

825 if z1 is not None:

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

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

828 p1, z1 = p2, z2

829

830 R = abs(R * _2_0)

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

832 R = abs(R - PI2)

833 if radius:

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

835 R *= _mean_radius(radius, a)**2

836 return float(R)

837

838

839 def _destination2(a, b, r, t):

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

841

842 @arg a: Latitude (C{radians}).

843 @arg b: Longitude (C{radians}).

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

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

846

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

848 '''

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

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

851 ca *= sr

852

853 a = asin1(ct * ca + cr * sa)

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

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

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

857

858

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

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

861 # and similar logic in .ellipsoidalBaseDI._intersect3

862 a1, b1 = s.philam

863

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

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

866 else: # must be a point

867 s.others(end, name=_end_ + _i_)

868 hs.append(end.height)

869 a2, b2 = end.philam

870 if wrap:

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

872

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

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

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

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

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

878 -(b1 + b2) * _0_5)

879 cb21 *= sin(a1 - a2) # sa21

880 sb21 *= sin(a1 + a2) # sa12

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

882 cb12 * cb21 + sb12 * sb21,

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

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

885

886

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

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

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

890 return fdot(ds, db, a - a1)

891

892

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

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

895 two points or as a point and bearing.

896

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

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

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

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

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

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

903 @kwarg radius_exact_height_wrap: Optional keyword arguments, see

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

905

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

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

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

909

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

911

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

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

914

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

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

917 '''

918 c = _T00.others(center=center)

919 p = _T00.others(point=point)

920 try:

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

922 except (TypeError, ValueError) as x:

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

924 **radius_exact_height_wrap)

925

926

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

928 LatLon=None, **LatLon_kwds):

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

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

931

932 s1, s2 = start1, start2

933 if wrap:

934 s2 = _Wrap.point(s2)

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

936

937 a1, b1 = s1.philam

938 a2, b2 = s2.philam

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

940 r12 = vincentys_(a2, a1, db)

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

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

943

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

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

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

947

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

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

950 raise IntersectionError(_parallel_)

951 # handle domain error for equivalent longitudes,

952 # see also functions asin_safe and acos_safe at

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

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

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

956 if sin(db) > 0:

957 t21 = PI2 - t21

958 else:

959 t12 = PI2 - t12

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

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

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

963 raise IntersectionError(_infinite_)

964 sx3 = sx1 * sx2

965# XXX if sx3 < 0:

966# XXX raise ValueError(_ambiguous_)

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

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

969

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

971 # like .ellipsoidalBaseDI,_intersect3, if this intersection

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

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

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

975

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

977 _N_vector_ = _MODS.nvectorBase._N_vector_

978

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

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

981 x = x1.cross(x2)

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

983 raise IntersectionError(_colinear_)

984 a, b = x.philam

985 # choose intersection similar to sphericalNvector

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

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

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

989

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

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

992 intersection, LatLon, LatLon_kwds)

993

994

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

996 LatLon=LatLon, **LatLon_kwds):

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

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

999

1000 @arg start1: Start point of the first line (L{LatLon}).

1001 @arg end1: End point of the first line (L{LatLon}) or

1002 the initial bearing at the first start point

1003 (compass C{degrees360}).

1004 @arg start2: Start point of the second line (L{LatLon}).

1005 @arg end2: End point of the second line (L{LatLon}) or

1006 the initial bearing at the second start point

1007 (compass C{degrees360}).

1008 @kwarg height: Optional height for the intersection point,

1009 overriding the mean height (C{meter}).

1010 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll

1011 B{C{start2}} and both B{C{end*}} points (C{bool}).

1012 @kwarg LatLon: Optional class to return the intersection

1013 point (L{LatLon}) or C{None}.

1014 @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword

1015 arguments, ignored if C{B{LatLon} is None}.

1016

1017 @return: The intersection point as a (B{C{LatLon}}) or if

1018 C{B{LatLon} is None} a L{LatLon3Tuple}C{(lat, lon,

1019 height)}. An alternate intersection point might

1020 be the L{antipode} to the returned result.

1021

1022 @raise IntersectionError: Ambiguous or infinite intersection

1023 or colinear, parallel or otherwise

1024 non-intersecting lines.

1025

1026 @raise TypeError: A B{C{start1}}, B{C{end1}}, B{C{start2}}

1027 or B{C{end2}} point not L{LatLon}.

1028

1029 @raise ValueError: Invalid B{C{height}} or C{null} line.

1030 '''

1031 s1 = _T00.others(start1=start1)

1032 s2 = _T00.others(start2=start2)

1033 try:

1034 return _intersect(s1, end1, s2, end2, height=height, wrap=wrap,

1035 LatLon=LatLon, **LatLon_kwds)

1036 except (TypeError, ValueError) as x:

1037 raise _xError(x, start1=start1, end1=end1, start2=start2, end2=end2)

1038

1039

1040 def intersections2(center1, rad1, center2, rad2, radius=R_M, eps=_0_0,

1041 height=None, wrap=False, # was=True

1042 LatLon=LatLon, **LatLon_kwds):

1043 '''Compute the intersection points of two circles each defined

1044 by a center point and a radius.

1045

1046 @arg center1: Center of the first circle (L{LatLon}).

1047 @arg rad1: Radius of the first circle (C{meter} or C{radians},

1048 see B{C{radius}}).

1049 @arg center2: Center of the second circle (L{LatLon}).

1050 @arg rad2: Radius of the second circle (C{meter} or C{radians},

1051 see B{C{radius}}).

1052 @kwarg radius: Mean earth radius (C{meter} or C{None} if B{C{rad1}},

1053 B{C{rad2}} and B{C{eps}} are given in C{radians}).

1054 @kwarg eps: Required overlap (C{meter} or C{radians}, see

1055 B{C{radius}}).

1056 @kwarg height: Optional height for the intersection points (C{meter},

1057 conventionally) or C{None} for the I{"radical height"}

1058 at the I{radical line} between both centers.

1059 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{center2}}

1060 (C{bool}).

1061 @kwarg LatLon: Optional class to return the intersection

1062 points (L{LatLon}) or C{None}.

1063 @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword

1064 arguments, ignored if C{B{LatLon} is None}.

1065

1066 @return: 2-Tuple of the intersection points, each a B{C{LatLon}}

1067 instance or if C{B{LatLon} is None} a L{LatLon3Tuple}C{(lat,

1068 lon, height)}. For abutting circles, both intersection

1069 points are the same instance, aka the I{radical center}.

1070

1071 @raise IntersectionError: Concentric, antipodal, invalid or

1072 non-intersecting circles.

1073

1074 @raise TypeError: If B{C{center1}} or B{C{center2}} not L{LatLon}.

1075

1076 @raise ValueError: Invalid B{C{rad1}}, B{C{rad2}}, B{C{radius}},

1077 B{C{eps}} or B{C{height}}.

1078

1079 @note: Courtesy of U{Samuel Čavoj<https://GitHub.com/mrJean1/PyGeodesy/issues/41>}.

1080

1081 @see: This U{Answer<https://StackOverflow.com/questions/53324667/

1082 find-intersection-coordinates-of-two-circles-on-earth/53331953>}.

1083 '''

1084 c1 = _T00.others(center1=center1)

1085 c2 = _T00.others(center2=center2)

1086 try:

1087 return _intersects2(c1, rad1, c2, rad2, radius=radius, eps=eps,

1088 height=height, wrap=wrap,

1089 LatLon=LatLon, **LatLon_kwds)

1090 except (TypeError, ValueError) as x:

1091 raise _xError(x, center1=center1, rad1=rad1,

1092 center2=center2, rad2=rad2, wrap=wrap)

1093

1094

1095 def _intersects2(c1, rad1, c2, rad2, radius=R_M, eps=_0_0, # in .ellipsoidalBaseDI._intersects2

1096 height=None, too_d=None, wrap=False, # was=True

1097 LatLon=LatLon, **LatLon_kwds):

1098 # (INTERNAL) Intersect two spherical circles, see L{intersections2}

1099 # above, separated to allow callers to embellish any exceptions

1100

1101 def _dest3(bearing, h):

1102 a, b = _destination2(a1, b1, r1, bearing)

1103 return _LL3Tuple(degrees90(a), degrees180(b), h,

1104 intersections2, LatLon, LatLon_kwds)

1105

1106 a1, b1 = c1.philam

1107 a2, b2 = c2.philam

1108 if wrap:

1109 a2, b2 = _Wrap.philam(a2, b2)

1110

1111 r1, r2, f = _rads3(rad1, rad2, radius)

1112 if f: # swapped radii, swap centers

1113 a1, a2 = a2, a1 # PYCHOK swap!

1114 b1, b2 = b2, b1 # PYCHOK swap!

1115

1116 db, b2 = unrollPI(b1, b2, wrap=wrap)

1117 d = vincentys_(a2, a1, db) # radians

1118 if d < max(r1 - r2, EPS):

1119 raise IntersectionError(_near_(_concentric_)) # XXX ConcentricError?

1120

1121 r = eps if radius is None else (m2radians(

1122 eps, radius=radius) if eps else _0_0)

1123 if r < _0_0:

1124 raise _ValueError(eps=r)

1125

1126 x = fsumf_(r1, r2, -d) # overlap

1127 if x > max(r, EPS):

1128 sd, cd, sr1, cr1, _, cr2 = sincos2_(d, r1, r2)

1129 x = sd * sr1

1130 if isnear0(x):

1131 raise _ValueError(_invalid_)

1132 x = acos1((cr2 - cd * cr1) / x) # 0 <= x <= PI

1133

1134 elif x < r: # PYCHOK no cover

1135 t = (d * radius) if too_d is None else too_d

1136 raise IntersectionError(_too_(_Fmt.distant(t)))

1137

1138 if height is None: # "radical height"

1139 f = _radical2(d, r1, r2).ratio

1140 h = Height(favg(c1.height, c2.height, f=f))

1141 else:

1142 h = Height(height)

1143

1144 b = bearing_(a1, b1, a2, b2, final=False, wrap=wrap)

1145 if x < EPS4: # externally ...

1146 r = _dest3(b, h)

1147 elif x > _PI_EPS4: # internally ...

1148 r = _dest3(b + PI, h)

1149 else:

1150 return _dest3(b + x, h), _dest3(b - x, h)

1151 return r, r # ... abutting circles

1152

1153

1154@deprecated_function

1155 def isPoleEnclosedBy(points, wrap=False): # PYCHOK no cover

1156 '''DEPRECATED, use function L{pygeodesy.ispolar}.

1157 '''

1158 return ispolar(points, wrap=wrap)

1159

1160

1161 def _LL3Tuple(lat, lon, height, where, LatLon, LatLon_kwds):

1162 '''(INTERNAL) Helper for L{intersection}, L{intersections2} and L{meanOf}.

1163 '''

1164 n = where.__name__

1165 if LatLon is None:

1166 r = LatLon3Tuple(lat, lon, height, name=n)

1167 else:

1168 kwds = _xkwds(LatLon_kwds, height=height, name=n)

1169 r = LatLon(lat, lon, **kwds)

1170 return r

1171

1172

1173 def meanOf(points, height=None, wrap=False, LatLon=LatLon, **LatLon_kwds):

1174 '''Compute the I{geographic} mean of several points.

1175

1176 @arg points: Points to be averaged (L{LatLon}[]).

1177 @kwarg height: Optional height at mean point, overriding the mean

1178 height (C{meter}).

1179 @kwarg wrap: If C{True}, wrap or I{normalize} the B{C{points}}

1180 (C{bool}).

1181 @kwarg LatLon: Optional class to return the mean point (L{LatLon})

1182 or C{None}.

1183 @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword

1184 arguments, ignored if C{B{LatLon} is None}.

1185

1186 @return: The geographic mean and height (B{C{LatLon}}) or a

1187 L{LatLon3Tuple}C{(lat, lon, height)} if B{C{LatLon}}

1188 is C{None}.

1189

1190 @raise TypeError: Some B{C{points}} are not L{LatLon}.

1191

1192 @raise ValueError: No B{C{points}} or invalid B{C{height}}.

1193 '''

1194 def _N_vs(ps, w):

1195 Ps = _T00.PointsIter(ps, wrap=w)

1196 for p in Ps.iterate(closed=False):

1197 yield p._N_vector

1198

1199 m = _MODS.nvectorBase

1200 # geographic, vectorial mean

1201 n = m.sumOf(_N_vs(points, wrap), h=height, Vector=m.NvectorBase)

1202 lat, lon, h = n.latlonheight

1203 return _LL3Tuple(lat, lon, h, meanOf, LatLon, LatLon_kwds)

1204

1205

1206@deprecated_function

1207 def nearestOn2(point, points, **closed_radius_LatLon_options): # PYCHOK no cover

1208 '''DEPRECATED, use function L{sphericalTrigonometry.nearestOn3}.

1209

1210 @return: ... 2-tuple C{(closest, distance)} of the C{closest}

1211 point (L{LatLon}) on the polygon and the C{distance}

1212 between the C{closest} and the given B{C{point}}. The

1213 C{closest} is a B{C{LatLon}} or a L{LatLon2Tuple}C{(lat,

1214 lon)} if B{C{LatLon}} is C{None} ...

1215 '''

1216 ll, d, _ = nearestOn3(point, points, **closed_radius_LatLon_options) # PYCHOK 3-tuple

1217 if _xkwds_get(closed_radius_LatLon_options, LatLon=LatLon) is None:

1218 ll = LatLon2Tuple(ll.lat, ll.lon)

1219 return ll, d

1220

1221

1222 def nearestOn3(point, points, closed=False, radius=R_M, wrap=False, adjust=True,

1223 limit=9, **LatLon_and_kwds):

1224 '''Locate the point on a path or polygon closest to a reference point.

1225

1226 Distances are I{approximated} using function L{pygeodesy.equirectangular_},

1227 subject to the supplied B{C{options}}.

1228

1229 @arg point: The reference point (L{LatLon}).

1230 @arg points: The path or polygon points (L{LatLon}[]).

1231 @kwarg closed: Optionally, close the polygon (C{bool}).

1232 @kwarg radius: Mean earth radius (C{meter}).

1233 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the

1234 B{C{points}} (C{bool}).

1235 @kwarg adjust: See function L{pygeodesy.equirectangular_} (C{bool}).

1236 @kwarg limit: See function L{pygeodesy.equirectangular_} (C{degrees}),

1237 default C{9 degrees} is about C{1,000 Kmeter} (for mean

1238 spherical earth radius L{R_KM}).

1239 @kwarg LatLon: Optional class to return the closest point (L{LatLon})

1240 or C{None}.

1241 @kwarg options: Optional keyword arguments for function

1242 L{pygeodesy.equirectangular_}.

1243

1244 @return: A L{NearestOn3Tuple}C{(closest, distance, angle)} with the

1245 C{closest} point as B{C{LatLon}} or L{LatLon3Tuple}C{(lat,

1246 lon, height)} if B{C{LatLon}} is C{None}. The C{distance}

1247 is the L{pygeodesy.equirectangular_} distance between the

1248 C{closest} and the given B{C{point}} converted to C{meter},

1249 same units as B{C{radius}}. The C{angle} from the given

1250 B{C{point}} to the C{closest} is in compass C{degrees360},

1251 like function L{pygeodesy.compassAngle}. The C{height} is

1252 the (interpolated) height at the C{closest} point.

1253

1254 @raise LimitError: Lat- and/or longitudinal delta exceeds the B{C{limit}},

1255 see function L{pygeodesy.equirectangular_}.

1256

1257 @raise PointsError: Insufficient number of B{C{points}}.

1258

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

1260

1261 @raise ValueError: Invalid B{C{radius}}.

1262

1263 @see: Functions L{pygeodesy.equirectangular_} and L{pygeodesy.nearestOn5}.

1264 '''

1265 t = _nearestOn5(point, points, closed=closed, wrap=wrap,

1266 adjust=adjust, limit=limit)

1267 d = degrees2m(t.distance, radius=radius)

1268 h = t.height

1269 n = nearestOn3.__name__

1270

1271 kwds = _xkwds(LatLon_and_kwds, height=h, name=n)

1272 LL = _xkwds_pop(kwds, LatLon=LatLon)

1273 r = LatLon3Tuple(t.lat, t.lon, h, name=n) if LL is None else \

1274 LL(t.lat, t.lon, **kwds)

1275 return NearestOn3Tuple(r, d, t.angle, name=n)

1276

1277

1278 def perimeterOf(points, closed=False, radius=R_M, wrap=True):

1279 '''Compute the perimeter of a (spherical) polygon or composite

1280 (with great circle arcs joining the points).

1281

1282 @arg points: The polygon points or clips (L{LatLon}[], L{BooleanFHP}

1283 or L{BooleanGH}).

1284 @kwarg closed: Optionally, close the polygon (C{bool}).

1285 @kwarg radius: Mean earth radius (C{meter}) or C{None}.

1286 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the

1287 B{C{points}} (C{bool}).

1288

1289 @return: Polygon perimeter (C{meter}, same units as B{C{radius}}

1290 or C{radians} if B{C{radius}} is C{None}).

1291

1292 @raise PointsError: Insufficient number of B{C{points}}.

1293

1294 @raise TypeError: Some B{C{points}} are not L{LatLon}.

1295

1296 @raise ValueError: Invalid B{C{radius}} or C{B{closed}=False} with

1297 C{B{points}} a composite.

1298

1299 @note: Distances are based on function L{pygeodesy.vincentys_}.

1300

1301 @see: Functions L{perimeterOf<pygeodesy.perimeterOf>},

1302 L{sphericalNvector.perimeterOf} and L{ellipsoidalKarney.perimeterOf}.

1303 '''

1304 def _rads(ps, c, w): # angular edge lengths in radians

1305 Ps = _T00.PointsIter(ps, loop=1, wrap=w)

1306 a1, b1 = Ps[0].philam

1307 for p in Ps.iterate(closed=c):

1308 a2, b2 = p.philam

1309 db, b2 = unrollPI(b1, b2, wrap=w and not (c and Ps.looped))

1310 yield vincentys_(a2, a1, db)

1311 a1, b1 = a2, b2

1312

1313 if _MODS.booleans.isBoolean(points):

1314 if not closed:

1315 raise _ValueError(closed=closed, points=_composite_)

1316 r = points._sum2(LatLon, perimeterOf, closed=True, radius=radius, wrap=wrap)

1317 else:

1318 r = fsum(_rads(points, closed, wrap), floats=True)

1319 return _radians2m(r, radius)

1320

1321

1322 def triangle7(latA, lonA, latB, lonB, latC, lonC, radius=R_M,

1323 excess=excessAbc_,

1324 wrap=False):

1325 '''Compute the angles, sides, and area of a (spherical) triangle.

1326

1327 @arg latA: First corner latitude (C{degrees}).

1328 @arg lonA: First corner longitude (C{degrees}).

1329 @arg latB: Second corner latitude (C{degrees}).

1330 @arg lonB: Second corner longitude (C{degrees}).

1331 @arg latC: Third corner latitude (C{degrees}).

1332 @arg lonC: Third corner longitude (C{degrees}).

1333 @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter},

1334 L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple})

1335 or C{None}.

1336 @kwarg excess: I{Spherical excess} callable (L{excessAbc_},

1337 L{excessGirard_} or L{excessLHuilier_}).

1338 @kwarg wrap: If C{True}, wrap and L{pygeodesy.unroll180}

1339 longitudes (C{bool}).

1340

1341 @return: A L{Triangle7Tuple}C{(A, a, B, b, C, c, area)} with

1342 spherical angles C{A}, C{B} and C{C}, angular sides

1343 C{a}, C{b} and C{c} all in C{degrees} and C{area}

1344 in I{square} C{meter} or same units as B{C{radius}}

1345 I{squared} or if C{B{radius}=0} or C{None}, a

1346 L{Triangle8Tuple}C{(A, a, B, b, C, c, D, E)} all in

1347 C{radians} with the I{spherical excess} C{E} as the

1348 C{unit area} in C{radians}.

1349 '''

1350 t = triangle8_(Phi_(latA=latA), Lam_(lonA=lonA),

1351 Phi_(latB=latB), Lam_(lonB=lonB),

1352 Phi_(latC=latC), Lam_(lonC=lonC),

1353 excess=excess, wrap=wrap)

1354 return _t7Tuple(t, radius)

1355

1356

1357 def triangle8_(phiA, lamA, phiB, lamB, phiC, lamC, excess=excessAbc_,

1358 wrap=False):

1359 '''Compute the angles, sides, I{spherical deficit} and I{spherical

1360 excess} of a (spherical) triangle.

1361

1362 @arg phiA: First corner latitude (C{radians}).

1363 @arg lamA: First corner longitude (C{radians}).

1364 @arg phiB: Second corner latitude (C{radians}).

1365 @arg lamB: Second corner longitude (C{radians}).

1366 @arg phiC: Third corner latitude (C{radians}).

1367 @arg lamC: Third corner longitude (C{radians}).

1368 @kwarg excess: I{Spherical excess} callable (L{excessAbc_},

1369 L{excessGirard_} or L{excessLHuilier_}).

1370 @kwarg wrap: If C{True}, L{pygeodesy.unrollPI} the

1371 longitudinal deltas (C{bool}).

1372

1373 @return: A L{Triangle8Tuple}C{(A, a, B, b, C, c, D, E)} with

1374 spherical angles C{A}, C{B} and C{C}, angular sides

1375 C{a}, C{b} and C{c}, I{spherical deficit} C{D} and

1376 I{spherical excess} C{E}, all in C{radians}.

1377 '''

1378 def _a_r(w, phiA, lamA, phiB, lamB, phiC, lamC):

1379 d, _ = unrollPI(lamB, lamC, wrap=w)

1380 a = vincentys_(phiC, phiB, d)

1381 return a, (phiB, lamB, phiC, lamC, phiA, lamA) # rotate A, B, C

1382

1383 def _A_r(a, sa, ca, sb, cb, sc, cc):

1384 s = sb * sc

1385 A = acos1((ca - cb * cc) / s) if isnon0(s) else a

1386 return A, (sb, cb, sc, cc, sa, ca) # rotate sincos2_'s

1387

1388 # notation: side C{a} is oposite to corner C{A}, etc.

1389 a, r = _a_r(wrap, phiA, lamA, phiB, lamB, phiC, lamC)

1390 b, r = _a_r(wrap, *r)

1391 c, _ = _a_r(wrap, *r)

1392

1393 A, r = _A_r(a, *sincos2_(a, b, c))

1394 B, r = _A_r(b, *r)

1395 C, _ = _A_r(c, *r)

1396

1397 D = fsumf_(PI2, -a, -b, -c) # deficit aka defect

1398 E = excessGirard_(A, B, C) if excess in (excessGirard_, True) else (

1399 excessLHuilier_(a, b, c) if excess in (excessLHuilier_, False) else

1400 excessAbc_(*max((A, b, c), (B, c, a), (C, a, b))))

1401

1402 return Triangle8Tuple(A, a, B, b, C, c, D, E)

1403

1404

1405 def _t7Tuple(t, radius):

1406 '''(INTERNAL) Convert a L{Triangle8Tuple} to L{Triangle7Tuple}.

1407 '''

1408 if radius: # not in (None, _0_0)

1409 r = radius if _isRadius(radius) else \

1410 _ellipsoidal_datum(radius).ellipsoid.Rmean

1411 A, B, C = map1(degrees, t.A, t.B, t.C)

1412 t = Triangle7Tuple(A, (r * t.a),

1413 B, (r * t.b),

1414 C, (r * t.c), t.E * r**2)

1415 return t

1416

1417

1418__all__ += _ALL_OTHER(Cartesian, LatLon, # classes

1419 areaOf, # functions

1420 intersecant2, intersection, intersections2, ispolar,

1421 isPoleEnclosedBy, # DEPRECATED, use ispolar

1422 meanOf,

1423 nearestOn2, nearestOn3,

1424 perimeterOf,

1425 sumOf, # XXX == vector3d.sumOf

1426 triangle7, triangle8_)

1427

1428# **) MIT License

1429#

1430# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved.

1431#

1432# Permission is hereby granted, free of charge, to any person obtaining a

1433# copy of this software and associated documentation files (the "Software"),

1434# to deal in the Software without restriction, including without limitation

1435# the rights to use, copy, modify, merge, publish, distribute, sublicense,

1436# and/or sell copies of the Software, and to permit persons to whom the

1437# Software is furnished to do so, subject to the following conditions:

1438#

1439# The above copyright notice and this permission notice shall be included

1440# in all copies or substantial portions of the Software.

1441#

1442# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS

1443# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

1444# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL

1445# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR

1446# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,

1447# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR

1448# OTHER DEALINGS IN THE SOFTWARE.