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_pop2

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.nvectorBase import NvectorBase, sumOf # _MODE

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

39 Triangle7Tuple, Triangle8Tuple

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

41 Fmt as _Fmt # XXX shadowed

42 from pygeodesy.props import deprecated_function, deprecated_method

43 from pygeodesy.sphericalBase import _m2radians, CartesianSphericalBase, \

44 _intersecant2, LatLonSphericalBase, \

45 _rads3, _radians2m, _trilaterate5

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

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

48 Phi_, Radius_, Scalar

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

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

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

52 from pygeodesy.vector3d import sumOf, Vector3d

53

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

55

56__all__ = _ALL_LAZY.sphericalTrigonometry

57__version__ = '24.04.07'

58

59_PI_EPS4 = PI - EPS4

60 if _PI_EPS4 >= PI:

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

62

63

64 class Cartesian(CartesianSphericalBase):

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

66 spherical, geodetic L{LatLon}.

67 '''

68

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

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

71

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

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

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

75

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

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

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

79

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

81 '''

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

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

84

85

86 class LatLon(LatLonSphericalBase):

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

88 '''

89

90 def _ab1_ab2_db5(self, other, wrap):

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

92 '''

93 a1, b1 = self.philam

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

95 if wrap:

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

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

98 else: # unrollPI shortcut

99 db = b2 - b1

100 return a1, b1, a2, b2, db

101

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

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

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

105 end point.

106

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

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

109 from the start point to the point where the perpendicular

110 crosses the line.

111

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

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

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

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

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

117

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

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

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

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

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

123 B{C{start}} point.

124

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

126

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

128 '''

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

130 cx = cos(x)

131 return _0_0 if isnear0(cx) else \

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

133

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

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

136 '''

137 s = self.others(start=start)

138 e = self.others(end=end)

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

140

141 r = Radius_(radius)

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

143

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

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

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

147 return r, x, -b

148

149 @deprecated_method

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

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

152 '''

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

154

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

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

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

158

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

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

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

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

163

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

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

166 '''

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

168 sa, ca, sa1, ca1, \

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

170 sa1 *= ca2 * ca

171

172 x = sa1 * sdb

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

174 z = ca1 * sdb * ca2 * sa

175

176 h = hypot(x, y)

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

178 return None # great circle doesn't reach latitude

179

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

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

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

183

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

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

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

187

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

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

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

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

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

193

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

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

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

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

198

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

200

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

202 '''

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

204 return _radians2m(x, radius)

205

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

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

208 travelled the given distance on the given initial bearing.

209

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

211 B{C{radius}}).

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

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

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

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

216

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

218

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

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

221 '''

222 a, b = self.philam

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

224

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

226 h = self._heigHt(height)

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

228

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

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

231

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

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

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

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

236

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

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

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

240

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

242

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

244 '''

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

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

247

248# @Property_RO

249# def Ecef(self):

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

251# '''

252# return _MODS.ecef.EcefKarney

253

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

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

256 on the given initial bearing from this point.

257

258 Direction of vector is such that initial bearing vector

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

260

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

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

263 overriding the default L{Vector3d}.

264 @kwarg Vector_kwds: Optional, additional keyword argunents

265 for B{C{Vector}}.

266

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

268

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

270 '''

271 a, b = self.philam

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

273

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

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

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

277

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

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

280 to an other point.

281

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

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

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

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

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

287 C{sphericalNvector.LatLon.initialBearingTo}.

288

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

290

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

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

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

294

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

296 '''

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

298 # XXX behavior like sphericalNvector.LatLon.initialBearingTo

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

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

301

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

303

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

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

306 and an other point.

307

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

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

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

311 @kwarg height: Optional height, overriding the intermediate

312 height (C{meter}).

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

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

315

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

317

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

319

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

321

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

323 '''

324 p = self

325 f = Scalar(fraction=fraction)

326 if not isnear0(f):

327 p = p.others(other)

328 if wrap:

329 p = _Wrap.point(p)

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

331 a1, b1 = self.philam

332 a2, b2 = p.philam

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

334 r = vincentys_(a2, a1, db)

335 sr = sin(r)

336 if isnon0(sr):

337 sa1, ca1, sa2, ca2, \

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

339

340 t = f * r

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

342 b = sin( t) # / sr superflous

343

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

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

346 z = a * sa1 + b * sa2

347

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

349 b = atan2d(y, x)

350

351 else: # PYCHOK no cover

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

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

354

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

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

357 return p

358

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

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

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

362

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

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

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

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

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

368 C{degrees360}).

369 @kwarg height: Optional height for intersection point,

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

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

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

373

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

375 intersection point might be the L{antipode} to

376 the returned result.

377

378 @raise IntersectionError: Ambiguous or infinite intersection

379 or colinear, parallel or otherwise

380 non-intersecting lines.

381

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

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

384

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

386 '''

387 try:

388 s2 = self.others(other)

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

390 LatLon=self.classof)

391 except (TypeError, ValueError) as x:

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

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

394

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

396 height=None, wrap=True):

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

398 by a center point and radius.

399

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

401 see B{C{radius}}).

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

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

404 see B{C{radius}}).

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

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

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

408 B{C{radius}}).

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

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

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

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

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

414

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

416 instance. For abutting circles, both intersection

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

418

419 @raise IntersectionError: Concentric, antipodal, invalid or

420 non-intersecting circles.

421

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

423

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

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

426 '''

427 try:

428 c2 = self.others(other)

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

430 height=height, wrap=wrap,

431 LatLon=self.classof)

432 except (TypeError, ValueError) as x:

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

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

435

436 @deprecated_method

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

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

439 return self.isenclosedBy(points)

440

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

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

443

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

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

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

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

448

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

450 composite, C{False} otherwise.

451

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

453

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

455

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

457

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

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

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

461 '''

462 if _MODS.booleans.isBoolean(points):

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

464

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

466 n0 = self._N_vector

467

468 v2 = Ps[0]._N_vector

469 p1 = Ps[1]

470 v1 = p1._N_vector

471 # check whether this point on same side of all

472 # polygon edges (to the left or right depending

473 # on the anti-/clockwise polygon direction)

474 gc1 = v2.cross(v1)

475 t0 = gc1.angleTo(n0) > PI_2

476 s0 = None

477 # get great-circle vector for each edge

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

479 if wrap and not Ps.looped:

480 p2 = _unrollon(p1, p2)

481 p1 = p2

482 v2 = p2._N_vector

483 gc = v1.cross(v2)

484 t = gc.angleTo(n0) > PI_2

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

486 return False # outside

487

488 # check for convex polygon: angle between

489 # gc vectors, signed by direction of n0

490 # (otherwise the test above is not reliable)

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

492 if s != s0:

493 if s0 is None:

494 s0 = s

495 else:

496 t = _Fmt.SQUARE(points=i)

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

498 gc1, v1 = gc, v2

499

500 return True # inside

501

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

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

504

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

506 @kwarg height: Optional height for midpoint, overriding

507 the mean height (C{meter}).

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

509 may be negative or greater than 1.0.

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

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

512

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

514

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

516

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

518

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

520 '''

521 if fraction is _0_5:

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

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

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

525

526 x = ca2 * cdb + ca1

527 y = ca2 * sdb

528

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

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

531

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

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

534 else:

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

536 return r

537

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

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

540

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

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

543

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

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

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

547 @kwarg wrap_adjust_limit: Optional keyword arguments for functions

548 L{sphericalTrigonometry.nearestOn3} and

549 L{pygeodesy.equirectangular_},

550

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

552

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

554 see function L{pygeodesy.equirectangular_}.

555

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

557 is not (yet) supported.

558

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

560

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

562

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

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

565 '''

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

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

568 if not w:

569 self._notImplemented(within=w)

570

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

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

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

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

575# return point1

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

577# if isnear0(d):

578# return point1 # or point2

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

580# return point2

581# f = a / d

582# if within:

583# if f > EPS1:

584# return point2

585# elif f < EPS:

586# return point1

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

588

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

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

591 **kwds)[0]

592

593 @deprecated_method

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

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

596

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

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

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

600 units of B{C{radius}}.

601 '''

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

603 return r.closest, r.distance

604

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

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

607

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

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

610

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

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

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

614 @kwarg wrap_adjust_limit: Optional keyword arguments for function

615 L{sphericalTrigonometry.nearestOn3} and

616 L{pygeodesy.equirectangular_},

617

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

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

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

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

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

623 function L{pygeodesy.compassAngle}.

624

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

626 see function L{pygeodesy.equirectangular_}.

627

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

629

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

631

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

633

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

635 and L{pygeodesy.nearestOn5}.

636 '''

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

638 LatLon=self.classof, **wrap_adjust_limit)

639

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

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

642 coordinates.

643

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

645 and other keyword arguments, ignored

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

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

648 this L{Cartesian} class or specify

649 C{B{Cartesian}=None}.

650

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

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

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

654

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

656 '''

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

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

659

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

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

662

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

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

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

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

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

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

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

670

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

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

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

674

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

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

677 '''

678 B = self.others(otherB=otherB)

679 C = self.others(otherC=otherC)

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

681

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

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

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

685

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

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

688 at this and at the other point.

689

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

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

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

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

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

695

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

697

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

699 '''

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

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

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

703

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

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

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

707 intersection} of three corresponding circles.

708

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

710 as B{C{radius}}).

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

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

713 B{C{radius}}).

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

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

716 B{C{radius}}).

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

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

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

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

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

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

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

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

725

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

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

728 the corresponding trilaterated points C{minPoint} and

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

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

731

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

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

734

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

736 respectively largest I{radial} overlap found.

737

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

739 nearest respectively farthest intersection margin.

740

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

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

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

744 largest B{C{distance#}}.

745

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

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

748 no intersection or all (near-)concentric for

749 C{B{area}=False}.

750

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

752

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

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

755 or B{C{radius}}.

756 '''

757 return _trilaterate5(self, distance1,

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

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

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

761

762

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

764

765

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

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

768 (with the pointsjoined by great circle arcs).

769

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

771 or L{BooleanGH}).

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

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

774 or C{None}.

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

776 (C{bool}).

777

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

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

780

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

782

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

784

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

786

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

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

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

790

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

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

793 L{ellipsoidalKarney.areaOf}.

794 '''

795 if _MODS.booleans.isBoolean(points):

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

797

798 _at2, _t_2 = atan2, tan_2

799 _un, _w180 = unrollPI, wrap180

800

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

802 p1 = p2 = Ps[0]

803 a1, b1 = p1.philam

804 ta1, z1 = _t_2(a1), None

805

806 A = Fsum() # mean phi

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

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

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

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

811 D = Fsum()

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

813 a2, b2 = p2.philam

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

815 A += a2

816 ta2 = _t_2(a2)

817 tdb = _t_2(db, points=i)

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

819 _1_0 + ta1 * ta2)

820 ta1, b1 = ta2, b2

821

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

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

824 if z1 is not None:

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

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

827 p1, z1 = p2, z2

828

829 R = abs(R * _2_0)

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

831 R = abs(R - PI2)

832 if radius:

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

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

835 return float(R)

836

837

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

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

840

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

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

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

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

845

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

847 '''

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

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

850 ca *= sr

851

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

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

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

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

856

857

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

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

860 # and similar logic in .ellipsoidalBaseDI._intersect3

861 a1, b1 = s.philam

862

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

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

865 else: # must be a point

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

867 hs.append(end.height)

868 a2, b2 = end.philam

869 if wrap:

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

871

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

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

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

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

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

877 -(b1 + b2) * _0_5)

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

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

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

881 cb12 * cb21 + sb12 * sb21,

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

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

884

885

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

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

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

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

890

891

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

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

894 two points or as a point and bearing.

895

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

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

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

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

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

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

902 @kwarg radius_exact_height_wrap: Optional keyword arguments, see

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

904

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

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

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

908

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

910

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

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

913

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

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

916 '''

917 c = _T00.others(center=center)

918 p = _T00.others(point=point)

919 try:

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

921 except (TypeError, ValueError) as x:

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

923 **radius_exact_height_wrap)

924

925

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

927 LatLon=None, **LatLon_kwds):

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

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

930

931 s1, s2 = start1, start2

932 if wrap:

933 s2 = _Wrap.point(s2)

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

935

936 a1, b1 = s1.philam

937 a2, b2 = s2.philam

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

939 r12 = vincentys_(a2, a1, db)

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

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

942

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

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

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

946

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

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

949 raise IntersectionError(_parallel_)

950 # handle domain error for equivalent longitudes,

951 # see also functions asin_safe and acos_safe at

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

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

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

955 if sin(db) > 0:

956 t21 = PI2 - t21

957 else:

958 t12 = PI2 - t12

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

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

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

962 raise IntersectionError(_infinite_)

963 sx3 = sx1 * sx2

964# XXX if sx3 < 0:

965# XXX raise ValueError(_ambiguous_)

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

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

968

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

970 # like .ellipsoidalBaseDI,_intersect3, if this intersection

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

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

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

974

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

976 _N_vector_ = _MODS.nvectorBase._N_vector_

977

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

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

980 x = x1.cross(x2)

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

982 raise IntersectionError(_colinear_)

983 a, b = x.philam

984 # choose intersection similar to sphericalNvector

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

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

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

988

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

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

991 intersection, LatLon, LatLon_kwds)

992

993

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

995 LatLon=LatLon, **LatLon_kwds):

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

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

998

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

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

1001 the initial bearing at the first start point

1002 (compass C{degrees360}).

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

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

1005 the initial bearing at the second start point

1006 (compass C{degrees360}).

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

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

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

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

1011 @kwarg LatLon: Optional class to return the intersection

1012 point (L{LatLon}) or C{None}.

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

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

1015

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

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

1018 height)}. An alternate intersection point might

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

1020

1021 @raise IntersectionError: Ambiguous or infinite intersection

1022 or colinear, parallel or otherwise

1023 non-intersecting lines.

1024

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

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

1027

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

1029 '''

1030 s1 = _T00.others(start1=start1)

1031 s2 = _T00.others(start2=start2)

1032 try:

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

1034 LatLon=LatLon, **LatLon_kwds)

1035 except (TypeError, ValueError) as x:

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

1037

1038

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

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

1041 LatLon=LatLon, **LatLon_kwds):

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

1043 by a center point and a radius.

1044

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

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

1047 see B{C{radius}}).

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

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

1050 see B{C{radius}}).

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

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

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

1054 B{C{radius}}).

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

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

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

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

1059 (C{bool}).

1060 @kwarg LatLon: Optional class to return the intersection

1061 points (L{LatLon}) or C{None}.

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

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

1064

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

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

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

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

1069

1070 @raise IntersectionError: Concentric, antipodal, invalid or

1071 non-intersecting circles.

1072

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

1074

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

1076 B{C{eps}} or B{C{height}}.

1077

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

1079

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

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

1082 '''

1083 c1 = _T00.others(center1=center1)

1084 c2 = _T00.others(center2=center2)

1085 try:

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

1087 height=height, wrap=wrap,

1088 LatLon=LatLon, **LatLon_kwds)

1089 except (TypeError, ValueError) as x:

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

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

1092

1093

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

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

1096 LatLon=LatLon, **LatLon_kwds):

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

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

1099

1100 def _dest3(bearing, h):

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

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

1103 intersections2, LatLon, LatLon_kwds)

1104

1105 a1, b1 = c1.philam

1106 a2, b2 = c2.philam

1107 if wrap:

1108 a2, b2 = _Wrap.philam(a2, b2)

1109

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

1111 if f: # swapped radii, swap centers

1112 a1, a2 = a2, a1 # PYCHOK swap!

1113 b1, b2 = b2, b1 # PYCHOK swap!

1114

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

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

1117 if d < max(r1 - r2, EPS):

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

1119

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

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

1122 if r < _0_0:

1123 raise _ValueError(eps=r)

1124

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

1126 if x > max(r, EPS):

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

1128 x = sd * sr1

1129 if isnear0(x):

1130 raise _ValueError(_invalid_)

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

1132

1133 elif x < r: # PYCHOK no cover

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

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

1136

1137 if height is None: # "radical height"

1138 f = _radical2(d, r1, r2).ratio

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

1140 else:

1141 h = Height(height)

1142

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

1144 if x < EPS4: # externally ...

1145 r = _dest3(b, h)

1146 elif x > _PI_EPS4: # internally ...

1147 r = _dest3(b + PI, h)

1148 else:

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

1150 return r, r # ... abutting circles

1151

1152

1153@deprecated_function

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

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

1156 '''

1157 return ispolar(points, wrap=wrap)

1158

1159

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

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

1162 '''

1163 n = where.__name__

1164 if LatLon is None:

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

1166 else:

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

1168 r = LatLon(lat, lon, **kwds)

1169 return r

1170

1171

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

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

1174

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

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

1177 height (C{meter}).

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

1179 (C{bool}).

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

1181 or C{None}.

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

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

1184

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

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

1187 is C{None}.

1188

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

1190

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

1192 '''

1193 def _N_vs(ps, w):

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

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

1196 yield p._N_vector

1197

1198 m = _MODS.nvectorBase

1199 # geographic, vectorial mean

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

1201 lat, lon, h = n.latlonheight

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

1203

1204

1205@deprecated_function

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

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

1208

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

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

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

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

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

1214 '''

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

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

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

1218 return ll, d

1219

1220

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

1222 limit=9, **LatLon_and_kwds):

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

1224

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

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

1227

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

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

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

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

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

1233 B{C{points}} (C{bool}).

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

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

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

1237 spherical earth radius L{R_KM}).

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

1239 or C{None}.

1240 @kwarg options: Optional keyword arguments for function

1241 L{pygeodesy.equirectangular_}.

1242

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

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

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

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

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

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

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

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

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

1252

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

1254 see function L{pygeodesy.equirectangular_}.

1255

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

1257

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

1259

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

1261

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

1263 '''

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

1265 adjust=adjust, limit=limit)

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

1267 h = t.height

1268 n = nearestOn3.__name__

1269

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

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

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

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

1274

1275

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

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

1278 (with great circle arcs joining the points).

1279

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

1281 or L{BooleanGH}).

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

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

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

1285 B{C{points}} (C{bool}).

1286

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

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

1289

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

1291

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

1293

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

1295 C{B{points}} a composite.

1296

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

1298

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

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

1301 '''

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

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

1304 a1, b1 = Ps[0].philam

1305 for p in Ps.iterate(closed=c):

1306 a2, b2 = p.philam

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

1308 yield vincentys_(a2, a1, db)

1309 a1, b1 = a2, b2

1310

1311 if _MODS.booleans.isBoolean(points):

1312 if not closed:

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

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

1315 else:

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

1317 return _radians2m(r, radius)

1318

1319

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

1321 excess=excessAbc_,

1322 wrap=False):

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

1324

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

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

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

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

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

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

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

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

1333 or C{None}.

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

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

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

1337 longitudes (C{bool}).

1338

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

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

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

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

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

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

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

1346 C{unit area} in C{radians}.

1347 '''

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

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

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

1351 excess=excess, wrap=wrap)

1352 return _t7Tuple(t, radius)

1353

1354

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

1356 wrap=False):

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

1358 excess} of a (spherical) triangle.

1359

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

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

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

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

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

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

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

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

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

1369 longitudinal deltas (C{bool}).

1370

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

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

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

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

1375 '''

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

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

1378 a = vincentys_(phiC, phiB, d)

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

1380

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

1382 s = sb * sc

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

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

1385

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

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

1388 b, r = _a_r(wrap, *r)

1389 c, _ = _a_r(wrap, *r)

1390

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

1392 B, r = _A_r(b, *r)

1393 C, _ = _A_r(c, *r)

1394

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

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

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

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

1399

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

1401

1402

1403 def _t7Tuple(t, radius):

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

1405 '''

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

1407 r = radius if _isRadius(radius) else \

1408 _ellipsoidal_datum(radius).ellipsoid.Rmean

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

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

1411 B, (r * t.b),

1412 C, (r * t.c), t.E * r**2)

1413 return t

1414

1415

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

1417 areaOf, # functions

1418 intersecant2, intersection, intersections2, ispolar,

1419 isPoleEnclosedBy, # DEPRECATED, use ispolar

1420 meanOf,

1421 nearestOn2, nearestOn3,

1422 perimeterOf,

1423 sumOf, # XXX == vector3d.sumOf

1424 triangle7, triangle8_)

1425

1426# **) MIT License

1427#

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

1429#

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

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

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

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

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

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

1436#

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

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

1439#

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

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

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

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

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

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

1446# OTHER DEALINGS IN THE SOFTWARE.