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 2011-2024} 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, fdot_, fmean, hypot

28 from pygeodesy.fsums import Fsum, fsum, fsumf_

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

30 excessGirard_, excessLHuilier_, opposing_, _radical2, \

31 vincentys_

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

33 _concentric_, _convex_, _end_, _infinite_, \

34 _invalid_, _line_, _near_, _null_, _parallel_, \

35 _point_, _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, Lamd, \

48 Phid, 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__ = '24.11.07'

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 C{B{LatLon} is 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 a great

186 circle from a start to 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 a bearing from North.

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 C{B{radius} is 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 from this point on the bearing from North.

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 sa *= st

275 return Vector(fdot_(sb, ct, -cb, sa),

276 -fdot_(cb, ct, sb, sa),

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 and if B{C{raiser}} and L{crosserrors

294 <pygeodesy.crosserrors>} are both 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 = fdot_(a, ca1 * cb1, b, ca2 * cb2)

346 y = fdot_(a, ca1 * sb1, b, ca2 * sb2)

347 z = fdot_(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 a 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 a 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 other points closest to this point.

541

542 Distances are approximated by function L{pygeodesy.equirectangular4},

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.equirectangular4},

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.equirectangular4}.

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.equirectangular4} and L{pygeodesy.nearestOn5}

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

566 '''

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

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

569 if not w:

570 self._notImplemented(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 **kwds)[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.equirectangular4},

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.equirectangular4},

618

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

620 C{closest} point (L{LatLon}), the L{pygeodesy.equirectangular4}

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.equirectangular4}.

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.equirectangular4}

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) coordinates.

643

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

645 keyword arguments, ignored if C{B{Cartesian} is

646 None}. Use C{B{Cartesian}=...} to override this

647 L{Cartesian} class or specify C{B{Cartesian}=None}.

648

649 @return: The cartesian point (L{Cartesian}) or if C{B{Cartesian} is None},

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

651 and C{M} if available.

652

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

654 '''

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

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

657

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

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

660

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

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

663 @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter}, L{Ellipsoid},

664 L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}) or C{None}.

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

666 and B{C{otherC}} (C{bool}).

667

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

669 None}, a L{Triangle8Tuple}C{(A, a, B, b, C, c, D, E)}.

670

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

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

673 '''

674 B = self.others(otherB=otherB)

675 C = self.others(otherC=otherC)

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

677

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

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

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

681

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

683 '''Locate a point given this, an other point and a bearing from

684 North at both points.

685

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

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

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

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

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

691

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

693

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

695 '''

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

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

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

699

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

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

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

703 of three corresponding circles.

704

705 @arg distance1: Distance to this point (C{meter}, same units as B{C{radius}}).

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

707 @arg distance2: Distance to point2 (C{meter}, same units as B{C{radius}}).

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

709 @arg distance3: Distance to point3 (C{meter}, same units as B{C{radius}}).

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

711 intersection of the circles (C{bool}).

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

713 I{intersection margin} if C{B{area}=False} (C{meter}, same

714 units as B{C{radius}}).

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

716 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{point2}} and

717 B{C{point3}} (C{bool}).

718

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

720 C{min} and C{max} in C{meter}, same units as B{C{eps}}, the

721 corresponding trilaterated points C{minPoint} and C{maxPoint}

722 as I{spherical} C{LatLon} and C{n}, the number of trilatered

723 points found for the given B{C{eps}}.

724

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

726 C{minPoint I{is} maxPoint} and C{n = 1}.

727

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

729 largest I{radial} overlap found.

730

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

732 respectively farthest intersection margin.

733

734 If C{B{area}=True} and all 3 circles are concentric, C{n=0} and

735 C{minPoint} and C{maxPoint} are both the B{C{point#}} with the

736 smallest B{C{distance#}} C{min} and C{max} the largest B{C{distance#}}.

737

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

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

740 no intersection or all (near-)concentric if

741 C{B{area}=False}.

742

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

744

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

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

747 or B{C{radius}}.

748 '''

749 return _trilaterate5(self, distance1,

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

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

752 area=area, radius=radius, eps=eps, wrap=wrap)

753

754

755_T00 = LatLon(0, 0, name='T00') # reference instance (L{LatLon})

756

757

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

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

760 points joined by great circle arcs).

761

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

763 or L{BooleanGH}).

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

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

766 or C{None}.

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

768 (C{bool}).

769

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

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

772

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

774

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

776

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

778

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

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

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

782

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

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

785 L{ellipsoidalKarney.areaOf}.

786 '''

787 if _MODS.booleans.isBoolean(points):

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

789

790 _at2, _t_2 = atan2, tan_2

791 _un, _w180 = unrollPI, wrap180

792

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

794 p1 = p2 = Ps[0]

795 a1, b1 = p1.philam

796 ta1, z1 = _t_2(a1), None

797

798 A = Fsum() # mean phi

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

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

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

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

803 D = Fsum()

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

805 a2, b2 = p2.philam

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

807 A += a2

808 ta2 = _t_2(a2)

809 tdb = _t_2(db, points=i)

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

811 _1_0 + ta1 * ta2)

812 ta1, b1 = ta2, b2

813

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

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

816 if z1 is not None:

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

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

819 p1, z1 = p2, z2

820

821 R = abs(R * _2_0)

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

823 R = abs(R - PI2)

824 if radius:

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

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

827 return float(R)

828

829

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

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

832

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

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

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

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

837

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

839 '''

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

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

842 ca *= sr

843

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

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

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

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

848

849

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

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

852 # and similar logic in .ellipsoidalBaseDI._intersect3

853 a1, b1 = s.philam

854

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

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

857 else: # must be a point

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

859 hs.append(end.height)

860 a2, b2 = end.philam

861 if wrap:

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

863

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

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

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

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

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

869 -(b1 + b2) * _0_5)

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

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

872 x = Vector(fdot_(sb12, cb21, -cb12, sb21),

873 fdot_(cb12, cb21, sb12, sb21),

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

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

876

877

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

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

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

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

882

883

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

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

886 two points or as a point and a bearing from North.

887

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

889 @arg circle: Radius of the circle (C{meter}, same units as the earth

890 B{C{radius}}) or a point on the circle (L{LatLon}).

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

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

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

894 @kwarg radius_exact_height_wrap: Optional keyword arguments, see method

895 L{intersecant2<pygeodesy.sphericalBase.LatLonSphericalBase.

896 intersecant2>} for further details.

897

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

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

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

901

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

903

904 @raise TypeError: If B{C{center}}, B{C{point}}, B{C{circle}} or B{C{other}}

905 not L{LatLon}.

906

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

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

909 '''

910 c = _T00.others(center=center)

911 p = _T00.others(point=point)

912 try:

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

914 except (TypeError, ValueError) as x:

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

916 **radius_exact_height_wrap)

917

918

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

920 LatLon=LatLon, **LatLon_kwds):

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

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

923

924 s1, s2 = start1, start2

925 if wrap:

926 s2 = _Wrap.point(s2)

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

928

929 a1, b1 = s1.philam

930 a2, b2 = s2.philam

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

932 r12 = vincentys_(a2, a1, db)

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

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

935

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

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

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

939

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

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

942 raise IntersectionError(_parallel_)

943 # handle domain error for equivalent longitudes,

944 # see also functions asin_safe and acos_safe at

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

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

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

948 if sin(db) > 0:

949 t21 = PI2 - t21

950 else:

951 t12 = PI2 - t12

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

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

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

955 raise IntersectionError(_infinite_)

956 sx3 = sx1 * sx2

957# XXX if sx3 < 0:

958# XXX raise ValueError(_ambiguous_)

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

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

961

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

963 # like .ellipsoidalBaseDI,_intersect3, if this intersection

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

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

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

967

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

969 _N_vector_ = _MODS.nvectorBase._N_vector_

970

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

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

973 x = x1.cross(x2)

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

975 raise IntersectionError(_colinear_)

976 a, b = x.philam

977 # choose intersection similar to sphericalNvector

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

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

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

981

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

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

984 intersection, LatLon, LatLon_kwds)

985

986

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

988 **LatLon_and_kwds):

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

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

991

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

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

994 at the first start point (compass C{degrees360}).

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

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

997 at the second start point (compass C{degrees360}).

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

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

1000 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{start2}}

1001 and both B{C{end*}} points (C{bool}).

1002 @kwarg LatLon_and_kwds: Optional class C{B{LatLon}=}L{LatLon} to use

1003 for the intersection point and optionally additional

1004 B{C{LatLon}} keyword arguments, ignored if C{B{LatLon}

1005 is None}.

1006

1007 @return: The intersection point as a (B{C{LatLon}}) or if C{B{LatLon}

1008 is None} a L{LatLon3Tuple}C{(lat, lon, height)}. An alternate

1009 intersection point might be the L{antipode} to the returned result.

1010

1011 @raise IntersectionError: Ambiguous or infinite intersection or colinear,

1012 parallel or otherwise non-intersecting lines.

1013

1014 @raise TypeError: A B{C{start1}}, B{C{end1}}, B{C{start2}} or B{C{end2}}

1015 point not L{LatLon}.

1016

1017 @raise ValueError: Invalid B{C{height}} or C{null} line.

1018 '''

1019 s1 = _T00.others(start1=start1)

1020 s2 = _T00.others(start2=start2)

1021 try:

1022 return _intersect(s1, end1, s2, end2, height=height, wrap=wrap, **LatLon_and_kwds)

1023 except (TypeError, ValueError) as x:

1024 raise _xError(x, start1=start1, end1=end1, start2=start2, end2=end2)

1025

1026

1027 def intersections2(center1, rad1, center2, rad2, radius=R_M, eps=_0_0,

1028 height=None, wrap=False, # was=True

1029 **LatLon_and_kwds):

1030 '''Compute the intersection points of two circles each defined by a

1031 center point and a radius.

1032

1033 @arg center1: Center of the first circle (L{LatLon}).

1034 @arg rad1: Radius of the first circle (C{meter} or C{radians}, see

1035 B{C{radius}}).

1036 @arg center2: Center of the second circle (L{LatLon}).

1037 @arg rad2: Radius of the second circle (C{meter} or C{radians}, see

1038 B{C{radius}}).

1039 @kwarg radius: Mean earth radius (C{meter} or C{None} if B{C{rad1}},

1040 B{C{rad2}} and B{C{eps}} are given in C{radians}).

1041 @kwarg eps: Required overlap (C{meter} or C{radians}, see B{C{radius}}).

1042 @kwarg height: Optional height for the intersection points (C{meter},

1043 conventionally) or C{None} for the I{"radical height"}

1044 at the I{radical line} between both centers.

1045 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{center2}}

1046 (C{bool}).

1047 @kwarg LatLon_and_kwds: Optional class C{B{LatLon}=}L{LatLon} to use for

1048 the intersection points and optionally additional B{C{LatLon}}

1049 keyword arguments, ignored if C{B{LatLon} is None}.

1050

1051 @return: 2-Tuple of the intersection points, each a B{C{LatLon}}

1052 instance or if C{B{LatLon} is None} a L{LatLon3Tuple}C{(lat,

1053 lon, height)}. For abutting circles, both intersection

1054 points are the same instance, aka the I{radical center}.

1055

1056 @raise IntersectionError: Concentric, antipodal, invalid or

1057 non-intersecting circles.

1058

1059 @raise TypeError: If B{C{center1}} or B{C{center2}} not L{LatLon}.

1060

1061 @raise ValueError: Invalid B{C{rad1}}, B{C{rad2}}, B{C{radius}},

1062 B{C{eps}} or B{C{height}}.

1063

1064 @note: Courtesy of U{Samuel Čavoj<https://GitHub.com/mrJean1/PyGeodesy/issues/41>}.

1065

1066 @see: This U{Answer<https://StackOverflow.com/questions/53324667/

1067 find-intersection-coordinates-of-two-circles-on-earth/53331953>}.

1068 '''

1069 c1 = _T00.others(center1=center1)

1070 c2 = _T00.others(center2=center2)

1071 try:

1072 return _intersects2(c1, rad1, c2, rad2, radius=radius, eps=eps,

1073 height=height, wrap=wrap,

1074 **LatLon_and_kwds)

1075 except (TypeError, ValueError) as x:

1076 raise _xError(x, center1=center1, rad1=rad1,

1077 center2=center2, rad2=rad2, wrap=wrap)

1078

1079

1080 def _intersects2(c1, rad1, c2, rad2, radius=R_M, eps=_0_0, # in .ellipsoidalBaseDI._intersects2

1081 height=None, too_d=None, wrap=False, # was=True

1082 LatLon=LatLon, **LatLon_kwds):

1083 # (INTERNAL) Intersect two spherical circles, see L{intersections2}

1084 # above, separated to allow callers to embellish any exceptions

1085

1086 def _dest3(bearing, h):

1087 a, b = _destination2(a1, b1, r1, bearing)

1088 return _LL3Tuple(degrees90(a), degrees180(b), h,

1089 intersections2, LatLon, LatLon_kwds)

1090

1091 a1, b1 = c1.philam

1092 a2, b2 = c2.philam

1093 if wrap:

1094 a2, b2 = _Wrap.philam(a2, b2)

1095

1096 r1, r2, f = _rads3(rad1, rad2, radius)

1097 if f: # swapped radii, swap centers

1098 a1, a2 = a2, a1 # PYCHOK swap!

1099 b1, b2 = b2, b1 # PYCHOK swap!

1100

1101 db, b2 = unrollPI(b1, b2, wrap=wrap)

1102 d = vincentys_(a2, a1, db) # radians

1103 if d < max(r1 - r2, EPS):

1104 raise IntersectionError(_near_(_concentric_)) # XXX ConcentricError?

1105

1106 r = eps if radius is None else (m2radians(

1107 eps, radius=radius) if eps else _0_0)

1108 if r < _0_0:

1109 raise _ValueError(eps=r)

1110

1111 x = fsumf_(r1, r2, -d) # overlap

1112 if x > max(r, EPS):

1113 sd, cd, sr1, cr1, _, cr2 = sincos2_(d, r1, r2)

1114 x = sd * sr1

1115 if isnear0(x):

1116 raise _ValueError(_invalid_)

1117 x = acos1((cr2 - cd * cr1) / x) # 0 <= x <= PI

1118

1119 elif x < r: # PYCHOK no cover

1120 t = (d * radius) if too_d is None else too_d

1121 raise IntersectionError(_too_(_Fmt.distant(t)))

1122

1123 if height is None: # "radical height"

1124 f = _radical2(d, r1, r2).ratio

1125 h = Height(favg(c1.height, c2.height, f=f))

1126 else:

1127 h = Height(height)

1128

1129 b = bearing_(a1, b1, a2, b2, final=False, wrap=wrap)

1130 if x < EPS4: # externally ...

1131 r = _dest3(b, h)

1132 elif x > _PI_EPS4: # internally ...

1133 r = _dest3(b + PI, h)

1134 else:

1135 return _dest3(b + x, h), _dest3(b - x, h)

1136 return r, r # ... abutting circles

1137

1138

1139@deprecated_function

1140 def isPoleEnclosedBy(points, wrap=False): # PYCHOK no cover

1141 '''DEPRECATED, use function L{pygeodesy.ispolar}.

1142 '''

1143 return ispolar(points, wrap=wrap)

1144

1145

1146 def _LL3Tuple(lat, lon, height, where, LatLon, LatLon_kwds):

1147 '''(INTERNAL) Helper for L{intersection}, L{intersections2} and L{meanOf}.

1148 '''

1149 n = where.__name__

1150 if LatLon is None:

1151 r = LatLon3Tuple(lat, lon, height, name=n)

1152 else:

1153 kwds = _xkwds(LatLon_kwds, height=height, name=n)

1154 r = LatLon(lat, lon, **kwds)

1155 return r

1156

1157

1158 def meanOf(points, height=None, wrap=False, LatLon=LatLon, **LatLon_kwds):

1159 '''Compute the I{geographic} mean of several points.

1160

1161 @arg points: Points to be averaged (L{LatLon}[]).

1162 @kwarg height: Optional height at mean point, overriding the mean height

1163 (C{meter}).

1164 @kwarg wrap: If C{True}, wrap or I{normalize} the B{C{points}} (C{bool}).

1165 @kwarg LatLon: Optional class to return the mean point (L{LatLon}) or C{None}.

1166 @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword arguments,

1167 ignored if C{B{LatLon} is None}.

1168

1169 @return: The geographic mean and height (B{C{LatLon}}) or if C{B{LatLon}

1170 is None}, a L{LatLon3Tuple}C{(lat, lon, height)}.

1171

1172 @raise TypeError: Some B{C{points}} are not L{LatLon}.

1173

1174 @raise ValueError: No B{C{points}} or invalid B{C{height}}.

1175 '''

1176 def _N_vs(ps, w):

1177 Ps = _T00.PointsIter(ps, wrap=w)

1178 for p in Ps.iterate(closed=False):

1179 yield p._N_vector

1180

1181 m = _MODS.nvectorBase

1182 # geographic, vectorial mean

1183 n = m.sumOf(_N_vs(points, wrap), h=height, Vector=m.NvectorBase)

1184 lat, lon, h = n.latlonheight

1185 return _LL3Tuple(lat, lon, h, meanOf, LatLon, LatLon_kwds)

1186

1187

1188@deprecated_function

1189 def nearestOn2(point, points, **closed_radius_LatLon_options): # PYCHOK no cover

1190 '''DEPRECATED, use function L{sphericalTrigonometry.nearestOn3}.

1191

1192 @return: ... 2-tuple C{(closest, distance)} of the C{closest}

1193 point (L{LatLon}) on the polygon and the C{distance}

1194 between the C{closest} and the given B{C{point}}. The

1195 C{closest} is a B{C{LatLon}} or a L{LatLon2Tuple}C{(lat,

1196 lon)} if C{B{LatLon} is None} ...

1197 '''

1198 ll, d, _ = nearestOn3(point, points, **closed_radius_LatLon_options) # PYCHOK 3-tuple

1199 if _xkwds_get(closed_radius_LatLon_options, LatLon=LatLon) is None:

1200 ll = LatLon2Tuple(ll.lat, ll.lon)

1201 return ll, d

1202

1203

1204 def nearestOn3(point, points, closed=False, radius=R_M, wrap=False, adjust=True,

1205 limit=9, **LatLon_and_kwds):

1206 '''Locate the point on a path or polygon closest to a reference point.

1207

1208 Distances are I{approximated} using function L{equirectangular4

1209 <pygeodesy.equirectangular4>}, subject to the supplied B{C{options}}.

1210

1211 @arg point: The reference point (L{LatLon}).

1212 @arg points: The path or polygon points (L{LatLon}[]).

1213 @kwarg closed: Optionally, close the polygon (C{bool}).

1214 @kwarg radius: Mean earth radius (C{meter}).

1215 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the

1216 B{C{points}} (C{bool}).

1217 @kwarg adjust: See function L{equirectangular4<pygeodesy.equirectangular4>}

1218 (C{bool}).

1219 @kwarg limit: See function L{equirectangular4<pygeodesy.equirectangular4>}

1220 (C{degrees}), default C{9 degrees} is about C{1,000 Km} (for

1221 (mean spherical earth radius L{R_KM}).

1222 @kwarg LatLon_and_kwds: Optional class C{B{LatLon}=L{LatLon}} to return the

1223 closest point and optionally additional C{B{LatLon}} keyword

1224 arguments or specify C{B{LatLon}=None}.

1225

1226 @return: A L{NearestOn3Tuple}C{(closest, distance, angle)} with the

1227 C{closest} point as B{C{LatLon}} or L{LatLon3Tuple}C{(lat,

1228 lon, height)} if C{B{LatLon} is None}. The C{distance} is

1229 the L{equirectangular4<pygeodesy.equirectangular4>} distance

1230 between the C{closest} and the given B{C{point}} converted to

1231 C{meter}, same units as B{C{radius}}. The C{angle} from the

1232 given B{C{point}} to the C{closest} is in compass C{degrees360},

1233 like function L{compassAngle<pygeodesy.compassAngle>}. The

1234 C{height} is the (interpolated) height at the C{closest} point.

1235

1236 @raise LimitError: Lat- and/or longitudinal delta exceeds the B{C{limit}},

1237 see function L{equirectangular4<pygeodesy.equirectangular4>}.

1238

1239 @raise PointsError: Insufficient number of B{C{points}}.

1240

1241 @raise TypeError: Some B{C{points}} are not C{LatLon}.

1242

1243 @raise ValueError: Invalid B{C{radius}}.

1244

1245 @see: Functions L{equirectangular4<pygeodesy.equirectangular4>} and

1246 L{nearestOn5<pygeodesy.nearestOn5>}.

1247 '''

1248 t = _nearestOn5(point, points, closed=closed, wrap=wrap,

1249 adjust=adjust, limit=limit)

1250 d = degrees2m(t.distance, radius=radius)

1251 h = t.height

1252 n = nearestOn3.__name__

1253

1254 LL, kwds = _xkwds_pop2(LatLon_and_kwds, LatLon=LatLon)

1255 r = LatLon3Tuple(t.lat, t.lon, h, name=n) if LL is None else \

1256 LL(t.lat, t.lon, **_xkwds(kwds, height=h, name=n))

1257 return NearestOn3Tuple(r, d, t.angle, name=n)

1258

1259

1260 def perimeterOf(points, closed=False, radius=R_M, wrap=True):

1261 '''Compute the perimeter of a (spherical) polygon or composite

1262 (with great circle arcs joining the points).

1263

1264 @arg points: The polygon points or clips (L{LatLon}[], L{BooleanFHP}

1265 or L{BooleanGH}).

1266 @kwarg closed: Optionally, close the polygon (C{bool}).

1267 @kwarg radius: Mean earth radius (C{meter}) or C{None}.

1268 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the

1269 B{C{points}} (C{bool}).

1270

1271 @return: Polygon perimeter (C{meter}, same units as B{C{radius}}

1272 or C{radians} if C{B{radius} is None}).

1273

1274 @raise PointsError: Insufficient number of B{C{points}}.

1275

1276 @raise TypeError: Some B{C{points}} are not L{LatLon}.

1277

1278 @raise ValueError: Invalid B{C{radius}} or C{B{closed}=False} with

1279 C{B{points}} a composite.

1280

1281 @note: Distances are based on function L{vincentys_<pygeodesy.vincentys_>}.

1282

1283 @see: Functions L{perimeterOf<pygeodesy.perimeterOf>},

1284 L{sphericalNvector.perimeterOf} and L{ellipsoidalKarney.perimeterOf}.

1285 '''

1286 def _rads(ps, c, w): # angular edge lengths in radians

1287 Ps = _T00.PointsIter(ps, loop=1, wrap=w)

1288 a1, b1 = Ps[0].philam

1289 for p in Ps.iterate(closed=c):

1290 a2, b2 = p.philam

1291 db, b2 = unrollPI(b1, b2, wrap=w and not (c and Ps.looped))

1292 yield vincentys_(a2, a1, db)

1293 a1, b1 = a2, b2

1294

1295 if _MODS.booleans.isBoolean(points):

1296 if not closed:

1297 raise _ValueError(closed=closed, points=_composite_)

1298 r = points._sum2(LatLon, perimeterOf, closed=True, radius=radius, wrap=wrap)

1299 else:

1300 r = fsum(_rads(points, closed, wrap))

1301 return _radians2m(r, radius)

1302

1303

1304 def triangle7(latA, lonA, latB, lonB, latC, lonC, radius=R_M,

1305 excess=excessAbc_,

1306 wrap=False):

1307 '''Compute the angles, sides, and area of a (spherical) triangle.

1308

1309 @arg latA: First corner latitude (C{degrees}).

1310 @arg lonA: First corner longitude (C{degrees}).

1311 @arg latB: Second corner latitude (C{degrees}).

1312 @arg lonB: Second corner longitude (C{degrees}).

1313 @arg latC: Third corner latitude (C{degrees}).

1314 @arg lonC: Third corner longitude (C{degrees}).

1315 @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter},

1316 L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple})

1317 or C{None}.

1318 @kwarg excess: I{Spherical excess} callable (L{excessAbc_},

1319 L{excessGirard_} or L{excessLHuilier_}).

1320 @kwarg wrap: If C{True}, wrap and L{unroll180<pygeodesy.unroll180>}

1321 longitudes (C{bool}).

1322

1323 @return: A L{Triangle7Tuple}C{(A, a, B, b, C, c, area)} with

1324 spherical angles C{A}, C{B} and C{C}, angular sides

1325 C{a}, C{b} and C{c} all in C{degrees} and C{area}

1326 in I{square} C{meter} or same units as B{C{radius}}

1327 I{squared} or if C{B{radius}=0} or C{None}, a

1328 L{Triangle8Tuple}C{(A, a, B, b, C, c, D, E)} with

1329 I{spherical deficit} C{D} and I{spherical excess}

1330 C{E} as the C{unit area}, all in C{radians}.

1331 '''

1332 t = triangle8_(Phid(latA=latA), Lamd(lonA=lonA),

1333 Phid(latB=latB), Lamd(lonB=lonB),

1334 Phid(latC=latC), Lamd(lonC=lonC),

1335 excess=excess, wrap=wrap)

1336 return _t7Tuple(t, radius)

1337

1338

1339 def triangle8_(phiA, lamA, phiB, lamB, phiC, lamC, excess=excessAbc_,

1340 wrap=False):

1341 '''Compute the angles, sides, I{spherical deficit} and I{spherical

1342 excess} of a (spherical) triangle.

1343

1344 @arg phiA: First corner latitude (C{radians}).

1345 @arg lamA: First corner longitude (C{radians}).

1346 @arg phiB: Second corner latitude (C{radians}).

1347 @arg lamB: Second corner longitude (C{radians}).

1348 @arg phiC: Third corner latitude (C{radians}).

1349 @arg lamC: Third corner longitude (C{radians}).

1350 @kwarg excess: I{Spherical excess} callable (L{excessAbc_},

1351 L{excessGirard_} or L{excessLHuilier_}).

1352 @kwarg wrap: If C{True}, L{unrollPI<pygeodesy.unrollPI>} the

1353 longitudinal deltas (C{bool}).

1354

1355 @return: A L{Triangle8Tuple}C{(A, a, B, b, C, c, D, E)} with

1356 spherical angles C{A}, C{B} and C{C}, angular sides

1357 C{a}, C{b} and C{c}, I{spherical deficit} C{D} and

1358 I{spherical excess} C{E}, all in C{radians}.

1359 '''

1360 def _a_r(w, phiA, lamA, phiB, lamB, phiC, lamC):

1361 d, _ = unrollPI(lamB, lamC, wrap=w)

1362 a = vincentys_(phiC, phiB, d)

1363 return a, (phiB, lamB, phiC, lamC, phiA, lamA) # rotate A, B, C

1364

1365 def _A_r(a, sa, ca, sb, cb, sc, cc):

1366 s = sb * sc

1367 A = acos1((ca - cb * cc) / s) if isnon0(s) else a

1368 return A, (sb, cb, sc, cc, sa, ca) # rotate sincos2_'s

1369

1370 # notation: side C{a} is oposite to corner C{A}, etc.

1371 a, r = _a_r(wrap, phiA, lamA, phiB, lamB, phiC, lamC)

1372 b, r = _a_r(wrap, *r)

1373 c, _ = _a_r(wrap, *r)

1374

1375 A, r = _A_r(a, *sincos2_(a, b, c))

1376 B, r = _A_r(b, *r)

1377 C, _ = _A_r(c, *r)

1378

1379 D = fsumf_(PI2, -a, -b, -c) # deficit aka defect

1380 E = excessGirard_(A, B, C) if excess in (excessGirard_, True) else (

1381 excessLHuilier_(a, b, c) if excess in (excessLHuilier_, False) else

1382 excessAbc_(*max((A, b, c), (B, c, a), (C, a, b))))

1383

1384 return Triangle8Tuple(A, a, B, b, C, c, D, E)

1385

1386

1387 def _t7Tuple(t, radius):

1388 '''(INTERNAL) Convert a L{Triangle8Tuple} to L{Triangle7Tuple}.

1389 '''

1390 if radius: # not in (None, _0_0)

1391 r = radius if _isRadius(radius) else \

1392 _ellipsoidal_datum(radius).ellipsoid.Rmean

1393 A, B, C = map1(degrees, t.A, t.B, t.C)

1394 t = Triangle7Tuple(A, (r * t.a),

1395 B, (r * t.b),

1396 C, (r * t.c), t.E * r**2)

1397 return t

1398

1399

1400__all__ += _ALL_OTHER(Cartesian, LatLon, # classes

1401 areaOf, # functions

1402 intersecant2, intersection, intersections2, ispolar,

1403 isPoleEnclosedBy, # DEPRECATED, use ispolar

1404 meanOf,

1405 nearestOn2, nearestOn3,

1406 perimeterOf,

1407 sumOf, # XXX == vector3d.sumOf

1408 triangle7, triangle8_)

1409

1410# **) MIT License

1411#

1412# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved.

1413#

1414# Permission is hereby granted, free of charge, to any person obtaining a

1415# copy of this software and associated documentation files (the "Software"),

1416# to deal in the Software without restriction, including without limitation

1417# the rights to use, copy, modify, merge, publish, distribute, sublicense,

1418# and/or sell copies of the Software, and to permit persons to whom the

1419# Software is furnished to do so, subject to the following conditions:

1420#

1421# The above copyright notice and this permission notice shall be included

1422# in all copies or substantial portions of the Software.

1423#

1424# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS

1425# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

1426# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL

1427# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR

1428# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,

1429# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR

1430# OTHER DEALINGS IN THE SOFTWARE.