Hide keyboard shortcuts

Hot-keys on this page

r m x p   toggle line displays

j k   next/prev highlighted chunk

0   (zero) top of page

1   (one) first highlighted chunk

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

176

177

178

179

180

181

182

183

184

185

186

187

188

189

190

191

192

193

194

195

196

197

198

199

200

201

202

203

204

205

206

207

208

209

210

211

212

213

214

215

216

217

218

219

220

221

222

223

224

225

226

227

228

229

230

231

232

233

234

235

236

237

238

239

240

241

242

243

244

245

246

247

248

249

250

251

252

253

254

255

256

257

258

259

260

261

262

263

264

265

266

267

268

269

270

271

272

273

274

275

276

277

278

279

280

281

282

283

284

285

286

287

288

289

290

291

292

293

294

295

296

297

298

299

300

301

302

303

304

305

306

307

308

309

310

311

312

313

314

315

316

317

318

319

320

321

322

323

324

325

326

327

328

329

330

331

332

333

334

335

336

337

338

339

340

341

342

343

344

345

346

347

348

349

350

351

352

353

354

355

356

357

358

359

360

361

362

363

364

365

366

367

368

369

370

371

372

373

374

375

376

377

378

379

380

381

382

383

384

385

386

387

388

389

390

391

392

393

394

395

396

397

398

399

400

401

402

403

404

405

406

407

408

409

410

411

412

413

414

415

416

417

418

419

420

421

422

423

424

425

426

427

428

429

430

431

432

433

434

435

436

437

438

439

440

441

442

443

444

445

446

447

448

449

450

451

452

453

454

455

456

457

458

459

460

461

462

463

464

465

466

467

468

469

470

471

472

473

474

475

476

477

478

479

480

481

482

483

484

485

486

487

488

489

490

491

492

493

494

495

496

497

498

499

500

501

502

503

504

505

506

507

508

509

510

511

512

513

514

515

516

517

518

519

520

521

522

523

524

525

526

527

528

529

530

531

532

533

534

535

536

537

538

539

540

541

542

543

544

545

546

547

548

549

550

551

552

553

554

555

556

557

558

559

560

561

562

563

564

565

566

567

568

569

570

571

572

573

574

575

576

577

578

579

580

581

582

583

584

585

586

587

588

589

590

591

592

593

594

595

596

597

598

599

600

601

602

603

604

605

606

607

608

609

610

611

612

613

614

615

616

617

618

619

620

621

622

623

624

625

626

627

628

629

630

631

632

633

634

635

636

637

638

639

640

641

642

643

644

645

646

647

648

649

650

651

652

653

654

655

656

657

658

659

660

661

662

663

664

665

666

667

668

669

670

671

672

673

674

675

676

677

678

679

680

681

682

683

684

685

686

687

688

689

690

691

692

693

694

695

696

697

698

699

700

701

702

703

704

705

706

707

708

709

710

711

712

713

714

715

716

717

718

719

720

721

722

723

724

725

726

727

728

729

730

731

732

733

734

735

736

737

738

739

740

741

742

743

744

745

746

747

748

749

750

751

752

753

754

755

756

757

758

759

760

761

762

763

764

765

766

767

768

769

770

771

772

773

774

775

776

777

778

779

780

781

782

783

784

785

786

787

788

789

790

791

792

793

794

795

796

797

798

799

800

801

802

803

804

805

806

807

808

809

810

811

812

813

814

815

816

817

818

819

820

821

822

823

824

825

826

827

828

829

830

831

832

833

834

835

836

837

838

839

840

841

842

843

844

845

846

847

848

849

850

851

852

853

854

855

856

857

858

859

860

861

862

863

864

865

866

867

868

869

870

871

872

873

874

875

876

877

878

879

880

881

882

883

884

885

886

887

888

889

890

891

892

893

894

895

896

897

898

899

900

901

902

903

904

905

906

907

908

909

910

911

912

913

914

915

916

917

918

919

920

921

922

923

924

925

926

927

928

929

930

931

932

933

934

935

936

937

938

939

940

941

942

943

944

945

946

947

948

949

950

951

952

953

954

955

956

957

958

959

960

961

962

963

964

965

966

967

968

969

970

971

972

973

974

975

976

977

978

979

980

981

982

983

984

985

986

987

988

989

990

991

992

993

994

995

996

997

998

999

1000

1001

1002

1003

1004

1005

1006

1007

1008

1009

1010

1011

1012

1013

1014

1015

1016

1017

1018

1019

1020

1021

1022

1023

1024

1025

1026

1027

1028

1029

1030

1031

1032

1033

1034

1035

1036

1037

1038

1039

1040

1041

1042

1043

1044

1045

1046

1047

1048

1049

1050

1051

1052

1053

1054

1055

1056

1057

1058

1059

1060

1061

1062

1063

1064

1065

1066

1067

1068

1069

1070

1071

1072

1073

1074

1075

1076

1077

1078

1079

1080

1081

1082

1083

1084

1085

1086

1087

1088

1089

1090

1091

1092

1093

1094

1095

1096

1097

1098

1099

1100

1101

1102

1103

1104

1105

1106

1107

1108

1109

1110

1111

1112

1113

1114

1115

1116

1117

1118

1119

1120

1121

1122

1123

1124

1125

1126

1127

1128

1129

1130

1131

1132

1133

1134

1135

1136

1137

1138

1139

1140

1141

1142

1143

1144

1145

1146

1147

1148

1149

1150

1151

1152

1153

1154

1155

1156

1157

1158

1159

1160

1161

1162

1163

1164

1165

1166

1167

1168

1169

1170

1171

1172

1173

1174

1175

1176

1177

1178

1179

1180

1181

1182

1183

1184

1185

1186

1187

1188

1189

1190

1191

1192

1193

1194

1195

1196

1197

1198

1199

1200

1201

1202

1203

1204

1205

1206

1207

1208

1209

1210

1211

1212

1213

1214

1215

1216

1217

1218

1219

1220

1221

1222

1223

1224

1225

1226

1227

1228

1229

1230

1231

1232

1233

1234

1235

1236

1237

1238

1239

1240

1241

1242

1243

1244

1245

1246

1247

1248

1249

1250

1251

1252

1253

1254

1255

1256

1257

1258

1259

1260

1261

1262

1263

1264

1265

1266

1267

1268

1269

1270

1271

1272

1273

1274

1275

1276

1277

1278

1279

1280

1281

1282

1283

1284

1285

1286

1287

1288

1289

1290

1291

1292

1293

1294

1295

1296

1297

1298

1299

1300

1301

1302

1303

1304

1305

1306

1307

1308

1309

1310

1311

1312

1313

1314

1315

1316

1317

1318

1319

1320

1321

1322

1323

1324

1325

1326

1327

1328

1329

1330

1331

1332

1333

1334

1335

1336

1337

1338

1339

1340

1341

1342

1343

1344

1345

1346

1347

1348

1349

1350

1351

1352

1353

1354

1355

1356

1357

1358

1359

1360

1361

1362

1363

1364

1365

1366

1367

1368

1369

1370

1371

1372

1373

1374

1375

1376

1377

1378

1379

1380

1381

1382

1383

1384

1385

1386

1387

1388

1389

1390

1391

1392

1393

1394

1395

1396

1397

1398

1399

1400

1401

1402

1403

1404

1405

1406

1407

1408

1409

1410

1411

1412

1413

1414

1415

1416

1417

1418

1419

1420

1421

1422

1423

1424

1425

1426

1427

1428

1429

1430

1431

1432

1433

1434

1435

 

# -*- coding: utf-8 -*- 

 

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

 

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

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

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

L{perimeterOf}, I{all spherical}. 

 

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

spherical trigonometry, transcoded from JavaScript originals by 

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

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

''' 

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

from __future__ import division 

 

from pygeodesy.basics import copysign0, isnear0, isnon0, isscalar, map1, signOf 

from pygeodesy.datums import _ellipsoidal_datum, _mean_radius 

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

_ValueError, IntersectionError, _xError, \ 

_xkwds, _xkwds_get 

from pygeodesy.fmath import favg, fdot, fmean, Fsum, fsum, fsum_, hypot 

from pygeodesy.formy import antipode_, bearing_, _bearingTo2, excessAbc, \ 

excessGirard, excessLHuilier, _radical2, vincentys_ 

from pygeodesy.interns import EPS, EPS1, PI, PI2, PI_2, PI_4, R_M, \ 

_EPS4, _coincident_, _colinear_, _convex_, \ 

_end_, _invalid_, _LatLon_, _near_concentric_, \ 

_not_, _points_, _SPACE_, _too_, _1_, _2_, \ 

_0_0, _0_5, _1_0, _2_0, _90_0 

from pygeodesy.lazily import _ALL_LAZY, _ALL_OTHER 

from pygeodesy.named import notImplemented, _xnamed 

from pygeodesy.namedTuples import LatLon2Tuple, LatLon3Tuple, \ 

NearestOn3Tuple, Triangle7Tuple, \ 

Triangle8Tuple 

from pygeodesy.nvectorBase import NvectorBase as _Nvector 

from pygeodesy.points import ispolar, nearestOn5 as _nearestOn5 

from pygeodesy.props import deprecated_function, deprecated_method 

from pygeodesy.sphericalBase import _angular, CartesianSphericalBase, \ 

LatLonSphericalBase, _rads3, _trilaterate5 

from pygeodesy.streprs import Fmt as _Fmt # XXX shadowed 

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

Radius_, Scalar 

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

m2radians, radiansPI2, sincos2, tan_2, \ 

unrollPI, wrap180, wrapPI 

from pygeodesy.vector3d import sumOf, Vector3d 

 

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

 

__all__ = _ALL_LAZY.sphericalTrigonometry 

__version__ = '21.06.30' 

 

_infinite_ = 'infinite' 

_null_ = 'null' 

_parallel_ = 'parallel' 

_path_ = 'path' 

 

_PI_EPS4 = PI - _EPS4 

if _PI_EPS4 >= PI: 

raise _AssertionError(EPS4=_EPS4, PI=PI, PI_EPS4=_PI_EPS4) 

 

 

def _destination2(a, b, r, t): 

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

 

@arg a: Latitude (C{radians}). 

@arg b: Longitude (C{radians}). 

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

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

 

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

''' 

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

sa, ca, sr, cr, st, ct = sincos2(a, r, t) 

 

a = asin1(ct * sr * ca + cr * sa) 

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

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

return a, b + d 

 

 

def _r2m(r, radius): 

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

''' 

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

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

return r 

 

 

class Cartesian(CartesianSphericalBase): 

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

spherical, geodetic L{LatLon}. 

''' 

 

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

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

geodetic point. 

 

@kwarg LatLon_datum_kwds: Optional L{LatLon}, B{C{datum}} and 

other keyword arguments, ignored if 

C{B{LatLon}=None}. Use 

C{B{LatLon}=...} to override this 

L{LatLon} class or specify 

C{B{LatLon}=None}. 

 

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

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

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

 

@raise TypeError: Invalid B{C{LatLon_datum_kwds}}. 

''' 

kwds = _xkwds(LatLon_datum_kwds, LatLon=LatLon, datum=self.datum) 

return CartesianSphericalBase.toLatLon(self, **kwds) 

 

 

class LatLon(LatLonSphericalBase): 

'''New point on spherical model earth model. 

 

@example: 

 

>>> p = LatLon(52.205, 0.119) # height=0 

''' 

 

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

'''Compute the (angular) distance (signed) from the start to 

the closest point on the great circle path defined by a 

start and an end point. 

 

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

great circle path, the along-track distance is the distance 

from the start point to the point where the perpendicular 

crosses the path. 

 

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

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

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

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

 

@return: Distance along the great circle path (C{meter}, 

same units as B{C{radius}}) or C{radians} if 

C{B{radius}=None}, positive if after the B{C{start}} 

toward the B{C{end}} point of the path or negative 

if before the B{C{start}} point. 

 

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

 

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

 

@example: 

 

>>> p = LatLon(53.2611, -0.7972) 

 

>>> s = LatLon(53.3206, -1.7297) 

>>> e = LatLon(53.1887, 0.1334) 

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

''' 

r, x, b = self._trackDistanceTo3(start, end, radius, wrap) 

cx = cos(x) 

return _0_0 if isnear0(cx) else \ 

_r2m(copysign0(acos1(cos(r) / cx), cos(b)), radius) 

 

@deprecated_method 

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

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

''' 

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

 

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

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

by this and an other point crosses the given latitude. 

 

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

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

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

 

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

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

''' 

self.others(other) 

 

a1, b1 = self.philam 

a2, b2 = other.philam 

 

a = radians(lat) 

db, b2 = unrollPI(b1, b2, wrap=wrap) 

 

sa, ca, sa1, ca1, \ 

sa2, ca2, sdb, cdb = sincos2(a, a1, a2, db) 

 

x = sa1 * ca2 * ca * sdb 

y = sa1 * ca2 * ca * cdb - ca1 * sa2 * ca 

z = ca1 * ca2 * sa * sdb 

 

h = hypot(x, y) 

if h < EPS or abs(z) > h: 

return None # great circle doesn't reach latitude 

 

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

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

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

 

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

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

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

 

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

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

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

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

 

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

or positive if to the right of the path) (C{meter}, 

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

B{C{radius}} is C{None}). 

 

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

 

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

 

@example: 

 

>>> p = LatLon(53.2611, -0.7972) 

 

>>> s = LatLon(53.3206, -1.7297) 

>>> e = LatLon(53.1887, 0.1334) 

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

''' 

_, x, _ = self._trackDistanceTo3(start, end, radius, wrap) 

return _r2m(x, radius) 

 

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

'''Locate the destination from this point after having 

travelled the given distance on the given initial bearing. 

 

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

B{C{radius}}). 

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

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

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

units a B{C{radius}}). 

 

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

 

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

B{C{radius}} or B{C{height}}. 

 

@example: 

 

>>> p1 = LatLon(51.4778, -0.0015) 

>>> p2 = p1.destination(7794, 300.7) 

>>> p2.toStr() # '51.5135°N, 000.0983°W' 

 

@JSname: I{destinationPoint}. 

''' 

a, b = self.philam 

r, t = _angular(distance, radius), Bearing_(bearing) 

 

a, b = _destination2(a, b, r, t) 

h = self.height if height is None else Height(height) 

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

 

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

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

 

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

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

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

 

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

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

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

 

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

 

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

 

@example: 

 

>>> p1 = LatLon(52.205, 0.119) 

>>> p2 = LatLon(48.857, 2.351); 

>>> d = p1.distanceTo(p2) # 404300 

''' 

self.others(other) 

 

a1, b1 = self.philam 

a2, b2 = other.philam 

 

db, _ = unrollPI(b1, b2, wrap=wrap) 

return _r2m(vincentys_(a2, a1, db), radius) 

 

# @Property_RO 

# def Ecef(self): 

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

# ''' 

# from pygeodesy.ecef import EcefKarney 

# return EcefKarney 

 

def greatCircle(self, bearing): 

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

on the given initial bearing from this point. 

 

Direction of vector is such that initial bearing vector 

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

 

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

 

@return: Vector representing great circle (L{Vector3d}). 

 

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

 

@example: 

 

>>> p = LatLon(53.3206, -1.7297) 

>>> g = p.greatCircle(96.0) 

>>> g.toStr() # (-0.794, 0.129, 0.594) 

''' 

a, b = self.philam 

 

t = Bearing_(bearing) 

sa, ca, sb, cb, st, ct = sincos2(a, b, t) 

 

return Vector3d(sb * ct - cb * sa * st, 

-cb * ct - sb * sa * st, 

ca * st) # XXX .unit()? 

 

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

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

to an other point. 

 

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

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

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

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

C{sphericalNvector.LatLon.initialBearingTo}. 

 

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

 

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

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

L{crosserrors} is C{True}. 

 

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

 

@example: 

 

>>> p1 = LatLon(52.205, 0.119) 

>>> p2 = LatLon(48.857, 2.351) 

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

 

@JSname: I{bearingTo}. 

''' 

self.others(other) 

 

a1, b1 = self.philam 

a2, b2 = other.philam 

 

# XXX behavior like sphericalNvector.LatLon.initialBearingTo 

if raiser and crosserrors() and max(abs(a2 - a1), abs(b2 - b1)) < EPS: 

raise CrossError(_points_, self, txt=_coincident_) 

 

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

 

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

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

other point. 

 

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

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

0.0 for this and 1.0 for the other point). 

@kwarg height: Optional height, overriding the fractional 

height (C{meter}). 

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

 

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

 

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

 

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

 

@example: 

 

>>> p1 = LatLon(52.205, 0.119) 

>>> p2 = LatLon(48.857, 2.351) 

>>> p = p1.intermediateTo(p2, 0.25) # 51.3721°N, 000.7073°E 

 

@JSname: I{intermediatePointTo}. 

''' 

self.others(other) 

 

f = Scalar(fraction=fraction) # not high=_1_0 

 

a1, b1 = self.philam 

a2, b2 = other.philam 

 

db, b2 = unrollPI(b1, b2, wrap=wrap) 

r = vincentys_(a2, a1, db) 

sr = sin(r) 

if isnon0(sr): 

sa1, ca1, sa2, ca2, \ 

sb1, cb1, sb2, cb2 = sincos2(a1, a2, b1, b2) 

 

A = sin((_1_0 - f) * r) / sr 

B = sin( f * r) / sr 

 

x = A * ca1 * cb1 + B * ca2 * cb2 

y = A * ca1 * sb1 + B * ca2 * sb2 

z = A * sa1 + B * sa2 

 

a = atan2(z, hypot(x, y)) 

b = atan2(y, x) 

 

else: # points too close 

a = favg(a1, a2, f=f) 

b = favg(b1, b2, f=f) 

 

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

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

 

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

'''Compute the intersection point of two paths, each defined 

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

 

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

initial bearing at this point (compass 

C{degrees360}). 

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

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

the initial bearing at the other start point 

(compass C{degrees360}). 

@kwarg height: Optional height for intersection point, 

overriding the mean height (C{meter}). 

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

 

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

intersection point might be the L{antipode} to 

the returned result. 

 

@raise IntersectionError: Ambiguous or infinite intersection 

or colinear, parallel or otherwise 

non-intersecting paths. 

 

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

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

 

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

 

@example: 

 

>>> p = LatLon(51.8853, 0.2545) 

>>> s = LatLon(49.0034, 2.5735) 

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

''' 

try: 

s2 = self.others(other) 

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

LatLon=self.classof) 

except (TypeError, ValueError) as x: 

raise _xError(x, start1=self, end1=end1, other=other, end2=end2) 

 

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

height=None, wrap=True): 

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

by a center point and radius. 

 

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

see B{C{radius}}). 

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

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

see B{C{radius}}). 

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

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

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

B{C{radius}}). 

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

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

at the I{radical line} between both centers. 

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

 

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

instance. For abutting circles, both intersection 

points are the same instance. 

 

@raise IntersectionError: Concentric, antipodal, invalid or 

non-intersecting circles. 

 

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

 

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

B{C{eps}} or B{C{height}}. 

''' 

try: 

c2 = self.others(other) 

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

height=height, wrap=wrap, 

LatLon=self.classof) 

except (TypeError, ValueError) as x: 

raise _xError(x, center=self, rad1=rad1, other=other, rad2=rad2) 

 

def isenclosedBy(self, points): 

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

 

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

 

@return: C{True} if the polygon encloses this point, 

C{False} otherwise. 

 

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

 

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

 

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

 

@example: 

 

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

>>> p = LatLon(45,1, 1.1) 

>>> inside = p.isEnclosedBy(b) # True 

''' 

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

n0 = self._N_vector 

 

v2 = Ps[0]._N_vector 

v1 = Ps[1]._N_vector 

gc1 = v2.cross(v1) 

# check whether this point on same side of all 

# polygon edges (to the left or right depending 

# on anti-/clockwise polygon direction) 

t0 = gc1.angleTo(n0) > PI_2 

s0 = None 

# get great-circle vector for each edge 

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

v2 = p._N_vector 

gc = v1.cross(v2) 

v1 = v2 

 

t = gc.angleTo(n0) > PI_2 

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

return False # outside 

 

# check for convex polygon: angle between 

# gc vectors, signed by direction of n0 

# (otherwise the test above is not reliable) 

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

if s != s0: 

if s0 is None: 

s0 = s 

else: 

t = _Fmt.SQUARE(points=i) 

raise _ValueError(t, p, txt=_not_(_convex_)) 

gc1 = gc 

 

return True # inside 

 

@deprecated_method 

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

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

return self.isenclosedBy(points) 

 

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

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

 

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

@kwarg height: Optional height for midpoint, overriding 

the mean height (C{meter}). 

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

 

@return: Midpoint (L{LatLon}). 

 

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

 

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

 

@example: 

 

>>> p1 = LatLon(52.205, 0.119) 

>>> p2 = LatLon(48.857, 2.351) 

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

''' 

self.others(other) 

 

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

a1, b1 = self.philam 

a2, b2 = other.philam 

 

db, b2 = unrollPI(b1, b2, wrap=wrap) 

 

sa1, ca1, sa2, ca2, sdb, cdb = sincos2(a1, a2, db) 

 

x = ca2 * cdb + ca1 

y = ca2 * sdb 

 

a = atan2(sa1 + sa2, hypot(x, y)) 

b = atan2(y, x) + b1 

 

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

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

 

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

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

 

Distances are approximated by function L{equirectangular_}, 

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

 

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

@arg point2: End point (L{LatLon}). 

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

@kwarg options: Optional keyword arguments for function 

L{equirectangular_}. 

 

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

 

@raise LimitError: Lat- and/or longitudinal delta exceeds 

B{C{limit}}, see function L{equirectangular_}. 

 

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

is not (yet) supported. 

 

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

 

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

 

@see: Functions L{equirectangular_} and L{nearestOn5} and 

method L{sphericalTrigonometry.LatLon.nearestOn3}. 

''' 

try: # remove kwarg B{C{within}} if present 

within = options.pop('within') 

if not within: 

notImplemented(self, within=within) 

 

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

# wrap = options.get('wrap', False) 

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

# if abs(a) < EPS or (within and a < EPS): 

# return point1 

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

# if isnear0(d): 

# return point1 # or point2 

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

# return point2 

# f = a / d 

# if within: 

# if f > EPS1: 

# return point2 

# elif f < EPS: 

# return point1 

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

 

except KeyError: 

pass 

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

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

**options)[0] 

 

@deprecated_method 

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

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

 

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

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

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

units of B{C{radius}}. 

''' 

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

return r.closest, r.distance 

 

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

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

 

Distances are approximated by function L{equirectangular_}, 

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

 

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

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

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

@kwarg options: Optional keyword arguments for function 

L{equirectangular_}. 

 

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

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

C{distance} between this and the C{closest} point in 

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

from this to the C{closest} point is in compass 

C{degrees360}, like function L{compassAngle}. 

 

@raise LimitError: Lat- and/or longitudinal delta exceeds 

B{C{limit}}, see function L{equirectangular_}. 

 

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

 

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

 

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

 

@see: Functions L{compassAngle}, L{equirectangular_} and 

L{nearestOn5}. 

''' 

if _LatLon_ in options: 

raise _ValueError(**options) 

 

lat, lon, d, c, h = _nearestOn5(self, points, closed=closed, **options) 

return NearestOn3Tuple(self.classof(lat, lon, height=h), 

degrees2m(d, radius=radius), c) 

 

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

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

coordinates. 

 

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

and other keyword arguments, ignored 

if C{B{Cartesian}=None}. Use 

C{B{Cartesian}=...} to override 

this L{Cartesian} class or specify 

C{B{Cartesian}=None}. 

 

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

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

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

 

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

''' 

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

return LatLonSphericalBase.toCartesian(self, **kwds) 

 

def _trackDistanceTo3(self, start, end, radius, wrap): 

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

''' 

self.others(start=start) 

self.others(end=end) 

 

r = Radius_(radius) 

r = start.distanceTo(self, r, wrap=wrap) / r 

 

b = radians(start.initialBearingTo(self, wrap=wrap)) 

e = radians(start.initialBearingTo(end, wrap=wrap)) 

x = asin(sin(r) * sin(b - e)) 

return r, x, (e - b) 

 

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

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

 

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

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

@kwarg radius: Mean earth radius, ellipsoid or datum 

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

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

@kwarg wrap: Wrap/unroll angular distances (C{bool}). 

 

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

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

a, B, b, C, c, D, E)}. 

 

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

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

''' 

self.others(otherB=otherB) 

self.others(otherC=otherC) 

 

r = self.philam + otherB.philam + otherC.philam 

t = triangle8_(*r, wrap=wrap) 

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

 

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

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

'''Trilaterate three points by area overlap or perimeter intersection 

of three corresponding circles. 

 

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

as B{C{radius}}). 

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

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

B{C{radius}}). 

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

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

B{C{radius}}). 

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

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

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

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

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

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

@kwarg wrap: Wrap/unroll angular distances (C{bool}). 

 

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

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

the corresponding trilaterated points C{minPoint} and 

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

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

 

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

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

 

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

respectively largest I{radial} overlap found. 

 

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

nearest respectively farthest intersection margin. 

 

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

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

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

largest B{C{distance#}}. 

 

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

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

no intersection or all (near-)concentric for 

C{B{area}=False}. 

 

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

 

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

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

''' 

return _trilaterate5(self, distance1, 

self.others(point2=point2), distance2, 

self.others(point3=point3), distance3, 

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

 

 

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

 

 

def areaOf(points, radius=R_M, wrap=True): 

'''Calculate the area of a (spherical) polygon (with the points 

joined by great circle arcs). 

 

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

@kwarg radius: Mean earth radius, ellipsoid or datum 

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

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

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

 

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

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

 

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

 

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

 

@raise ValueError: Invalid B{C{radius}} or semi-circular 

polygon edge. 

 

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

polygon'<https://OSGeo-org.1560.x6.nabble.com/ 

Area-of-a-spherical-polygon-td3841625.html>}. 

 

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

L{ellipsoidalKarney.areaOf} and L{excessKarney}. 

 

@example: 

 

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

>>> areaOf(b) # 8666058750.718977 

 

>>> c = LatLon(0, 0), LatLon(1, 0), LatLon(0, 1) 

>>> areaOf(c) # 6.18e9 

''' 

Ps = _T00.PointsIter(points, loop=1) 

p1 = p2 = Ps[0] 

 

A = Fsum() # mean phi 

 

E = Fsum() # see L{excessKarney_} 

a1, b1 = p1.philam 

ta1 = tan_2(a1) 

 

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

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

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

D = Fsum() 

z1 = _0_0 # z1, _ = _bearingTo2(p2, p1, wrap=wrap) 

 

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

a2, b2 = p2.philam 

db, b2 = unrollPI(b1, b2, wrap=wrap if i else False) 

ta2 = tan_2(a2) 

E += atan2(tan_2(db, points=i) * (ta1 + ta2), _1_0 + ta1 * ta2) 

ta1, b1 = ta2, b2 

 

if not p2.isequalTo(p1, EPS): 

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

D.fadd_(wrap180(z - z1), # (z - z1 + 540) % 360 - 180 

wrap180(z2 - z)) # (z2 - z + 540) % 360 - 180 

p1, z1 = p2, z2 

A += a2 

 

r = abs(E.fsum()) * _2_0 

if abs(D.fsum()) < _90_0: # ispolar(points) 

r = abs(r - PI2) 

if radius: 

a = degrees(A.fsum() / len(A)) # mean lat 

r *= _mean_radius(radius, a)**2 

return r 

 

 

def _ib(a1, b1, end, a, b, wrap): 

# difference between the bearing to (a, b) and the given 

# bearing is negative if both are in opposite directions 

r = bearing_(a1, b1, a, b, wrap=wrap) 

return abs(wrapPI(r - radians(end))) > PI_2 

 

 

def _i3d2(start, end, wrap, _i_, hs): 

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

a1, b1 = start.philam 

 

if isscalar(end): # bearing, make a point 

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

else: # must be a point 

start.others(end, name=_end_ + _i_) 

hs.append(end.height) 

a2, b2 = end.philam 

 

db, b2 = unrollPI(b1, b2, wrap=wrap) 

if max(abs(db), abs(a2 - a1)) < EPS: 

raise _ValueError(_SPACE_(_path_ + _i_, _null_)) 

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

b21, b12 = db * _0_5, -(b1 + b2) * _0_5 

 

sb21, cb21, sb12, cb12, \ 

sa21, _, sa12, _ = sincos2(b21, b12, a1 - a2, a1 + a2) 

 

x = _Nvector(sa21 * sb12 * cb21 - sa12 * cb12 * sb21, 

sa21 * cb12 * cb21 + sa12 * sb12 * sb21, 

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

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

 

 

def _idot(ds, a1, b1, a, b, wrap): 

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

db, _ = unrollPI(b1, b, wrap=wrap) 

return fdot(ds, db, a - a1) 

 

 

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

LatLon=LatLon, **LatLon_kwds): 

'''Compute the intersection point of two paths, each defined 

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

 

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

@arg end1: End point ofthe first path (L{LatLon}) or 

the initial bearing at the first start point 

(compass C{degrees360}). 

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

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

the initial bearing at the second start point 

(compass C{degrees360}). 

@kwarg height: Optional height for the intersection point, 

overriding the mean height (C{meter}). 

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

@kwarg LatLon: Optional class to return the intersection 

point (L{LatLon}) or C{None}. 

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

arguments, ignored if C{B{LatLon}=None}. 

 

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

if C{B{LatLon}=None} a L{LatLon3Tuple}C{(lat, lon, 

height)}. An alternate intersection point might 

be the L{antipode} to the returned result. 

 

@raise IntersectionError: Ambiguous or infinite intersection 

or colinear, parallel or otherwise 

non-intersecting paths. 

 

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

 

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

 

@example: 

 

>>> p = LatLon(51.8853, 0.2545) 

>>> s = LatLon(49.0034, 2.5735) 

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

''' 

_T00.others(start1=start1) 

_T00.others(start2=start2) 

 

try: 

return _intersect(start1, end1, start2, end2, height=height, wrap=wrap, 

LatLon=LatLon, **LatLon_kwds) 

except (TypeError, ValueError) as x: 

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

 

 

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

LatLon=None, **LatLon_kwds): 

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

# above, separated to allow callers to embellish any exceptions 

 

hs = [start1.height, start2.height] 

 

a1, b1 = start1.philam 

a2, b2 = start2.philam 

 

db, b2 = unrollPI(b1, b2, wrap=wrap) 

r12 = vincentys_(a2, a1, db) 

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

a, b = favg(a1, a2), favg(b1, b2) 

 

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

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

sa1, ca1, sa2, ca2, sr12, cr12 = sincos2(a1, a2, r12) 

 

x1, x2 = (sr12 * ca1), (sr12 * ca2) 

if isnear0(x1) or isnear0(x2): 

raise IntersectionError(_parallel_) 

# handle domain error for equivalent longitudes, 

# see also functions asin_safe and acos_safe at 

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

t1, t2 = map1(acos1, (sa2 - sa1 * cr12) / x1, 

(sa1 - sa2 * cr12) / x2) 

if sin(db) > 0: 

t12, t21 = t1, PI2 - t2 

else: 

t12, t21 = PI2 - t1, t2 

 

t13, t23 = map1(radiansPI2, end1, end2) 

x1, x2 = map1(wrapPI, t13 - t12, # angle 2-1-3 

t21 - t23) # angle 1-2-3 

sx1, cx1, sx2, cx2 = sincos2(x1, x2) 

if sx1 == 0 and sx2 == 0: # max(abs(sx1), abs(sx2)) < EPS 

raise IntersectionError(_infinite_) 

sx3 = sx1 * sx2 

# XXX if sx3 < 0: 

# XXX raise ValueError(_ambiguous_) 

x3 = acos1(cr12 * sx3 - cx2 * cx1) 

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

 

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

# choose antipode for opposing bearings 

if _ib(a1, b1, end1, a, b, wrap) or \ 

_ib(a2, b2, end2, a, b, wrap): 

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

 

else: # end point(s) or bearing(s) 

x1, d1 = _i3d2(start1, end1, wrap, _1_, hs) 

x2, d2 = _i3d2(start2, end2, wrap, _2_, hs) 

x = x1.cross(x2) 

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

raise IntersectionError(_colinear_) 

a, b = x.philam 

# choose intersection similar to sphericalNvector 

d1 = _idot(d1, a1, b1, a, b, wrap) 

if d1: 

d2 = _idot(d2, a2, b2, a, b, wrap) 

if (d2 < 0 and d1 > 0) or (d2 > 0 and d1 < 0): 

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

 

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

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

intersection, LatLon, LatLon_kwds) 

 

 

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

height=None, wrap=True, 

LatLon=LatLon, **LatLon_kwds): 

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

by a center point and a radius. 

 

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

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

see B{C{radius}}). 

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

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

see B{C{radius}}). 

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

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

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

B{C{radius}}). 

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

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

at the I{radical line} between both centers. 

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

@kwarg LatLon: Optional class to return the intersection 

points (L{LatLon}) or C{None}. 

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

arguments, ignored if C{B{LatLon}=None}. 

 

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

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

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

points are the same instance. 

 

@raise IntersectionError: Concentric, antipodal, invalid or 

non-intersecting circles. 

 

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

 

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

B{C{eps}} or B{C{height}}. 

 

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

 

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

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

''' 

c1 = _T00.others(center1=center1) 

c2 = _T00.others(center2=center2) 

 

try: 

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

height=height, wrap=wrap, 

LatLon=LatLon, **LatLon_kwds) 

except (TypeError, ValueError) as x: 

raise _xError(x, center1=center1, rad1=rad1, center2=center2, rad2=rad2) 

 

 

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

height=None, too_d=None, wrap=True, 

LatLon=LatLon, **LatLon_kwds): 

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

# above, separated to allow callers to embellish any exceptions 

 

def _dest3(bearing, h): 

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

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

intersections2, LatLon, LatLon_kwds) 

 

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

if f: # swapped 

c1, c2 = c2, c1 # PYCHOK swap 

 

a1, b1 = c1.philam 

a2, b2 = c2.philam 

 

db, b2 = unrollPI(b1, b2, wrap=wrap) 

d = vincentys_(a2, a1, db) # radians 

if d < max(r1 - r2, EPS): 

raise IntersectionError(_near_concentric_) 

 

r = eps if radius is None else (m2radians( 

eps, radius=radius) if eps else _0_0) 

if r < _0_0: 

raise _ValueError(eps=r) 

 

x = fsum_(r1, r2, -d) # overlap 

if x > max(r, EPS): 

sd, cd, sr1, cr1, _, cr2 = sincos2(d, r1, r2) 

x = sd * sr1 

if isnear0(x): 

raise _ValueError(_invalid_) 

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

 

elif x < r: 

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

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

 

if height is None: # "radical height" 

f = _radical2(d, r1, r2).ratio 

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

else: 

h = Height(height) 

 

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

if x < _EPS4: # externally ... 

r = _dest3(b, h) 

elif x > _PI_EPS4: # internally ... 

r = _dest3(b + PI, h) 

else: 

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

return r, r # ... abutting circles 

 

 

@deprecated_function 

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

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

''' 

return ispolar(points, wrap=wrap) 

 

 

def _LL3Tuple(lat, lon, height, func, LatLon, LatLon_kwds): 

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

''' 

n = func.__name__ 

if LatLon is None: 

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

else: 

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

r = LatLon(lat, lon, **kwds) 

return r 

 

 

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

'''Compute the geographic mean of several points. 

 

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

@kwarg height: Optional height at mean point, overriding 

the mean height (C{meter}). 

@kwarg LatLon: Optional class to return the mean point 

(L{LatLon}) or C{None}. 

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

keyword arguments, ignored if 

C{B{LatLon}=None}. 

 

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

or a L{LatLon3Tuple}C{(lat, lon, height)} if 

B{C{LatLon}} is C{None}. 

 

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

 

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

''' 

# geographic mean 

m = sumOf(p._N_vector for p in _T00.PointsIter(points).iterate(closed=False)) 

lat, lon = m._N_vector.latlon 

 

if height is None: 

h = fmean(p.height for p in _T00.PointsIter(points).iterate(closed=False)) 

else: 

h = Height(height) 

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

 

 

@deprecated_function 

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

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

 

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

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

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

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

lon)} if B{C{LatLon}} is C{None} ... 

''' 

ll, d, _ = nearestOn3(point, points, **closed_radius_LatLon_options) 

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

ll = LatLon2Tuple(ll.lat, ll.lon) 

return ll, d 

 

 

def nearestOn3(point, points, closed=False, radius=R_M, 

LatLon=LatLon, **options): 

'''Locate the point on a polygon closest to an other, reference point. 

 

Distances are approximated by function L{equirectangular_}, 

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

 

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

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

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

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

@kwarg LatLon: Optional class to return the closest point 

(L{LatLon}) or C{None}. 

@kwarg options: Optional keyword arguments for function 

L{equirectangular_}. 

 

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

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

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

is the L{equirectangular_} distance between the C{closest} 

and the given B{C{point}} in C{meter}, same units as 

B{C{radius}}. The C{angle} from the given B{C{point}} 

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

L{compassAngle}. The C{height} is the (interpolated) height 

at the C{closest} point. 

 

@raise LimitError: Lat- and/or longitudinal delta exceeds the 

B{C{limit}}, see function L{equirectangular_}. 

 

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

 

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

 

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

 

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

''' 

lat, lon, d, c, h = _nearestOn5(point, points, closed=closed, 

LatLon=None, **options) 

r = LatLon3Tuple(lat, lon, h) if LatLon is None else \ 

LatLon(lat, lon, height=h) 

r = NearestOn3Tuple(r, degrees2m(d, radius=radius), c) 

return _xnamed(r, nearestOn3.__name__) 

 

 

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

'''Compute the perimeter of a (spherical) polygon (with great circle 

arcs joining the points). 

 

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

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

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

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

 

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

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

 

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

 

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

 

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

 

@note: This perimeter is based on the L{haversine} formula. 

 

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

and L{ellipsoidalKarney.perimeterOf}. 

''' 

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

a1, b1 = Ps[0].philam 

for i, p in Ps.enumerate(closed=closed): 

a2, b2 = p.philam 

db, b2 = unrollPI(b1, b2, wrap=wrap if i else False) 

yield vincentys_(a2, a1, db) 

a1, b1 = a2, b2 

 

r = fsum(_rads(_T00.PointsIter(points, loop=1), closed, wrap)) 

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

 

 

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

excess=excessAbc, 

wrap=False): 

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

 

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

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

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

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

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

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

@kwarg radius: Mean earth radius, ellipsoid or datum 

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

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

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

L{excessGirard} or L{excessLHuilier}). 

@kwarg wrap: Wrap and L{unroll180} longitudes (C{bool}). 

 

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

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

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

in I{square} C{meter} or units of B{C{radius}} 

I{squared} or if B{C{radius}} is C{None}, a 

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

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

C{unit area} in C{radians}. 

''' 

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

Phi_(latB=latB), Lam_(lonB=lonB), 

Phi_(latC=latC), Lam_(lonC=lonC), 

excess=excess, wrap=wrap) 

return _t7Tuple(t, radius) 

 

 

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

wrap=False): 

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

I{spherical excess} of a (spherical) triangle. 

 

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

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

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

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

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

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

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

L{excessGirard} or L{excessLHuilier}). 

@kwarg wrap: Wrap and L{unrollPI} longitudes (C{bool}). 

 

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

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

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

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

''' 

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

d, _ = unrollPI(lamB, lamC, wrap=w) 

a = vincentys_(phiC, phiB, d) 

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

 

def _A_r(a, sa, ca, sb, cb, sc, cc): 

s = sb * sc 

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

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

 

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

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

b, r = _a_r(wrap, *r) 

c, _ = _a_r(wrap, *r) 

 

A, r = _A_r(a, *sincos2(a, b, c)) 

B, r = _A_r(b, *r) 

C, _ = _A_r(c, *r) 

 

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

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

excessLHuilier(a, b, c) if excess in (excessLHuilier, False) else 

excessAbc(*max((A, b, c), (B, c, a), (C, a, b)))) 

 

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

 

 

def _t7Tuple(t, radius): 

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

''' 

if radius: # not in (None, _0_0) 

r = radius if isscalar(radius) else \ 

_ellipsoidal_datum(radius).ellipsoid.Rmean 

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

t = Triangle7Tuple(A, (r * t.a), 

B, (r * t.b), 

C, (r * t.c), t.E * r**2) 

return t 

 

 

__all__ += _ALL_OTHER(Cartesian, LatLon, # classes 

areaOf, # functions 

intersection, intersections2, ispolar, 

isPoleEnclosedBy, # DEPRECATED, use ispolar 

meanOf, 

nearestOn2, nearestOn3, 

perimeterOf, 

sumOf, # == vector3d.sumOf 

triangle7, triangle8_) 

 

# **) MIT License 

# 

# Copyright (C) 2016-2021 -- mrJean1 at Gmail -- All Rights Reserved. 

# 

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

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

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

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

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

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

# 

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

# in all copies or substantial portions of the Software. 

# 

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

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

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

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

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

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

# OTHER DEALINGS IN THE SOFTWARE.