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

 

# -*- coding: utf-8 -*- 

 

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

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

L{isPoleEnclosedBy}, L{meanOf}, L{nearestOn2} and L{perimeterOf}, 

I{all spherical}. 

 

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

spherical trigonometry, transcribed 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>}. 

 

@newfield example: Example, Examples 

''' 

 

from pygeodesy.datum import R_M 

from pygeodesy.fmath import EPS, acos1, favg, fdot, fmean, fsum, \ 

isscalar, map1 

from pygeodesy.formy import antipode, bearing_, haversine_ 

from pygeodesy.lazily import _ALL_LAZY, _2kwds 

from pygeodesy.named import LatLon3Tuple, NearestOn3Tuple, _xnamed 

from pygeodesy.points import _imdex2, ispolar, nearestOn5 as _nearestOn5 

from pygeodesy.sphericalBase import CartesianSphericalBase, LatLonSphericalBase 

from pygeodesy.utily import PI2, PI_2, PI_4, degrees90, degrees180, \ 

degrees2m, iterNumpy2, radiansPI2, \ 

sincos2, tan_2, unrollPI, wrapPI 

from pygeodesy.vector3d import CrossError, crosserrors, Vector3d, sumOf 

 

from math import asin, atan2, copysign, cos, degrees, hypot, \ 

radians, sin 

 

# all public contants, classes and functions 

__all__ = _ALL_LAZY.sphericalTrigonometry + ( 

'Cartesian', 'LatLon', # classes 

'areaOf', # functions 

'intersection', 'ispolar', 'isPoleEnclosedBy', # DEPRECATED, use ispolar 

'meanOf', 

'nearestOn2', 'nearestOn3', 

'perimeterOf', 

'sumOf') # == vector3d.sumOf 

__version__ = '19.10.21' 

 

 

class Cartesian(CartesianSphericalBase): 

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

spherical, geodetic L{LatLon}. 

''' 

 

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

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

geodetic point. 

 

@keyword kwds: Optional, additional B{C{LatLon}} keyword 

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

For example, use C{LatLon=...} to override 

the L{LatLon} (sub-)class or specify 

 

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

C{B{LatLon}=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}}, B{C{datum}} 

or B{C{kwds}}. 

''' 

kwds = _2kwds(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 _trackDistanceTo3(self, start, end, radius, wrap): 

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

''' 

self.others(start, name='start') 

self.others(end, name='end') 

 

r = float(radius) 

if r < EPS: 

raise ValueError('%s invalid: %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 alongTrackDistanceTo(self, start, end, radius=R_M, wrap=False): 

'''Compute the (signed) distance 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. 

 

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

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

@keyword radius: Optional, mean earth radius (C{meter}). 

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

 

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

same units as B{C{radius}}), 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: The B{C{start}} or B{C{end}} point is not L{LatLon}. 

 

@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) 

if abs(cx) > EPS: 

return copysign(acos1(cos(r) / cx), cos(b)) * radius 

else: 

return 0.0 

 

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

'''DEPRECATED, use method C{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. 

 

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

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

@keyword 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.to2ab() 

a2, b2 = other.to2ab() 

 

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 (signed) distance from this point to the great 

circle defined by a start and an end point. 

 

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

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

@keyword radius: Optional, mean earth radius (C{meter}). 

@keyword 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). 

 

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

 

@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 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. 

 

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

B{C{radius}}). 

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

@keyword radius: Optional, mean earth radius (C{meter}). 

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

same units a B{C{radius}}). 

 

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

 

@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.to2ab() 

 

r = float(distance) / float(radius) # angular distance in radians 

t = radians(bearing) 

 

a, b = _destination2(a, b, r, t) 

h = self.height if height is None else height 

return self.classof(a, b, height=h) 

 

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

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

 

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

@keyword radius: Optional, mean earth radius (C{meter}). 

@keyword 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}}). 

 

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

 

@example: 

 

>>> p1 = LatLon(52.205, 0.119) 

>>> p2 = LatLon(48.857, 2.351); 

>>> d = p1.distanceTo(p2) # 404300 

''' 

self.others(other) 

 

a1, b1 = self.to2ab() 

a2, b2 = other.to2ab() 

 

db, b2 = unrollPI(b1, b2, wrap=wrap) 

r = haversine_(a2, a1, db) 

return r * float(radius) 

 

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. 

 

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

 

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

 

@example: 

 

>>> p = LatLon(53.3206, -1.7297) 

>>> g = p.greatCircle(96.0) 

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

''' 

a, b = self.to2ab() 

t = radians(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. 

 

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

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

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

use B{C{raiser}}=C{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 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.to2ab() 

a2, b2 = other.to2ab() 

 

# XXX behavior like sphericalNvector.LatLon.initialBearingTo 

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

raise CrossError('%s %s: %r' % ('coincident', 'points', other)) 

 

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. 

 

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

@param fraction: Fraction between both points (float, 0.0 = 

this point, 1.0 = the other point). 

@keyword height: Optional height, overriding the fractional 

height (C{meter}). 

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

 

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

 

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

 

@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) 

 

a1, b1 = self.to2ab() 

a2, b2 = other.to2ab() 

 

db, b2 = unrollPI(b1, b2, wrap=wrap) 

r = haversine_(a2, a1, db) 

sr = sin(r) 

if abs(sr) > EPS: 

sa1, ca1, sa2, ca2, \ 

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

 

A = sin((1 - fraction) * r) / sr 

B = sin( fraction * 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=fraction) 

b = favg(b1, b2, f=fraction) 

 

if height is None: 

h = self._havg(other, f=fraction) 

else: 

h = height 

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

 

def intersection(self, end1, start2, end2, 

height=None, wrap=False): 

'''Locate the intersection point of two paths both defined 

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

 

@param end1: End point of the first path (L{LatLon}) or 

the initial bearing at this point (compass 

C{degrees360}). 

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

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

the initial bearing at the second point 

(compass C{degrees360}). 

@keyword height: Optional height for intersection point, 

overriding the mean height (C{meter}). 

@keyword 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 TypeError: The B{C{start2}}, B{C{end1}} or B{C{end2}} is 

not L{LatLon}. 

 

@raise ValueError: Intersection is ambiguous or infinite or 

the paths are parallel, coincident or null. 

 

@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' 

''' 

return intersection(self, end1, start2, end2, 

height=height, wrap=wrap, 

LatLon=self.classof) 

 

def isenclosedBy(self, points): 

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

 

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

 

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

C{False} otherwise. 

 

@raise ValueError: Insufficient number of B{C{points}} or 

non-convex polygon. 

 

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

 

@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 

''' 

n, points = self.points2(points, closed=True) 

 

n0 = self.toVector3d() 

 

if iterNumpy2(points): 

 

v1 = points[n-1].toVector3d() 

v2 = points[n-2].toVector3d() 

gc1 = v2.cross(v1) 

t0 = gc1.angleTo(n0) > PI_2 

for i in range(n): 

v2 = points[i].toVector3d() 

gc = v1.cross(v2) 

v1 = v2 

 

ti = gc.angleTo(n0) > PI_2 

if ti != t0: 

return False # outside 

 

if gc1.angleTo(gc, vSign=n0) < 0: 

raise ValueError('non-convex: %r...' % (points[:2],)) 

gc1 = gc 

 

else: 

# get great-circle vector for each edge 

gc, v1 = [], points[n-1].toVector3d() 

for i in range(n): 

v2 = points[i].toVector3d() 

gc.append(v1.cross(v2)) 

v1 = v2 

 

# check whether this point on same side of all 

# polygon edges (to the left or right depending 

# on anti-/clockwise polygon direction) 

t0 = gc[0].angleTo(n0) > PI_2 # True if on the right 

for i in range(1, n): 

ti = gc[i].angleTo(n0) > PI_2 

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

return False # outside 

 

# check for convex polygon (otherwise 

# the test above is not reliable) 

gc1 = gc[n-1] 

for gc2 in gc: 

# angle between gc vectors, signed by direction of n0 

if gc1.angleTo(gc2, vSign=n0) < 0: 

raise ValueError('non-convex: %r...' % (points[:2],)) 

gc1 = gc2 

 

return True # inside 

 

def isEnclosedBy(self, points): 

'''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. 

 

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

@keyword height: Optional height for midpoint, overriding 

the mean height (C{meter}). 

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

 

@return: Midpoint (L{LatLon}). 

 

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

 

@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.to2ab() 

a2, b2 = other.to2ab() 

 

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 

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 and this point. 

 

Distances are approximated by function L{equirectangular_}, 

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

 

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

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

@keyword radius: Optional, mean earth radius (C{meter}). 

@keyword options: Optional keyword arguments for function 

L{equirectangular_}. 

 

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

 

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

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

 

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

 

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

method L{sphericalTrigonometry.LatLon.nearestOn3}. 

''' 

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

**options)[0] 

 

def nearestOn2(self, points, closed=False, radius=R_M, **options): 

'''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 ... 

''' 

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

**options)[:2] 

 

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}}. 

 

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

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

@keyword radius: Optional, mean earth radius (C{meter}). 

@keyword options: Optional keyword arguments for function 

L{equirectangular_}. 

 

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

where C{distance} is the L{equirectangular_} 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 TypeError: Some B{C{points}} are not C{LatLon}. 

 

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

 

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

L{nearestOn5}. 

''' 

a, b, d, c, h = _nearestOn5(self, points, closed=closed, **options) 

return NearestOn3Tuple(self.classof(a, b, height=h), 

degrees2m(d, radius=radius), c) 

 

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

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

coordinates. 

 

@keyword kwds: Optional, additional B{C{Cartesian}} keyword 

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

For example, use C{Cartesian=...} to override 

the L{Cartesian} (sub-)class or specify 

 

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

C{B{Cartesian}=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}}, B{C{datum}} 

or B{C{kwds}}. 

''' 

kwds = _2kwds(kwds, Cartesian=Cartesian, datum=self.datum) 

return LatLonSphericalBase.toCartesian(self, **kwds) 

 

 

_Trll = LatLon(0, 0) #: (INTERNAL) Reference instance (L{LatLon}). 

 

 

def _destination2_(a, b, r, t): 

'''(INTERNAL) Computes destination lat- and longitude. 

 

@param a: Latitude (C{radians}). 

@param b: Longitude (C{radians}). 

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

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

 

@return: 2-Tuple (lat, lon) of (radians, radians). 

''' 

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

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

 

a = asin(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 _destination2(a, b, r, t): 

'''(INTERNAL) Computes destination lat- and longitude. 

 

@param a: Latitude (C{radians}). 

@param b: Longitude (C{radians}). 

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

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

 

@return: 2-Tuple (lat, lon) of (C{degrees90}, C{degrees180}). 

''' 

a, b = _destination2_(a, b, r, t) 

return degrees90(a), degrees180(b) 

 

 

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

'''Calculate the area of a (spherical) polygon (with great circle 

arcs joining the points). 

 

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

@keyword radius: Optional, mean earth radius (C{meter}). 

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

 

@return: Polygon area (C{meter}, same units as B{C{radius}}, squared). 

 

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

 

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

 

@note: The area is based on 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} and 

L{ellipsoidalKarney.areaOf}. 

 

@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 

''' 

n, points = _Trll.points2(points, closed=True) 

 

# Area method due to Karney: for each edge of the polygon, 

# 

# tan(Δλ/2) · (tan(φ1/2) + tan(φ2/2)) 

# tan(E/2) = ------------------------------------ 

# 1 + tan(φ1/2) · tan(φ2/2) 

# 

# where E is the spherical excess of the trapezium obtained by 

# extending the edge to the equator-circle vector for each edge 

 

def _exs(n, points): # iterate over spherical edge excess 

a1, b1 = points[n-1].to2ab() 

ta1 = tan_2(a1) 

for i in range(n): 

a2, b2 = points[i].to2ab() 

db, b2 = unrollPI(b1, b2, wrap=wrap) 

ta2, tdb = map1(tan_2, a2, db) 

yield atan2(tdb * (ta1 + ta2), 1 + ta1 * ta2) 

ta1, b1 = ta2, b2 

 

s = fsum(_exs(n, points)) * 2 

 

if isPoleEnclosedBy(points): 

s = abs(s) - PI2 

 

return abs(s * radius**2) 

 

 

def _x3d2(start, end, wrap, n, hs): 

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

a1, b1 = start.to2ab() 

 

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

a2, b2 = _destination2_(a1, b1, PI_4, radians(end)) 

else: # must be a point 

_Trll.others(end, name='end' + n) 

hs.append(end.height) 

a2, b2 = end.to2ab() 

 

db, b2 = unrollPI(b1, b2, wrap=wrap) 

if max(abs(db), abs(a2 - a1)) < EPS: 

raise ValueError('intersection %s%s null: %r' % ('path', n, (start, end))) 

 

# 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 = Vector3d(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 _xb(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, radians(a), radians(b), wrap=wrap) 

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

 

 

def _xdot(d, a1, b1, a, b, wrap): 

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

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

return fdot(d, db, radians(a) - a1) 

 

 

def intersection(start1, end1, start2, end2, 

height=None, wrap=False, LatLon=LatLon): 

'''Compute the intersection point of two paths both defined 

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

 

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

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

the initial bearing at the first start point 

(compass C{degrees360}). 

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

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

the initial bearing at the second start point 

(compass C{degrees360}). 

@keyword height: Optional height for the intersection point, 

overriding the mean height (C{meter}). 

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

@keyword LatLon: Optional (sub-)class to return the intersection 

point (L{LatLon}) or C{None}. 

 

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

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

is C{None}. An alternate intersection point might 

be the L{antipode} to the returned result. 

 

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

 

@raise ValueError: Intersection is ambiguous or infinite or 

the paths are parallel, coincident or null. 

 

@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' 

''' 

_Trll.others(start1, name='start1') 

_Trll.others(start2, name='start2') 

 

hs = [start1.height, start2. height] 

 

a1, b1 = start1.to2ab() 

a2, b2 = start2.to2ab() 

 

db, b2 = unrollPI(b1, b2, wrap=wrap) 

r12 = haversine_(a2, a1, db) 

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

a, b = map1(degrees, 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 abs(x1) < EPS or abs(x2) < EPS: 

raise ValueError('intersection %s: %r vs %r' % ('parallel', 

(start1, end1), (start2, end2))) 

 

# 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 ValueError('intersection %s: %r vs %r' % ('infinite', 

(start1, end1), (start2, end2))) 

sx3 = sx1 * sx2 

# if sx3 < 0: 

# raise ValueError('intersection %s: %r vs %r' % ('ambiguous', 

# (start1, end1), (start2, end2))) 

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 _xb(a1, b1, end1, a, b, wrap) < 0 or \ 

_xb(a2, b2, end2, a, b, wrap) < 0: 

a, b = antipode(a, b) 

 

else: # end point(s) or bearing(s) 

x1, d1 = _x3d2(start1, end1, wrap, '1', hs) 

x2, d2 = _x3d2(start2, end2, wrap, '2', hs) 

x = x1.cross(x2) 

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

raise ValueError('intersection %s: %r vs %r' % ('colinear', 

(start1, end1), (start2, end2))) 

a, b = x.to2ll() 

# choose intersection similar to sphericalNvector 

d1 = _xdot(d1, a1, b1, a, b, wrap) 

d2 = _xdot(d2, a2, b2, a, b, wrap) 

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

a, b = antipode(a, b) 

 

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

r = LatLon3Tuple(a, b, h) if LatLon is None else \ 

LatLon(a, b, height=h) 

return _xnamed(r, intersection.__name__) 

 

 

def isPoleEnclosedBy(points, wrap=False): 

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

''' 

return ispolar(points, wrap=wrap) 

 

 

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

'''Compute the geographic mean of several points. 

 

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

@keyword height: Optional height at mean point, overriding 

the mean height (C{meter}). 

@keyword LatLon: Optional (sub-)class to return the mean 

point (L{LatLon}) or C{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}}. 

''' 

# geographic mean 

n, points = _Trll.points2(points, closed=False) 

 

m = sumOf(points[i].toVector3d() for i in range(n)) 

a, b = m.to2ll() 

 

if height is None: 

h = fmean(points[i].height for i in range(n)) 

else: 

h = height 

r = LatLon3Tuple(a, b, h) if LatLon is None else \ 

LatLon(a, b, height=h) 

return _xnamed(r, meanOf.__name__) 

 

 

def nearestOn2(point, points, closed=False, radius=R_M, 

LatLon=LatLon, **options): 

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

 

@return: ... C{closest} as B{C{LatLon}} or a 2-tuple C{(lat, lon)} 

without the height if B{C{LatLon}} is C{None} ... 

''' 

a, b, d, _, h = _nearestOn5(point, points, closed=closed, **options) 

ll = (a, b) if LatLon is None else LatLon(a, b, height=h) 

return ll, degrees2m(d, radius=radius) 

 

 

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}}. 

 

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

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

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

@keyword radius: Optional, mean earth radius (C{meter}). 

@keyword LatLon: Optional (sub-)class to return the closest 

point (L{LatLon}) or C{None}. 

@keyword options: Optional keyword arguments for function 

L{equirectangular_}. 

 

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

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

the C{closest} and reference B{C{point}} in C{meter}, same 

units as B{C{radius}}. The C{angle} from the reference 

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 TypeError: Some I{points} are not C{LatLon}. 

 

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

 

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

''' 

a, b, d, c, h = _nearestOn5(point, points, closed=closed, 

LatLon=None, **options) 

r = LatLon3Tuple(a, b, h) if LatLon is None else \ 

LatLon(a, b, 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). 

 

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

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

@keyword radius: Optional, mean earth radius (C{meter}). 

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

 

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

 

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

 

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

 

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

 

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

and L{ellipsoidalKarney.perimeterOf}. 

''' 

n, points = _Trll.points2(points, closed=closed) 

 

def _rads(n, points, closed): # angular edge lengths in radians 

i, m = _imdex2(closed, n) 

a1, b1 = points[i].to2ab() 

for i in range(m, n): 

a2, b2 = points[i].to2ab() 

db, b2 = unrollPI(b1, b2, wrap=wrap) 

yield haversine_(a2, a1, db) 

a1, b1 = a2, b2 

 

r = fsum(_rads(n, points, closed)) 

return r * float(radius) 

 

# **) MIT License 

# 

# Copyright (C) 2016-2020 -- 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.