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

 

# -*- coding: utf-8 -*- 

 

u'''A pure Python version of I{Karney}'s C++ classes U{Rhumb 

<https://GeographicLib.SourceForge.io/C++/classGeographicLib_1_1Rhumb.html>} and U{RhumbLine 

<https://GeographicLib.SourceForge.io/C++/classGeographicLib_1_1RhumbLine.html>}. 

 

For more details, see the C++ U{GeographicLib<https://GeographicLib.SourceForge.io/C++/doc/index.html>} 

documentation, especially the U{Class List<https://GeographicLib.SourceForge.io/C++/doc/annotated.html>}, 

the background information on U{Rhumb lines<https://GeographicLib.SourceForge.io/C++/doc/rhumb.html>}, 

the utily U{RhumbSolve<https://GeographicLib.SourceForge.io/C++/doc/RhumbSolve.1.html>} and U{Online 

rhumb line calculations<https://GeographicLib.SourceForge.io/cgi-bin/RhumbSolve>}. 

 

Copyright (C) U{Charles Karney<mailto:Charles@Karney.com>} (2014-2022) 

and licensed under the MIT/X11 License. For more information, see the 

U{GeographicLib<https://GeographicLib.SourceForge.io>} documentation. 

''' 

# make sure int/int division yields float quotient 

from __future__ import division as _; del _ # PYCHOK semicolon 

 

from pygeodesy.basics import isnan, istuplist, _xinstanceof 

from pygeodesy.errors import _or, _ValueError, _xkwds 

from pygeodesy.fmath import hypot, hypot1 

from pygeodesy.fsums import fsum1_, pairs 

from pygeodesy.interns import NAN, NN, PI_2, _COMMASPACE_, _lat1_, _lat2_, \ 

_lon1_, _lon2_, _not_, _s12_, _S12_, _UNDER_, \ 

_0_0, _0_5, _1_0, _2_0, _4_0, _90_0, _180_0, \ 

_720_0 # PYCHOK used! 

from pygeodesy.karney import Caps, _CapsBase, _atan2d, _diff182, _ellipsoid, \ 

Direct9Tuple, _EWGS84, _fix90, GDict, _GTuple, \ 

Inverse10Tuple, _norm180, _polynomial, _tand, \ 

LatLon2Tuple 

from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY 

# from pygeodesy.namedTuples import LatLon2Tuple # from .karney 

from pygeodesy.props import Property, Property_RO, property_RO, _update_all 

# from pygeodesy.streprs import pairs # from .fsums 

from pygeodesy.units import Bearing as _Azi, Int, Lat, Lon, \ 

Meter as _M, Meter2 as _M2 

from pygeodesy.utily import sincos2_, sincos2d 

 

from math import asinh, atan, cos, cosh, degrees, fabs, radians, sin, sinh, sqrt 

 

__all__ = _ALL_LAZY.rhumbx + _ALL_DOCS(Caps) 

__version__ = '22.05.22' 

 

_azi12_ = 'azi12' 

 

_rls = [] # instances of C{RbumbLine} to be updated 

 

 

def _Lat(*lat, **Error_name): 

'''(INTERNAL) Latitude B{C{lat}}. 

''' 

kwds = _xkwds(Error_name, clip=0, Error=RhumbError) 

return Lat(*lat, **kwds) 

 

 

def _Lon(*lon, **Error_name): 

'''(INTERNAL) Longitude B{C{lon}}. 

''' 

kwds = _xkwds(Error_name, clip=0, Error=RhumbError) 

return Lon(*lon, **kwds) 

 

 

def _update_all_rls(r): 

'''(INTERNAL) Zap cached/memoized C{Property[_RO]}s 

of any L{RhumbLine} instances tied to the given 

L{Rhumb} instance B{C{r}}. 

''' 

_xinstanceof(r, Rhumb) 

_update_all(r) 

for rl in _rls: # PYCHOK use weakref? 

if rl._rhumb is r: 

_update_all(rl) 

 

 

class Rhumb(_CapsBase): 

'''Class to solve of the I{direct} and I{inverse rhumb} problems, accurately. 

 

@see: The U{Detailed Description<https://GeographicLib.SourceForge.io/C++/doc/ 

classGeographicLib_1_1Rhumb.html>} of I{Karney}'s C++ C{Rhumb Class}. 

''' 

_E = _EWGS84 

_exact = True 

_mRA = 6 

_mTM = 6 

 

def __init__(self, a_earth=_EWGS84, f=None, exact=True, name=NN, **orders): 

'''New L{Rhumb}. 

 

@kwarg a_earth: This rhumb's earth (L{Ellipsoid}, L{Ellipsoid2}, 

L{a_f2Tuple}, L{Datum}, 2-tuple (C{a, f})) or the 

equatorial radius (C{scalar}). 

@kwarg f: The ellipsoid's flattening (C{scalar}), iff B{C{a_earth}} is 

a C{scalar}, ignored otherwise. 

@kwarg exact: If C{True}, use an addition theorem for elliptic integrals 

to compute I{Divided differences}, otherwise use the Krüger 

series expansion (C{bool}), see also property C{exact}. 

@kwarg name: Optional name (C{str}). 

@kwarg orders: Optional keyword arguments B{C{RAorder}} and B{C{TMorder}} 

to set the respective C{order}, see properties C{RAorder} 

and C{TMorder} and method C{orders}. 

 

@raise RhumbError: Invalid B{C{a_earth}}, B{C{f}} or B{C{orders}}. 

''' 

if f is not None: 

self.ellipsoid = a_earth, f 

elif a_earth not in (_EWGS84, None): 

self.ellipsoid = a_earth 

if not exact: 

self._exact = False 

if name: 

self.name = name 

if orders: 

self.orders(**orders) 

 

def _DConformal2Rectifying(self, chix, chiy): 

return _1_0 + _sincosSeries(True, chix, chiy, 

*self._DRectifying2ConformalAc2) 

 

@Property_RO 

def _DRectifying2ConformalAc2(self): 

return self._Xs2(_AlpCoeffs, self.TMorder) 

 

def Direct(self, lat1, lon1, azi12, s12, outmask=Caps.LATITUDE_LONGITUDE): 

'''Solve the I{direct rhumb} problem, optionally with the area. 

 

@arg lat1: Latitude of the first point (C{degrees90}). 

@arg lon1: Longitude of the first point (C{degrees180}). 

@arg azi12: Azimuth of the rhumb line (compass C{degrees}). 

@arg s12: Distance along the rhumb line from points 1 to 

the second point (C{meter}), can be negative. 

 

@return: L{GDict} with 2 up to 9 items C{azi12, s12, S12, lat2, 

lon2, lat1, lon1} with latitude C{lat2} and longitude 

C{lon2} of the other point in C{degrees} and the area 

C{S12} under the rhumb line in C{meter} I{squared}. 

 

@note: If B{C{s12}} is large enough that the rhumb line crosses 

a pole, the longitude of the second point is indeterminate 

and C{NAN} is returned for C{lon2} and area C{S12}. 

 

If the first point is a pole, the cosine of its latitude is 

taken to be C{epsilon}**-2 (where C{epsilon} is 2.0**-52. 

This position is extremely close to the actual pole and 

allows the calculation to be carried out in finite terms. 

''' 

rl = _RhumbLine(self, lat1, lon1, azi12, caps=Caps.LINE_OFF, 

name=self.name) 

return rl.Position(s12, outmask | self._debug) # lat2, lon2, S12 

 

def Direct7(self, lat1, lon1, azi12, s12, outmask=Caps.LATITUDE_LONGITUDE_AREA): 

'''Like method L{Rhumb.Direct} but returning a L{Rhumb7Tuple} with area C{S12}. 

''' 

return self.Direct(lat1, lon1, azi12, s12, outmask=outmask).toRhumb7Tuple() 

 

def DirectLine(self, lat1, lon1, azi12, name=NN): # caps=Caps.STANDARD 

'''Define a L{RhumbLine} in terms of the I{direct} rhumb problem. 

 

@arg lat1: Latitude of the first point (C{degrees90}). 

@arg lon1: Longitude of the first point (C{degrees180}). 

@arg azi12: Azimuth of the rhumb line (compass C{degrees}). 

 

@return: A L{RhumbLine} instance and invoke its method 

L{RhumbLine.Position} to compute each point. 

 

@note: Updates to this rhumb are reflected in the returned 

rhumb line. 

''' 

return RhumbLine(self, lat1=lat1, lon1=lon1, azi12=azi12, 

caps=self._debug, name=name or self.name) 

 

def _DIsometric(self, latx, laty): 

phix = radians(latx) 

phiy = radians(laty) 

return (_Dtand(latx, laty) * _Dasinh(_tand(latx), _tand(laty)) - 

_Dsin(phix, phiy) * _DeatanhE(sin(phix), sin(phiy), self._E)) 

 

def _DIsometric2Rectifying(self, psix, psiy): # degrees 

if self.exact: 

E = self._E 

latx = E.auxIsometric(psix, inverse=True) 

laty = E.auxIsometric(psiy, inverse=True) 

r = self._DRectifying(latx, laty) / self._DIsometric(latx, laty) 

else: 

x, y = radians(psix), radians(psiy) 

r = self._DConformal2Rectifying(_gd(x), _gd(y)) * _Dgd(x, y) 

return r 

 

def _DRectifying(self, latx, laty): 

E, eF = self._E, self._eF 

tbetx = E.f1 * _tand(latx) 

tbety = E.f1 * _tand(laty) 

return (E.f1 * _DfEt(tbetx, tbety, eF) * E.b 

* _Dtand(latx, laty) * PI_2 

* _Datan(tbetx, tbety)) / E.L 

 

def _DRectifying2Conformal(self, mux, muy): 

return _1_0 - _sincosSeries(True, mux, muy, 

*self._DRectifying2ConformalBc2) 

 

@Property_RO 

def _DRectifying2ConformalBc2(self): 

return self._Xs2(_BetCoeffs, self.TMorder) 

 

def _DRectifying2Isometric(self, mux, muy): # radians 

E = self._E 

latx = E.auxRectifying(degrees(mux), inverse=True) 

laty = E.auxRectifying(degrees(muy), inverse=True) 

if self.exact: 

r = self._DIsometric(latx, laty) / self._DRectifying(latx, laty) 

else: 

r = self._DRectifying2Conformal(mux, muy) * \ 

_Dgdinv(E.es_taupf(_tand(latx)), 

E.es_taupf(_tand(laty))) 

return r 

 

@Property_RO 

def _eF(self): 

'''(INTERNAL) Get the ellipsoid's elliptic function. 

''' 

# .k2 = 0.006739496742276434 

return self._E._elliptic_e12 # _MODS.elliptic.Elliptic(-self._E._e12) 

 

@Property 

def ellipsoid(self): 

'''Get this rhumb's ellipsoid (L{Ellipsoid}). 

''' 

return self._E 

 

@ellipsoid.setter # PYCHOK setter! 

def ellipsoid(self, a_earth_f): 

'''Set this rhumb's ellipsoid (L{Ellipsoid}, L{Ellipsoid2}, L{Datum}, 

L{a_f2Tuple} or 2-tuple C{(a, f)}). 

''' 

try: 

E = _ellipsoid(*a_earth_f[:2]) if istuplist(a_earth_f, 2) else \ 

_ellipsoid( a_earth_f, None) 

except Exception as x: 

raise RhumbError(str(x)) 

if E != self._E: 

_update_all_rls(self) 

self._E = E 

 

@Property 

def exact(self): 

'''Get the I{exact} option (C{bool}). 

''' 

return self._exact 

 

@exact.setter # PYCHOK setter! 

def exact(self, exact): 

'''Set the I{exact} option (C{bool}). 

 

@arg exact: If C{True}, use I{exact} rhumb calculations, if C{False} 

results are less precise for more oblate or more prolate 

ellipsoids with M{abs(flattening) > 0.01} (C{bool}). 

 

@see: Option U{B{-s}<https://GeographicLib.SourceForge.io/C++/doc/RhumbSolve.1.html>} 

and U{ACCURACY<https://GeographicLib.SourceForge.io/C++/doc/RhumbSolve.1.html#ACCURACY>}. 

''' 

x = bool(exact) 

if x != self.exact: 

_update_all_rls(self) 

self._exact = x 

 

def Inverse(self, lat1, lon1, lat2, lon2, outmask=Caps.AZIMUTH_DISTANCE): 

'''Solve the I{inverse rhumb} problem, optionally with the area. 

 

@arg lat1: Latitude of the first point (C{degrees90}). 

@arg lon1: Longitude of the first point (C{degrees180}). 

@arg lat2: Latitude of the second point (C{degrees90}). 

@arg lon2: Longitude of the second point (C{degrees180}). 

 

@return: L{GDict} with 4 to 7 itens C{lat1, lon2, lat2, lon2, 

azi12, s12, S12}, the azimuth C{azi12} of the rhumb 

line in compass C{degrees} between C{-180} and C{+180}, 

the rhumb C{s12} distance between both points in C{meter} 

and the area C{S12} under the rhumb line in C{meter} 

I{squared}. 

 

@note: The shortest rhumb line is found. If the end points are 

on opposite meridians, there are two shortest rhumb lines 

and the East-going one is chosen. 

 

If either point is a pole, the cosine of its latitude is 

taken to be C{epsilon}**-2 (where C{epsilon} is 2.0**-52). 

This position is extremely close to the actual pole and 

allows the calculation to be carried out in finite terms. 

''' 

r = GDict(S12=NAN, name=self.name) 

if (outmask & Caps.AZIMUTH_DISTANCE_AREA): 

r.set_(lat1=lat1, lon1=lon1, lat2=lat2, lon2=lon2) 

E = self._E 

psi1 = E.auxIsometric(lat1) 

psi2 = E.auxIsometric(lat2) 

psi12 = psi2 - psi1 

lon12, _ = _diff182(lon1, lon2) 

if (outmask & Caps.AZIMUTH): 

r.set_(azi12=_atan2d(lon12, psi12)) 

if (outmask & Caps.DISTANCE): 

h = hypot(lon12, psi12) 

d = self._DIsometric2Rectifying(psi2, psi1) 

r.set_(s12=h * d * E.L / _90_0) 

if (outmask & Caps.AREA): 

r.set_(S12=self._S12d(lon12, psi2, psi1)) 

if ((outmask | self._debug) & Caps._DEBUG_INVERSE): # PYCHOK no cover 

r.set_(a=E.a, f=E.f, f1=E.f1, L=E.L, 

b=E.b, e=E.e, e2=E.e2, k2=self._eF.k2, 

lon12=lon12, psi1=psi1, exact=self.exact, 

psi12=psi12, psi2=psi2) 

return r 

 

def Inverse7(self, lat1, lon1, azi12, s12, outmask=Caps.AZIMUTH_DISTANCE_AREA): 

'''Like method L{Rhumb.Inverse} but returning a L{Rhumb7Tuple} with area C{S12}. 

''' 

return self.Inverse(lat1, lon1, azi12, s12, outmask=outmask).toRhumb7Tuple() 

 

def InverseLine(self, lat1, lon1, lat2, lon2, name=NN): # caps=Caps.STANDARD 

'''Define a L{RhumbLine} in terms of the I{inverse} rhumb problem. 

 

@arg lat1: Latitude of the first point (C{degrees90}). 

@arg lon1: Longitude of the first point (C{degrees180}). 

@arg lat2: Latitude of the second point (C{degrees90}). 

@arg lon2: Longitude of the second point (C{degrees180}). 

 

@return: A L{RhumbLine} instance and invoke its method 

L{RhumbLine.Position} to compute each point. 

 

@note: Updates to this rhumb are reflected in the returned 

rhumb line. 

''' 

r = self.Inverse(lat1, lon1, lat2, lon2, outmask=Caps.AZIMUTH) 

return RhumbLine(self, lat1=lat1, lon1=lon1, azi12=r.azi12, 

caps=self._debug, name=name or self.name) 

 

Line = DirectLine # synonimous 

 

def _MeanSinXid(self, psix, psiy): # psix, psiy in degrees 

x, y = radians(psix), radians(psiy) 

t = _sincosSeries(False, _gd(x), _gd(y), *self._Rc2) 

return _Dlog(cosh(x), cosh(y)) * _Dcosh(x, y) + t * _Dgd(x, y) 

 

def orders(self, RAorder=None, TMorder=None): 

'''Get and set the I{RAorder} and/or I{TMorder}. 

 

@kwarg RAorder: I{Rhumb Area} order (C{int}, 

4, 5, 6, 7 or 8). 

@kwarg TMorder: I{Transverse Mercator} order (C{int}, 

4, 5, 6, 7 or 8). 

 

@return: L{RhumbOrder2Tuple}C{(RAorder, TMorder)} with the 

previous C{RAorder} and C{TMorder} setting. 

''' 

t = RhumbOrder2Tuple(self.RAorder, self.TMorder) 

if RAorder not in (None, self._mRA): 

self.RAorder = RAorder 

if TMorder not in (None, self._mTM): 

self.TMorder = TMorder 

return t 

 

@Property 

def RAorder(self): 

'''Get the I{Rhumb Area} order (C{int}, 4, 5, 6, 7 or 8). 

''' 

return self._mRA 

 

@RAorder.setter # PYCHOK setter! 

def RAorder(self, order): 

'''Get the I{Rhumb Area} order (C{int}, 4, 5, 6, 7 or 8). 

''' 

n = self._Xorder(_RCoeffs, RAorder=order) 

if n != self.RAorder: 

_update_all_rls(self) 

self._mRA = n 

 

@Property_RO 

def _Rc2(self): 

# for WGS84: (0, -0.0005583633519275459, -3.743803759172812e-07, -4.633682270824446e-10, 

# RAorder 6: -7.709197397676237e-13, -1.5323287106694307e-15, -3.462875359099873e-18) 

return self._Xs2(_RCoeffs, self.RAorder, RA=True) 

 

def _S12d(self, lon12, psi2, psi1): 

'''(INTERNAL) Compute the area C{S12}. 

''' 

r = self._E.areax if self.exact else self._E.area 

# for WGS84: r = 510065621724088.44 

# a = 6378137.0, f = 0.0033528106647474805 

# b = 6356752.314245179, c2 = 40589732499314.76 

r *= lon12 / _720_0 

return r * self._MeanSinXid(psi2, psi1) 

 

@Property 

def TMorder(self): 

'''Get the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8). 

''' 

return self._mTM 

 

@TMorder.setter # PYCHOK setter! 

def TMorder(self, order): 

'''Set the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8). 

 

@note: Setting C{TMorder} turns C{exact} off. 

''' 

n = self._Xorder(_AlpCoeffs, TMorder=order) 

if n != self.TMorder: 

_update_all_rls(self) 

self._mTM = n 

self.exact = False 

 

def toStr(self, prec=6, sep=_COMMASPACE_, **unused): # PYCHOK signature 

'''Return this C{Rhumb} as string. 

 

@kwarg prec: The C{float} precision, number of decimal digits (0..9). 

Trailing zero decimals are stripped for B{C{prec}} values 

of 1 and above, but kept for negative B{C{prec}} values. 

@kwarg sep: Separator to join (C{str}). 

 

@return: Tuple items (C{str}). 

''' 

d = dict(ellipsoid=self.ellipsoid, RAorder=self.RAorder, 

exact=self.exact, TMorder=self.TMorder) 

return sep.join(pairs(d, prec=prec)) 

 

def _Xorder(self, _Coeffs, **Xorder): 

'''(INTERNAL) Validate C{RAorder} or C{TMorder}. 

''' 

try: 

Xorder, order = Xorder.popitem() 

n = int(order) 

if n not in _Coeffs: 

t = sorted(map(str, _Coeffs.keys())) 

raise ValueError(_not_(_or(*t))) 

except (IndexError, TypeError, ValueError) as x: 

raise RhumbError(Xorder, order, txt=str(x)) 

return n 

 

def _Xs2(self, _Coeffs, m, RA=False): 

'''(INTERNAL) Compute the C{R}, C{A} or C{B} terms for 

I{_sincosSeries}, return 2-tuple C{(terms, order)}. 

''' 

cs = _Coeffs[m] 

assert len(cs) == (((m + 1) * (m + 4)) if RA else 

(m * (m + 3))) // 2 

n, i = self._E.n, ((m + 2) if RA else 0) 

n_, X = n, [0] # X[0] never used, it ... 

# ... is just an integration constant, so 

# it cancels when evaluating a definite 

# integral. Don't bother computing it, it's 

# not used when invoking C{_sincosSeries}. 

for r in range(m - 1, -1, -1): # [m-1 ... 0] 

j = i + r + 1 

X.append(_polynomial(n, cs, i, j) * n_ / cs[j]) 

i = j + 1 

n_ *= n 

return tuple(X), m 

 

 

class RhumbError(_ValueError): 

'''Raised for an L{Rhumb} or L{RhumbLine} issue. 

''' 

pass 

 

 

class _RhumbLine(_CapsBase): 

'''(INTERNAL) Class L{RhumbLine} 

''' 

_azi12 = _0_0 

# _caps = 0 

# _lat1 = _0_0 

# _lon1 = _0_0 

_salp = _0_0 

_calp = _1_0 

_rhumb = None # L{Rhumb} instance 

 

def __init__(self, rhumb, lat1, lon1, azi12, caps=0, name=NN): # case=Caps.? 

'''New C{RhumbLine}. 

''' 

if (caps & Caps._DEBUG_DIRECT_LINE): 

self._debug |= caps & Caps._DEBUG_DIRECT_LINE 

_xinstanceof(Rhumb, rhumb=rhumb) 

self._lat1 = _Lat(lat1=_fix90(lat1)) 

self._lon1 = _Lon(lon1= lon1) 

if azi12: # non-zero 

self.azi12 = azi12 

self._caps = caps 

if not (caps & Caps.LINE_OFF): 

_rls.append(self) 

if name: 

self.name=name or rhumb.name 

self._rhumb = rhumb # last 

 

def __del__(self): # XXX use weakref? 

if _rls: # may be empty or None 

try: # PYCHOK no cover 

_rls.remove(self) 

except (TypeError, ValueError): 

pass 

self._rhumb = None 

# _update_all(self) # throws TypeError during Python 2 cleanup 

 

def _update(self, updated, *attrs, **unused): 

if updated: 

_update_all(self, *attrs) 

 

@Property 

def azi12(self): 

'''Get this rhumb line's I{azimuth} (compass C{degrees}). 

''' 

return self._azi12 

 

@azi12.setter # PYCHOK setter! 

def azi12(self, azi12): 

'''Set this rhumb line's I{azimuth}. 

 

@arg azi12: The new I{azimuth} (compass C{degrees}). 

''' 

z = _norm180(azi12) 

if z != self._azi12: 

if self._rhumb: 

_update_all(self) 

self._azi12 = z 

self._salp, self._calp = sincos2d(z) # no NEG0 

 

@Property_RO 

def ellipsoid(self): 

'''Get this rhumb line's ellipsoid (L{Ellipsoid}). 

''' 

return self._rhumb._E 

 

@property_RO 

def exact(self): 

'''Get this rhumb line's I{exact} option (C{bool}). 

''' 

return self.rhumb.exact 

 

@property_RO 

def lat1(self): 

'''Get this rhumb line's latitude (C{degrees90}). 

''' 

return self._lat1 

 

@property_RO 

def lon1(self): 

'''Get this rhumb line's longitude (C{degrees180}). 

''' 

return self._lon1 

 

@Property_RO 

def latlon1(self): 

'''Get this rhumb line's lat- and longitude (L{LatLon2Tuple}C{(lat, lon)}). 

''' 

return LatLon2Tuple(self.lat1, self.lon1) 

 

@Property_RO 

def _mu1(self): 

'''(INTERNAL) Get the I{rectifying auxiliary} latitude C{mu} (C{degrees}).. 

''' 

return self.ellipsoid.auxRectifying(self._lat1) 

 

@Property_RO 

def _psi1(self): 

'''(INTERNAL) Get the I{isometric auxiliary} latitude C{psi} (C{degrees}). 

''' 

return self.ellipsoid.auxIsometric(self._lat1) 

 

@property_RO 

def RAorder(self): 

'''Get this rhumb line's I{Rhumb Area} order (C{int}, 4, 5, 6, 7 or 8). 

''' 

return self.rhumb.RAorder 

 

@Property_RO 

def _r1rad(self): 

'''(INTERNAL) Get this rhumb line's I{icircle radius} (C{meter}). 

''' 

return radians(self.ellipsoid.circle4(self._lat1).radius) 

 

@Property_RO 

def rhumb(self): 

'''Get this rhumb line's rhumb (L{Rhumb}). 

''' 

return self._rhumb 

 

def Position(self, s12, outmask=Caps.LATITUDE_LONGITUDE): 

'''Compute a position at a distance, optionally the area. 

 

@arg s12: The distance along this rhumb between its point and 

the other point (C{meters}), can be negative. 

 

@return: L{GDict} with 2 up to 9 items C{azi12, s12, S12, lat2, 

lon2, lat1, lon1} with latitude C{lat2} and longitude 

C{lon2} of the other point in C{degrees} and the area 

C{S12} under the rhumb line in C{meter} I{squared}. 

 

@note: If B{C{s12}} is large enough that the rhumb line crosses 

a pole, the longitude of the second point is indeterminate 

and C{NAN} is returned for C{lon2} and area C{S12}. 

 

If the first point is a pole, the cosine of its latitude is 

taken to be C{epsilon}**-2 (where C{epsilon} is 2**-52). 

This position is extremely close to the actual pole and 

allows the calculation to be carried out in finite terms. 

''' 

r = GDict(S12=NAN, name=self.name) 

if (outmask & Caps.LATITUDE_LONGITUDE_AREA): 

r.set_(s12=s12, azi12=self.azi12) 

E, R = self.ellipsoid, self.rhumb 

mu12 = s12 * self._calp * _90_0 / E.L 

mu2 = mu12 + self._mu1 

if fabs(mu2) > 90: # PYCHOK no cover 

mu2 = _norm180(mu2) # reduce to [-180, 180) 

if fabs(mu2) > 90: # point on anti-meridian 

mu2 = _norm180(_180_0 - mu2) 

lat2x = E.auxRectifying(mu2, inverse=True) 

lon2x = NAN 

else: 

psi2 = self._psi1 

if self._calp: 

lat2x = E.auxRectifying(mu2, inverse=True) 

psi12 = R._DRectifying2Isometric(radians(mu2), 

radians(self._mu1)) * mu12 

lon2x = psi12 * self._salp / self._calp 

psi2 += psi12 

else: # PYCHOK no cover 

lat2x = self.lat1 

lon2x = self._salp * s12 / self._r1rad 

 

if (outmask & Caps.AREA): 

r.set_(S12=R._S12d(lon2x, self._psi1, psi2)) 

 

if (outmask & Caps.LATITUDE): 

r.set_(lat2=lat2x, lat1=self.lat1) 

if (outmask & Caps.LONGITUDE): 

if (outmask & Caps.LONG_UNROLL) and not isnan(lat2x): 

lon2x += self.lon1 

else: 

lon2x = _norm180(_norm180(self.lon1) + lon2x) 

r.set_(lon2=lon2x, lon1=self.lon1) 

if ((outmask | self._debug) & Caps._DEBUG_DIRECT_LINE): # PYCHOK no cover 

r.set_(a=E.a, f=E.f, f1=E.f1, L=E.L, exact=R.exact, 

b=E.b, e=E.e, e2=E.e2, k2=R._eF.k2, 

calp=self._calp, mu1 =self._mu1, mu12=mu12, 

salp=self._salp, psi1=self._psi1, mu2=mu2) 

return r 

 

def toStr(self, prec=6, sep=_COMMASPACE_, **unused): # PYCHOK signature 

'''Return this C{RhumbLine} as string. 

 

@kwarg prec: The C{float} precision, number of decimal digits (0..9). 

Trailing zero decimals are stripped for B{C{prec}} values 

of 1 and above, but kept for negative B{C{prec}} values. 

@kwarg sep: Separator to join (C{str}). 

 

@return: C{RhumbLine} (C{str}). 

''' 

d = dict(rhumb=self.rhumb, lat1=self.lat1, lon1=self.lon1, 

azi12=self.azi12, exact=self.exact) 

return sep.join(pairs(d, prec=prec)) 

 

@property_RO 

def TMorder(self): 

'''Get this rhumb line's I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8). 

''' 

return self.rhumb.TMorder 

 

 

class RhumbLine(_RhumbLine): 

'''Compute one or more points on a single rhumb line. 

 

Class L{RhumbLine} facilitates the determination of points 

on a single rhumb line. The starting point (C{lat1}, C{lon1}) 

and the azimuth C{azi12} are specified once. Calls to method 

L{RhumbLine.Position} return the location of an other point 

and optionally the distance C{s12} along the rhumb line and 

the corresponding area C{S12} under the rhumb line. 

''' 

def __init__(self, rhumb, lat1=0, lon1=0, azi12=None, caps=0, name=NN): # case=Caps.? 

'''New L{RhumbLine}. 

 

@arg rhumb: The rhumb reference (L{Rhumb}). 

@kwarg lat1: The latitude of the starting point (C{degrees90}). 

@kwarg lon1: The longitude of the starting point (C{degrees180}). 

@kwarg azi12: The azimuth of this rhumb line (compass C{degrees}). 

@kwarg caps: Bit-or'ed combination of L{Caps} values specifying 

the capabilities. Use C{Caps.LINE_OFF} if updates 

to the B{C{rhumb}} should I{not} be reflected in 

this rhumb line. 

@kwarg name: Optional name (C{str}). 

''' 

if (caps & Caps.LINE_OFF): # copy to avoid updates 

rhumb = rhumb.copy(deep=False, name=NN(_UNDER_, rhumb.name)) 

_RhumbLine.__init__(self, rhumb, lat1, lon1, azi12, caps=caps, name=name) 

 

 

class RhumbOrder2Tuple(_GTuple): 

'''2-Tuple C{(RAorder, TMorder)} with a I{Rhumb Area} and 

I{Transverse Mercator} order, both C{int}. 

''' 

_Names_ = (Rhumb.RAorder.name, Rhumb.TMorder.name) 

_Units_ = ( Int, Int) 

 

 

class Rhumb7Tuple(_GTuple): 

'''7-Tuple C{(lat1, lon1, lat2, lon2, azi12, s12, S12)} with lat- C{lat1}, C{lat2} 

and longitudes C{lon1_init__}, C{lon2} of both points, the azimuth of the rhumb line 

C{azi12}, the rhumb distance C{s12} and the area C{S12} under the rhumb line. 

''' 

_Names_ = (_lat1_, _lon1_, _lat2_, _lon2_, _azi12_, _s12_, _S12_) 

_Units_ = (_Lat, _Lon, _Lat, _Lon, _Azi, _M, _M2) 

 

def toDirect9Tuple(self, dflt=NAN, **azi1_azi2_m12_M12_M21): 

'''Convert this L{Rhumb7Tuple} result to a 9-tuple, like I{Karney}'s 

method C{geographiclib.geodesic.Geodesic._GenDirect}. 

 

@kwarg dflt: Default value for missing items (C{any}). 

@kwarg azi1_azi2_m12_M12_M21: Optional keyword arguments 

to specify or override L{Inverse10Tuple} items. 

 

@return: L{Direct9Tuple}C{(a12, lat2, lon2, azi2, s12, 

m12, M12, M21, S12)} 

''' 

d = dict(azi1=self.azi12, azi2=self.azi12, m12=self.s12, M12=_1_0, M21=_1_0) # PYCHOK attr 

d.update(azi1_azi2_m12_M12_M21) 

return self._toTuple(Direct9Tuple, dflt, d) 

 

def toInverse10Tuple(self, dflt=NAN, **a12_m12_M12_M21_salp1_calp1_salp2_calp2): 

'''Convert this L{Rhumb7Tuple} to a 10-tuple, like I{Karney}'s 

method C{geographiclib.geodesic.Geodesic._GenInverse}. 

 

@kwarg dflt: Default value for missing items (C{any}). 

@kwarg a12_m12_M12_M21_salp1_calp1_salp2_calp2: Optional keyword 

arguments to specify or override L{Inverse10Tuple} items. 

 

@return: L{Inverse10Tuple}C{(a12, s12, salp1, calp1, salp2, calp2, 

m12, M12, M21, S12)}. 

''' 

s, c = sincos2d(self.azi12) # PYCHOK attr 

d = dict(a12=self.azi12, m12=self.s12, M12=_1_0, M21=_1_0, # PYCHOK attr 

salp1=s, calp1=c, salp2=s, calp2=c) 

d.update(a12_m12_M12_M21_salp1_calp1_salp2_calp2) 

return self._toTuple(Inverse10Tuple, dflt, d) 

 

def _toTuple(self, nTuple, dflt, updates={}): 

'''(INTERNAL) Convert this C{Rhumb7Tuple} to an B{C{nTuple}}. 

''' 

r = self.toGDict(**updates) 

t = tuple(r.get(n, dflt) for n in nTuple._Names_) 

return nTuple(t, name=self.name) 

 

 

# Use I{Divided Differences} to determine (mu2 - mu1) / (psi2 - psi1) accurately. 

# Definition: _Df(x,y,d) = (f(x) - f(y)) / (x - y), @see W. M. Kahan & R. J. 

# Fateman, "Symbolic computation of Divided Differences", SIGSAM Bull. 33(3), 

# 7-28 (1999). U{ACM<https://DL.ACM.org/doi/pdf/10.1145/334714.334716> and 

# U{UCB<https://www.CS.bBerkeley.edu/~fateman/papers/divdiff.pdf>} 

 

def _Datan(x, y): 

xy = x * y 

r = xy + _1_0 

d = x - y 

if d: # 2 * xy > -1 == 2 * xy + 1 > 0 == xy + r > 0 == xy > -r 

r = (atan(d / r) if xy > -r else (atan(x) - atan(y))) / d 

else: 

r = _1_0 / r 

return r 

 

 

def _Dasinh(x, y): 

hx = hypot1(x) 

d = x - y 

if d: 

hx *= y 

hy = x * hypot1(y) 

t = (d * (x + y) / (hy + hx)) if (x * y) > 0 else (hy - hx) 

r = asinh(t) / d 

else: 

r = _1_0 / hx 

return r 

 

 

def _Dcosh(x, y): 

return _Dsincos(x, y, sinh, sinh) 

 

 

def _DeatanhE(x, y, E): 

# Deatanhe(x, y) = eatanhe((x - y) / (1 - e^2 * x * y)) / (x - y) 

e = _1_0 - E.e2 * x * y 

# assert not isnear0(e) 

d = x - y 

return (E._es_atanh(d / e) / d) if d else (E.e2 / e) 

 

 

def _DfEt(x, y, eF): # x, y are tangents 

# eF = Elliptic(-E.e12) # -E.e2 / (1 - E.e2) 

x, y, r = atan(x), atan(y), _1_0 

d = x - y 

if (x * y) > 0: 

# See U{DLMF<https://DLMF.NIST.gov/19.11>}: 19.11.2 and 19.11.4 

# letting theta -> x, phi -> -y, psi -> z 

# (E(x) - E(y)) / d = E(z)/d - k2 * sin(x) * sin(y) * sin(z)/d 

# tan(z/2) = (sin(x)*Delta(y) - sin(y)*Delta(x)) / (cos(x) + cos(y)) 

# = d * Dsin(x,y) * (sin(x) + sin(y))/(cos(x) + cos(y)) / 

# (sin(x)*Delta(y) + sin(y)*Delta(x)) 

# = t = d * Dt 

# sin(z) = 2*t/(1+t^2); cos(z) = (1-t^2)/(1+t^2) 

# Alt (this only works for |z| <= pi/2 -- however, this conditions 

# holds if x*y > 0): 

# sin(z) = d * Dsin(x,y) * (sin(x) + sin(y)) / 

# (sin(x)*cos(y)*Delta(y) + sin(y)*cos(x)*Delta(x)) 

# cos(z) = sqrt((1-sin(z))*(1+sin(z))) 

sx, cx, sy, cy = sincos2_(x, y) 

D = (cx + cy) * (eF.fDelta(sy, cy) * sx + 

eF.fDelta(sx, cx) * sy) 

D = (sx + sy) * _Dsin(x, y) / D 

t = D * d 

t2 = t**2 + _1_0 

D *= _2_0 / t2 

s = D * d 

if s: 

c = -(t + _1_0) * (t - _1_0) / t2 

r = eF.fE(s, c, eF.fDelta(s, c)) / s 

r = D * (r - eF.k2 * sx * sy) 

elif d: 

r = (eF.fE(x) - eF.fE(y)) / d 

return r 

 

 

def _Dgd(x, y): 

return _Datan(sinh(x), sinh(y)) * _Dsinh(x, y) 

 

 

def _Dgdinv(x, y): # x, y are tangents 

return _Dasinh(x, y) / _Datan(x, y) 

 

 

def _Dlog(x, y): 

d = x - y 

# Changed atanh(t / (x + y)) to asinh(t / (2 * sqrt(x*y))) to avoid 

# taking atanh(1) when x is large and y is 1. N.B. this routine is 

# invoked with positive x and y, so no need to guard against taking 

# the sqrt of a negative quantity. This also fixes bogus results 

# being returned for the area when an endpoint is at a pole. 

return (asinh(d / (sqrt(x * y) * _2_0)) * _2_0 / d) if d else (_1_0 / x) 

 

 

def _Dsin(x, y): 

return _Dsincos(x, y, sin, cos) 

 

 

def _Dsincos(x, y, s_, c_): 

r = c_((x + y) * _0_5) 

d = (x - y) * _0_5 

if d: 

r *= s_(d) / d 

return r 

 

 

def _Dsinh(x, y): 

return _Dsincos(x, y, sinh, cosh) 

 

 

def _Dtand(x, y): # x, y in degrees 

tx, ty = _tand(x), _tand(y) 

txy = tx * ty 

r = txy + _1_0 

d = x - y 

if d: # 2 * txy > -1 == 2 * txy + 1 > 0 == txy + r > 0 == txy > -r 

r = ((_tand(d) * r) if txy > -r else (tx - ty)) / radians(d) 

return r 

 

 

def _gd(x): 

return atan(sinh(x)) 

 

 

def _sincosSeries(sinp, x, y, c, n): 

# N.B. n >= 0 and c[] has n+1 elements 0..n, 

# of which c[0] is ignored. 

# Use Clenshaw summation to evaluate 

# m = (g(x) + g(y)) / 2 -- mean value 

# s = (g(x) - g(y)) / (x - y) -- average slope 

# where 

# g(x) = sum(c[j] * SC(2 * j * x), j = 1..n) 

# SC = sinp ? sin : cos 

# CS = sinp ? cos : sin 

# ... 

d = x - y 

sp, cp, sd, cd = sincos2_(x + y, d) 

m = cp * cd * _2_0 

s = (sp * sd / d) if d else sp 

# 2x2 matrices in row-major order 

A0, A1 = m, (-s * d**2) 

A2, A3 = (-s * _4_0), m 

b2 = b1 = (_0_0,) * 4 

if n > 0: 

b1 = c[n], _0_0, _0_0, c[n] 

for j in range(n - 1, 0, -1): 

b1, b2, cj = b2, b1, c[j] # c[0] unused 

# b1 = A * b2 - b1 + c[j] * I 

B0, B1, B2, B3 = b2 

b1 = (fsum1_(A0 * B0, A1 * B2, -b1[0], cj), 

fsum1_(A0 * B1, A1 * B3, -b1[1]), 

fsum1_(A2 * B0, A3 * B2, -b1[2]), 

fsum1_(A2 * B1, A3 * B3, -b1[3], cj)) 

# Here are the full expressions for m and s 

# f01, f02, f11, f12 = (0, 0, cd * sp, 2 * sd * cp) if sinp else \ 

# (1, 0, cd * cp, -2 * sd * sp) 

# m = -b2[1] * f02 + (c[0] - b2[0]) * f01 + b1[0] * f11 + b1[1] * f12 

# s = -b2[2] * f01 + (c[0] - b2[3]) * f02 + b1[2] * f11 + b1[3] * f12 

s = fsum1_(b1[2] * cd * sp, b1[3] * sd * cp * _2_0) if sinp else \ 

fsum1_(b1[2] * cd * cp, -b1[3] * sd * sp * _2_0, -b2[2]) 

return s 

 

 

# _Alp- and _BetCoeffs copied from I{Karney}'s U{TransverseMercator.cpp 

# <https://GeographicLib.SourceForge.io/C++/doc/TransverseMercator_8cpp_source.html>} 

_AlpCoeffs = { # Generated by Maxima on 2015-05-14 22:55:13-04:00 

4: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 4 

164, 225, -480, 360, 720, # Alp[1]/n^1, polynomial(n), order 3 

557, -864, 390, 1440, # Alp[2]/n^2, polynomial(n), order 2 

-1236, 427, 1680, # PYCHOK Alp[3]/n^3, polynomial(n), order 1 

49561, 161280), # Alp[4]/n^4, polynomial(n), order 0, count = 14 

5: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 5 

-635, 328, 450, -960, 720, 1440, # Alp[1]/n^1, polynomial(n), order 4 

4496, 3899, -6048, 2730, 10080, # PYCHOK Alp[2]/n^2, polynomial(n), order 3 

15061, -19776, 6832, 26880, # PYCHOK Alp[3]/n^3, polynomial(n), order 2 

-171840, 49561, 161280, # Alp[4]/n^4, polynomial(n), order 1 

34729, 80640), # PYCHOK Alp[5]/n^5, polynomial(n), order 0, count = 20 

6: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 6 

31564, -66675, 34440, 47250, -100800, 75600, 151200, # Alp[1]/n^1, polynomial(n), order 5 

-1983433, 863232, 748608, -1161216, 524160, 1935360, # PYCHOK Alp[2]/n^2, polynomial(n), order 4 

670412, 406647, -533952, 184464, 725760, # Alp[3]/n^3, polynomial(n), order 3 

6601661, -7732800, 2230245, 7257600, # Alp[4]/n^4, polynomial(n), order 2 

-13675556, 3438171, 7983360, # PYCHOK Alp[5]/n^5, polynomial(n), order 1 

212378941, 319334400), # Alp[6]/n^6, polynomial(n), order 0, count = 27 

7: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 7 

1804025, 2020096, -4267200, 2204160, 3024000, -6451200, 4838400, 9676800, # Alp[1]/n^1, polynomial(n), order 6 

4626384, -9917165, 4316160, 3743040, -5806080, 2620800, 9676800, # Alp[2]/n^2, polynomial(n), order 5 

-67102379, 26816480, 16265880, -21358080, 7378560, 29030400, # PYCHOK Alp[3]/n^3, polynomial(n), order 4 

155912000, 72618271, -85060800, 24532695, 79833600, # Alp[4]/n^4, polynomial(n), order 3 

102508609, -109404448, 27505368, 63866880, # Alp[5]/n^5, polynomial(n), order 2 

-12282192400, 2760926233, 4151347200, # PYCHOK Alp[6]/n^6, polynomial(n), order 1 

1522256789, 1383782400), # Alp[7]/n^7, polynomial(n), order 0, count = 35 

8: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 8 

-75900428, 37884525, 42422016, -89611200, 46287360, 63504000, -135475200, 101606400, 203212800, # Alp[1]/n^1, polynomial(n), order 7 

148003883, 83274912, -178508970, 77690880, 67374720, -104509440, 47174400, 174182400, # PYCHOK Alp[2]/n^2, polynomial(n), order 6 

318729724, -738126169, 294981280, 178924680, -234938880, 81164160, 319334400, # PYCHOK Alp[3]/n^3, polynomial(n), order 5 

-40176129013, 14967552000, 6971354016, -8165836800, 2355138720, 7664025600, # Alp[4]/n^4, polynomial(n), order 4 

10421654396, 3997835751, -4266773472, 1072709352, 2490808320, # PYCHOK Alp[5]/n^5, polynomial(n), order 3 

175214326799, -171950693600, 38652967262, 58118860800, # PYCHOK Alp[6]/n^6, polynomial(n), order 2 

-67039739596, 13700311101, 12454041600, # PYCHOK Alp[7]/n^7, polynomial(n), order 1 

1424729850961, 743921418240) # PYCHOK Alp[8]/n^8, polynomial(n), order 0, count = 44 

} 

_BetCoeffs = { # Generated by Maxima on 2015-05-14 22:55:13-04:00 

4: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 4 

-4, 555, -960, 720, 1440, # Bet[1]/n^1, polynomial(n), order 3 

-437, 96, 30, 1440, # Bet[2]/n^2, polynomial(n), order 2 

-148, 119, 3360, # Bet[3]/n^3, polynomial(n), order 1 

4397, 161280), # PYCHOK Bet[4]/n^4, polynomial(n), order 0, count = 14 

5: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 5 

-3645, -64, 8880, -15360, 11520, 23040, # Bet[1]/n^1, polynomial(n), order 4 

4416, -3059, 672, 210, 10080, # PYCHOK Bet[2]/n^2, polynomial(n), order 3 

-627, -592, 476, 13440, # Bet[3]/n^3, polynomial(n), order 2 

-3520, 4397, 161280, # Bet[4]/n^4, polynomial(n), order 1 

4583, 161280), # PYCHOK Bet[5]/n^5, polynomial(n), order 0, count = 20 

6: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 6 

384796, -382725, -6720, 932400, -1612800, 1209600, 2419200, # Bet[1]/n^1, polynomial(n), order 5 

-1118711, 1695744, -1174656, 258048, 80640, 3870720, # PYCHOK Bet[2]/n^2, polynomial(n), order 4 

22276, -16929, -15984, 12852, 362880, # Bet[3]/n^3, polynomial(n), order 3 

-830251, -158400, 197865, 7257600, # PYCHOK Bet[4]/n^4, polynomial(n), order 2 

-435388, 453717, 15966720, # PYCHOK Bet[5]/n^5, polynomial(n), order 1 

20648693, 638668800), # Bet[6]/n^6, polynomial(n), order 0, count = 27 

7: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 7 

-5406467, 6156736, -6123600, -107520, 14918400, -25804800, 19353600, 38707200, # Bet[1]/n^1, polynomial(n), order 6 

829456, -5593555, 8478720, -5873280, 1290240, 403200, 19353600, # PYCHOK Bet[2]/n^2, polynomial(n), order 5 

9261899, 3564160, -2708640, -2557440, 2056320, 58060800, # PYCHOK Bet[3]/n^3, polynomial(n), order 4 

14928352, -9132761, -1742400, 2176515, 79833600, # PYCHOK Bet[4]/n^4, polynomial(n), order 3 

-8005831, -1741552, 1814868, 63866880, # Bet[5]/n^5, polynomial(n), order 2 

-261810608, 268433009, 8302694400, # Bet[6]/n^6, polynomial(n), order 1 

219941297, 5535129600), # PYCHOK Bet[7]/n^7, polynomial(n), order 0, count = 35 

8: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 8 

31777436, -37845269, 43097152, -42865200, -752640, 104428800, -180633600, 135475200, 270950400, # Bet[1]/n^1, polynomial(n), order 7 

24749483, 14930208, -100683990, 152616960, -105719040, 23224320, 7257600, 348364800, # Bet[2]/n^2, polynomial(n), order 6 

-232468668, 101880889, 39205760, -29795040, -28131840, 22619520, 638668800, # PYCHOK Bet[3]/n^3, polynomial(n), order 5 

324154477, 1433121792, -876745056, -167270400, 208945440, 7664025600, # Bet[4]/n^4, polynomial(n), order 4 

457888660, -312227409, -67920528, 70779852, 2490808320, # Bet[5]/n^5, polynomial(n), order 3 

-19841813847, -3665348512, 3758062126, 116237721600, # PYCHOK Bet[6]/n^6, polynomial(n), order 2 

-1989295244, 1979471673, 49816166400, # PYCHOK Bet[7]/n^7, polynomial(n), order 1 

191773887257, 3719607091200) # Bet[8]/n^8, polynomial(n), order 0, count = 44 

} 

_RCoeffs = { # Generated by Maxima on 2015-05-15 08:24:04-04:00 

4: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 4 

691, 7860, -20160, 18900, 0, 56700, # R[0]/n^0, polynomial(n), order 4 

1772, -5340, 6930, -4725, 14175, # R[1]/n^1, polynomial(n), order 3 

-1747, 1590, -630, 4725, # PYCHOK R[2]/n^2, polynomial(n), order 2 

104, -31, 315, # R[3]/n^3, polynomial(n), order 1 

-41, 420), # PYCHOK R[4]/n^4, polynomial(n), order 0, count = 20 

5: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 5 

-79036, 22803, 259380, -665280, 623700, 0, 1871100, # PYCHOK R[0]/n^0, polynomial(n), order 5 

41662, 58476, -176220, 228690, -155925, 467775, # PYCHOK R[1]/n^1, polynomial(n), order 4 

18118, -57651, 52470, -20790, 155925, # PYCHOK R[2]/n^2, polynomial(n), order 3 

-23011, 17160, -5115, 51975, # PYCHOK R[3]/n^3, polynomial(n), order 2 

5480, -1353, 13860, # PYCHOK R[4]/n^4, polynomial(n), order 1 

-668, 5775), # PYCHOK R[5]/n^5, polynomial(n), order 0, count = 27 

6: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 6 

128346268, -107884140, 31126095, 354053700, -908107200, 851350500, 0, 2554051500, # R[0]/n^0, polynomial(n), order 6 

-114456994, 56868630, 79819740, -240540300, 312161850, -212837625, 638512875, # PYCHOK R[1]/n^1, polynomial(n), order 5 

51304574, 24731070, -78693615, 71621550, -28378350, 212837625, # R[2]/n^2, polynomial(n), order 4 

1554472, -6282003, 4684680, -1396395, 14189175, # R[3]/n^3, polynomial(n), order 3 

-4913956, 3205800, -791505, 8108100, # PYCHOK R[4]/n^4, polynomial(n), order 2 

1092376, -234468, 2027025, # R[5]/n^5, polynomial(n), order 1 

-313076, 2027025), # PYCHOK R[6]/n^6, polynomial(n), order 0, count = 35 

7: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 7 

-317195588, 385038804, -323652420, 93378285, 1062161100, -2724321600, 2554051500, 0, 7662154500, # PYCHOK R[0]/n^0, polynomial(n), order 7 

258618446, -343370982, 170605890, 239459220, -721620900, 936485550, -638512875, 1915538625, # PYCHOK R[1]/n^1, polynomial(n), order 6 

-248174686, 153913722, 74193210, -236080845, 214864650, -85135050, 638512875, # PYCHOK R[2]/n^2, polynomial(n), order 5 

114450437, 23317080, -94230045, 70270200, -20945925, 212837625, # PYCHOK R[3]/n^3, polynomial(n), order 4 

15445736, -103193076, 67321800, -16621605, 170270100, # PYCHOK R[4]/n^4, polynomial(n), order 3 

-27766753, 16385640, -3517020, 30405375, # PYCHOK R[4]/n^4, polynomial(n), order 3 

4892722, -939228, 6081075, # PYCHOK R[4]/n^4, polynomial(n), order 3 

-3189007, 14189175), # PYCHOK R[7]/n^7, polynomial(n), order 0, count = 44 

8: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 8 

71374704821, -161769749880, 196369790040, -165062734200, 47622925350, 541702161000, -1389404016000, 1302566265000, 0, 3907698795000, # R[0]/n^0, polynomial(n), order 8 

-13691187484, 65947703730, -87559600410, 43504501950, 61062101100, -184013329500, 238803815250, -162820783125, 488462349375, # PYCHOK R[1]/n^1, polynomial(n), order 7 

30802104839, -63284544930, 39247999110, 18919268550, -60200615475, 54790485750, -21709437750, 162820783125, # R[2]/n^2, polynomial(n), order 6 

-8934064508, 5836972287, 1189171080, -4805732295, 3583780200, -1068242175, 10854718875, # PYCHOK R[3]/n^3, polynomial(n), order 5 

50072287748, 3938662680, -26314234380, 17167059000, -4238509275, 43418875500, # R[4]/n^4, polynomial(n), order 4 

359094172, -9912730821, 5849673480, -1255576140, 10854718875, # R[5]/n^5, polynomial(n), order 3 

-16053944387, 8733508770, -1676521980, 10854718875, # PYCHOK R[6]/n^6, polynomial(n), order 2 

930092876, -162639357, 723647925, # R[7]/n^7, polynomial(n), order 1 

-673429061, 1929727800) # PYCHOK R[8]/n^8, polynomial(n), order 0, count = 54 

} 

 

assert set(_AlpCoeffs.keys()) == set(_BetCoeffs.keys()) 

 

__all__ += _ALL_DOCS(_RhumbLine) 

 

if __name__ == '__main__': 

 

def _re(fmt, r3, x3): 

e3 = [] 

for r, x in zip(r3, x3): 

e = abs(r - x) / abs(x) 

e3.append('%.g' % (e,)) 

print((fmt % r3) + ' errors: ' + ', '.join(e3)) 

 

# <https://GeographicLib.SourceForge.io/cgi-bin/RhumbSolve> 

rhumb = Rhumb(exact=True) # WGS84 default 

r = rhumb.Direct7(40.6, -73.8, 51, 5.5e6) # from JFK about NE 

_re('# lat2=%.8f, lon2=%.8f, S12=%.1f', (r.lat2, r.lon2, r.S12), (71.68889988, 0.25551982, 44095641862956)) 

r = rhumb.Inverse7(40.6, -73.8, 51.6, -0.5) # JFK to LHR 

_re('# azi12=%.8f, s12=%.3f S12=%.1f', (r.azi12, r.s12, r.S12), (77.76838971, 5771083.383, 37395209100030)) 

r = rhumb.Inverse7(40.6, -73.8, 35.8, 140.3) # JFK to Tokyo Narita 

_re('# azi12=%.8f, s12=%.3f S12=%.1f', (r.azi12, r.s12, r.S12), (-92.38888798, 12782581.068, -63760642939073)) 

 

# % python3 -m pygeodesy.rhumbx 

 

# lat2=71.68889988, lon2=0.25551982, S12=44054189889953.1 errors: 4e-11, 2e-08, 0.0009 

# azi12=77.76838971, s12=5771083.383 S12=37362984299663.1 errors: 3e-12, 6e-11, 0.0009 

# azi12=-92.38888798, s12=12782581.068 S12=-63665156875078.1 errors: 2e-11, 2e-11, 0.001 

 

# **) MIT License 

# 

# Copyright (C) 2022-2022 -- 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.