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

 

# -*- coding: utf-8 -*- 

 

u'''I{Geocentric} Earth-Centered, Earth-Fixed (ECEF) coordinates. 

 

Geocentric conversions transcoded from I{Charles Karney}'s C++ class U{Geocentric 

<https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1Geocentric.html>} 

into pure Python class L{EcefKarney}, class L{EcefSudano} based on I{John Sudano}'s 

U{paper<https://www.ResearchGate.net/publication/ 

3709199_An_exact_conversion_from_an_Earth-centered_coordinate_system_to_latitude_longitude_and_altitude>}, 

class L{EcefVeness} transcoded from I{Chris Veness}' JavaScript classes U{LatLonEllipsoidal, 

Cartesian<https://www.Movable-Type.co.UK/scripts/geodesy/docs/latlon-ellipsoidal.js.html>}, class L{EcefYou} 

implementing I{Rey-Jer You}'s U{transformations <https://www.ResearchGate.net/publication/240359424>} and 

classes L{EcefFarrell22} and L{EcefFarrell22} from I{Jay A. Farrell}'s U{Table 2.1 and 2.2 

<https://Books.Google.com/books?id=fW4foWASY6wC>}, page 29-30. 

 

Following is a copy of I{Karney}'s U{Detailed Description 

<https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1Geocentric.html>}. 

 

Convert between geodetic coordinates C{lat}-, C{lon}gitude and height C{h} 

(measured vertically from the surface of the ellipsoid) to geocentric C{x}, 

C{y} and C{z} coordinates, also known as I{Earth-Centered, Earth-Fixed} 

(U{ECEF<https://WikiPedia.org/wiki/ECEF>}). 

 

The origin of geocentric coordinates is at the center of the earth. The C{z} 

axis goes thru the North pole, C{lat} = 90°. The C{x} axis goes thru C{lat} 

= 0°, C{lon} = 0°. 

 

The I{local (cartesian) origin} is at (C{lat0}, C{lon0}, C{height0}). The I{local} C{x} 

axis points East, the I{local} C{y} axis points North and the I{local} C{z} axis is 

normal to the ellipsoid. The plane C{z = -height0} is tangent to the ellipsoid, hence 

the alternate name I{local tangent plane}. 

 

Forward conversion from geodetic to geocentric (ECEF) coordinates is straightforward. 

 

For the reverse transformation we use Hugues Vermeille's U{Direct transformation 

from geocentric coordinates to geodetic coordinates 

<https://DOI.org/10.1007/s00190-002-0273-6>}, J. Geodesy (2002) 76, 451-454. 

 

Several changes have been made to ensure that the method returns accurate 

results for all finite inputs (even if h is infinite). The changes are 

described in Appendix B of C. F. F. Karney U{Geodesics on an ellipsoid of 

revolution<https://ArXiv.org/abs/1102.1215v1>}, Feb. 2011, 85, 105-117 

(U{preprint<https://ArXiv.org/abs/1102.1215v1>}). Vermeille similarly updated 

his method in U{An analytical method to transform geocentric into geodetic 

coordinates<https://DOI.org/10.1007/s00190-010-0419-x>}, J. Geodesy (2011) 85, 

105-117. See U{Geocentric coordinates 

<https://GeographicLib.SourceForge.io/html/geocentric.html>} for more information. 

 

The errors in these routines are close to round-off. Specifically, for points 

within 5,000 km of the surface of the ellipsoid (either inside or outside the 

ellipsoid), the error is bounded by 7 nm (7 nanometers) for the WGS84 ellipsoid. 

See U{Geocentric coordinates<https://GeographicLib.SourceForge.io/html/geocentric.html>} 

for further information on the errors. 

 

@see: Module L{ltp} and class L{LocalCartesian}, a transcription of I{Charles Karney}'s 

C++ class U{LocalCartesian 

<https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1LocalCartesian.html>}, 

providing conversion to and from I{local} cartesian cordinates in a I{local tangent 

plane} as opposed to I{geocentric} (ECEF) ones. 

''' 

 

from pygeodesy.basics import copysign0, isnon0, isscalar, issubclassof, \ 

neg, map1, _xinstanceof, _xsubclassof 

from pygeodesy.datums import _ellipsoidal_datum 

from pygeodesy.ellipsoids import a_f2Tuple 

from pygeodesy.errors import _datum_datum, _IndexError, LenError, \ 

_ValueError, _TypesError, _xkwds 

from pygeodesy.fmath import cbrt, fdot, Fsum, fsum_, hypot, hypot1, hypot2_ 

from pygeodesy.interns import EPS, EPS0, EPS02, EPS1, EPS_2, NN, PI, PI_2, \ 

_a_, _C_, _convergence_, _datum_, _ellipsoid_, \ 

_f_, _h_, _height_, _lat_, _lon_, _M_, _name_, \ 

_no_, _singular_, _SPACE_, _x_, _xyz_, _y_, _z_, \ 

_0_0, _0_5, _1_0, _1_0_T, _2_0, _3_0, _4_0, \ 

_6_0, _90_0 

from pygeodesy.interns import _N_2_0 # PYCHOK used! 

from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY 

from pygeodesy.named import _NamedBase, _NamedTuple, notOverloaded, \ 

_Pass, _xnamed 

from pygeodesy.namedTuples import LatLon2Tuple, LatLon3Tuple, \ 

PhiLam2Tuple, Vector3Tuple, Vector4Tuple 

from pygeodesy.props import deprecated_method, Property_RO 

from pygeodesy.streprs import unstr 

from pygeodesy.units import Height, Int, Lam, Lat, Lon, Meter, Phi, Scalar 

from pygeodesy.utily import atan2d, degrees90, degrees180, \ 

sincos2, sincos2_, sincos2d_ 

 

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

 

__all__ = _ALL_LAZY.ecef 

__version__ = '21.11.18' 

 

_Ecef_ = 'Ecef' 

_prolate_ = 'prolate' 

_TRIPS = 17 # 8..9 sufficient, EcefSudano.reverse 

 

 

class EcefError(_ValueError): 

'''An ECEF or C{Ecef*} related issue. 

''' 

pass 

 

 

def _llhn4(latlonh, lon, height, suffix=NN, Error=EcefError, name=NN): # in .ltp.LocalCartesian.forward and -.reset 

'''(INTERNAL) Get C{lat, lon, h, name} as C{4-tuple}. 

''' 

try: 

lat, lon = latlonh.lat, latlonh.lon 

h = getattr(latlonh, _height_, 

getattr(latlonh, _h_, height)) 

n = getattr(latlonh, _name_, NN) 

except AttributeError: 

lat, h, n = latlonh, height, NN 

 

try: 

llhn = Lat(lat), Lon(lon), Height(h), (name or n) 

except (TypeError, ValueError) as x: 

t = _lat_, _lon_, _height_ 

if suffix: 

t = (_ + suffix for _ in t) 

d = dict(zip(t, (lat, lon, h))) 

raise Error(txt=str(x), **d) 

return llhn 

 

 

def _sch3(y, x): 

'''(INTERNAL) Compute sin, cos and hypotenuse. 

''' 

h = hypot(y, x) 

if h > 0: # EPS_2 

s, c = y / h, x / h 

else: 

s, c = _0_0, _1_0 

return s, c, h 

 

 

def _xyzn4(xyz, y, z, Types, Error=EcefError, name=NN): # in .ltp, @see: .vector3d 

'''(INTERNAL) Get an C{(x, y, z, name)} 4-tuple. 

''' 

try: 

try: 

t = xyz.x, xyz.y, xyz.z, getattr(xyz, _name_, name) 

if not isinstance(xyz, Types): 

raise _TypesError(_xyz_, xyz, *Types) 

except AttributeError: 

t = map1(float, xyz, y, z) + (name,) 

 

except (TypeError, ValueError) as x: 

d = dict(zip((_xyz_, _y_, _z_), (xyz, y, z))) 

raise Error(txt=str(x), **d) 

 

return t 

 

 

class _EcefBase(_NamedBase): 

'''(INTERNAL) Base class for L{EcefKarney}, L{EcefVeness} and L{EcefYou}. 

''' 

_datum = None 

_E = None 

 

def __init__(self, a_ellipsoid, f=None, name=NN): 

'''New C{Ecef*} converter. 

 

@arg a_ellipsoid: A (non-prolate) ellipsoid (L{Ellipsoid}, L{Ellipsoid2}, 

L{Datum} or L{a_f2Tuple}) or C{scalar} ellipsoid's 

equatorial radius (C{meter}). 

@kwarg f: C{None} or the ellipsoid flattening (C{scalar}), required 

for C{scalar} B{C{a_ellipsoid}}, C{B{f}=0} represents a 

sphere, negative B{C{f}} a prolate ellipsoid. 

@kwarg name: Optional name (C{str}). 

 

@raise EcefError: If B{C{a_ellipsoid}} not L{Ellipsoid}, L{Ellipsoid2}, 

L{Datum} or L{a_f2Tuple} or C{scalar} or B{C{f}} not 

C{scalar} or if C{scalar} B{C{a_ellipsoid}} not positive 

or B{C{f}} not less than 1.0. 

''' 

if name: 

self.name = name 

try: 

E = a_ellipsoid 

if f is None: 

pass 

elif isscalar(E) and isscalar(f): 

E = a_f2Tuple(E, f) 

else: 

raise ValueError 

 

d = _ellipsoidal_datum(E, name=name) 

E = d.ellipsoid 

if E.a < EPS or E.f > EPS1: 

raise ValueError 

 

except (TypeError, ValueError) as x: 

t = unstr(self.classname, a=a_ellipsoid, f=f) 

raise EcefError(_SPACE_(t, _ellipsoid_), txt=str(x)) 

 

self._datum = d 

self._E = E 

 

def __eq__(self, other): 

'''Compare this and an other Ecef. 

 

@arg other: The other ecef (C{Ecef*}). 

 

@return: C{True} if equal, C{False} otherwise. 

''' 

return other is self or (isinstance(other, _EcefBase) and 

other.ellipsoid == self.ellipsoid) 

 

@Property_RO 

def equatoradius(self): 

'''Get the I{equatorial} radius, semi-axis (C{meter}). 

''' 

return self._E.a 

 

a = equatorialRadius = equatoradius # Karney property 

 

@Property_RO 

def datum(self): 

'''Get the datum (L{Datum}). 

''' 

return self._datum 

 

@Property_RO 

def ellipsoid(self): 

'''Get the ellipsoid (L{Ellipsoid} or L{Ellipsoid2}). 

''' 

return self._E 

 

@Property_RO 

def flattening(self): # Karney property 

'''Get the I{flattening} (C{float}), M{(a - b) / a}, positive for 

I{oblate}, negative for I{prolate} or C{0} for I{near-spherical}. 

''' 

return self._E.f 

 

f = flattening 

 

def _forward(self, lat, lon, h, name, M=False, _philam=False): # in .ltp.LocalCartesian.forward and -.reset 

'''(INTERNAL) Common for all C{Ecef*}. 

''' 

E = self.ellipsoid 

 

if _philam: 

sa, ca, sb, cb = sincos2_(lat, lon) 

lat = Lat(degrees90( lat)) 

lon = Lon(degrees180(lon)) 

else: 

sa, ca, sb, cb = sincos2d_(lat, lon) 

 

n = E.roc1_(sa, ca) if self._isYou else E.roc1_(sa) 

z = (h + n * E.e12) * sa 

x = (h + n) * ca 

 

m = self._Matrix(sa, ca, sb, cb) if M else None 

return Ecef9Tuple(x * cb, x * sb, z, lat, lon, h, 

0, m, self.datum, 

name=name or self.name) 

 

def forward(self, latlonh, lon=None, height=0, M=False, name=NN): 

'''Convert from geodetic C{(lat, lon, height)} to geocentric C{(x, y, z)}. 

 

@arg latlonh: Either a C{LatLon}, an L{Ecef9Tuple} or C{scalar} 

latitude (C{degrees}). 

@kwarg lon: Optional C{scalar} longitude for C{scalar} B{C{latlonh}} 

(C{degrees}). 

@kwarg height: Optional height (C{meter}), vertically above (or below) 

the surface of the ellipsoid. 

@kwarg M: Optionally, return the rotation L{EcefMatrix} (C{bool}). 

@kwarg name: Optional name (C{str}). 

 

@return: An L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)} with 

geocentric C{(x, y, z)} coordinates for the given geodetic ones 

C{(lat, lon, height)}, case C{C} 0, optional C{M} (L{EcefMatrix}) 

and C{datum} if available. 

 

@raise EcefError: If B{C{latlonh}} not C{LatLon}, L{Ecef9Tuple} or 

C{scalar} or B{C{lon}} not C{scalar} for C{scalar} 

B{C{latlonh}} or C{abs(lat)} exceeds 90°. 

 

@note: Use method C{.forward_} to specify C{lat} and C{lon} in C{radians} 

and avoid double angle conversions. 

''' 

llhn = _llhn4(latlonh, lon, height, name=name) 

return _EcefBase._forward(self, *llhn, M=M) 

 

def forward_(self, phi, lam, height=0, M=False, name=NN): 

'''Like method C{.forward} except with geodetic lat- and longitude given 

in I{radians}. 

 

@arg phi: Latitude in I{radians} (C{scalar}). 

@arg lam: Longitude in I{radians} (C{scalar}). 

@kwarg height: Optional height (C{meter}), vertically above (or below) 

the surface of the ellipsoid. 

@kwarg M: Optionally, return the rotation L{EcefMatrix} (C{bool}). 

@kwarg name: Optional name (C{str}). 

 

@return: An L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)} 

with C{lat} set to C{degrees90(B{phi})} and C{lon} to 

C{degrees180(B{lam})}. 

 

@raise EcefError: If B{C{phi}} or B{C{lam}} invalid or not C{scalar}. 

''' 

try: # like function C{_llhn4} above 

plhn = Phi(phi), Lam(lam), Height(height), name 

except (TypeError, ValueError) as x: 

raise EcefError(phi=phi, lam=lam, height=height, txt=str(x)) 

return self._forward(*plhn, M=M, _philam=True) 

 

@Property_RO 

def _Geocentrics(self): 

'''(INTERNAL) Valid geocentric classes. 

''' 

from pygeodesy.cartesianBase import CartesianBase 

return Ecef9Tuple, CartesianBase 

 

@Property_RO 

def _isYou(self): 

'''(INTERNAL) Is this an C{EcefYou}?. 

''' 

return isinstance(self, EcefYou) 

 

def _Matrix(self, sa, ca, sb, cb): 

'''Creation a rotation matrix. 

 

@arg sa: C{sin(phi)} (C{float}). 

@arg ca: C{cos(phi)} (C{float}). 

@arg sb: C{sin(lambda)} (C{float}). 

@arg cb: C{cos(lambda)} (C{float}). 

 

@return: An L{EcefMatrix}. 

''' 

return self._xnamed(EcefMatrix(sa, ca, sb, cb)) 

 

def reverse(self, xyz, y=None, z=None, M=False, name=NN): # PYCHOK no cover 

'''(INTERNAL) I{Must be overloaded}, see function C{notOverloaded}. 

''' 

notOverloaded(self, xyz, y=y, z=z, M=M, name=name) 

 

def toStr(self, prec=9, **unused): # PYCHOK signature 

'''Return this C{Ecef*} as a string. 

 

@kwarg prec: Optional precision, number of decimal digits (0..9). 

 

@return: This C{Ecef*} representation (C{str}). 

''' 

return self.attrs(_a_, _f_, _datum_, _ellipsoid_, _name_, prec=prec) 

 

 

class EcefKarney(_EcefBase): 

'''Conversion between geodetic and geocentric, aka I{Earth-Centered, 

Earth-Fixed} (ECEF) coordinates transcoded from I{Karney}'s C++ U{Geocentric 

<https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1Geocentric.html>} 

methods. 

 

@note: On methods C{.forward} and C{.forwar_}, let C{v} be a unit vector located 

at C{(lat, lon, h)}. We can express C{v} as column vectors in one of two 

ways, C{v1} in east, north, up coordinates (where the components are 

relative to a local coordinate system at C{C(lat0, lon0, h0)}) or as C{v0} 

in geocentric C{x, y, z} coordinates. Then, M{v0 = M ⋅ v1} where C{M} is 

the rotation matrix. 

''' 

 

@Property_RO 

def hmax(self): 

'''Get the distance or height limit (C{meter}, conventionally). 

''' 

return self.equatoradius / EPS_2 # self.equatoradius * _2_EPS, 12M lighyears 

 

def reverse(self, xyz, y=None, z=None, M=False, name=NN): 

'''Convert from geocentric C{(x, y, z)} to geodetic C{(lat, lon, height)}. 

 

@arg xyz: A geocentric (C{Cartesian}, L{Ecef9Tuple}) or C{scalar} ECEF C{x} 

coordinate (C{meter}). 

@kwarg y: ECEF C{y} coordinate for C{scalar} B{C{xyz}} and B{C{z}} (C{meter}). 

@kwarg z: ECEF C{z} coordinate for C{scalar} B{C{xyz}} and B{C{y}} (C{meter}). 

@kwarg M: Optionally, return the rotation L{EcefMatrix} (C{bool}). 

@kwarg name: Optional name (C{str}). 

 

@return: An L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)} with 

geodetic coordinates C{(lat, lon, height)} for the given geocentric 

ones C{(x, y, z)}, case C{C}, optional C{M} (L{EcefMatrix}) and 

C{datum} if available. 

 

@raise EcefError: Invalid B{C{xyz}} or C{scalar} C{x} or B{C{y}} and/or B{C{z}} 

not C{scalar} for C{scalar} B{C{xyz}}. 

 

@note: In general, there are multiple solutions and the result which minimizes 

C{height} is returned, i.e., C{(lat, lon)} corresponds to the closest 

point on the ellipsoid. If there are still multiple solutions with 

different latitudes (applies only if C{z} = 0), then the solution with 

C{lat} > 0 is returned. If there are still multiple solutions with 

different longitudes (applies only if C{x} = C{y} = 0) then C{lon} = 0 

is returned. The returned C{height} value is not below M{−E.a * (1 − 

E.e2) / sqrt(1 − E.e2 * sin(lat)**2)}. The returned C{lon} is in the 

range [−180°, 180°]. Like C{forward} above, M{v1 = Transpose(M) ⋅ v0}. 

''' 

x, y, z, name = _xyzn4(xyz, y, z, self._Geocentrics, name=name) 

 

E = self.ellipsoid 

 

sb, cb, R = _sch3(y, x) 

h = hypot(R, z) # distance to earth center 

if h > self.hmax: # PYCHOK no cover 

# We are really far away (> 12M light years). Treat the earth 

# as a point and h, above as an acceptable approximation to the 

# height. This avoids overflow, e.g., in the computation of disc 

# below. It's possible that h has overflowed to INF, that's OK. 

# Treat finite x, y, but R overflows to +INF by scaling by 2. 

sb, cb, R = _sch3(y * _0_5, x * _0_5) 

sa, ca, _ = _sch3(z * _0_5, R) 

C = 1 

 

elif E.e4: # E.isEllipsoidal 

# Treat prolate spheroids by swapping R and Z here and by 

# switching the arguments to phi = atan2(...) at the end. 

p = (R / E.a)**2 

q = E.e12 * (z / E.a)**2 

if E.isProlate: 

p, q = q, p 

r = p + q - E.e4 

e = E.e4 * q 

if e or r > 0: 

# Avoid possible division by zero when r = 0 by multiplying 

# equations for s and t by r^3 and r, respectively. 

s = e * p / _4_0 # s = r^3 * s 

u = r = r / _6_0 

r2 = r**2 

r3 = r * r2 

t3 = s + r3 

disc = s * (r3 + t3) 

if disc < 0: 

# t is complex, but the way u is defined, the result is real. 

# There are three possible cube roots. We choose the root 

# which avoids cancellation. Note, disc < 0 implies r < 0. 

u += _2_0 * r * cos(atan2(sqrt(-disc), -t3) / _3_0) 

else: 

# Pick the sign on the sqrt to maximize abs(T3). This 

# minimizes loss of precision due to cancellation. The 

# result is unchanged because of the way the t is used 

# in definition of u. 

if disc > 0: 

t3 += copysign0(sqrt(disc), t3) # t3 = (r * t)^3 

# N.B. cbrt always returns the real root, cbrt(-8) = -2. 

t = cbrt(t3) # t = r * t 

# t can be zero; but then r2 / t -> 0. 

if t: 

u = fsum_(u, t, r2 / t) 

v = sqrt(e + u**2) # guaranteed positive 

# Avoid loss of accuracy when u < 0. Underflow doesn't occur in 

# E.e4 * q / (v - u) because u ~ e^4 when q is small and u < 0. 

uv = (e / (v - u)) if u < 0 else (u + v) # u+v, guaranteed positive 

# Need to guard against w going negative due to roundoff in uv - q. 

w = max(_0_0, E.e2abs * (uv - q) / (_2_0 * v)) 

# Rearrange expression for k to avoid loss of accuracy due to 

# subtraction. Division by 0 not possible because uv > 0, w >= 0. 

k1 = k2 = uv / (sqrt(uv + w**2) + w) 

if E.isProlate: 

k1 -= E.e2 

else: 

k2 += E.e2 

sa, ca, h = _sch3(z / k1, R / k2) 

h *= k1 - E.e12 

C = 2 

 

else: # e = E.e4 * q == 0 and r <= 0 

# This leads to k = 0 (oblate, equatorial plane) and k + E.e^2 = 0 

# (prolate, rotation axis) and the generation of 0/0 in the general 

# formulas for phi and h, using the general formula and division 

# by 0 in formula for h. Handle this case by taking the limits: 

# f > 0: z -> 0, k -> E.e2 * sqrt(q) / sqrt(E.e4 - p) 

# f < 0: r -> 0, k + E.e2 -> -E.e2 * sqrt(q) / sqrt(E.e4 - p) 

q = E.e4 - p 

if E.isProlate: 

p, q = q, p 

e = E.a 

else: 

e = E.b2_a 

sa, ca, h = _sch3(sqrt(q / E.e12), sqrt(p)) 

if z < 0: 

sa = neg(sa) # for tiny negative z, not for prolate 

h *= neg(e / E.e2abs) 

C = 3 

 

else: # E.e4 == 0, spherical case 

# Dealing with underflow in the general case with E.e2 = 0 is 

# difficult. Origin maps to North pole, same as with ellipsoid. 

sa, ca, _ = _sch3((z if h else _1_0), R) 

h -= E.a 

C = 4 

 

m = self._Matrix(sa, ca, sb, cb) if M else None 

return Ecef9Tuple(x, y, z, atan2d(sa, ca), 

atan2d(sb, cb), h, 

C, m, self.datum, 

name=name or self.name) 

 

 

class EcefFarrell21(_EcefBase): 

'''Conversion between geodetic and geocentric, aka I{Earth-Centered, 

Earth-Fixed} (ECEF) coordinates based on I{Jay A. Farrell}'s U{Table 

2.1<https://Books.Google.com/books?id=fW4foWASY6wC>}, page 29. 

''' 

 

def reverse(self, xyz, y=None, z=None, M=None, name=NN): # PYCHOK unused M 

'''Convert from geocentric C{(x, y, z)} to geodetic C{(lat, lon, height)} using 

I{Farrell}'s U{Table 2.1<https://Books.Google.com/books?id=fW4foWASY6wC>}, 

page 29. 

 

@arg xyz: A geocentric (C{Cartesian}, L{Ecef9Tuple}) or C{scalar} ECEF C{x} 

coordinate (C{meter}). 

@kwarg y: ECEF C{y} coordinate for C{scalar} B{C{xyz}} and B{C{z}} (C{meter}). 

@kwarg z: ECEF C{z} coordinate for C{scalar} B{C{xyz}} and B{C{y}} (C{meter}). 

@kwarg M: I{Ignored}, rotation matrix C{M} not available. 

@kwarg name: Optional name (C{str}). 

 

@return: An L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)} with 

geodetic coordinates C{(lat, lon, height)} for the given geocentric 

ones C{(x, y, z)}, case C{C=1}, C{M=None} always and C{datum} 

if available. 

 

@raise EcefError: Invalid B{C{xyz}} or C{scalar} C{x} or B{C{y}} and/or B{C{z}} 

not C{scalar} for C{scalar} B{C{xyz}} or C{sqrt} domain or 

zero division error. 

 

@see: L{EcefFarrell22} and L{EcefVeness}. 

''' 

x, y, z, name = _xyzn4(xyz, y, z, self._Geocentrics, name=name) 

 

E = self.ellipsoid 

a = E.a 

a2 = E.a2 

b2 = E.b2 

e_ = E.a_b * E.e # 0.0820944... WGS84 

e2 = E.e2 

e4 = E.e4 

 

try: 

z2 = z**2 

ez = (_1_0 - e2) * z2 # E.e2s2(z) 

 

p = hypot(x, y) 

p2 = p**2 

F = 54 * b2 * z2 

G = p2 + ez - e2 * (a2 - b2) # p2 + ez - e4 * a2 

c = e4 * F * p2 / G**3 

s = cbrt(_1_0 + c + sqrt(c**2 + c + c)) 

P = F / (_3_0 * fsum_(_1_0, s, _1_0 / s)**2 * G**2) 

Q = sqrt(_1_0 + _2_0 * e4 * P) 

Q1 = Q + _1_0 

r0 = P * e2 * p / Q1 - sqrt(fsum_(a2 * (Q1 / Q) * _0_5, 

-P * ez / (Q * Q1), 

-P * p2 * _0_5)) 

r = p + e2 * r0 

v = b2 / (a * sqrt(r**2 + ez)) 

 

h = hypot(r, z) * (_1_0 - v) 

t = atan2(z + e_**2 * v * z, p) 

 

except (ValueError, ZeroDivisionError) as e: 

raise EcefError(x=x, y=y, z=z, txt=str(e)) 

 

return Ecef9Tuple(x, y, z, degrees90(t), atan2d(y, x), h, 

1, None, self.datum, 

name=name or self.name) 

 

 

class EcefFarrell22(_EcefBase): 

'''Conversion between geodetic and geocentric, aka I{Earth-Centered, 

Earth-Fixed} (ECEF) coordinates based on I{Jay A. Farrell}'s U{Table 

2.2<https://Books.Google.com/books?id=fW4foWASY6wC>}, page 30. 

''' 

 

def reverse(self, xyz, y=None, z=None, M=None, name=NN): # PYCHOK unused M 

'''Convert from geocentric C{(x, y, z)} to geodetic C{(lat, lon, height)} using 

I{Farrell}'s U{Table 2.2<https://Books.Google.com/books?id=fW4foWASY6wC>}, 

page 30. 

 

@arg xyz: A geocentric (C{Cartesian}, L{Ecef9Tuple}) or C{scalar} ECEF C{x} 

coordinate (C{meter}). 

@kwarg y: ECEF C{y} coordinate for C{scalar} B{C{xyz}} and B{C{z}} (C{meter}). 

@kwarg z: ECEF C{z} coordinate for C{scalar} B{C{xyz}} and B{C{y}} (C{meter}). 

@kwarg M: I{Ignored}, rotation matrix C{M} not available. 

@kwarg name: Optional name (C{str}). 

 

@return: An L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)} with 

geodetic coordinates C{(lat, lon, height)} for the given geocentric 

ones C{(x, y, z)}, case C{C=1}, C{M=None} always and C{datum} 

if available. 

 

@raise EcefError: Invalid B{C{xyz}} or C{scalar} C{x} or B{C{y}} and/or B{C{z}} 

not C{scalar} for C{scalar} B{C{xyz}} or C{sqrt} domain or 

zero division error. 

 

@see: L{EcefFarrell21} and L{EcefVeness}. 

''' 

x, y, z, name = _xyzn4(xyz, y, z, self._Geocentrics, name=name) 

 

E = self.ellipsoid 

a = E.a 

b = E.b 

 

try: # see EcefVeness.reverse 

p = hypot(x, y) 

s, c = sincos2(atan2(z * a, p * b)) 

 

t = atan2(z + E.e22 * b * s**3, 

p - E.e2 * a * c**3) 

 

s, c = sincos2(t) 

h = p / c - E.roc1_(s) # E.a / sqrt(1 - e2 * s**2) 

 

except (ValueError, ZeroDivisionError) as e: 

raise EcefError(x=x, y=y, z=z, txt=str(e)) 

 

return Ecef9Tuple(x, y, z, degrees90(t), atan2d(y, x), h, 

1, None, self.datum, 

name=name or self.name) 

 

 

class EcefSudano(_EcefBase): 

'''Conversion between geodetic and geocentric, aka I{Earth-Centered, 

Earth-Fixed} (ECEF) coordinates based on I{John J. Sudano}'s U{paper 

<https://www.ResearchGate.net/publication/ 

3709199_An_exact_conversion_from_an_Earth-centered_coordinate_system_to_latitude_longitude_and_altitude>}. 

''' 

 

def reverse(self, xyz, y=None, z=None, M=None, name=NN): # PYCHOK unused M 

'''Convert from geocentric C{(x, y, z)} to geodetic C{(lat, lon, height)} using 

I{Sudano}'s U{iterative method<https://www.ResearchGate.net/publication/ 

3709199_An_exact_conversion_from_an_Earth-centered_coordinate_system_to_latitude_longitude_and_altitude>}. 

 

@arg xyz: A geocentric (C{Cartesian}, L{Ecef9Tuple}) or C{scalar} ECEF C{x} 

coordinate (C{meter}). 

@kwarg y: ECEF C{y} coordinate for C{scalar} B{C{xyz}} and B{C{z}} (C{meter}). 

@kwarg z: ECEF C{z} coordinate for C{scalar} B{C{xyz}} and B{C{y}} (C{meter}). 

@kwarg M: I{Ignored}, rotation matrix C{M} not available. 

@kwarg name: Optional name (C{str}). 

 

@return: An L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)} with geodetic 

coordinates C{(lat, lon, height)} for the given geocentric ones C{(x, y, z)}, 

iteration C{C}, C{M=None} always and C{datum} if available. 

 

@raise EcefError: Invalid B{C{xyz}} or C{scalar} C{x} or B{C{y}} and/or B{C{z}} 

not C{scalar} for C{scalar} B{C{xyz}} or no convergence. 

''' 

x, y, z, name = _xyzn4(xyz, y, z, self._Geocentrics, name=name) 

 

E = self.ellipsoid 

e = E.e2 * E.a 

h = hypot(x, y) # Rh 

d = e - h 

 

a = atan2(z, h * E.e12) 

sa, ca = sincos2(abs(a)) 

# Sudano's Eq (A-6) and (A-7) refactored/reduced, 

# replacing Rn from Eq (A-4) with n = E.a / ca: 

# N = ca**2 * ((z + E.e2 * n * sa) * ca - h * sa) 

# = ca**2 * (z * ca + E.e2 * E.a * sa - h * sa) 

# = ca**2 * (z * ca + (E.e2 * E.a - h) * sa) 

# D = ca**3 * (E.e2 * n / E.e2s2(sa)) - h 

# = ca**2 * (E.e2 * E.a / E.e2s2(sa) - h / ca**2) 

# N / D = (z * ca + (E.e2 * E.a - h) * sa) / 

# (E.e2 * E.a / E.e2s2(sa) - h / ca**2) 

S = Fsum(sa) 

for C in range(1, _TRIPS): 

ca2 = _1_0 - sa**2 

if ca2 < EPS_2: # PYCHOK no cover 

ca = _0_0 

break 

ca = sqrt(ca2) 

t = e / E.e2s2(sa) - h / ca2 

if abs(t) < EPS_2: 

break 

a = None 

sa, t = S.fsum2_(-(z * ca + d * sa) / t) 

if abs(t) < EPS: 

break 

else: 

t = unstr(self.reverse.__name__, x=x, y=y, z=z) 

raise EcefError(t, txt=_no_(_convergence_)) 

 

if a is None: 

a = copysign0(asin(sa), z) 

h = fsum_(h * ca, abs(z * sa), -E.a * E.e2s(sa)) # use Veness', 

# since Sudano's Eq (7) doesn't provide the correct height 

# h = (abs(z) + h - E.a * cos(a + E.e12) * sa / ca) / (ca + sa) 

 

r = Ecef9Tuple(x, y, z, degrees90(a), atan2d(y, x), h, 

C, None, self.datum, 

name=name or self.name) 

r._iteration = C 

return r 

 

 

class EcefVeness(_EcefBase): 

'''Conversion between geodetic and geocentric, aka I{Earth-Centered, 

Earth-Fixed} (ECEF) coordinates transcoded from I{Chris Veness}' 

JavaScript classes U{LatLonEllipsoidal, Cartesian 

<https://www.Movable-Type.co.UK/scripts/geodesy/docs/latlon-ellipsoidal.js.html>}. 

 

@see: U{A Guide to Coordinate Systems in Great Britain 

<https://www.OrdnanceSurvey.co.UK/documents/resources/guide-coordinate-systems-great-britain.pdf>}, 

section I{B) Converting between 3D Cartesian and ellipsoidal 

latitude, longitude and height coordinates}. 

''' 

 

def reverse(self, xyz, y=None, z=None, M=None, name=NN): # PYCHOK unused M 

'''Convert from geocentric C{(x, y, z)} to geodetic C{(lat, lon, height)} 

transcoded from I{Chris Veness}' U{JavaScript 

<https://www.Movable-Type.co.UK/scripts/geodesy/docs/latlon-ellipsoidal.js.html>}. 

 

Uses B. R. Bowring’s formulation for μm precision in concise 

form: U{'The accuracy of geodetic latitude and height equations' 

<https://www.ResearchGate.net/publication/ 

233668213_The_Accuracy_of_Geodetic_Latitude_and_Height_Equations>}, 

Survey Review, Vol 28, 218, Oct 1985. 

 

@arg xyz: A geocentric (C{Cartesian}, L{Ecef9Tuple}) or C{scalar} ECEF C{x} 

coordinate (C{meter}). 

@kwarg y: ECEF C{y} coordinate for C{scalar} B{C{xyz}} and B{C{z}} (C{meter}). 

@kwarg z: ECEF C{z} coordinate for C{scalar} B{C{xyz}} and B{C{y}} (C{meter}). 

@kwarg M: I{Ignored}, rotation matrix C{M} not available. 

@kwarg name: Optional name (C{str}). 

 

@return: An L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)} with 

geodetic coordinates C{(lat, lon, height)} for the given geocentric 

ones C{(x, y, z)}, case C{C}, C{M=None} always and C{datum} if available. 

 

@raise EcefError: Invalid B{C{xyz}} or C{scalar} C{x} or B{C{y}} and/or B{C{z}} 

not C{scalar} for C{scalar} B{C{xyz}}. 

 

@see: Ralph M. Toms U{'An Efficient Algorithm for Geocentric to Geodetic 

Coordinate Conversion'<https://www.OSTI.gov/scitech/biblio/110235>}, 

Sept 1995 and U{'An Improved Algorithm for Geocentric to Geodetic 

Coordinate Conversion'<https://www.OSTI.gov/scitech/servlets/purl/231228>}, 

Apr 1996, both from Lawrence Livermore National Laboratory (LLNL) and 

John J. Sudano U{An exact conversion from an Earth-centered coordinate 

system to latitude longitude and altitude<https://www.ResearchGate.net/ 

publication/3709199_An_exact_conversion_from_an_Earth-centered_coordinate_system_to_latitude_longitude_and_altitude>}. 

''' 

x, y, z, name = _xyzn4(xyz, y, z, self._Geocentrics, name=name) 

 

E = self.ellipsoid 

 

p = hypot(x, y) # distance from minor axis 

r = hypot(p, z) # polar radius 

if min(p, r) > EPS0: 

# parametric latitude (Bowring eqn 17, replaced) 

t = (E.b * z) / (E.a * p) * (_1_0 + E.e22 * E.b / r) 

c = _1_0 / hypot1(t) 

s = t * c 

 

# geodetic latitude (Bowring eqn 18) 

a = atan2(z + E.e22 * E.b * s**3, 

p - E.e2 * E.a * c**3) 

 

# height above ellipsoid (Bowring eqn 7) 

sa, ca = sincos2(a) 

# r = E.a / E.e2s(sa) # length of normal terminated by minor axis 

# h = p * ca + z * sa - (E.a * E.a / r) 

h = fsum_(p * ca, z * sa, -E.a * E.e2s(sa)) 

 

C, lat, lon = 1, degrees90(a), atan2d(y, x) 

 

# see <https://GIS.StackExchange.com/questions/28446> 

elif p > EPS: # lat arbitrarily zero 

C, lat, lon, h = 2, _0_0, atan2d(y, x), p - E.a 

 

else: # polar lat, lon arbitrarily zero 

C, lat, lon, h = 3, copysign0(_90_0, z), _0_0, abs(z) - E.b 

 

return Ecef9Tuple(x, y, z, lat, lon, h, 

C, None, # M=None 

self.datum, name=name or self.name) 

 

 

class EcefYou(_EcefBase): 

'''Conversion between geodetic and geocentric, aka I{Earth-Centered, 

Earth-Fixed} (ECEF) coordinates using I{Rey-Jer You}'s U{transformation 

<https://www.ResearchGate.net/publication/240359424>}. 

 

@see: W.E. Featherstone, S.J. (Sten) Claessens U{Closed-form transformation 

between geodetic and ellipsoidal coordinates 

<https://Espace.Curtin.edu.AU/bitstream/handle/20.500.11937/11589/115114_9021_geod2ellip_final.pdf>} 

Studia Geophysica et Geodaetica, 2008, 52, 1-18 and U{PyMap3D 

<https://PyPI.org/project/pymap3d>}. 

''' 

 

def __init__(self, a_ellipsoid, f=None, name=NN): 

_EcefBase.__init__(self, a_ellipsoid, f=f, name=name) # inherited documentation 

E = self.ellipsoid 

if E.isProlate or (E.a2 - E.b2) < 0: 

raise EcefError(ellipsoid=E, txt=_prolate_) 

 

def reverse(self, xyz, y=None, z=None, M=None, name=NN): # PYCHOK unused M 

'''Convert geocentric C{(x, y, z)} to geodetic C{(lat, lon, height)} 

using I{Rey-Jer You}'s transformation. 

 

@arg xyz: A geocentric (C{Cartesian}, L{Ecef9Tuple}) or C{scalar} ECEF C{x} 

coordinate (C{meter}). 

@kwarg y: ECEF C{y} coordinate for C{scalar} B{C{xyz}} and B{C{z}} (C{meter}). 

@kwarg z: ECEF C{z} coordinate for C{scalar} B{C{xyz}} and B{C{y}} (C{meter}). 

@kwarg M: I{Ignored}, rotation matrix C{M} not available. 

@kwarg name: Optional name (C{str}). 

 

@return: An L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)} with 

geodetic coordinates C{(lat, lon, height)} for the given geocentric 

ones C{(x, y, z)}, case C{C=1}, C{M=None} always and C{datum} if 

available. 

 

@raise EcefError: Invalid B{C{xyz}} or C{scalar} C{x} or B{C{y}} and/or B{C{z}} 

not C{scalar} for C{scalar} B{C{xyz}}. 

''' 

x, y, z, name = _xyzn4(xyz, y, z, self._Geocentrics, name=name) 

 

r2 = hypot2_(x, y, z) 

 

E = self.ellipsoid 

e2 = E.a2 - E.b2 # == E.e2 * E.a2 

if e2 < 0: 

raise EcefError(ellipsoid=E, txt=_prolate_) 

e = sqrt(e2) # XXX sqrt0(e2)? 

 

q = hypot(x, y) 

u = fsum_(r2, -e2, hypot(r2 - e2, 2 * e * z)) * _0_5 

if u > EPS02: 

u = sqrt(u) 

p = hypot(u, e) 

B = atan2(p * z, u * q) # beta0 = atan(p / u * z / q) 

sB, cB = sincos2(B) 

if cB and sB: 

p *= E.a 

d = (p / cB - e2 * cB) / sB 

if isnon0(d): 

B += fsum_(u * E.b, -p, e2) / d 

sB, cB = sincos2(B) 

elif u < 0: 

raise EcefError(x=x, y=y, z=z, txt=_singular_) 

else: 

sB, cB = copysign0(_1_0, z), _0_0 

 

h = hypot(z - E.b * sB, q - E.a * cB) 

if hypot2_(x, y, z * E.a_b) < E.a2: 

h = neg(h) # inside ellipsoid 

 

return Ecef9Tuple(x, y, z, atan2d(E.a * sB, E.b * cB), # atan(E.a_b * tan(B)) 

atan2d(y, x), h, 

1, None, # C=1, M=None 

self.datum, name=name or self.name) 

 

 

class EcefMatrix(_NamedTuple): 

'''A rotation matrix. 

''' 

_Names_ = ('_0_0_', '_0_1_', '_0_2_', # row-order 

'_1_0_', '_1_1_', '_1_2_', 

'_2_0_', '_2_1_', '_2_2_') 

_Units_ = (Scalar,) * len(_Names_) 

 

def _validate(self, **_OK): # PYCHOK unused 

'''(INTERNAL) Allow C{_Names_} with leading underscore. 

''' 

_NamedTuple._validate(self, _OK=True) 

 

def __new__(cls, sa, ca, sb, cb, *_more): 

'''New L{EcefMatrix} matrix. 

 

@arg sa: C{sin(phi)} (C{float}). 

@arg ca: C{cos(phi)} (C{float}). 

@arg sb: C{sin(lambda)} (C{float}). 

@arg cb: C{cos(lambda)} (C{float}). 

@arg _more: (INTERNAL) from C{.multiply}. 

 

@raise EcefError: If B{C{sa}}, B{C{ca}}, B{C{sb}} or 

B{C{cb}} outside M{[-1.0, +1.0]}. 

''' 

t = sa, ca, sb, cb 

if _more: # all 9 matrix elements ... 

t += _more # ... from .multiply 

 

elif max(map(abs, t)) > _1_0: 

raise EcefError(unstr(EcefMatrix.__name__, *t)) 

 

else: # build matrix from the following quaternion operations 

# qrot(lam, [0,0,1]) * qrot(phi, [0,-1,0]) * [1,1,1,1]/2 

# or 

# qrot(pi/2 + lam, [0,0,1]) * qrot(-pi/2 + phi, [-1,0,0]) 

# where 

# qrot(t,v) = [cos(t/2), sin(t/2)*v[1], sin(t/2)*v[2], sin(t/2)*v[3]] 

 

# Local X axis (east) in geocentric coords 

# M[0] = -slam; M[3] = clam; M[6] = 0; 

# Local Y axis (north) in geocentric coords 

# M[1] = -clam * sphi; M[4] = -slam * sphi; M[7] = cphi; 

# Local Z axis (up) in geocentric coords 

# M[2] = clam * cphi; M[5] = slam * cphi; M[8] = sphi; 

t = (-sb, -cb * sa, cb * ca, 

cb, -sb * sa, sb * ca, 

_0_0, ca, sa) 

 

return _NamedTuple.__new__(cls, *t) 

 

def column(self, column): 

'''Get matrix B{C{column}} as 3-tuple. 

''' 

if 0 <= column < 3: 

return self[column::3] 

raise _IndexError(column=column) 

 

@Property_RO 

def _column_0(self): 

return self.column(0) 

 

@Property_RO 

def _column_1(self): 

return self.column(1) 

 

@Property_RO 

def _column_2(self): 

return self.column(2) 

 

def copy(self, **unused): # PYCHOK signature 

'''Make a shallow or deep copy of this instance. 

 

@return: The copy (C{This class} or subclass thereof). 

''' 

return self.classof(*self) 

 

__copy__ = __deepcopy__ = copy 

 

def multiply(self, other): 

'''Matrix multiply M{M0' ⋅ M} this matrix transposed with 

an other matrix. 

 

@arg other: The other matrix (L{EcefMatrix}). 

 

@return: The matrix product (L{EcefMatrix}). 

 

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

''' 

_xinstanceof(EcefMatrix, other=other) 

 

# like LocalCartesian.MatrixMultiply, transposed(self) X other 

# <https://GeographicLib.SourceForge.io/html/LocalCartesian_8cpp_source.html> 

M = (fdot(self[r::3], *other[c::3]) for r in range(3) for c in range(3)) 

return _xnamed(EcefMatrix(*M), EcefMatrix.multiply.__name__) 

 

def rotate(self, xyz, *xyz0): 

'''Forward rotation M{M0' ⋅ ([x, y, z] - [x0, y0, z0])'}. 

 

@arg xyz: Local C{(x, y, z)} coordinates (C{3-tuple}). 

@arg xyz0: Optional, local C{(x0, y0, z0)} origin (C{3-tuple}). 

 

@return: Rotated C{(x, y, z)} location (C{3-tuple}). 

 

@raise LenError: Unequal C{len(B{xyz})} and C{len(B{xyz0})}. 

''' 

if xyz0: 

if len(xyz0) != len(xyz): 

raise LenError(self.rotate, xyz0=len(xyz0), xyz=len(xyz)) 

 

xyz = tuple(c - c0 for c, c0 in zip(xyz, xyz0)) 

 

# x' = M[0] * x + M[3] * y + M[6] * z 

# y' = M[1] * x + M[4] * y + M[7] * z 

# z' = M[2] * x + M[5] * y + M[8] * z 

return (fdot(xyz, *self._column_0), 

fdot(xyz, *self._column_1), 

fdot(xyz, *self._column_2)) 

 

def row(self, row): 

'''Get matrix B{C{row}} as 3-tuple. 

''' 

if 0 <= row < 3: 

r = row * 3 

return self[r:r+3] 

raise _IndexError(row=row) 

 

@Property_RO 

def _row_0(self): 

return self.row(0) 

 

@Property_RO 

def _row_1(self): 

return self.row(1) 

 

@Property_RO 

def _row_2(self): 

return self.row(2) 

 

def unrotate(self, xyz, *xyz0): 

'''Inverse rotation M{[x0, y0, z0] + M0 ⋅ [x,y,z]'}. 

 

@arg xyz: Local C{(x, y, z)} coordinates (C{3-tuple}). 

@arg xyz0: Optional, local C{(x0, y0, z0)} origin (C{3-tuple}). 

 

@return: Unrotated C{(x, y, z)} location (C{3-tuple}). 

 

@raise LenError: Unequal C{len(B{xyz})} and C{len(B{xyz0})}. 

''' 

if xyz0: 

if len(xyz0) != len(xyz): 

raise LenError(self.unrotate, xyz0=len(xyz0), xyz=len(xyz)) 

 

_xyz = _1_0_T + xyz 

# x' = x0 + M[0] * x + M[1] * y + M[2] * z 

# y' = y0 + M[3] * x + M[4] * y + M[5] * z 

# z' = z0 + M[6] * x + M[7] * y + M[8] * z 

xyz_ = (fdot(_xyz, xyz0[0], *self._row_0), 

fdot(_xyz, xyz0[1], *self._row_1), 

fdot(_xyz, xyz0[2], *self._row_2)) 

else: 

# x' = M[0] * x + M[1] * y + M[2] * z 

# y' = M[3] * x + M[4] * y + M[5] * z 

# z' = M[6] * x + M[7] * y + M[8] * z 

xyz_ = (fdot(xyz, *self._row_0), 

fdot(xyz, *self._row_1), 

fdot(xyz, *self._row_2)) 

return xyz_ 

 

 

class Ecef9Tuple(_NamedTuple): 

'''9-Tuple C{(x, y, z, lat, lon, height, C, M, datum)} with I{geocentric} 

C{x}, C{y} and C{z} plus I{geodetic} C{lat}, C{lon} and C{height}, case 

C{C} (see the C{Ecef*.reverse} methods) and optionally, the rotation 

matrix C{M} (L{EcefMatrix}) and C{datum}, with C{lat} and C{lon} in 

C{degrees} and C{x}, C{y}, C{z} and C{height} in C{meter}, conventionally. 

''' 

_Names_ = (_x_, _y_, _z_, _lat_, _lon_, _height_, _C_, _M_, _datum_) 

_Units_ = ( Meter, Meter, Meter, Lat, Lon, Height, Int, _Pass, _Pass) 

 

_IndexM = _Names_.index(_M_) # for ._M_x_M 

 

@deprecated_method 

def convertDatum(self, datum2): # for backward compatibility 

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

return self.toDatum(datum2) 

 

@Property_RO 

def lam(self): 

'''Get the longitude in C{radians} (C{float}). 

''' 

return self.philam.lam 

 

@Property_RO 

def lamVermeille(self): 

'''Get the longitude in C{radians [-PI*3/2..+PI*3/2]} after U{Vermeille 

<https://Search.ProQuest.com/docview/639493848>} (2004), p 95. 

 

@see: U{Karney<https://GeographicLib.SourceForge.io/html/geocentric.html>}, 

U{Vermeille<https://Search.ProQuest.com/docview/847292978>} 2011, pp 112-113, 116 

and U{Featherstone, et.al.<https://Search.ProQuest.com/docview/872827242>}, p 7. 

''' 

x, y = self.x, self.y 

if y > EPS0: 

r = _N_2_0 * atan2(x, hypot(y, x) + y) + PI_2 

elif y < -EPS0: 

r = _2_0 * atan2(x, hypot(y, x) - y) - PI_2 

else: # y == 0 

r = PI if x < 0 else _0_0 

return Lam(Vermeille=r) 

 

@Property_RO 

def latlon(self): 

'''Get the lat-, longitude in C{degrees} (L{LatLon2Tuple}C{(lat, lon)}). 

''' 

return LatLon2Tuple(self.lat, self.lon, name=self.name) 

 

@Property_RO 

def latlonheight(self): 

'''Get the lat-, longitude in C{degrees} and height (L{LatLon3Tuple}C{(lat, lon, height)}). 

''' 

return self.latlon.to3Tuple(self.height) 

 

@Property_RO 

def latlonheightdatum(self): 

'''Get the lat-, longitude in C{degrees} with height and datum (L{LatLon4Tuple}C{(lat, lon, height, datum)}). 

''' 

return self.latlonheight.to4Tuple(self.datum) 

 

@Property_RO 

def latlonVermeille(self): 

'''Get the latitude and I{Vermeille} longitude in C{degrees [-225..+225]} (L{LatLon2Tuple}C{(lat, lon)}). 

 

@see: Property C{lonVermeille}. 

''' 

return LatLon2Tuple(self.lat, self.lonVermeille, name=self.name) 

 

@Property_RO 

def lonVermeille(self): 

'''Get the longitude in C{degrees [-225..+225]} after U{Vermeille 

<https://Search.ProQuest.com/docview/639493848>} (2004), p 95. 

 

@see: Property C{lamVermeille}. 

''' 

return Lon(Vermeille=degrees(self.lamVermeille)) 

 

def _T_x_M(self, T): 

'''(INTERNAL) Update M{self.M = T.multiply(self.M)}. 

''' 

t = list(self) 

M = self._IndexM 

t[M] = T.multiply(t[M]) 

return self.classof(*t) 

 

@Property_RO 

def phi(self): 

'''Get the latitude in C{radians} (C{float}). 

''' 

return self.philam.phi 

 

@Property_RO 

def philam(self): 

'''Get the lat-, longitude in C{radians} (L{PhiLam2Tuple}C{(phi, lam)}). 

''' 

return PhiLam2Tuple(radians(self.lat), radians(self.lon), name=self.name) 

 

@Property_RO 

def philamheight(self): 

'''Get the lat-, longitude in C{radians} and height (L{PhiLam3Tuple}C{(phi, lam, height)}). 

''' 

return self.philam.to3Tuple(self.height) 

 

@Property_RO 

def philamheightdatum(self): 

'''Get the lat-, longitude in C{radians} with height and datum (L{PhiLam4Tuple}C{(phi, lam, height, datum)}). 

''' 

return self.philamheight.to4Tuple(self.datum) 

 

@Property_RO 

def philamVermeille(self): 

'''Get the latitude and I{Vermeille} longitude in C{radians [-PI*3/2..+PI*3/2]} (L{PhiLam2Tuple}C{(phi, lam)}). 

 

@see: Property C{lamVermeille}. 

''' 

return PhiLam2Tuple(radians(self.lat), self.lamVermeille, name=self.name) 

 

def toCartesian(self, Cartesian=None, **Cartesian_kwds): 

'''Return the geocentric C{(x, y, z)} coordinates as an ellipsoidal or spherical 

C{Cartesian}. 

 

@kwarg Cartesian: Optional class to return C{(x, y, z)} (L{ellipsoidalKarney.Cartesian}, 

L{ellipsoidalNvector.Cartesian}, L{ellipsoidalVincenty.Cartesian}, 

L{sphericalNvector.Cartesian} or L{sphericalTrigonometry.Cartesian}) 

or C{None}. 

@kwarg Cartesian_kwds: Optional, additional B{C{Cartesian}} keyword arguments, ignored 

if C{B{Cartesian} is None}. 

 

@return: A C{B{Cartesian}(x, y, z, **B{Cartesian_kwds})} instance or 

a L{Vector4Tuple}C{(x, y, z, h)} if C{B{Cartesian} is None}. 

 

@raise TypeError: Invalid B{C{Cartesian}} or B{C{Cartesian_kwds}}. 

''' 

if Cartesian in (None, Vector4Tuple): 

r = self.xyzh 

elif Cartesian is Vector3Tuple: 

r = self.xyz 

else: 

from pygeodesy.cartesianBase import CartesianBase 

_xsubclassof(CartesianBase, Cartesian=Cartesian) 

r = Cartesian(self, **_xkwds(Cartesian_kwds, name=self.name)) 

return r 

 

def toDatum(self, datum2): 

'''Convert this C{Ecef9Tuple} to an other datum. 

 

@arg datum2: Datum to convert I{to} (L{Datum}). 

 

@return: The converted 9-Tuple (C{Ecef9Tuple}). 

 

@raise TypeError: The B{C{datum2}} is not a L{Datum}. 

''' 

if self.datum in (None, datum2): # PYCHOK _Names_ 

r = self.copy() 

else: 

from pygeodesy.cartesianBase import CartesianBase 

c = CartesianBase(self, datum=self.datum, name=self.name) # PYCHOK _Names_ 

# c.toLatLon converts datum, x, y, z, lat, lon, etc. 

# and returns another Ecef9Tuple iff LatLon is None 

r = c.toLatLon(datum=datum2, LatLon=None) 

return r 

 

def toLatLon(self, LatLon=None, **LatLon_kwds): 

'''Return the geodetic C{(lat, lon, height[, datum])} coordinates. 

 

@kwarg LatLon: Optional class to return C{(lat, lon, height[, datum])} 

or C{None}. 

@kwarg LatLon_kwds: Optional B{C{height}}, B{C{datum}} and other 

B{C{LatLon}} keyword arguments. 

 

@return: An instance of C{B{LatLon}(lat, lon, **B{LatLon_kwds})} 

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

height)} respectively L{LatLon4Tuple}C{(lat, lon, height, 

datum)} depending on whether C{datum} is un-/specified. 

 

@raise TypeError: Invalid B{C{LatLon}} or B{C{LatLon_kwds}}. 

''' 

kwds = _xkwds(LatLon_kwds, height=self.height, datum=self.datum, name=self.name) # PYCHOK Ecef9Tuple 

d = kwds[_datum_] 

if LatLon is None: 

r = LatLon3Tuple(self.lat, self.lon, kwds[_height_], name=kwds[_name_]) # PYCHOK Ecef9Tuple 

if d: 

r = r.to4Tuple(d) # checks type(d) 

else: 

if d is None: # remove None datum 

_ = kwds.pop[_datum_] 

r = LatLon(self.lat, self.lon, **kwds) # PYCHOK Ecef9Tuple 

_datum_datum(getattr(r, _datum_, self.datum), self.datum) # PYCHOK Ecef9Tuple 

return r 

 

def toLocal(self, ltp, Xyz=None, **Xyz_kwds): 

'''Convert this geocentric to I{local} C{x}, C{y} and C{z}. 

 

@kwarg ltp: The I{local tangent plane} (LTP) to use (L{Ltp}). 

@kwarg Xyz: Optional class to return C{x}, C{y} and C{z} 

(L{XyzLocal}, L{Enu}, L{Ned}) or C{None}. 

@kwarg Xyz_kwds: Optional, additional B{C{Xyz}} keyword 

arguments, ignored if C{B{Xyz} is None}. 

 

@return: An B{C{Xyz}} instance or if C{B{Xyz} is None}, 

a L{Local9Tuple}C{(x, y, z, lat, lon, height, 

ltp, ecef, M)} with C{M=None}, always. 

 

@raise TypeError: Invalid B{C{ltp}}. 

''' 

from pygeodesy.ltp import _xLtp 

return _xLtp(ltp)._ecef2local(self, Xyz, Xyz_kwds) 

 

def toVector(self, Vector=None, **Vector_kwds): 

'''Return the geocentric C{(x, y, z)} coordinates as vector. 

 

@kwarg Vector: Optional vector class to return C{(x, y, z)} or 

C{None}. 

@kwarg Vector_kwds: Optional, additional B{C{Vector}} keyword 

arguments, ignored if C{B{Vector} is None}. 

 

@return: A C{Vector}C{(x, y, z, **Vector_kwds)} instance or a 

L{Vector3Tuple}C{(x, y, z)} if B{C{Vector}} is C{None}. 

 

@see: Propertes C{xyz} and C{xyzh} 

''' 

return self.xyz if Vector is None else self._xnamed( 

Vector(self.x, self.y, self.z, **Vector_kwds)) # PYCHOK Ecef9Tuple 

 

@Property_RO 

def xyz(self): 

'''Get the geocentric C{(x, y, z)} coordinates (L{Vector3Tuple}C{(x, y, z)}). 

''' 

return Vector3Tuple(self.x, self.y, self.z, name=self.name) 

 

@Property_RO 

def xyzh(self): 

'''Get the geocentric C{(x, y, z)} coordinates and C{height} (L{Vector4Tuple}C{(x, y, z, h)}) 

''' 

return self.xyz.to4Tuple(self.height) 

 

 

def _xEcef(Ecef): # PYCHOK .latlonBase.py 

'''(INTERNAL) Validate B{C{Ecef}} I{class}. 

''' 

if issubclassof(Ecef, _EcefBase): 

return Ecef 

raise _TypesError(_Ecef_, Ecef, EcefFarrell21, EcefFarrell22, EcefKarney, EcefSudano, EcefVeness, EcefYou) 

 

 

__all__ += _ALL_DOCS(_EcefBase) 

 

# **) MIT License 

# 

# Copyright (C) 2016-2022 -- 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.