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

 

# -*- coding: utf-8 -*- 

 

u'''Class L{JacobiConformal}, Jacobi's conformal projection of a triaxial ellipsoid, trancoded 

from I{Charles Karney}'s C++ class U{JacobiConformal<https://GeographicLib.SourceForge.io/C++/ 

doc/classGeographicLib_1_1JacobiConformal.html#details>} to pure Python, class L{Triaxial} and 

classes L{BetaOmega3Tuple},L{JacobiError}, L{Jacobi2Tuple} and L{TriaxialError}. 

 

@see: U{Geodesics on a triaxial ellipsoid<https://WikiPedia.org/wiki/Geodesics_on_an_ellipsoid# 

Geodesics_on_a_triaxial_ellipsoid>} and U{Triaxial coordinate systems and their geometrical 

interpretation<https://www.Topo.Auth.GR/wp-content/uploads/sites/111/2021/12/09_Panou.pdf>}. 

''' 

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

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

 

# from pygeodesy.basics import isscalar # from .fsums 

from pygeodesy.constants import EPS, EPS0, EPS02, PI2, PI_3, PI4, _0_0, _0_5, \ 

_1_0, _3_0, _4_0, isfinite 

from pygeodesy.elliptic import Elliptic, Property_RO 

# from pygeodesy.errors import _ValueError # from .fsums 

from pygeodesy.fmath import fdot, Fmt, hypot, hypot_, hypot2, hypot2_, norm2 

from pygeodesy.fsums import Fsum, fsum_, isscalar, _ValueError 

from pygeodesy.interns import NN, _distant_, _height_, _inside_, _near_, _not_, _null_, _opposite_, \ 

_outside_, _spherical_, _too_, _x_, _y_ 

# from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS # from .vector3d 

from pygeodesy.named import _NamedBase, _NamedTuple, _Pass 

from pygeodesy.namedTuples import LatLon3Tuple, Vector3Tuple, Vector4Tuple 

# from pygeodesy.streprs import Fmt # from .fmath 

from pygeodesy.units import Degrees, Height_, Meter, Meter2, Meter3, Radians, Radius, Scalar 

from pygeodesy.utily import asin1, atan2d, sincos2, sincos2d, sincos2d_ 

from pygeodesy.vector3d import _ALL_LAZY, _MODS, _otherV3d, Vector3d 

 

from math import atan2, fabs, sqrt 

 

__all__ = _ALL_LAZY.triaxials 

__version__ = '22.10.22' 

 

_not_ordered_ = _not_('ordered') 

_TRIPS = 1074 # Eberly root finder 

 

 

class BetaOmega2Tuple(_NamedTuple): 

'''2-Tuple C{(beta, omega)} with I{ellipsoidal} lat- and 

longitude C{beta} and C{omega} both in C{Radians} (or 

C{Degrees}). 

''' 

_Names_ = ('beta', 'omega') 

_Units_ = (_Pass, _Pass) 

 

def toDegrees(self, **toDMS_kwds): 

'''Convert this L{BetaOmega2Tuple} to C{Degrees} or C{toDMS}. 

 

@return: L{BetaOmega2Tuple}C{(beta, omega)} with 

C{beta} and C{omega} both in C{Degrees} 

or as an L{toDMS} string provided some 

B{C{toDMS_kwds}} are supplied. 

''' 

return _toDegrees(self, *self, **toDMS_kwds) 

 

def toRadians(self): 

'''Convert this L{BetaOmega2Tuple} to C{Radians}. 

 

@return: L{BetaOmega2Tuple}C{(beta, omega)} with 

C{beta} and C{omega} both in C{Radians}. 

''' 

return _toRadians(self, *self) 

 

 

class BetaOmega3Tuple(_NamedTuple): 

'''3-Tuple C{(beta, omega, height)} with I{ellipsoidal} lat- and 

longitude C{beta} and C{omega} both in C{Radians} (or C{Degrees}) 

and the C{height}, rather the (signed) I{distance} to the triaxial's 

surface (measured along the radial line to the triaxial's center) 

in C{meter}, conventionally. 

''' 

_Names_ = BetaOmega2Tuple._Names_ + (_height_,) 

_Units_ = BetaOmega2Tuple._Units_ + ( Meter,) 

 

def toDegrees(self, **toDMS_kwds): 

'''Convert this L{BetaOmega3Tuple} to C{Degrees} or C{toDMS}. 

 

@return: L{BetaOmega3Tuple}C{(beta, omega, height)} with 

C{beta} and C{omega} both in C{Degrees} or as an 

L{toDMS} string provided some B{C{toDMS_kwds}} 

are supplied. 

''' 

return _toDegrees(self, *self, **toDMS_kwds) 

 

def toRadians(self): 

'''Convert this L{BetaOmega3Tuple} to C{Radians}. 

 

@return: L{BetaOmega3Tuple}C{(beta, omega, height)} with 

C{beta} and C{omega} both in C{Radians}. 

''' 

return _toRadians(self, *self) 

 

def to2Tuple(self): 

'''Reduce this L{BetaOmega3Tuple} to a L{BetaOmega2Tuple}. 

''' 

return BetaOmega2Tuple(*self[:2]) 

 

 

class JacobiError(_ValueError): 

'''Raised for L{JacobiConformal} issues. 

''' 

pass # ... 

 

 

class Jacobi2Tuple(_NamedTuple): 

'''2-Tuple C{(x, y)} with a Jacobi Conformal C{x} and C{y} 

projection, both in C{Radians} (or C{Degrees}). 

''' 

_Names_ = (_x_, _y_) 

_Units_ = (_Pass, _Pass) 

 

def toDegrees(self, **toDMS_kwds): 

'''Convert this L{Jacobi2Tuple} to C{Degrees} or C{toDMS}. 

 

@return: L{Jacobi2Tuple}C{(x, y)} with C{x} and C{y} 

both in C{Degrees} or as an L{toDMS} string 

provided some B{C{toDMS_kwds}} are supplied. 

''' 

return _toDegrees(self, *self, **toDMS_kwds) 

 

def toRadians(self): 

'''Convert this L{Jacobi2Tuple} to C{Radians}. 

 

@return: L{Jacobi2Tuple}C{(x, y)} with C{x} 

and C{y} both in C{Radians}. 

''' 

return _toRadians(self, *self) 

 

 

class TriaxialError(_ValueError): 

'''Raised for L{Triaxial} issues. 

''' 

pass # ... 

 

 

class Triaxial(_NamedBase): # _NamedEnumItem 

'''Triaxial ellipsoid with semi-axes C{a}, C{b} and C{c}, oriented such that 

the large principal ellipse C{ab} is the equator I{Z}=0, I{beta}=0, while 

the small principal ellipse C{ac} is the prime meridian, I{Y}=0, I{omega}=0. 

The four umbilic points, C{abs}(I{omega}) = C{abs}(I{beta}) = C{PI/2}, lie 

on the middle principal ellipse C{bc} in plane I{X}=0, I{omega}=C{PI/2}. 

 

@note: Geodetic lat- and longitudes are in C{degrees}, I{ellipsoidal} lat- 

and longitude C{beta} and C{omega} are in C{Radians} by default 

(or C{Degrees}. 

''' 

_Error = TriaxialError 

 

def __init__(self, a, b, c, name=NN): 

'''New L{Triaxial}. 

 

@arg a: The largest semi-axis (C{meter}, conventionally). 

@arg b: The middle semi-axis (C{meter}, same units as B{C{a}}). 

@arg c: The smallest semi-axis (C{meter}, same units as B{C{a}}). 

@kwarg name: Optional name (C{str}). 

 

@note: The semi-axes must be ordered as C{B{a} >= B{b} >= B{c} > 0} 

and be ellipsoidal, C{B{a} > B{c}}. 

 

@raise TriaxialError: Semi-axes not ordered, spherical or invalid. 

''' 

if name: 

self.name = name 

E = self._Error 

 

self._a = a = Radius(a=a, Error=E) 

self._b = b = Radius(b=b, Error=E) 

self._c = c = Radius(c=c, Error=E) 

if not (isfinite(a) and a >= b >= c > 0): 

raise E(a=a, b=b, c=c, txt=_not_ordered_) 

 

self._a2b2 = (a - b) * (a + b) # == a**2 - b**2 == E_sub_e**2 

self._b2c2 = (b - c) * (b + c) # == b**2 - c**2 == E_sub_y**2 

self._a2c2 = (a - c) * (a + c) # == a**2 - c**2 == E_sub_x**2 

if not (a > c and self._a2c2 > 0 and self.e2ac > 0): 

raise E(a=a, c=c, e2ac=self.e2ac, txt=_spherical_) 

 

self._a2_b2 = _1_0 if a == b else (a / b)**2 

self._c2_b2 = _1_0 if c == b else (c / b)**2 

 

def __str__(self): 

return self.toStr() 

 

@Property_RO 

def a(self): 

'''Get the largest semi-axis (C{meter}, conventionally). 

''' 

return self._a 

 

@Property_RO 

def area(self): 

'''Get the surface area (C{meter} I{squared}). 

 

@see: U{Surface area<https://WikiPedia.org/wiki/Ellipsoid#Surface_area>}. 

''' 

kp2, k2 = self._k2_kp2 # swapped! 

aE = Elliptic(k2, _0_0, kp2, _1_0) 

c2 = self._1e2ac # cos(phi)**2 == (c / a)**2 

s2 = self. e2ac # sin(phi)**2 == 1 - c2 

s = sqrt(s2) 

r = asin1(s) # phi == atan2(sqrt(c2), s) 

a = self.c**2 + (aE.fE(r) * s2 + aE.fF(r) * c2) / s * self.a * self.b 

return Meter2(area=PI2 * a) 

 

def area_p(self, p=1.6075): 

'''I{Approximate} the surface area (C{meter} I{squared}). 

 

@kwarg p: Exponent (C{scalar}), 1.6 for near-spherical or 1.5849625007 

for "near-flat" triaxials 

 

@see: U{Surface area<https://WikiPedia.org/wiki/Ellipsoid#Approximate_formula>}. 

''' 

a, b, c, _p = self.a, self.b, self.c, pow 

a = _p(fsum_(_p(a * b, p), _p(a * c, p), _p(b * c, p)) / _3_0, _1_0 / p) 

return Meter2(area_p=PI4 * a) 

 

@Property_RO 

def b(self): 

'''Get the middle semi-axis (C{meter}, same units as B{C{a}}). 

''' 

return self._b 

 

@Property_RO 

def c(self): 

'''Get the smallest semi-axis (C{meter}, same units as B{C{a}}). 

''' 

return self._c 

 

@Property_RO 

def e2ab(self): 

'''Get the C{ab} ellipse' I{(1st) eccentricity squared} (C{scalar}), M{1 - (b / a)**2}. 

''' 

return Scalar(e2ab=(_1_0 - self._1e2ab) if self.b != self.a else _0_0) 

 

@Property_RO 

def _1e2ab(self): 

'''(INTERNAL) Get C{1 - e2ab}. 

''' 

return (_1_0 / self._a2_b2) if self.b != self.a else _1_0 

 

@Property_RO 

def e2bc(self): 

'''Get the C{bc} ellipse' I{(1st) eccentricity squared} (C{scalar}), M{1 - (c / b)**2}. 

''' 

return Scalar(e2bc=(_1_0 - self._1e2bc) if self.c != self.b else _0_0) 

 

@Property_RO 

def _1e2bc(self): 

'''(INTERNAL) Get C{1 - e2bc}. 

''' 

return self._c2_b2 

 

@Property_RO 

def e2ac(self): 

'''Get the C{ac} ellipse' I{(1st) eccentricity squared} (C{scalar}), M{1 - (c / a)**2}. 

''' 

return Scalar(e2ac=(_1_0 - self._1e2ac) if self.c != self.a else _0_0) 

 

@Property_RO 

def _1e2ac(self): 

'''(INTERNAL) Get C{1 - e2ac}. 

''' 

return (self.c / self.a)**2 if self.c != self.a else _1_0 

 

@Property_RO 

def _Exycr4(self): 

'''(INTERNAL) Helper for C{.forwardBetOmg}. 

''' 

return self._Exyur4(self.c) 

 

def _Exyur4(self, u): 

'''(INTERNAL) Helper for C{.forwardBetOmg}. 

''' 

if u > 0: 

u2 = u**2 

x = self._sqrt(_1_0 + self._a2c2 / u2) * u 

y = self._sqrt(_1_0 + self._b2c2 / u2) * u 

else: 

x = y = u = _0_0 

return x, y, u, (self._a2b2 / self._a2c2) 

 

def forwardBetaOmega(self, beta, omega, height=0, name=NN): 

'''Convert I{ellipsoidal} lat- and longitude C{beta}, C{omega} 

and height to cartesian. 

 

@arg beta: Ellipsoidal latitude (C{radians} or L{Degrees}). 

@arg omega: Ellipsoidal longitude (C{radians} or L{Degrees}). 

@arg height: Height above or below the ellipsoid's surface 

(C{meter}, same units as this triaxial's C{a}, 

C{b} and C{c} semi-axes). 

@kwarg name: Optional name (C{str}). 

 

@return: A L{Vector3Tuple}C{(x, y, z)}. 

 

@see: U{Expressions (23-25)<https://www.Topo.Auth.GR/wp-content/ 

uploads/sites/111/2021/12/09_Panou.pdf>}. 

''' 

sa, ca = _sincos2(beta) 

sb, cb = _sincos2(omega) 

 

if height: 

h = Height_(height=height, low=-self.c, Error=self._Error) 

x, y, z, r = self._Exyur4(h + self.c) 

else: 

x, y, z, r = self._Exycr4 

if z: # and x and y 

x *= cb * self._sqrt(ca**2 + r * sa**2) 

y *= ca * sb 

z *= sa * self._sqrt(_1_0 - r * cb**2) 

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

 

def forwardBetaOmega_(self, sbeta, cbeta, somega, comega, name=NN): 

'''Convert I{ellipsoidal} lat- and longitude C{beta} and C{omega} to 

cartesian coordinates I{on the ellipsoid's surface}. 

 

@arg sbeta: Ellipsoidal latitude C{sin(beta)} (C{scalar}). 

@arg cbeta: Ellipsoidal latitude C{cos(beta)} (C{scalar}). 

@arg somega: Ellipsoidal longitude C{sin(omega)} (C{scalar}). 

@arg comega: Ellipsoidal longitude C{cos(omega)} (C{scalar}). 

@kwarg name: Optional name (C{str}). 

 

@return: A L{Vector3Tuple}C{(x, y, z)}. 

 

@see: U{Triaxial ellipsoid coordinate system<https://WikiPedia.org/wiki/ 

Geodesics_on_an_ellipsoid#Triaxial_ellipsoid_coordinate_system>} 

and U{expressions (23-25)<https://www.Topo.Auth.GR/wp-content/ 

uploads/sites/111/2021/12/09_Panou.pdf>}. 

''' 

sa, ca = self._norm2(sbeta, cbeta) 

sb, cb = self._norm2(somega, comega) 

 

b2_a2 = self._1e2ab # == (b / a)**2 

c2_a2 = -self._1e2ac # == (c / a)**2 

a2c2_a2 = self. e2ac # (a**2 - c**2) / a**2 == 1 - (c / a)**2 

 

x = Fsum(_1_0, -b2_a2 * sa**2, c2_a2 * ca**2).fover(a2c2_a2) 

x = self.a * cb * self._sqrt(x) 

y = self.b * ca * sb 

z = Fsum(c2_a2, sb**2, b2_a2 * cb**2).fover(a2c2_a2) 

z = self.c * sa * self._sqrt(z) 

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

 

def forwardLatLon(self, lat, lon, height=0, name=NN): 

'''Convert I{geodetic} lat-, longitude and heigth to cartesian. 

 

@arg lat: Geodetic latitude (C{degrees}). 

@arg lon: Geodetic longitude (C{degrees}). 

@arg height: Height above the ellipsoid (C{meter}, same units 

as this triaxial's C{a}, C{b} and C{c} axes). 

@kwarg name: Optional name (C{str}). 

 

@return: A L{Vector3Tuple}C{(x, y, z)}. 

 

@see: U{Expressions (9-11)<https://www.Topo.Auth.GR/wp-content/ 

uploads/sites/111/2021/12/09_Panou.pdf>}. 

''' 

return self._forward3(height, name, *sincos2d_(lat, lon)) 

 

def forwardLatLon_(self, slat, clat, slon, clon, height=0, name=NN): 

'''Convert I{geodetic} lat-, longitude and heigth to cartesian. 

 

@arg slat: Geodetic latitude C{sin(lat)} (C{scalar}). 

@arg clat: Geodetic latitude C{cos(lat)} (C{scalar}). 

@arg slon: Geodetic longitude C{sin(lon)} (C{scalar}). 

@arg clon: Geodetic longitude C{cos(lon)} (C{scalar}). 

@arg height: Height above the ellipsoid (C{meter}, same units 

as this triaxial's axes C{a}, C{b} and C{c}). 

@kwarg name: Optional name (C{str}). 

 

@return: A L{Vector3Tuple}C{(x, y, z)}. 

 

@see: U{Expressions (9-11)<https://www.Topo.Auth.GR/wp-content/ 

uploads/sites/111/2021/12/09_Panou.pdf>}. 

''' 

sa, ca = self._norm2(slat, clat) 

sb, cb = self._norm2(slon, clon) 

return self._forward3(height, name, sa, ca, sb, cb) 

 

def _forward3(self, h, name, sa, ca, sb, cb): 

'''(INTERNAL) Helper for C{.forwardLatLon} and C{.forwardLatLon_}. 

''' 

ca_sb = ca * sb 

# 1 - (1 - (c/a)**2) * sa**2 - (1 - (b / a)**2) * ca**2 * sb**2 

t = fsum_(_1_0, -self.e2ac * sa**2, -self.e2ab * ca_sb**2) 

N = self.a / self._sqrt(t) # prime vertical 

x = (h + N) * ca * cb 

y = (h + N * self._1e2ab) * ca_sb 

z = (h + N * self._1e2ac) * sa 

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

 

def hartzell(self, pov, los=None, name=NN): 

'''Compute the intersection of a Line-Of-Sight from a Point-Of-View in space 

with the surface of this triaxial. 

 

@arg pov: Point-Of-View outside this triaxial (C{Cartesian}, L{Ecef9Tuple} 

or L{Vector3d}). 

@kwarg los: Line-Of-Sight, I{direction} to this triaxial (L{Vector3d}) or 

C{None} to point to this triaxial's center. 

@kwarg name: Optional name (C{str}). 

 

@return: L{Vector4Tuple}C{(x, y, z, h)} on this triaxial's surface. 

 

@raise TriaxialError: Null B{C{pov}} or B{C{los}} vector, B{C{pov}} is 

inside this triaxial or B{C{los}} points outside 

this triaxial or points in an opposite direction. 

 

@raise TypeError: Invalid B{C{pov}} or B{C{los}}. 

 

@see: Function L{pygeodesy.tyr3d} for B{C{los}} and Hartzell, S. U{I{Satellite 

Line-of-Sight Intersection with Earth}<https://StephenHartzell.Medium.com/ 

satellite-line-of-sight-intersection-with-earth-d786b4a6a9b6>}. 

''' 

def _Error(txt): 

return TriaxialError(pov=pov, los=los, triaxial=self, txt=txt) 

 

v, h = _hartzell3d2(pov, los, self.a, self.b, self.c, _Error) 

return Vector4Tuple(v.x, v.y, v.z, h, name=name or self.hartzell.__name__) 

 

def height4(self, x_xyz, y=None, z=None, normal=True, eps=EPS): 

'''Compute the projection on and the height of a cartesian above or below 

this triaxial's surface. 

 

@arg x_xyz: X component (C{scalar}) or a cartesian (C{Cartesian}, 

L{Ecef9Tuple}, L{Vector3d}, L{Vector3Tuple} or L{Vector4Tuple}). 

@kwarg y: Y component (C{scalar}), needed if B{C{x_xyz}} if C{scalar}. 

@kwarg z: Z component (C{scalar}), needed if B{C{x_xyz}} if C{scalar}. 

@kwarg normal: If C{True} the projection is the nearest point on this 

triaxial's surface, otherwise the intersection of the 

radial line to the center and this triaxial's surface. 

@kwarg eps: Tolerance for root finding (C{scalar}). 

 

@return: L{Vector4Tuple}C{(x, y, z, h)} with the cartesian coordinates 

C{x}, C{y} and C{z} of the projection on or the intersection 

with and with the height C{h} above or below the triaxial's 

surface in C{meter}, conventionally. 

 

@raise TriaxialError: Non-cartesian B{C{xyz}}, invalid B{C{eps}} or 

no convergence in root finding. 

 

@see: U{Eberly<https://www.GeometricTools.com/Documentation/DistancePointEllipseEllipsoid.pdf>}. 

''' 

v = Vector3d(x_xyz, y, z) if isscalar(x_xyz) else \ 

_otherV3d(x_xyz=x_xyz) 

 

if normal: # perpendicular to triaxial 

x, y, z, h, i = _normalTo5(v.x, v.y, v.z, self, eps=eps) 

else: # radial to triaxial's center 

x, y, z = self.forwardBetaOmega_(v.z, hypot(v.x, v.y), v.y, v.x) 

h, i = v.minus_(x, y, z).length, None 

 

if h and hypot2_(v.x / self.a, v.y / self.b, v.z / self.c) < _1_0: 

h = -h # below the surface 

return Vector4Tuple(x, y, z, h, iteration=i, name=self.height4.__name__) 

 

@Property_RO 

def _k2_kp2(self): 

'''(INTERNAL) Get C{k2} and C{kp2} for C{._xE}, C{._yE} and C{.area}. 

''' 

# k2 = a2b2 / a2c2 * c2_b2 

# kp2 = b2c2 / a2c2 * a2_b2 

# b2 = b**2 

# xE = Elliptic(k2, -a2b2 / b2, kp2, a2_b2) 

# yE = Elliptic(kp2, +b2c2 / b2, k2, c2_b2) 

# aE = Elliptic(kp2, 0, k2, 1) 

return (self._a2b2 / self._a2c2 * self._c2_b2, 

self._b2c2 / self._a2c2 * self._a2_b2) 

 

def _norm2(self, s, c, *a): 

'''(INTERNAL) Normalize C{s} and C{c} iff needed. 

''' 

if fabs(s) > _1_0 or fabs(c) > _1_0: 

s, c = norm2(s, c) 

if a: 

s, c = norm2(s * self.b, c * a[0]) 

return (s or _0_0), (c or _0_0) 

 

def reverseBetaOmega(self, x, y, z, name=NN): 

'''Convert cartesian to I{ellipsoidal} lat- and longitude, C{beta}, C{omega} 

and height. 

 

@arg x: X coordinate along C{a}-axis (C{meter}, same units as the axes). 

@arg y: Y coordinate along C{b}-axis (C{meter}, same units as the axes). 

@arg z: Z coordinate along C{c}-axis (C{meter}, same units as the axes). 

@kwarg name: Optional name (C{str}). 

 

@return: A L{BetaOmega3Tuple}C{(beta, omega, height)} with C{beta} and 

C{omega} in C{Radians} and C{height} in C{meter}, same units 

as this triaxial's axes. 

 

@see: U{Expressions (21-22)<https://www.Topo.Auth.GR/wp-content/uploads/ 

sites/111/2021/12/09_Panou.pdf>}. 

''' 

v = Vector3d(x, y, z) 

a, b, h = self._reverse3(x, y, z, atan2, v, self.forwardBetaOmega_) 

return BetaOmega3Tuple(Radians(beta=a), Radians(omega=b), h, name=name) 

 

def reverseLatLon(self, x, y, z, name=NN): 

'''Convert cartesian to I{geodetic} lat-, longitude and height. 

 

@arg x: X coordinate along C{a}-axis (C{meter}, same units as the axes). 

@arg y: Y coordinate along C{b}-axis (C{meter}, same units as the axes). 

@arg z: Z coordinate along C{c}-axis (C{meter}, same units as the axes). 

@kwarg name: Optional name (C{str}). 

 

@return: A L{LatLon3Tuple}C{(lat, lon, height)} with C{lat} and C{lon} 

in C{degrees} and C{height} in C{meter}, same units as this 

triaxial's axes. 

 

@see: U{Expressions (4-5)<https://www.Topo.Auth.GR/wp-content/uploads/ 

sites/111/2021/12/09_Panou.pdf>}. 

''' 

v = Vector3d(x, y, z) 

x *= self._1e2ac # == 1 - e_sub_x**2 

y *= self._1e2bc # == 1 - e_sub_y**2 

t = self._reverse3(x, y, z, atan2d, v, self.forwardLatLon_) 

return LatLon3Tuple(*t, name=name) 

 

def _reverse3(self, x, y, z, atan2_, v, forward_): 

'''(INTERNAL) Helper for C{.reverseBetOmg} and C{.reverseLatLon}. 

''' 

d = hypot( x, y) 

a = atan2_(z, d) 

b = atan2_(y, x) 

h = v.minus_(*forward_(z, d, y, x)).length 

return a, b, h 

 

def _sqrt(self, x): 

'''(INTERNAL) Helper. 

''' 

if x < 0: 

raise self._Error(Fmt.PAREN(sqrt=x)) 

return sqrt(x) if x > EPS02 else _0_0 

 

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

'''Return this C{Triaxial} as a string. 

 

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

@kwarg name: Override name (C{str}) or C{None} to exclude 

this ellipsoid's name. 

 

@return: This C{Triaxial}'s attributes (C{str}). 

''' 

T = self.__class__ 

try: 

t = (T.xyQ2.name,) 

except AttributeError: 

t = () 

return self._instr(name, prec, T.a.name, T.b.name, T.c.name, 

T.e2ab.name, T.e2bc.name, T.e2ac.name, 

T.area.name, T.volume.name, *t) 

 

@Property_RO 

def volume(self): 

'''Get the volume (C{meter**3}), M{4 / 3 * PI * a * b * c}. 

''' 

return Meter3(volume=self.a * self.b * self.c * PI_3 * _4_0) 

 

 

class JacobiConformal(Triaxial): 

'''This is a conformal projection of the ellipsoid to a plane in which the grid 

lines are straight, see Jacobi, C. G. J. U{I{Vorlesungen über Dynamik} 

<https://Books.Google.com/books?id=ryEOAAAAQAAJ&pg=PA212>}, page 212ff. 

 

Ellipsoidal coordinates I{beta} and I{omega} are converted to Jacobi Conformal 

I{y} respectively I{x} separately. Jacobi's coordinates have been multiplied 

by C{sqrt(B{a}**2 - B{c}**2) / (2 * B{b})} so that the customary results are 

returned in the case of an ellipsoid of revolution (or a sphere, I{currently 

not supported}). 

 

Copyright (C) U{Charles Karney<mailto:Charles@Karney.com>} (2014-2020) and 

licensed under the MIT/X11 License. 

 

@note: This constructor can not be used to specify a sphere. 

 

@see: L{Triaxial}, C++ class U{JacobiConformal<https://GeographicLib.SourceForge.io/ 

C++/doc/classGeographicLib_1_1JacobiConformal.html#details>} and U{Jacobi's 

conformal projection<https://GeographicLib.SourceForge.io/C++/doc/jacobi.html>}. 

''' 

_Error = JacobiError 

 

# @Property_RO 

# def ab(self): 

# '''Get relative magnitude C{ab} (C{None} or C{meter}, same units as B{C{a}}). 

# ''' 

# return self._ab 

 

# @Property_RO 

# def bc(self): 

# '''Get relative magnitude C{bc} (C{None} or C{meter}, same units as B{C{a}}). 

# ''' 

# return self._bc 

 

@Property_RO 

def _xE(self): 

'''(INTERNAL) Get the x-elliptic function. 

''' 

k2, kp2 = self._k2_kp2 

# -a2b2 / b2 == (b2 - a2) / b2 == 1 - a2 / b2 == 1 - a2_b2 

return Elliptic(k2, _1_0 - self._a2_b2, kp2, self._a2_b2) 

 

def xR(self, omega): 

'''Compute a Jacobi Conformal C{x} projection. 

 

@arg omega: Ellipsoidal longitude (C{radians} or L{Degrees}). 

 

@return: The C{x} projection (C{Radians}). 

''' 

return self.xR_(*_sincos2(omega)) 

 

def xR_(self, somega, comega): 

'''Compute a Jacobi Conformal C{x} projection. 

 

@arg somega: Ellipsoidal longitude C{sin(omega)} (C{scalar}). 

@arg comega: Ellipsoidal longitude C{cos(omega)} (C{scalar}). 

 

@return: The C{x} projection (C{Radians}). 

''' 

s, c = self._norm2(somega, comega, self.a) 

return Radians(x=self._xE.fPi(s, c) * self._a2_b2) 

 

def xyR2(self, beta, omega, name=NN): 

'''Compute a Jacobi Conformal C{x} and C{y} projection. 

 

@arg beta: Ellipsoidal latitude (C{radians} or L{Degrees}). 

@arg omega: Ellipsoidal longitude (C{radians} or L{Degrees}). 

@kwarg name: Optional name (C{str}). 

 

@return: A L{Jacobi2Tuple}C{(x, y)}. 

''' 

return self.xyR2_(*(_sincos2(beta) + _sincos2(omega)), 

name=name or self.xyR2.__name__) 

 

def xyR2_(self, sbeta, cbeta, somega, comega, name=NN): 

'''Compute a Jacobi Conformal C{x} and C{y} projection. 

 

@arg sbeta: Ellipsoidal latitude C{sin(beta)} (C{scalar}). 

@arg cbeta: Ellipsoidal latitude C{cos(beta)} (C{scalar}). 

@arg somega: Ellipsoidal longitude C{sin(omega)} (C{scalar}). 

@arg comega: Ellipsoidal longitude C{cos(omega)} (C{scalar}). 

@kwarg name: Optional name (C{str}). 

 

@return: A L{Jacobi2Tuple}C{(x, y)}. 

''' 

return Jacobi2Tuple(self.xR_(somega, comega), 

self.yR_(sbeta, cbeta), 

name=name or self.xyR2_.__name__) 

 

@Property_RO 

def xyQ2(self): 

'''Get the Jacobi Conformal quadrant size (L{Jacobi2Tuple}C{(x, y)}). 

''' 

return Jacobi2Tuple(Radians(x=self._a2_b2 * self._xE.cPi), 

Radians(y=self._c2_b2 * self._yE.cPi), 

name=JacobiConformal.xyQ2.name) 

 

@Property_RO 

def _yE(self): 

'''(INTERNAL) Get the x-elliptic function. 

''' 

kp2, k2 = self._k2_kp2 # swapped! 

# b2c2 / b2 == (b2 - c2) / b2 == 1 - c2 / b2 == e2bc 

return Elliptic(k2, self.e2bc, kp2, self._c2_b2) 

 

def yR(self, beta): 

'''Compute a Jacobi Conformal C{y} projection. 

 

@arg beta: Ellipsoidal latitude (C{radians} or L{Degrees}). 

 

@return: The C{y} projection (C{Radians}). 

''' 

return self.yR_(*_sincos2(beta)) 

 

def yR_(self, sbeta, cbeta): 

'''Compute a Jacobi Conformal C{y} projection. 

 

@arg sbeta: Ellipsoidal latitude C{sin(beta)} (C{scalar}). 

@arg cbeta: Ellipsoidal latitude C{cos(beta)} (C{scalar}). 

 

@return: The C{y} projection (C{Radians}). 

''' 

s, c = self._norm2(sbeta, cbeta, self.c) 

return Radians(y=self._yE.fPi(s, c) * self._c2_b2) 

 

 

def _hartzell3d2(pov, los, a, b, c, Error): 

'''(INTERNAL) Hartzell's "Satellite Lin-of-Sight Intersection ..." 

''' 

a2 = a**2 

if a == b: 

b2 = a2 

p2 = _1_0 

else: 

b2 = b**2 

p2 = b2 / a2 

c2 = c**2 

q2 = c2 / a2 

 

V3 = _MODS.vector3d._otherV3d 

p3 = V3(pov=pov) 

u3 = V3(los=los) if los else p3.negate() 

u3 = u3.unit() # unit vector, opposing signs 

 

x2, y2, z2 = p3.x2y2z2 # p3.times_(p3).xyz 

ux, vy, wz = u3.times_(p3).xyz 

u2, v2, w2 = u3.x2y2z2 # u3.times_(u3).xyz 

 

t = p2 * c2, c2, b2 

m = fdot(t, u2, v2, w2) # a2 factored out 

if m < EPS0: # zero or near-null LOS vector 

raise Error(_near_(_null_)) 

 

r = fsum_(b2 * w2, c2 * v2, -v2 * z2, vy * wz * 2, 

-w2 * y2, b2 * u2 * q2, -u2 * z2 * p2, ux * wz * 2 * p2, 

-w2 * x2 * p2, -u2 * y2 * q2, -v2 * x2 * q2, ux * vy * 2 * q2, floats=True) 

if r > 0: # a2 factored out 

r = sqrt(r) * b * c # == a * a * b * c / a2 

elif r < 0: # LOS pointing away from or missing the triaxial 

raise Error(_opposite_ if max(ux, vy, wz) > 0 else _outside_) 

 

n = fdot(t, ux, vy, wz) # a2 factored out 

d = (n + r) / m # (n - r) / m for antipode 

if d > 0: # POV inside or LOS missing the triaxial 

raise Error(_inside_ if min(x2 - a2, y2 - b2, z2 - c2) < EPS else _outside_) 

elif fsum_(x2, y2, z2) < d**2: # d past triaxial's center 

raise Error(_too_(_distant_)) 

 

v = p3.minus(u3.times(d)) # Vector3d 

h = p3.minus(v).length 

return v, h 

 

 

def _normalTo4(x, y, a, b, eps=EPS): 

'''(INTERNAL) Nearest point on and distance to a 2-D ellipse. 

 

@see: Function C{.ellipsoids._normalTo3} and U{Eberly<https:// 

www.GeometricTools.com/Documentation/DistancePointEllipseEllipsoid.pdf>}. 

''' 

def _root2d(r, u, v, g, eps): 

# robust root finder 

u *= r 

t0 = v - _1_0 

t1 = _0_0 if g < 0 else (hypot(u, v) - _1_0) 

_f = fabs 

_h = hypot2 

for i in range(1, _TRIPS): 

t = (t0 + t1) * _0_5 

e = _f(t0 - t1) 

if e < eps or t in (t0, t1): 

break 

g = _h(u / (t + r), v / (t + _1_0)) - _1_0 

if g > 0: 

t0 = t 

elif g < 0: 

t1 = t 

else: 

break 

else: # PYCHOK no cover 

t = _root2d.__name__ 

raise TriaxialError(Fmt.no_convergence(e, eps), txt=t) 

return t, i 

 

if not (a >= b > 0 and eps > 0): 

raise TriaxialError(a=a, b=b, eps=eps) 

 

i = None 

if y: 

if x: 

u = fabs(x / a) 

v = fabs(y / b) 

g = hypot2(u, v) - _1_0 

if g: # on the ellipse 

r = (a / b)**2 

t, i = _root2d(r, u, v, g, eps) 

a = x / (t / r + _1_0) 

b = y / (t + _1_0) 

d = hypot(a - x, b - y) 

else: 

a, b, d = x, y, _0_0 

else: # x == 0 

a, b, d = _0_0, y, fabs(b - y) 

 

else: # PYCHOK no cover 

n = a * x 

d = (a + b) * (a - b) 

if d > fabs(n): 

r = n / d 

a *= r 

b *= sqrt(_1_0 - r**2) 

d = hypot(a - x, b) 

else: 

a, b, d = x, _0_0, fabs(a - x) 

return a, b, d, i 

 

 

def _normalTo5(x, y, z, T, eps=EPS): # MCCABE 16 

'''(INTERNAL) Nearest point on and distance to a 3-D triaxial. 

 

@see: U{Eberly<https://www.GeometricTools.com/Documentation/ 

DistancePointEllipseEllipsoid.pdf>}. 

''' 

def _root3d(r, s, u, v, w, g, eps): 

# robust root finder 

u *= r 

v *= s 

t0 = w - _1_0 

t1 = _0_0 if g < 0 else (hypot_(u, v, z) - _1_0) 

_f = fabs 

_h = hypot2_ 

for i in range(1, _TRIPS): 

t = (t0 + t1) * _0_5 

e = _f(t0 - t1) 

if e < eps or t in (t0, t1): 

break 

g = _h(u / (t + r), v / (t + s), w / (t + _1_0)) - _1_0 

if g > 0: 

t0 = t 

elif g < 0: 

t1 = t 

else: 

break 

else: # PYCHOK no cover 

t = _root3d.__name__ 

raise TriaxialError(Fmt.no_convergence(e, eps), txt=t) 

return t, i 

 

a, b, c = T.a, T.b, T.c 

if not (a >= b >= c > 0 and eps > 0): 

raise TriaxialError(a=a, b=b, c=c, eps=eps) 

 

i = None 

if z: 

if y: 

if x: 

u = fabs(x / a) 

v = fabs(y / b) 

w = fabs(z / c) 

g = hypot2_(u, v, w) - _1_0 

if g: 

r = (a / c)**2 

s = (b / c)**2 

t, i = _root3d(r, s, u, v, w, g, eps) 

a = x / (t / r + _1_0) 

b = y / (t / s + _1_0) 

c = z / (t + _1_0) 

d = hypot_(a - x, b - y, c - z) 

else: # on the ellipsoid 

a, b, c, d = x, y, z, _0_0 

else: # x == 0 

a = _0_0 

b, c, d, i = _normalTo4(y, z, b, c, eps=eps) 

elif x: # y == 0 

b = _0_0 

a, c, d, i = _normalTo4(x, z, a, c, eps=eps) 

else: # x == y == 0 

a, b, c, d = x, y, z, fabs(c - z) 

 

else: # z == 0 

t = False 

n = a * x 

d = (a + c) * (a - c) 

if d > fabs(n): 

u = n / d 

n = b * y 

d = (b + c) * (b - c) 

if d > fabs(n): # PYCHOK no cover 

v = n / d 

d = _1_0 - hypot2(u, v) 

if d > 0: 

a *= u 

b *= v 

c *= sqrt(d) 

d = hypot_(a - x, b - y, c) 

t = True 

if not t: 

c = _0_0 

a, b, d, i = _normalTo4(x, y, a, b, eps=eps) 

 

e = hypot2_(a / T.a, b / T.b, c / T.c) - _1_0 

if fabs(e) > eps: 

raise TriaxialError(x=a, y=b, z=c, d=d, triaxial=T, 

e=e, eps=eps, txt=_not_('on')) 

return a, b, c, d, i 

 

 

def _sincos2(x): 

'''Get C{sin} and C{cos} of C{x} in C{radians} or {degrees}. 

''' 

return sincos2d(x) if isinstance(x, Degrees) else ( 

sincos2(x) if isinstance(x, Radians) else 

sincos2(float(x))) # assume C{radians} 

 

 

def _toDegrees(inst, a, b, *c, **toDMS_kwds): 

'''Helper for L{BetaOmega3Tuple} and L{Jacobi2Tuple} 

''' 

if toDMS_kwds: 

toDMS = _MODS.dms.toDMS 

a = toDMS(a.toDegrees(), **toDMS_kwds) 

b = toDMS(b.toDegrees(), **toDMS_kwds) 

elif isinstance(a, Degrees) and \ 

isinstance(b, Degrees): 

return inst 

else: 

a, b = a.toDegrees(), b.toDegrees() 

return inst.classof(a, b, *c, name=inst.name) 

 

 

def _toRadians(inst, a, b, *c): 

'''Helper for L{BetaOmega3Tuple} and L{Jacobi2Tuple} 

''' 

return inst if isinstance(a, Radians) and \ 

isinstance(b, Radians) else \ 

inst.classof(a.toRadians(), b.toRadians(), 

*c, name=inst.name) 

 

 

if __name__ == '__main__': 

 

from pygeodesy.lazily import printf 

 

def _km2m(*abc): 

for m in abc: 

yield m * 1e3 

 

# <https://ArxIV.org/pdf/1909.06452.pdf> Table 1 Semi-axes in km # Planet 

# <https://www.JPS.NASA.gov/education/images/pdf/ss-moons.pdf> 

# <https://link.Springer.com/article/10.1007/s00190-022-01650-9> 

for n, a, b, c in (('Amalthea', 125.0, 73.0, 64), # Jupiter 

('Ariel', 581.1, 577.9, 577.7), # Uranus 

('Earth', 6378.173435, 6378.1039, 6356.7544), 

('Enceladus', 256.6, 251.4, 248.3), # Saturn 

('Europa', 1564.13, 1561.23, 1560.93), # Jupiter 

('Io', 1829.4, 1819.3, 1815.7), # Jupiter 

('Mars', 3394.6, 3393.3, 3376.3), 

('Mimas', 207.4, 196.8, 190.6), # Saturn 

('Miranda', 240.4, 234.2, 232.9), # Uranus 

('Moon', 1735.55, 1735.324, 1734.898), # Earth 

('Tethys', 535.6, 528.2, 525.8)): # Saturn 

t = Triaxial(*_km2m(a, b, c), name=n) 

printf('%r, area_p=%g', t, t.area_p()) 

 

# **) MIT License 

# 

# Copyright (C) 2022-2023 -- 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.