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

 

# -*- coding: utf-8 -*- 

 

u'''Fréchet distances. 

 

Classes L{Frechet}, L{FrechetDegrees}, L{FrechetRadians}, 

L{FrechetCosineAndoyerLambert}, L{FrechetCosineForsytheAndoyerLambert}, 

L{FrechetCosineLaw}, L{FrechetDistanceTo}< L{FrechetEquirectangular}, 

L{FrechetEuclidean}, L{FrechetExact}, L{FrechetFlatLocal}, L{FrechetFlatPolar}, 

L{FrechetHaversine}, L{FrechetHubeny}, L{FrechetKarney}, L{FrechetThomas} 

and L{FrechetVincentys} to compute I{discrete} U{Fréchet 

<https://WikiPedia.org/wiki/Frechet_distance>} distances between two sets 

of C{LatLon}, C{NumPy}, C{tuples} or other types of points. 

 

Only L{FrechetDistanceTo} -iff used with L{ellipsoidalKarney.LatLon} 

points- and L{FrechetKarney} requires installation of I{Charles Karney}'s 

U{geographiclib<https://PyPI.org/project/geographiclib>}. 

 

Typical usage is as follows. First, create a C{Frechet} calculator 

from one set of C{LatLon} points. 

 

C{f = FrechetXyz(points1, ...)} 

 

Get the I{discrete} Fréchet distance to another set of C{LatLon} points 

by 

 

C{t6 = f.discrete(points2)} 

 

Or, use function C{frechet_} with a proper C{distance} function passed 

as keyword arguments as follows 

 

C{t6 = frechet_(points1, points2, ..., distance=...)}. 

 

In both cases, the returned result C{t6} is a L{Frechet6Tuple}. 

 

For C{(lat, lon, ...)} points in a C{NumPy} array or plain C{tuples}, 

wrap the points in a L{Numpy2LatLon} respectively L{Tuple2LatLon} 

instance, more details in the documentation thereof. 

 

For other points, create a L{Frechet} sub-class with the appropriate 

C{distance} method overloading L{Frechet.distance} as in this example. 

 

>>> from pygeodesy import Frechet, hypot_ 

>>> 

>>> class F3D(Frechet): 

>>> """Custom Frechet example. 

>>> """ 

>>> def distance(self, p1, p2): 

>>> return hypot_(p1.x - p2.x, p1.y - p2.y, p1.z - p2.z) 

>>> 

>>> f3D = F3D(xyz1, ..., units="...") 

>>> t6 = f3D.discrete(xyz2) 

 

Transcribed from the original U{Computing Discrete Fréchet Distance 

<https://www.kr.TUWien.ac.AT/staff/eiter/et-archive/cdtr9464.pdf>} by 

Eiter, T. and Mannila, H., 1994, April 25, Technical Report CD-TR 94/64, 

Information Systems Department/Christian Doppler Laboratory for Expert 

Systems, Technical University Vienna, Austria. 

 

This L{Frechet.discrete} implementation optionally generates intermediate 

points for each point set separately. For example, using keyword argument 

C{fraction=0.5} adds one additional point halfway between each pair of 

points. Or using C{fraction=0.1} interpolates nine additional points 

between each points pair. 

 

The L{Frechet6Tuple} attributes C{fi1} and/or C{fi2} will be I{fractional} 

indices of type C{float} if keyword argument C{fraction} is used. Otherwise, 

C{fi1} and/or C{fi2} are simply type C{int} indices into the respective 

points set. 

 

For example, C{fractional} index value 2.5 means an intermediate point 

halfway between points[2] and points[3]. Use function L{fractional} 

to obtain the intermediate point for a I{fractional} index in the 

corresponding set of points. 

 

The C{Fréchet} distance was introduced in 1906 by U{Maurice Fréchet 

<https://WikiPedia.org/wiki/Maurice_Rene_Frechet>}, see U{reference 

[6]<https://www.kr.TUWien.ac.AT/staff/eiter/et-archive/cdtr9464.pdf>}. 

It is a measure of similarity between curves that takes into account the 

location and ordering of the points. Therefore, it is often a better metric 

than the well-known C{Hausdorff} distance, see the L{hausdorff} module. 

''' 

 

from pygeodesy.basics import isscalar, _xinstanceof 

from pygeodesy.datums import Datum, _WGS84 

from pygeodesy.errors import _IsnotError, PointsError 

from pygeodesy.fmath import hypot2 

from pygeodesy.formy import cosineAndoyerLambert_, cosineForsytheAndoyerLambert_, \ 

cosineLaw_, euclidean_, flatPolar_, haversine_, \ 

thomas_, vincentys_, _scale_rad 

from pygeodesy.interns import EPS, EPS1, INF, NN, _datum_, _distanceTo_, _DOT_, \ 

_n_, _points_, _units_ 

from pygeodesy.iters import points2 as _points2 

from pygeodesy.lazily import _ALL_LAZY, _FOR_DOCS 

from pygeodesy.named import _Named, _NamedTuple, notOverloaded, _Pass 

from pygeodesy.namedTuples import PhiLam2Tuple 

from pygeodesy.points import _fractional 

from pygeodesy.props import Property_RO, property_doc_ 

from pygeodesy.streprs import _boolkwds, Fmt 

from pygeodesy.units import FIx, Float, Number_, _Str_degrees, _Str_meter, \ 

_Str_NN, _Str_radians, _Str_radians2, _xUnit, _xUnits 

from pygeodesy.utily import unrollPI 

 

from collections import defaultdict as _defaultdict 

from math import radians 

 

__all__ = _ALL_LAZY.frechet 

__version__ = '21.10.05' 

 

 

def _fraction(fraction, n): 

f = 1 # int, no fractional indices 

if fraction in (None, 1): 

pass 

elif not (isscalar(fraction) and EPS < fraction < EPS1 

and (float(n) - fraction) < n): 

raise FrechetError(fraction=fraction) 

elif fraction < EPS1: 

f = float(fraction) 

return f 

 

 

class FrechetError(PointsError): 

'''Fréchet issue. 

''' 

pass 

 

 

class Frechet(_Named): 

'''Frechet base class, requires method L{Frechet.distance} to 

be overloaded. 

''' 

_adjust = None # not applicable 

_datum = None # not applicable 

_f1 = 1 

_n1 = 0 

_ps1 = None 

_units = _Str_NN # XXX Str to _Pass and for backward compatibility 

_wrap = None # not applicable 

 

def __init__(self, points, fraction=None, name=NN, units=NN, **wrap_adjust): 

'''New C{Frechet...} calculator/interpolator. 

 

@arg points: First set of points (C{LatLon}[], L{Numpy2LatLon}[], 

L{Tuple2LatLon}[] or C{other}[]). 

@kwarg fraction: Index fraction (C{float} in L{EPS}..L{EPS1}) to 

interpolate intermediate B{C{points}} or use 

C{None}, C{0} or C{1} for no intermediate 

B{C{points}} and no I{fractional} indices. 

@kwarg name: Optional calculator/interpolator name (C{str}). 

@kwarg units: Optional, the distance units (C{Unit} or C{str}). 

@kwarg wrap_adjust: Optionally, C{wrap} and unroll longitudes, iff 

applicable (C{bool}) and C{adjust} wrapped, 

unrolled longitudinal delta by the cosine 

of the mean latitude, iff applicable. 

 

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

invalid B{C{fraction}} or B{C{wrap}} or 

B{C{ajust}} not applicable. 

 

''' 

self._n1, self._ps1 = self._points2(points) 

if fraction: 

self.fraction = fraction 

if name: 

self.name = name 

if units: # and not self.units: 

self.units = units 

if wrap_adjust: 

_boolkwds(self, **wrap_adjust) 

 

@Property_RO 

def adjust(self): 

'''Get the adjust setting (C{bool} or C{None} if not applicable). 

''' 

return self._adjust 

 

@Property_RO 

def datum(self): 

'''Get the datum (L{Datum} or C{None} if not applicable). 

''' 

return self._datum 

 

def _datum_setter(self, datum): 

'''(INTERNAL) Set the datum. 

''' 

d = datum or getattr(self._ps1[0], _datum_, datum) 

if d and d != self.datum: # PYCHOK no cover 

_xinstanceof(Datum, datum=d) 

self._datum = d 

 

def discrete(self, points, fraction=None): 

'''Compute the C{forward, discrete Fréchet} distance. 

 

@arg points: Second set of points (C{LatLon}[], L{Numpy2LatLon}[], 

L{Tuple2LatLon}[] or C{other}[]). 

@kwarg fraction: Index fraction (C{float} in L{EPS}..L{EPS1}) to 

interpolate intermediate B{C{points}} or use 

C{None}, C{0} or C{1} for no intermediate 

B{C{points}} and no I{fractional} indices. 

 

@return: A L{Frechet6Tuple}C{(fd, fi1, fi2, r, n, units)}. 

 

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

invalid B{C{point}} or B{C{fraction}}. 

 

@raise RecursionError: Recursion depth exceeded, see U{sys.getrecursionlimit() 

<https://docs.Python.org/3/library/sys.html#sys.getrecursionlimit>}. 

''' 

n2, ps2 = self._points2(points) 

 

f2 = _fraction(fraction, n2) 

p2 = self.points_fraction if f2 < EPS1 else self.points_ # PYCHOK expected 

 

f1 = self.fraction 

p1 = self.points_fraction if f1 < EPS1 else self.points_ # PYCHOK expected 

 

def dF(fi1, fi2): 

return self.distance(p1(self._ps1, fi1), p2(ps2, fi2)) 

 

try: 

return _frechet_(self._n1, f1, n2, f2, dF, self.units) 

except TypeError as x: 

t = _DOT_(self.classname, self.discrete.__name__) 

raise FrechetError(t, txt=str(x)) 

 

def distance(self, point1, point2): # PYCHOK no cover 

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

''' 

notOverloaded(self, point1, point2) 

 

@property_doc_(''' the index fraction (C{float}).''') 

def fraction(self): 

'''Get the index fraction (C{float} or C{1}). 

''' 

return self._f1 

 

@fraction.setter # PYCHOK setter! 

def fraction(self, fraction): 

'''Set the index fraction (C{float} or C{1}). 

 

@arg fraction: Index fraction (C{float} in L{EPS}..L{EPS1}) to 

interpolate intermediate B{C{points}} or use 

C{None}, C{0} or C{1} for no intermediate 

B{C{points}} and no I{fractional} indices. 

 

@raise FrechetError: Invalid B{C{fraction}}. 

''' 

self._f1 = _fraction(fraction, self._n1) 

 

def point(self, point): 

'''Convert a point for the C{.distance} method. 

 

@arg point: The point to convert ((C{LatLon}, L{Numpy2LatLon}, 

L{Tuple2LatLon} or C{other}). 

 

@return: The converted B{C{point}}. 

''' 

return point # pass thru 

 

def points_(self, points, i): 

'''Get and convert a point for the C{.distance} method. 

 

@arg points: The orignal B{C{points}} to convert ((C{LatLon}[], 

L{Numpy2LatLon}[], L{Tuple2LatLon}[] or C{other}[]). 

@arg i: The B{C{points}} index (C{int}). 

 

@return: The converted B{C{points[i]}}. 

''' 

return self.point(points[i]) 

 

def _points2(self, points): 

'''(INTERNAL) Check a set of points. 

''' 

return _points2(points, closed=False, Error=FrechetError) 

 

def points_fraction(self, points, fi): 

'''Get and convert a I{fractional} point for the C{.distance} method. 

 

@arg points: The orignal B{C{points}} to convert ((C{LatLon}[], 

L{Numpy2LatLon}[], L{Tuple2LatLon}[] or C{other}[]). 

@arg fi: The I{fractional} index in B{C{points}} (C{float} or C{int}). 

 

@return: The interpolated, converted, intermediate B{C{points[fi]}}. 

''' 

return self.point(_fractional(points, fi, None)) # wrap=None? 

 

@property_doc_(''' the distance units (C{Unit} or C{str}).''') 

def units(self): 

'''Get the distance units (C{Unit} or C{str}). 

''' 

return self._units 

 

@units.setter # PYCHOK setter! 

def units(self, units): 

'''Set the distance units. 

 

@arg units: New units (C{Unit} or C{str}). 

 

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

''' 

self._units = _xUnits(units, Base=Float) 

 

@Property_RO 

def wrap(self): 

'''Get the wrap setting (C{bool} or C{None} if not applicable). 

''' 

return self._wrap 

 

 

class FrechetDegrees(Frechet): 

'''L{Frechet} base class for distances in C{degrees} from 

C{LatLon} points in C{degrees}. 

''' 

_units = _Str_degrees 

 

if _FOR_DOCS: 

__init__ = Frechet.__init__ 

discrete = Frechet.discrete 

 

 

class FrechetRadians(Frechet): 

'''L{Frechet} base class for distances in C{radians} or C{radians 

squared} from C{LatLon} points converted from C{degrees} to 

C{radians}. 

''' 

_units = _Str_radians 

 

if _FOR_DOCS: 

__init__ = Frechet.__init__ 

discrete = Frechet.discrete 

 

def point(self, point): 

'''Convert C{(lat, lon)} point in degrees to C{(a, b)} 

in radians. 

 

@return: An L{PhiLam2Tuple}C{(phi, lam)}. 

''' 

try: 

return point.philam 

except AttributeError: 

return PhiLam2Tuple(radians(point.lat), radians(point.lon)) 

 

 

class FrechetCosineAndoyerLambert(FrechetRadians): 

'''Compute the C{Frechet} distance based on the I{angular} distance 

in C{radians} from function L{pygeodesy.cosineAndoyerLambert_}. 

 

@see: L{FrechetCosineForsytheAndoyerLambert}, L{FrechetDistanceTo}, 

L{FrechetExact}, L{FrechetFlatLocal}, L{FrechetHubeny}, 

L{FrechetThomas} and L{FrechetKarney}. 

''' 

_datum = _WGS84 

_wrap = False 

 

def __init__(self, points, datum=None, wrap=False, fraction=None, name=NN): 

'''New L{FrechetCosineAndoyerLambert} calculator/interpolator. 

 

@arg points: First set of points (C{LatLon}[], L{Numpy2LatLon}[], 

L{Tuple2LatLon}[] or C{other}[]). 

@kwarg datum: Optional datum overriding the default C{Datums.WGS84} 

and first B{C{points}}' datum (L{Datum}). 

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

@kwarg fraction: Index fraction (C{float} in L{EPS}..L{EPS1}) to 

interpolate intermediate B{C{points}} or use C{0}, 

C{1} or C{None} to avoid intermediate B{C{points}} 

and L{pygeodesy.fractional} indices. 

@kwarg name: Optional calculator/interpolator name (C{str}). 

 

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

invalid B{C{fraction}}. 

 

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

''' 

FrechetRadians.__init__(self, points, fraction=fraction, name=name, 

wrap=wrap) 

self._datum_setter(datum) 

 

if _FOR_DOCS: 

discrete = Frechet.discrete 

 

def distance(self, p1, p2): 

'''Return the L{pygeodesy.cosineAndoyerLambert_} distance in C{radians}. 

''' 

r, _ = unrollPI(p1.lam, p2.lam, wrap=self._wrap) 

return cosineAndoyerLambert_(p2.phi, p1.phi, r, datum=self._datum) 

 

 

class FrechetCosineForsytheAndoyerLambert(FrechetRadians): 

'''Compute the C{Frechet} distance based on the I{angular} distance 

in C{radians} from function L{pygeodesy.cosineForsytheAndoyerLambert_}. 

 

@see: L{FrechetCosineAndoyerLambert}, L{FrechetDistanceTo}, 

L{FrechetExact}, L{FrechetFlatLocal}, L{FrechetHubeny}, 

L{FrechetThomas} and L{FrechetKarney}. 

''' 

_datum = _WGS84 

_wrap = False 

 

def __init__(self, points, datum=None, wrap=False, fraction=None, name=NN): 

'''New L{FrechetCosineForsytheAndoyerLambert} calculator/interpolator. 

 

@arg points: First set of points (C{LatLon}[], L{Numpy2LatLon}[], 

L{Tuple2LatLon}[] or C{other}[]). 

@kwarg datum: Optional datum overriding the default C{Datums.WGS84} 

and first B{C{points}}' datum (L{Datum}). 

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

@kwarg fraction: Index fraction (C{float} in L{EPS}..L{EPS1}) to 

interpolate intermediate B{C{points}} or use C{0}, 

C{1} or C{None} to avoid intermediate B{C{points}} 

and L{pygeodesy.fractional} indices. 

@kwarg name: Optional calculator/interpolator name (C{str}). 

 

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

invalid B{C{fraction}}. 

 

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

''' 

FrechetRadians.__init__(self, points, fraction=fraction, name=name, 

wrap=wrap) 

self._datum_setter(datum) 

 

if _FOR_DOCS: 

discrete = Frechet.discrete 

 

def distance(self, p1, p2): 

'''Return the L{pygeodesy.cosineForsytheAndoyerLambert_} distance in C{radians}. 

''' 

r, _ = unrollPI(p1.lam, p2.lam, wrap=self._wrap) 

return cosineForsytheAndoyerLambert_(p2.phi, p1.phi, r, datum=self._datum) 

 

 

class FrechetCosineLaw(FrechetRadians): 

'''Compute the C{Frechet} distance based on the I{angular} distance 

in C{radians} from function L{pygeodesy.cosineLaw_}. 

 

@see: L{FrechetEquirectangular}, L{FrechetEuclidean}, 

L{FrechetExact}, L{FrechetFlatPolar}, L{FrechetHaversine} 

and L{FrechetVincentys}. 

 

@note: See note at function L{pygeodesy.vincentys_}. 

''' 

_wrap = False 

 

def __init__(self, points, wrap=False, fraction=None, name=NN): 

'''New L{FrechetCosineLaw} calculator/interpolator. 

 

@arg points: First set of points (C{LatLon}[], L{Numpy2LatLon}[], 

L{Tuple2LatLon}[] or C{other}[]). 

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

@kwarg fraction: Index fraction (C{float} in L{EPS}..L{EPS1}) to 

interpolate intermediate B{C{points}} or use C{0}, 

C{1} or C{None} to avoid intermediate B{C{points}} 

and L{pygeodesy.fractional} indices. 

@kwarg name: Optional calculator/interpolator name (C{str}). 

 

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

invalid B{C{fraction}}. 

''' 

FrechetRadians.__init__(self, points, fraction=fraction, name=name, 

wrap=wrap) 

 

if _FOR_DOCS: 

discrete = Frechet.discrete 

 

def distance(self, p1, p2): 

'''Return the L{pygeodesy.cosineLaw_} distance in C{radians}. 

''' 

r, _ = unrollPI(p1.lam, p2.lam, wrap=self._wrap) 

return cosineLaw_(p2.phi, p1.phi, r) 

 

 

class FrechetDistanceTo(Frechet): 

'''Compute the C{Frechet} distance based on the distance from the 

points' C{LatLon.distanceTo} method, conventionally in C{meter}. 

 

@see: L{FrechetCosineAndoyerLambert}, L{FrechetCosineForsytheAndoyerLambert}, 

L{FrechetExact}, L{FrechetFlatLocal}, L{FrechetHubeny}, L{FrechetThomas} 

and L{FrechetKarney}. 

''' 

_distanceTo_kwds = {} 

_units = _Str_meter 

 

def __init__(self, points, fraction=None, name=NN, **distanceTo_kwds): 

'''New L{FrechetDistanceTo} calculator/interpolator. 

 

@arg points: First set of points (C{LatLon}[]). 

@kwarg fraction: Index fraction (C{float} in L{EPS}..L{EPS1}) to 

interpolate intermediate B{C{points}} or use C{0}, 

C{1} or C{None} to avoid intermediate B{C{points}} 

and L{pygeodesy.fractional} indices. 

@kwarg name: Optional calculator/interpolator name (C{str}). 

@kwarg distanceTo_kwds: Optional keyword arguments for the 

B{C{points}}' C{LatLon.distanceTo} 

method. 

 

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

invalid B{C{point}} or B{C{fraction}}. 

 

@raise ImportError: Package U{geographiclib 

<https://PyPI.org/project/geographiclib>} missing 

iff B{C{points}} are L{ellipsoidalKarney.LatLon}s. 

 

@note: All B{C{points}} I{must} be instances of the same 

ellipsoidal or spherical C{LatLon} class. 

''' 

FrechetRadians.__init__(self, points, fraction=fraction, name=name) 

if distanceTo_kwds: 

self._distanceTo_kwds = distanceTo_kwds 

 

if _FOR_DOCS: 

discrete = Frechet.discrete 

 

def distance(self, p1, p2): 

'''Return the distance in C{meter}. 

''' 

return p1.distanceTo(p2, **self._distanceTo_kwds) 

 

def _points2(self, points): 

'''(INTERNAL) Check a set of points. 

''' 

np, ps = Frechet._points2(self, points) 

for i, p in enumerate(ps): 

if not callable(getattr(p, _distanceTo_, None)): 

i = Fmt.SQUARE(_points_, i) 

raise FrechetError(i, p, txt=_distanceTo_) 

return np, ps 

 

 

class FrechetEquirectangular(FrechetRadians): 

'''Compute the C{Frechet} distance based on the I{equirectangular} 

distance in C{radians squared} like function L{pygeodesy.equirectangular_}. 

 

@see: L{FrechetCosineLaw}, L{FrechetEuclidean}, L{FrechetExact}, 

L{FrechetFlatPolar}, L{FrechetHaversine} and L{FrechetVincentys}. 

''' 

_adjust = True 

_units = _Str_radians2 

_wrap = False 

 

def __init__(self, points, adjust=True, wrap=False, fraction=None, name=NN): 

'''New L{FrechetEquirectangular} calculator/interpolator. 

 

@arg points: First set of points (C{LatLon}[], L{Numpy2LatLon}[], 

L{Tuple2LatLon}[] or C{other}[]). 

@kwarg adjust: Adjust the wrapped, unrolled longitudinal 

delta by the cosine of the mean latitude (C{bool}). 

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

@kwarg fraction: Index fraction (C{float} in L{EPS}..L{EPS1}) to 

interpolate intermediate B{C{points}} or use C{0}, 

C{1} or C{None} to avoid intermediate B{C{points}} 

and L{pygeodesy.fractional} indices. 

@kwarg name: Optional calculator/interpolator name (C{str}). 

 

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

invalid B{C{adjust}} or B{C{seed}}. 

''' 

FrechetRadians.__init__(self, points, fraction=fraction, name=name, 

adjust=adjust, wrap=wrap) 

 

if _FOR_DOCS: 

discrete = Frechet.discrete 

 

def distance(self, p1, p2): 

'''Return the L{pygeodesy.equirectangular_} distance in C{radians squared}. 

''' 

r, _ = unrollPI(p1.lam, p2.lam, wrap=self._wrap) 

if self._adjust: 

r *= _scale_rad(p1.phi, p2.phi) 

return hypot2(r, p2.phi - p1.phi) # like equirectangular_ d2 

 

 

class FrechetEuclidean(FrechetRadians): 

'''Compute the C{Frechet} distance based on the I{Euclidean} 

distance in C{radians} from function L{pygeodesy.euclidean_}. 

 

@see: L{FrechetCosineLaw}, L{FrechetEquirectangular}, 

L{FrechetExact}, L{FrechetFlatPolar}, L{FrechetHaversine} 

and L{FrechetVincentys}. 

''' 

_adjust = True 

_wrap = True # fixed 

 

def __init__(self, points, adjust=True, fraction=None, name=NN): 

'''New L{FrechetEuclidean} calculator/interpolator. 

 

@arg points: First set of points (C{LatLon}[], L{Numpy2LatLon}[], 

L{Tuple2LatLon}[] or C{other}[]). 

@kwarg adjust: Adjust the wrapped, unrolled longitudinal 

delta by the cosine of the mean latitude (C{bool}). 

@kwarg fraction: Index fraction (C{float} in L{EPS}..L{EPS1}) to 

interpolate intermediate B{C{points}} or use C{0}, 

C{1} or C{None} to avoid intermediate B{C{points}} 

and L{pygeodesy.fractional} indices. 

@kwarg name: Optional calculator/interpolator name (C{str}). 

 

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

invalid B{C{fraction}}. 

''' 

FrechetRadians.__init__(self, points, fraction=fraction, name=name, 

adjust=adjust) 

 

if _FOR_DOCS: 

discrete = Frechet.discrete 

 

def distance(self, p1, p2): 

'''Return the L{pygeodesy.euclidean_} distance in C{radians}. 

''' 

return euclidean_(p2.phi, p1.phi, p2.lam - p1.lam, adjust=self._adjust) 

 

 

class FrechetExact(FrechetDegrees): 

'''Compute the C{Frechet} distance based on the I{angular} 

distance in C{degrees} from method L{GeodesicExact}C{.Inverse}. 

 

@see: L{FrechetCosineAndoyerLambert}, L{FrechetCosineForsytheAndoyerLambert}, 

L{FrechetDistanceTo}, L{FrechetFlatLocal}, L{FrechetHubeny}, 

L{FrechetKarney} and L{FrechetThomas}. 

''' 

_datum = _WGS84 

_Inverse1 = None 

_wrap = False 

 

def __init__(self, points, datum=None, wrap=False, fraction=None, name=NN): 

'''New L{FrechetExact} calculator/interpolator. 

 

@arg points: First set of points (C{LatLon}[], L{Numpy2LatLon}[], 

L{Tuple2LatLon}[] or C{other}[]). 

@kwarg datum: Optional datum overriding the default C{Datums.WGS84} 

and first B{C{points}}' datum (L{Datum}). 

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

@kwarg fraction: Index fraction (C{float} in L{EPS}..L{EPS1}) to 

interpolate intermediate B{C{points}} or use C{0}, 

C{1} or C{None} to avoid intermediate B{C{points}} 

and L{pygeodesy.fractional} indices. 

@kwarg name: Optional calculator/interpolator name (C{str}). 

 

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

invalid B{C{fraction}}. 

 

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

''' 

FrechetDegrees.__init__(self, points, fraction=fraction, name=name, 

wrap=wrap) 

self._datum_setter(datum) 

self._Inverse1 = self.datum.ellipsoid.geodesicx.Inverse1 # note -x 

 

if _FOR_DOCS: 

discrete = Frechet.discrete 

 

def distance(self, p1, p2): 

'''Return the non-negative I{angular} distance in C{degrees}. 

''' 

return self._Inverse1(p1.lat, p1.lon, p2.lat, p2.lon, wrap=self._wrap) 

 

 

class FrechetFlatLocal(FrechetRadians): 

'''Compute the C{Frechet} distance based on the I{angular} distance in 

C{radians squared} like function L{pygeodesy.flatLocal_}/L{pygeodesy.hubeny_}. 

 

@see: L{FrechetCosineAndoyerLambert}, L{FrechetCosineForsytheAndoyerLambert}, 

L{FrechetDistanceTo}, L{FrechetExact}, L{FrechetHubeny}, 

L{FrechetKarney} and L{FrechetThomas}. 

''' 

_datum = _WGS84 

_hubeny2_ = None 

_units = _Str_radians2 

_wrap = False 

 

def __init__(self, points, datum=None, wrap=False, fraction=None, name=NN): 

'''New L{FrechetFlatLocal}/L{FrechetHubeny} calculator/interpolator. 

 

@arg points: First set of points (C{LatLon}[], L{Numpy2LatLon}[], 

L{Tuple2LatLon}[] or C{other}[]). 

@kwarg datum: Optional datum overriding the default C{Datums.WGS84} 

and first B{C{points}}' datum (L{Datum}). 

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

@kwarg fraction: Index fraction (C{float} in L{EPS}..L{EPS1}) to 

interpolate intermediate B{C{points}} or use C{0}, 

C{1} or C{None} to avoid intermediate B{C{points}} 

and L{pygeodesy.fractional} indices. 

@kwarg name: Optional calculator/interpolator name (C{str}). 

 

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

invalid B{C{fraction}}. 

 

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

''' 

FrechetRadians.__init__(self, points, fraction=fraction, name=name, 

wrap=wrap) 

self._datum_setter(datum) 

self._hubeny2_ = self.datum.ellipsoid._hubeny2_ 

 

if _FOR_DOCS: 

discrete = Frechet.discrete 

 

def distance(self, p1, p2): 

'''Return the L{pygeodesy.flatLocal_}/L{pygeodesy.hubeny_} distance 

in C{radians squared}. 

''' 

d, _ = unrollPI(p1.lam, p2.lam, wrap=self._wrap) 

return self._hubeny2_(p2.phi, p1.phi, d) 

 

 

class FrechetFlatPolar(FrechetRadians): 

'''Compute the C{Frechet} distance based on the I{angular} distance 

in C{radians} from function L{flatPolar_}. 

 

@see: L{FrechetCosineLaw}, L{FrechetEquirectangular}, 

L{FrechetEuclidean}, L{FrechetExact}, L{FrechetHaversine} 

and L{FrechetVincentys}. 

''' 

_wrap = False 

 

def __init__(self, points, wrap=False, fraction=None, name=NN): 

'''New L{FrechetFlatPolar} calculator/interpolator. 

 

@arg points: First set of points (C{LatLon}[], L{Numpy2LatLon}[], 

L{Tuple2LatLon}[] or C{other}[]). 

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

@kwarg fraction: Index fraction (C{float} in L{EPS}..L{EPS1}) to 

interpolate intermediate B{C{points}} or use C{0}, 

C{1} or C{None} to avoid intermediate B{C{points}} 

and L{pygeodesy.fractional} indices. 

@kwarg name: Optional calculator/interpolator name (C{str}). 

 

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

invalid B{C{fraction}}. 

''' 

FrechetRadians.__init__(self, points, fraction=fraction, name=name, 

wrap=wrap) 

 

if _FOR_DOCS: 

discrete = Frechet.discrete 

 

def distance(self, p1, p2): 

'''Return the L{flatPolar_} distance in C{radians}. 

''' 

d, _ = unrollPI(p1.lam, p2.lam, wrap=self._wrap) 

return flatPolar_(p2.phi, p1.phi, d) 

 

 

class FrechetHaversine(FrechetRadians): 

'''Compute the C{Frechet} distance based on the I{angular} 

distance in C{radians} from function L{pygeodesy.haversine_}. 

 

@see: L{FrechetCosineLaw}, L{FrechetEquirectangular}, 

L{FrechetEuclidean}, L{FrechetExact}, L{FrechetFlatPolar} 

and L{FrechetVincentys}. 

 

@note: See note at function L{pygeodesy.vincentys_}. 

''' 

_wrap = False 

 

def __init__(self, points, wrap=False, fraction=None, name=NN): 

'''New L{FrechetHaversine} calculator/interpolator. 

 

@arg points: First set of points (C{LatLon}[], L{Numpy2LatLon}[], 

L{Tuple2LatLon}[] or C{other}[]). 

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

@kwarg fraction: Index fraction (C{float} in L{EPS}..L{EPS1}) to 

interpolate intermediate B{C{points}} or use C{0}, 

C{1} or C{None} to avoid intermediate B{C{points}} 

and L{pygeodesy.fractional} indices. 

@kwarg name: Optional calculator/interpolator name (C{str}). 

 

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

invalid B{C{fraction}}. 

''' 

FrechetRadians.__init__(self, points, fraction=fraction, name=name, 

wrap=wrap) 

 

if _FOR_DOCS: 

discrete = Frechet.discrete 

 

def distance(self, p1, p2): 

'''Return the L{pygeodesy.haversine_} distance in C{radians}. 

''' 

d, _ = unrollPI(p1.lam, p2.lam, wrap=self._wrap) 

return haversine_(p2.phi, p1.phi, d) 

 

 

class FrechetHubeny(FrechetFlatLocal): # for Karl Hubeny 

if _FOR_DOCS: 

__doc__ = FrechetFlatLocal.__doc__ 

__init__ = FrechetFlatLocal.__init__ 

discrete = FrechetFlatLocal.discrete 

distance = FrechetFlatLocal.discrete 

 

 

class FrechetKarney(FrechetExact): 

'''Compute the C{Frechet} distance based on the I{angular} 

distance in C{degrees} from I{Karney}'s U{geographiclib 

<https://PyPI.org/project/geographiclib>} U{Geodesic 

<https://GeographicLib.SourceForge.io/html/python/code.html>} 

Inverse method. 

 

@see: L{FrechetCosineAndoyerLambert}, L{FrechetCosineForsytheAndoyerLambert}, 

L{FrechetDistanceTo}, L{FrechetExact}, L{FrechetFlatLocal}, 

L{FrechetHubeny} and L{FrechetThomas}. 

''' 

def __init__(self, points, datum=None, wrap=False, fraction=None, name=NN): 

'''New L{FrechetKarney} calculator/interpolator. 

 

@arg points: First set of points (C{LatLon}[], L{Numpy2LatLon}[], 

L{Tuple2LatLon}[] or C{other}[]). 

@kwarg datum: Optional datum overriding the default C{Datums.WGS84} 

and first B{C{points}}' datum (L{Datum}). 

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

@kwarg fraction: Index fraction (C{float} in L{EPS}..L{EPS1}) to 

interpolate intermediate B{C{points}} or use C{0}, 

C{1} or C{None} to avoid intermediate B{C{points}} 

and L{pygeodesy.fractional} indices. 

@kwarg name: Optional calculator/interpolator name (C{str}). 

 

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

invalid B{C{fraction}}. 

 

@raise ImportError: Package U{geographiclib 

<https://PyPI.org/project/geographiclib>} missing. 

 

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

''' 

FrechetDegrees.__init__(self, points, fraction=fraction, name=name, 

wrap=wrap) 

self._datum_setter(datum) 

self._Inverse1 = self.datum.ellipsoid.geodesic.Inverse1 

 

 

class FrechetThomas(FrechetRadians): 

'''Compute the C{Frechet} distance based on the I{angular} distance 

in C{radians} from function L{pygeodesy.thomas_}. 

 

@see: L{FrechetCosineAndoyerLambert}, L{FrechetCosineForsytheAndoyerLambert}, 

L{FrechetDistanceTo}, L{FrechetExact}, L{FrechetFlatLocal}, 

L{FrechetHubeny} and L{FrechetKarney}. 

''' 

_datum = _WGS84 

_wrap = False 

 

def __init__(self, points, datum=None, wrap=False, fraction=None, name=NN): 

'''New L{FrechetThomas} calculator/interpolator. 

 

@arg points: First set of points (C{LatLon}[], L{Numpy2LatLon}[], 

L{Tuple2LatLon}[] or C{other}[]). 

@kwarg datum: Optional datum overriding the default C{Datums.WGS84} 

and first B{C{points}}' datum (L{Datum}). 

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

@kwarg fraction: Index fraction (C{float} in L{EPS}..L{EPS1}) to 

interpolate intermediate B{C{points}} or use C{0}, 

C{1} or C{None} to avoid intermediate B{C{points}} 

and L{pygeodesy.fractional} indices. 

@kwarg name: Optional calculator/interpolator name (C{str}). 

 

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

invalid B{C{fraction}}. 

 

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

''' 

FrechetRadians.__init__(self, points, fraction=fraction, name=name, 

wrap=wrap) 

self._datum_setter(datum) 

 

if _FOR_DOCS: 

discrete = Frechet.discrete 

 

def distance(self, p1, p2): 

'''Return the L{pygeodesy.thomas_} distance in C{radians}. 

''' 

r, _ = unrollPI(p1.lam, p2.lam, wrap=self._wrap) 

return thomas_(p2.phi, p1.phi, r, datum=self._datum) 

 

 

class FrechetVincentys(FrechetRadians): 

'''Compute the C{Frechet} distance based on the I{angular} 

distance in C{radians} from function L{pygeodesy.vincentys_}. 

 

@see: L{FrechetCosineLaw}, L{FrechetEquirectangular}, 

L{FrechetEuclidean}, L{FrechetExact}, L{FrechetFlatPolar} 

and L{FrechetHaversine}. 

 

@note: See note at function L{pygeodesy.vincentys_}. 

''' 

_wrap = False 

 

def __init__(self, points, wrap=False, fraction=None, name=NN): 

'''New L{FrechetVincentys} calculator/interpolator. 

 

@arg points: First set of points (C{LatLon}[], L{Numpy2LatLon}[], 

L{Tuple2LatLon}[] or C{other}[]). 

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

@kwarg fraction: Index fraction (C{float} in L{EPS}..L{EPS1}) to 

interpolate intermediate B{C{points}} or use C{0}, 

C{1} or C{None} to avoid intermediate B{C{points}} 

and L{pygeodesy.fractional} indices. 

@kwarg name: Optional calculator/interpolator name (C{str}). 

 

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

invalid B{C{fraction}}. 

''' 

FrechetRadians.__init__(self, points, fraction=fraction, name=name, 

wrap=wrap) 

 

if _FOR_DOCS: 

discrete = Frechet.discrete 

 

def distance(self, p1, p2): 

'''Return the L{pygeodesy.vincentys_} distance in C{radians}. 

''' 

d, _ = unrollPI(p1.lam, p2.lam, wrap=self._wrap) 

return vincentys_(p2.phi, p1.phi, d) 

 

 

def _frechet_(ni, fi, nj, fj, dF, units): # MCCABE 14 

'''(INTERNAL) Recursive core of function L{frechet_} 

and method C{discrete} of C{Frechet...} classes. 

''' 

iFs = {} 

 

def iF(i): # cache index, depth ints and floats 

return iFs.setdefault(i, i) 

 

cF = _defaultdict(dict) 

 

def rF(i, j, r): # recursive Fréchet 

i = iF(i) 

j = iF(j) 

try: 

t = cF[i][j] 

except KeyError: 

r = iF(r + 1) 

try: 

if i > 0: 

if j > 0: 

t = min(rF(i - fi, j, r), 

rF(i - fi, j - fj, r), 

rF(i, j - fj, r)) 

elif j < 0: 

raise IndexError 

else: # j == 0 

t = rF(i - fi, 0, r) 

elif i < 0: 

raise IndexError 

 

elif j > 0: # i == 0 

t = rF(0, j - fj, r) 

elif j < 0: # i == 0 

raise IndexError 

else: # i == j == 0 

t = (-INF, i, j, r) 

 

d = dF(i, j) 

if d > t[0]: 

t = (d, i, j, r) 

except IndexError: 

t = (INF, i, j, r) 

cF[i][j] = t 

return t 

 

t = rF(ni - 1, nj - 1, 0) 

t += (sum(map(len, cF.values())), units) 

# del cF, iFs 

return Frechet6Tuple(*t) 

 

 

def frechet_(points1, points2, distance=None, units=NN): 

'''Compute the I{discrete} U{Fréchet<https://WikiPedia.org/wiki/Frechet_distance>} 

distance between two paths given as sets of points. 

 

@arg points1: First set of points (C{LatLon}[], L{Numpy2LatLon}[], 

L{Tuple2LatLon}[] or C{other}[]). 

@arg points2: Second set of points (C{LatLon}[], L{Numpy2LatLon}[], 

L{Tuple2LatLon}[] or C{other}[]). 

@kwarg distance: Callable returning the distance between a B{C{points1}} 

and a B{C{points2}} point (signature C{(point1, point2)}). 

@kwarg units: Optional, the distance units (C{Unit} or C{str}). 

 

@return: A L{Frechet6Tuple}C{(fd, fi1, fi2, r, n, units)} where C{fi1} 

and C{fi2} are type C{int} indices into B{C{points1}} respectively 

B{C{points2}}. 

 

@raise FrechetError: Insufficient number of B{C{points1}} or B{C{points2}}. 

 

@raise RecursionError: Recursion depth exceeded, see U{sys.getrecursionlimit() 

<https://docs.Python.org/3/library/sys.html#sys.getrecursionlimit>}. 

 

@raise TypeError: If B{C{distance}} is not a callable. 

 

@note: Function L{frechet_} does not support I{fractional} indices for 

intermediate B{C{points1}} and B{C{points2}}. 

''' 

if not callable(distance): 

raise _IsnotError(callable.__name__, distance=distance) 

 

n1, ps1 = _points2(points1, closed=False, Error=FrechetError) 

n2, ps2 = _points2(points2, closed=False, Error=FrechetError) 

 

def dF(i1, i2): 

return distance(ps1[i1], ps2[i2]) 

 

return _frechet_(n1, 1, n2, 1, dF, units) 

 

 

class Frechet6Tuple(_NamedTuple): 

'''6-Tuple C{(fd, fi1, fi2, r, n, units)} with the I{discrete} 

U{Fréchet<https://WikiPedia.org/wiki/Frechet_distance>} distance 

C{fd}, I{fractional} indices C{fi1} and C{fi2} as C{FIx}, the 

recursion depth C{r}, the number of distances computed C{n} and 

the L{units} class or class or name of the distance C{units}. 

 

If I{fractional} indices C{fi1} and C{fi2} are C{int}, the 

returned C{fd} is the distance between C{points1[fi1]} and 

C{points2[fi2]}. For C{float} indices, the distance is 

between an intermediate point along C{points1[int(fi1)]} and 

C{points1[int(fi1) + 1]} respectively an intermediate point 

along C{points2[int(fi2)]} and C{points2[int(fi2) + 1]}. 

 

Use function L{fractional} to compute the point at a 

I{fractional} index. 

''' 

_Names_ = ('fd', 'fi1', 'fi2', 'r', _n_, _units_) 

_Units_ = (_Pass, FIx, FIx, Number_, Number_, _Pass) 

 

def toUnits(self, **Error): # PYCHOK expected 

'''Overloaded C{_NamedTuple.toUnits} for C{fd} units. 

''' 

U = _xUnit(self.units, Float) # PYCHOK expected 

self._Units_ = (U,) + Frechet6Tuple._Units_[1:] 

return _NamedTuple.toUnits(self, **Error) 

 

# def __gt__(self, other): 

# _xinstanceof(Frechet6Tuple, other=other) 

# return self if self.fd > other.fd else other # PYCHOK .fd=[0] 

# 

# def __lt__(self, other): 

# _xinstanceof(Frechet6Tuple, other=other) 

# return self if self.fd < other.fd else other # PYCHOK .fd=[0] 

 

# **) 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.