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

 

# -*- coding: utf-8 -*- 

 

u'''Formulary of basic geodesy functions and approximations. 

 

@newfield example: Example, Examples 

''' 

from pygeodesy.fmath import EPS, fStr, fsum_, len2, map1 

from pygeodesy.lazily import _ALL_LAZY 

from pygeodesy.named import Distance4Tuple, LatLon2Tuple, Points2Tuple 

from pygeodesy.utily import PI, PI2, PI_2, R_M, degrees2m, degrees360, \ 

isNumpy2, isTuple2, LimitError, _limiterrors, \ 

sincos2, unroll180, unrollPI, wrap90, wrap180 

 

from math import atan2, cos, degrees, hypot, radians, sin, sqrt # pow 

 

# all public contants, classes and functions 

__all__ = _ALL_LAZY.formy 

__version__ = '19.10.19' 

 

 

def _scaled(lat1, lat2): # degrees 

# scale delta lon by cos(mean of lats) 

m = (lat1 + lat2) * 0.5 

return cos(radians(m)) if abs(m) < 90 else 0 

 

 

def _scaler(rad1, rad2): # radians, imported by heights.HeighIDW2 

# scale delta lon by cos(mean of lats) 

m = (rad1 + rad2) * 0.5 

return cos(m) if abs(m) < PI_2 else 0 

 

 

def antipode(lat, lon): 

'''Return the antipode, the point diametrically opposite 

to a given point. 

 

@param lat: Latitude (C{degrees}). 

@param lon: Longitude (C{degrees}). 

 

@return: A L{LatLon2Tuple}C{(lat, lon)}. 

 

@see: U{Geosphere<https://CRAN.R-Project.org/web/packages/geosphere/geosphere.pdf>}. 

''' 

return LatLon2Tuple(-wrap90(lat), wrap180(lon + 180)) 

 

 

def bearing(lat1, lon1, lat2, lon2, **options): 

'''Compute the initial or final bearing (forward or reverse 

azimuth) between a (spherical) start and end point. 

 

@param lat1: Start latitude (C{degrees}). 

@param lon1: Start longitude (C{degrees}). 

@param lat2: End latitude (C{degrees}). 

@param lon2: End longitude (C{degrees}). 

@keyword options: Optional keyword arguments for function 

L{bearing_}. 

 

@return: Initial or final bearing (compass C{degrees360}) or 

zero if start and end point coincide. 

''' 

ab4 = map1(radians, lat1, lon1, lat2, lon2) 

return degrees(bearing_(*ab4, **options)) 

 

 

def bearing_(a1, b1, a2, b2, final=False, wrap=False): 

'''Compute the initial or final bearing (forward or reverse 

azimuth) between a (spherical) start and end point. 

 

@param a1: Start latitude (C{radians}). 

@param b1: Start longitude (C{radians}). 

@param a2: End latitude (C{radians}). 

@param b2: End longitude (C{radians}). 

@keyword final: Return final bearing if C{True}, initial 

otherwise (C{bool}). 

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

 

@return: Initial or final bearing (compass C{radiansPI2}) or 

zero if start and end point coincide. 

''' 

if final: 

a1, b1, a2, b2 = a2, b2, a1, b1 

r = PI + PI2 

else: 

r = PI2 

 

db, _ = unrollPI(b1, b2, wrap=wrap) 

sa1, ca1, sa2, ca2, sdb, cdb = sincos2(a1, a2, db) 

 

# see <https://MathForum.org/library/drmath/view/55417.html> 

x = ca1 * sa2 - sa1 * ca2 * cdb 

y = sdb * ca2 

 

return (atan2(y, x) + r) % PI2 # wrapPI2 

 

 

def compassAngle(lat1, lon1, lat2, lon2, adjust=True, wrap=False): 

'''Return the angle from North for the direction vector 

M{(lon2 - lon1, lat2 - lat1)} between two points. 

 

Suitable only for short, non-near-polar vectors up to a few 

hundred Km or Miles. Use function L{bearing} for longer 

vectors. 

 

@param lat1: From latitude (C{degrees}). 

@param lon1: From longitude (C{degrees}). 

@param lat2: To latitude (C{degrees}). 

@param lon2: To longitude (C{degrees}). 

@keyword adjust: Adjust the longitudinal delta by the 

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

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

 

@return: Compass angle from North (C{degrees360}). 

 

@note: Courtesy Martin Schultz. 

 

@see: U{Local, flat earth approximation 

<https://www.EdWilliams.org/avform.htm#flat>}. 

''' 

d_lon, _ = unroll180(lon1, lon2, wrap=wrap) 

if adjust: # scale delta lon 

d_lon *= _scaled(lat1, lat2) 

return degrees360(atan2(d_lon, lat2 - lat1)) 

 

 

def equirectangular(lat1, lon1, lat2, lon2, radius=R_M, **options): 

'''Compute the distance between two points using 

the U{Equirectangular Approximation / Projection 

<https://www.Movable-Type.co.UK/scripts/latlong.html>}. 

 

@param lat1: Start latitude (C{degrees}). 

@param lon1: Start longitude (C{degrees}). 

@param lat2: End latitude (C{degrees}). 

@param lon2: End longitude (C{degrees}). 

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

@keyword options: Optional keyword arguments for function 

L{equirectangular_}. 

 

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

 

@see: Function L{equirectangular_} for more details, the 

available B{C{options}}, errors, restrictions and other, 

approximate or accurate distance functions. 

''' 

_, dy, dx, _ = equirectangular_(lat1, lon1, lat2, lon2, **options) # PYCHOK Distance4Tuple 

return degrees2m(hypot(dx, dy), radius=radius) 

 

 

def equirectangular_(lat1, lon1, lat2, lon2, 

adjust=True, limit=45, wrap=False): 

'''Compute the distance between two points using 

the U{Equirectangular Approximation / Projection 

<https://www.Movable-Type.co.UK/scripts/latlong.html>}. 

 

This approximation is valid for short distance of several 

hundred Km or Miles, see the B{C{limit}} keyword argument and 

the L{LimitError}. 

 

@param lat1: Start latitude (C{degrees}). 

@param lon1: Start longitude (C{degrees}). 

@param lat2: End latitude (C{degrees}). 

@param lon2: End longitude (C{degrees}). 

@keyword adjust: Adjust the wrapped, unrolled longitudinal 

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

@keyword limit: Optional limit for lat- and longitudinal deltas 

(C{degrees}) or C{None} or C{0} for unlimited. 

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

 

@return: A L{Distance4Tuple}C{(distance2, delta_lat, delta_lon, 

unroll_lon2)}. 

 

@raise LimitError: If the lat- and/or longitudinal delta exceeds 

the B{C{-limit..+limit}} range and L{limiterrors} 

set to C{True}. 

 

@see: U{Local, flat earth approximation 

<https://www.EdWilliams.org/avform.htm#flat>}, functions 

L{equirectangular}, L{euclidean}, L{haversine} and 

L{vincentys} and methods L{Ellipsoid.distance2}, 

C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}. 

''' 

d_lat = lat2 - lat1 

d_lon, ulon2 = unroll180(lon1, lon2, wrap=wrap) 

 

if limit and _limiterrors \ 

and max(abs(d_lat), abs(d_lon)) > limit > 0: 

t = fStr((lat1, lon1, lat2, lon2), prec=4) 

raise LimitError('%s(%s, limit=%s) delta exceeds limit' % 

('equirectangular_', t, fStr(limit, prec=2))) 

 

if adjust: # scale delta lon 

d_lon *= _scaled(lat1, lat2) 

 

d2 = d_lat**2 + d_lon**2 # degrees squared! 

return Distance4Tuple(d2, d_lat, d_lon, ulon2 - lon2) 

 

 

def euclidean(lat1, lon1, lat2, lon2, radius=R_M, adjust=True, wrap=False): 

'''Approximate the C{Euclidian} distance between two (spherical) points. 

 

@param lat1: Start latitude (C{degrees}). 

@param lon1: Start longitude (C{degrees}). 

@param lat2: End latitude (C{degrees}). 

@param lon2: End longitude (C{degrees}). 

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

@keyword adjust: Adjust the longitudinal delta by the 

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

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

 

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

 

@see: U{Distance between two (spherical) points 

<https://www.EdWilliams.org/avform.htm#Dist>}, functions 

L{equirectangular}, L{haversine} and L{vincentys} and 

methods L{Ellipsoid.distance2}, C{LatLon.distanceTo*} 

and C{LatLon.equirectangularTo}. 

''' 

d, _ = unroll180(lon1, lon2, wrap=wrap) 

r = euclidean_(radians(lat2), radians(lat1), radians(d), adjust=adjust) 

return r * float(radius) 

 

 

def euclidean_(a2, a1, b21, adjust=True): 

'''Approximate the I{angular} C{Euclidean} distance between two 

(spherical) points. 

 

@param a2: End latitude (C{radians}). 

@param a1: Start latitude (C{radians}). 

@param b21: Longitudinal delta, M{end-start} (C{radians}). 

@keyword adjust: Adjust the longitudinal delta by the 

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

 

@return: Angular distance (C{radians}). 

 

@see: Functions L{euclidean}, L{equirectangular_}, L{haversine_} 

and L{vincentys_}. 

''' 

a, b = abs(a2 - a1), abs(b21) 

if adjust: 

b *= _scaler(a2, a1) 

if a < b: 

a, b = b, a 

return a + b * 0.5 # 0.4142135623731 

 

 

def haversine(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

'''Compute the distance between two (spherical) points using the 

U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>} 

formula. 

 

@param lat1: Start latitude (C{degrees}). 

@param lon1: Start longitude (C{degrees}). 

@param lat2: End latitude (C{degrees}). 

@param lon2: End longitude (C{degrees}). 

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

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

 

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

 

@see: U{Distance between two (spherical) points 

<https://www.EdWilliams.org/avform.htm#Dist>}, functions 

L{equirectangular}, L{euclidean} and L{vincentys} and 

methods L{Ellipsoid.distance2}, C{LatLon.distanceTo*} 

and C{LatLon.equirectangularTo}. 

 

@note: See note under L{vincentys_}. 

''' 

d, _ = unroll180(lon1, lon2, wrap=wrap) 

r = haversine_(radians(lat2), radians(lat1), radians(d)) 

return r * float(radius) 

 

 

def haversine_(a2, a1, b21): 

'''Compute the I{angular} distance between two (spherical) points 

using the U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>} 

formula. 

 

@param a2: End latitude (C{radians}). 

@param a1: Start latitude (C{radians}). 

@param b21: Longitudinal delta, M{end-start} (C{radians}). 

 

@return: Angular distance (C{radians}). 

 

@see: Functions L{haversine}, L{equirectangular_}, L{euclidean_} 

and L{vincentys_}. 

 

@note: See note under L{vincentys_}. 

''' 

def _hsin(rad): 

return sin(rad * 0.5)**2 

 

h = _hsin(a2 - a1) + cos(a1) * cos(a2) * _hsin(b21) # haversine 

try: 

r = atan2(sqrt(h), sqrt(1 - h)) * 2 # == asin(sqrt(h)) * 2 

except ValueError: 

r = 0 if h < 0.5 else PI 

return r 

 

 

def heightOf(angle, distance, radius=R_M): 

'''Determine the height above the (spherical) earth after 

traveling along a straight line at a given tilt. 

 

@param angle: Tilt angle above horizontal (C{degrees}). 

@param distance: Distance along the line (C{meter} or same units 

as B{C{radius}}). 

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

 

@return: Height (C{meter}, same units as B{C{distance}} and B{C{radius}}). 

 

@raise ValueError: Invalid B{C{angle}}, B{C{distance}} or B{C{radius}}. 

 

@see: U{MultiDop geog_lib.GeogBeamHt<https://GitHub.com/NASA/MultiDop>} 

(U{Shapiro et al. 2009, JTECH 

<https://Journals.AMetSoc.org/doi/abs/10.1175/2009JTECHA1256.1>} 

and U{Potvin et al. 2012, JTECH 

<https://Journals.AMetSoc.org/doi/abs/10.1175/JTECH-D-11-00019.1>}). 

''' 

d, r = distance, radius 

if d > r: 

d, r = r, d 

 

if d > EPS: 

d = d / float(r) 

s = sin(radians(angle)) 

s = fsum_(1, 2 * s * d, d**2) 

if s > 0: 

return r * sqrt(s) - float(radius) 

 

raise ValueError('%s%r' % ('heightOf', (angle, distance, radius))) 

 

 

def horizon(height, radius=R_M, refraction=False): 

'''Determine the distance to the horizon from a given altitude 

above the (spherical) earth. 

 

@param height: Altitude (C{meter} or same units as B{C{radius}}). 

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

@keyword refraction: Consider atmospheric refraction (C{bool}). 

 

@return: Distance (C{meter}, same units as B{C{height}} and B{C{radius}}). 

 

@raise ValueError: Invalid B{C{height}} or B{C{radius}}. 

 

@see: U{Distance to horizon<https://www.EdWilliams.org/avform.htm#Horizon>}. 

''' 

if min(height, radius) < 0: 

raise ValueError('%s%r' % ('horizon', (height, radius))) 

 

if refraction: 

d2 = 2.415750694528 * height * radius # 2.0 / 0.8279 

else: 

d2 = height * fsum_(radius, radius, height) 

return sqrt(d2) 

 

 

def isantipode(lat1, lon1, lat2, lon2, eps=EPS): 

'''Check whether two points are antipodal, on diametrically 

opposite sides of the earth. 

 

@param lat1: Latitude of one point (C{degrees}). 

@param lon1: Longitude of one point (C{degrees}). 

@param lat2: Latitude of the other point (C{degrees}). 

@param lon2: Longitude of the other point (C{degrees}). 

@keyword eps: Tolerance for near-equality (C{degrees}). 

 

@return: C{True} if points are antipodal within the 

B{C{eps}} tolerance, C{False} otherwise. 

 

@see: U{Geosphere<https://CRAN.R-Project.org/web/packages/geosphere/geosphere.pdf>}. 

''' 

return abs(wrap90(lat1) + wrap90(lat2)) < eps and \ 

abs(abs(wrap180(lon1) - wrap180(lon2)) % 360 - 180) < eps 

 

 

def points2(points, closed=True, base=None, Error=ValueError): 

'''Check a path or polygon represented by points. 

 

@param points: The path or polygon points (C{LatLon}[]) 

@keyword closed: Optionally, consider the polygon closed, 

ignoring any duplicate or closing final 

B{C{points}} (C{bool}). 

@keyword base: Optionally, check all B{C{points}} against 

this base class, if C{None} don't check. 

@keyword Error: Exception to raise (C{ValueError}). 

 

@return: A L{Points2Tuple}C{(number, points)} with the number 

of points and the points C{list} or C{tuple}. 

 

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

 

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

''' 

n, points = len2(points) 

 

if closed: 

# remove duplicate or closing final points 

while n > 1 and (points[n-1] == points[0] or 

points[n-1] == points[n-2]): 

n -= 1 

# XXX following line is unneeded if points 

# are always indexed as ... i in range(n) 

points = points[:n] # XXX numpy.array slice is a view! 

 

if n < (3 if closed else 1): 

raise Error('too few %s: %s' % ('points', n)) 

 

if base and not (isNumpy2(points) or isTuple2(points)): 

for i in range(n): 

base.others(points[i], name='%s[%s]' % ('points', i)) 

 

return Points2Tuple(n, points) 

 

 

def vincentys(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): 

'''Compute the distance between two (spherical) points using 

U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>} 

spherical formula. 

 

@param lat1: Start latitude (C{degrees}). 

@param lon1: Start longitude (C{degrees}). 

@param lat2: End latitude (C{degrees}). 

@param lon2: End longitude (C{degrees}). 

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

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

 

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

 

@see: Functions L{equirectangular}, L{euclidean} and L{haversine} 

and methods L{Ellipsoid.distance2}, C{LatLon.distanceTo*} 

and C{LatLon.equirectangularTo}. 

 

@note: See note under L{vincentys_}. 

''' 

d, _ = unroll180(lon1, lon2, wrap=wrap) 

r = vincentys_(radians(lat2), radians(lat1), radians(d)) 

return r * float(radius) 

 

 

def vincentys_(a2, a1, b21): 

'''Compute the I{angular} distance between two (spherical) points using 

U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>} 

spherical formula. 

 

@param a2: End latitude (C{radians}). 

@param a1: Start latitude (C{radians}). 

@param b21: Longitudinal delta, M{end-start} (C{radians}). 

 

@return: Angular distance (C{radians}). 

 

@see: Functions L{vincentys}, L{equirectangular_}, L{euclidean_} 

and L{haversine_}. 

 

@note: Functions L{vincentys_} and L{haversine_} produce equivalent 

results, but L{vincentys_} is suitable for antipodal points 

and slightly more expensive than L{haversine_} (M{3 cos, 

3 sin, 1 hypot, 1 atan2, 7 mul, 2 add} versus M{2 cos, 2 

sin, 2 sqrt, 1 atan2, 5 mul, 1 add}). 

''' 

sa1, ca1, sa2, ca2, sb21, cb21 = sincos2(a1, a2, b21) 

 

x = sa1 * sa2 + ca1 * ca2 * cb21 

y = ca1 * sa2 - sa1 * ca2 * cb21 

y = hypot(ca2 * sb21, y) 

return atan2(y, x) 

 

# **) MIT License 

# 

# Copyright (C) 2016-2020 -- mrJean1 at Gmail -- All Rights Reserved. 

# 

# Permission is hereby granted, free of charge, to any person obtaining a 

# copy of this software and associated documentation files (the "Software"), 

# to deal in the Software without restriction, including without limitation 

# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

# and/or sell copies of the Software, and to permit persons to whom the 

# Software is furnished to do so, subject to the following conditions: 

# 

# The above copyright notice and this permission notice shall be included 

# in all copies or substantial portions of the Software. 

# 

# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

# OTHER DEALINGS IN THE SOFTWARE.