Coverage for pygeodesy/sphericalBase.py: 94%

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1 

2# -*- coding: utf-8 -*- 

3 

4u'''(INTERNAL) Private spherical base classes C{CartesianSphericalBase} and 

5C{LatLonSphericalBase} for L{sphericalNvector} and L{sphericalTrigonometry}. 

6 

7A pure Python implementation of geodetic (lat-/longitude) functions, 

8transcoded in part from JavaScript originals by I{(C) Chris Veness 2011-2016} 

9and published under the same MIT Licence**, see 

10U{Latitude/Longitude<https://www.Movable-Type.co.UK/scripts/latlong.html>}. 

11''' 

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

13from __future__ import division as _; del _ # PYCHOK semicolon 

14 

15from pygeodesy.basics import _copysign, isbool, isinstanceof, map1 

16from pygeodesy.cartesianBase import CartesianBase, Bearing2Tuple 

17from pygeodesy.constants import EPS, EPS0, PI, PI2, PI_2, R_M, \ 

18 _0_0, _0_5, _1_0, _180_0, _360_0, \ 

19 _over, isnear0, isnon0 

20from pygeodesy.datums import Datums, _earth_ellipsoid, _spherical_datum 

21from pygeodesy.errors import IntersectionError, _ValueError, \ 

22 _xattr, _xError 

23from pygeodesy.fmath import favg, fdot, hypot, sqrt_a 

24from pygeodesy.interns import _COMMA_, _concentric_, _datum_, _distant_, \ 

25 _exceed_PI_radians_, _name_, _near_, \ 

26 _radius_, _too_ 

27from pygeodesy.latlonBase import LatLonBase, _trilaterate5 # PYCHOK passed 

28from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY, _ALL_MODS as _MODS 

29# from pygeodesy.namedTuples import Bearing2Tuple # from .cartesianBase 

30from pygeodesy.nvectorBase import NvectorBase, Fmt, _xattrs 

31from pygeodesy.props import deprecated_method, property_doc_, property_RO, \ 

32 _update_all 

33# from pygeodesy.streprs import Fmt, _xattrs # from .nvectorBase 

34from pygeodesy.units import Bearing, Bearing_, _isRadius, Radians_, Radius, \ 

35 Radius_, Scalar_, _100km 

36from pygeodesy.utily import acos1, asin1, atan2b, atan2d, degrees90, \ 

37 degrees180, sincos2, sincos2d, _unrollon, \ 

38 tanPI_2_2, wrapPI 

39 

40from math import cos, fabs, log, sin, sqrt 

41 

42__all__ = _ALL_LAZY.sphericalBase 

43__version__ = '24.05.31' 

44 

45 

46class CartesianSphericalBase(CartesianBase): 

47 '''(INTERNAL) Base class for spherical C{Cartesian}s. 

48 ''' 

49 _datum = Datums.Sphere # L{Datum} 

50 

51 def intersections2(self, rad1, other, rad2, radius=R_M): 

52 '''Compute the intersection points of two circles each defined 

53 by a center point and a radius. 

54 

55 @arg rad1: Radius of the this circle (C{meter} or C{radians}, 

56 see B{C{radius}}). 

57 @arg other: Center of the other circle (C{Cartesian}). 

58 @arg rad2: Radius of the other circle (C{meter} or C{radians}, 

59 see B{C{radius}}). 

60 @kwarg radius: Mean earth radius (C{meter} or C{None} if both 

61 B{C{rad1}} and B{C{rad2}} are given in C{radians}). 

62 

63 @return: 2-Tuple of the intersection points, each C{Cartesian}. 

64 For abutting circles, the intersection points are the 

65 same C{Cartesian} instance, aka the I{radical center}. 

66 

67 @raise IntersectionError: Concentric, antipodal, invalid or 

68 non-intersecting circles. 

69 

70 @raise TypeError: If B{C{other}} is not C{Cartesian}. 

71 

72 @raise ValueError: Invalid B{C{rad1}}, B{C{rad2}} or B{C{radius}}. 

73 

74 @see: U{Calculating intersection of two Circles 

75 <https://GIS.StackExchange.com/questions/48937/ 

76 calculating-intersection-of-two-circles>} and method 

77 or function C{trilaterate3d2}. 

78 ''' 

79 x1, x2 = self, self.others(other) 

80 r1, r2, x = _rads3(rad1, rad2, radius) 

81 if x: 

82 x1, x2 = x2, x1 

83 try: 

84 n, q = x1.cross(x2), x1.dot(x2) 

85 n2, q1 = n.length2, (_1_0 - q**2) 

86 if n2 < EPS or isnear0(q1): 

87 raise ValueError(_near_(_concentric_)) 

88 c1, c2 = cos(r1), cos(r2) 

89 x0 = x1.times((c1 - q * c2) / q1).plus( 

90 x2.times((c2 - q * c1) / q1)) 

91 n1 = _1_0 - x0.length2 

92 if n1 < EPS: 

93 raise ValueError(_too_(_distant_)) 

94 except ValueError as x: 

95 raise IntersectionError(center=self, rad1=rad1, 

96 other=other, rad2=rad2, cause=x) 

97 n = n.times(sqrt(n1 / n2)) 

98 if n.length > EPS: 

99 x1 = x0.plus(n) 

100 x2 = x0.minus(n) 

101 else: # abutting circles 

102 x1 = x2 = x0 

103 

104 return (_xattrs(x1, self, _datum_, _name_), 

105 _xattrs(x2, self, _datum_, _name_)) 

106 

107 @property_RO 

108 def sphericalCartesian(self): 

109 '''Get this C{Cartesian}'s spherical class. 

110 ''' 

111 return type(self) 

112 

113 

114class LatLonSphericalBase(LatLonBase): 

115 '''(INTERNAL) Base class for spherical C{LatLon}s. 

116 ''' 

117 _datum = Datums.Sphere # spherical L{Datum} 

118 _napieradius = _100km 

119 

120 def __init__(self, latlonh, lon=None, height=0, datum=None, wrap=False, **name): 

121 '''Create a spherical C{LatLon} point frome the given lat-, longitude and 

122 height on the given datum. 

123 

124 @arg latlonh: Latitude (C{degrees} or DMS C{str} with N or S suffix) or 

125 a previous C{LatLon} instance provided C{B{lon}=None}. 

126 @kwarg lon: Longitude (C{degrees} or DMS C{str} with E or W suffix) or 

127 C(None), indicating B{C{latlonh}} is a C{LatLon}. 

128 @kwarg height: Optional height above (or below) the earth surface (C{meter}, 

129 same units as the datum's ellipsoid axes or radius). 

130 @kwarg datum: Optional, spherical datum to use (L{Datum}, L{Ellipsoid}, 

131 L{Ellipsoid2}, L{a_f2Tuple}) or earth radius in C{meter}, 

132 conventionally). 

133 @kwarg wrap: If C{True}, wrap or I{normalize} B{C{lat}} and B{C{lon}} 

134 (C{bool}). 

135 @kwarg name: Optional C{B{name}=NN} (C{str}). 

136 

137 @raise TypeError: If B{C{latlonh}} is not a C{LatLon} or B{C{datum}} not 

138 spherical. 

139 ''' 

140 LatLonBase.__init__(self, latlonh, lon=lon, height=height, wrap=wrap, **name) 

141 if datum not in (None, self.datum): 

142 self.datum = datum 

143 

144 def bearingTo2(self, other, wrap=False, raiser=False): 

145 '''Return the initial and final bearing (forward and reverse 

146 azimuth) from this to an other point. 

147 

148 @arg other: The other point (C{LatLon}). 

149 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the 

150 B{C{other}} point (C{bool}). 

151 

152 @return: A L{Bearing2Tuple}C{(initial, final)}. 

153 

154 @raise TypeError: The B{C{other}} point is not spherical. 

155 

156 @see: Methods C{initialBearingTo} and C{finalBearingTo}. 

157 ''' 

158 # .initialBearingTo is inside .-Nvector and .-Trigonometry 

159 i = self.initialBearingTo(other, wrap=wrap, raiser=raiser) # PYCHOK .initialBearingTo 

160 f = self.finalBearingTo( other, wrap=wrap, raiser=raiser) 

161 return Bearing2Tuple(i, f, name=self.name) 

162 

163 @property_doc_(''' this point's datum (L{Datum}).''') 

164 def datum(self): 

165 '''Get this point's datum (L{Datum}). 

166 ''' 

167 return self._datum 

168 

169 @datum.setter # PYCHOK setter! 

170 def datum(self, datum): 

171 '''Set this point's datum I{without conversion} (L{Datum}, L{Ellipsoid}, 

172 L{Ellipsoid2}, L{a_f2Tuple}) or C{scalar} spherical earth radius). 

173 

174 @raise TypeError: If B{C{datum}} invalid or not not spherical. 

175 ''' 

176 d = _spherical_datum(datum, name=self.name, raiser=_datum_) 

177 if self._datum != d: 

178 _update_all(self) 

179 self._datum = d 

180 

181 def finalBearingTo(self, other, wrap=False, raiser=False): 

182 '''Return the final bearing (reverse azimuth) from this to 

183 an other point. 

184 

185 @arg other: The other point (spherical C{LatLon}). 

186 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll 

187 the B{C{other}} point (C{bool}). 

188 

189 @return: Final bearing (compass C{degrees360}). 

190 

191 @raise TypeError: The B{C{other}} point is not spherical. 

192 ''' 

193 p = self.others(other) 

194 if wrap: 

195 p = _unrollon(self, p, wrap=wrap) 

196 # final bearing is the reverse of the other, initial one 

197 b = p.initialBearingTo(self, wrap=False, raiser=raiser) + _180_0 

198 return b if b < 360 else (b - _360_0) 

199 

200 def intersecant2(self, circle, point, other, radius=R_M, exact=False, # PYCHOK signature 

201 height=None, wrap=False): 

202 '''Compute the intersections of a circle and a (great circle) line 

203 given as two points or as a point and bearing. 

204 

205 @arg circle: Radius of the circle centered at this location (C{meter}, 

206 same units as B{C{radius}}) or a point on the circle 

207 (this C{LatLon}). 

208 @arg point: A point on the (great circle) line (this C{LatLon}). 

209 @arg other: An other point I{on} (this {LatLon}) or the bearing at 

210 B{C{point}} I{of} the (great circle) line (compass 

211 C{degrees}). 

212 @kwarg radius: Mean earth radius (C{meter}, conventionally). 

213 @kwarg exact: If C{True} use the I{exact} rhumb methods for azimuth, 

214 destination and distance, if C{False} use the basic 

215 rhumb methods (C{bool}) or if C{None} use the I{great 

216 circle} methods. 

217 @kwarg height: Optional height for the intersection points (C{meter}, 

218 conventionally) or C{None} for interpolated heights. 

219 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the points 

220 B{C{circle}}, B{C{point}} and/or B{C{other}} (C{bool}). 

221 

222 @return: 2-Tuple of the intersection points (representing a chord), each 

223 an instance of the B{C{point}} class. Both points are the same 

224 instance if the (great circle) line is tangent to the circle. 

225 

226 @raise IntersectionError: The circle and line do not intersect. 

227 

228 @raise TypeError: If B{C{point}} is not this C{LatLon} or B{C{circle}} 

229 or B{C{other}} invalid. 

230 

231 @raise UnitError: Invalid B{C{circle}}, B{C{other}}, B{C{radius}}, 

232 B{C{exact}}, B{C{height}} or B{C{napieradius}}. 

233 ''' 

234 p = self.others(point=point) 

235 try: 

236 return _intersecant2(self, circle, p, other, radius=radius, exact=exact, 

237 height=height, wrap=wrap) 

238 except (TypeError, ValueError) as x: 

239 raise _xError(x, center=self, circle=circle, point=point, other=other, 

240 radius=radius, exact=exact, height=height, wrap=wrap) 

241 

242 def maxLat(self, bearing): 

243 '''Return the maximum latitude reached when travelling on a great circle 

244 on given bearing from this point based on Clairaut's formula. 

245 

246 The maximum latitude is independent of longitude and the same for all 

247 points on a given latitude. 

248 

249 Negate the result for the minimum latitude (on the Southern hemisphere). 

250 

251 @arg bearing: Initial bearing (compass C{degrees360}). 

252 

253 @return: Maximum latitude (C{degrees90}). 

254 

255 @raise ValueError: Invalid B{C{bearing}}. 

256 ''' 

257 r = acos1(fabs(sin(Bearing_(bearing)) * cos(self.phi))) 

258 return degrees90(r) 

259 

260 def minLat(self, bearing): 

261 '''Return the minimum latitude reached when travelling on a great circle 

262 on given bearing from this point. 

263 

264 @arg bearing: Initial bearing (compass C{degrees360}). 

265 

266 @return: Minimum latitude (C{degrees90}). 

267 

268 @see: Method L{maxLat} for more details. 

269 

270 @raise ValueError: Invalid B{C{bearing}}. 

271 ''' 

272 return -self.maxLat(bearing) 

273 

274 def _mpr(self, radius=R_M, exact=None): # meter per radian 

275 if exact and not _isRadius(radius): # see .rhumb.ekx.Rhumb._mpr 

276 radius = _earth_ellipsoid(radius)._Lpr 

277 return radius 

278 

279 @property_doc_(''' the I{Napier} radius to apply spherical trigonometry.''') 

280 def napieradius(self): 

281 '''Get the I{Napier} radius (C{meter}, conventionally). 

282 ''' 

283 return self._napieradius 

284 

285 @napieradius.setter # PYCHOK setter! 

286 def napieradius(self, radius): 

287 '''Set this I{Napier} radius (C{meter}, conventionally) or C{0}. 

288 

289 In methods L{intersecant2} and L{rhumbIntersecant2}, I{Napier}'s 

290 spherical trigonometry is applied if the circle radius exceeds 

291 the I{Napier} radius, otherwise planar trigonometry is used. 

292 

293 @raise UnitError: Invalid B{C{radius}}. 

294 ''' 

295 self._napieradius = Radius(napieradius=radius or 0) 

296 

297# def nearestTo(self, point, other, **radius_exact_height_wrap): # PYCHOK signature 

298# p = self.others(point=point) 

299# try: 

300# p, q = _intersecant2(self, p, p, other, **radius_exact_height_wrap) 

301# except (TypeError, ValueError) as x: 

302# raise _xError(x, this=self, point=point, other=other, **radius_exact_height_wrap) 

303# return p.midpointTo(q) 

304 

305 def parse(self, strllh, height=0, sep=_COMMA_, **name): 

306 '''Parse a string representing a similar, spherical C{LatLon} 

307 point, consisting of C{"lat, lon[, height]"}. 

308 

309 @arg strllh: Lat, lon and optional height (C{str}), see function 

310 L{pygeodesy.parse3llh}. 

311 @kwarg height: Optional, default height (C{meter}). 

312 @kwarg sep: Optional separator (C{str}). 

313 @kwarg name: Optional C{B{name}=NN} (C{str}), overriding this name. 

314 

315 @return: The similar point (spherical C{LatLon}). 

316 

317 @raise ParseError: Invalid B{C{strllh}}. 

318 ''' 

319 llh = _MODS.dms.parse3llh(strllh, height=height, sep=sep) 

320 return self.classof(*llh, **name) 

321 

322 @property_RO 

323 def _radius(self): 

324 '''(INTERNAL) Get this sphere's radius. 

325 ''' 

326 return self.datum.ellipsoid.equatoradius 

327 

328 def _rhumbs3(self, other, wrap, r=False): # != .latlonBase._rhumbx3 

329 '''(INTERNAL) Rhumb_ helper function. 

330 

331 @arg other: The other point (spherical C{LatLon}). 

332 ''' 

333 p = self.others(other, up=2) 

334 if wrap: 

335 p = _unrollon(self, p, wrap=wrap) 

336 a2, b2 = p.philam 

337 a1, b1 = self.philam 

338 # if |db| > 180 take shorter rhumb 

339 # line across the anti-meridian 

340 db = wrapPI(b2 - b1) 

341 dp = _logPI_2_2(a2, a1) 

342 da = a2 - a1 

343 if r: 

344 # on Mercator projection, longitude distances shrink 

345 # by latitude; the 'stretch factor' q becomes ill- 

346 # conditioned along E-W line (0/0); use an empirical 

347 # tolerance to avoid it 

348 q = (da / dp) if fabs(dp) > EPS else cos(a1) 

349 da = hypot(da, q * db) # angular distance radians 

350 return da, db, dp 

351 

352 def rhumbAzimuthTo(self, other, radius=R_M, exact=False, wrap=False, b360=False): 

353 '''Return the azimuth (bearing) of a rhumb line (loxodrome) between 

354 this and an other (spherical) point. 

355 

356 @arg other: The other point (spherical C{LatLon}). 

357 @kwarg radius: Earth radius (C{meter}) or earth model (L{Datum}, 

358 L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}). 

359 @kwarg exact: If C{True}, use I{Elliptic, Krüger} L{Rhumb} (C{bool}), 

360 default C{False} for backward compatibility. 

361 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the 

362 B{C{other}} point (C{bool}). 

363 @kwarg b360: If C{True}, return the azimuth in the bearing range. 

364 

365 @return: Rhumb azimuth (compass C{degrees180} or C{degrees360}). 

366 

367 @raise TypeError: The B{C{other}} point is incompatible or 

368 B{C{radius}} is invalid. 

369 ''' 

370 if exact: # use series, always 

371 z = LatLonBase.rhumbAzimuthTo(self, other, exact=False, # Krüger 

372 radius=radius, wrap=wrap, b360=b360) 

373 else: 

374 _, db, dp = self._rhumbs3(other, wrap) 

375 z = (atan2b if b360 else atan2d)(db, dp) # see .rhumbBase.RhumbBase.Inverse 

376 return z 

377 

378 @deprecated_method 

379 def rhumbBearingTo(self, other): # unwrapped 

380 '''DEPRECATED, use method C{.rhumbAzimuthTo}.''' 

381 return self.rhumbAzimuthTo(other, b360=True) # [0..360) 

382 

383 def rhumbDestination(self, distance, azimuth, radius=R_M, height=None, exact=False): 

384 '''Return the destination point having travelled the given distance from 

385 this point along a rhumb line (loxodrome) of the given azimuth. 

386 

387 @arg distance: Distance travelled (C{meter}, same units as B{C{radius}}), 

388 may be negative if C{B{exact}=True}. 

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

390 @kwarg radius: Earth radius (C{meter}) or earth model (L{Datum}, 

391 L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}) if 

392 C{B{exact}=True}. 

393 @kwarg height: Optional height, overriding the default height (C{meter}. 

394 @kwarg exact: If C{True}, use I{Elliptic, Krüger} L{Rhumb} (C{bool}), 

395 default C{False} for backward compatibility. 

396 

397 @return: The destination point (spherical C{LatLon}). 

398 

399 @raise ValueError: Invalid B{C{distance}}, B{C{azimuth}}, B{C{radius}} 

400 or B{C{height}}. 

401 ''' 

402 if exact: # use series, always 

403 r = LatLonBase.rhumbDestination(self, distance, azimuth, exact=False, # Krüger 

404 radius=radius, height=height) 

405 else: # radius=None from .rhumbMidpointTo 

406 if radius in (None, self._radius): 

407 d, r = self.datum, radius 

408 else: 

409 d = _spherical_datum(radius, raiser=_radius_) # spherical only 

410 r = d.ellipsoid.equatoradius 

411 r = _m2radians(distance, r, low=-EPS) # distance=0 from .rhumbMidpointTo 

412 

413 a1, b1 = self.philam 

414 sb, cb = sincos2(Bearing_(azimuth)) # radians 

415 

416 da = r * cb 

417 a2 = a1 + da 

418 # normalize latitude if past pole 

419 if fabs(a2) > PI_2: 

420 a2 = _copysign(PI, a2) - a2 

421 

422 dp = _logPI_2_2(a2, a1) 

423 # q becomes ill-conditioned on E-W course 0/0 

424 q = cos(a1) if isnear0(dp) else (da / dp) 

425 b2 = b1 if isnear0(q) else (b1 + r * sb / q) 

426 

427 h = self._heigHt(height) 

428 r = self.classof(degrees90(a2), degrees180(b2), datum=d, height=h) 

429 return r 

430 

431 def rhumbDistanceTo(self, other, radius=R_M, exact=False, wrap=False): 

432 '''Return the distance from this to an other point along 

433 a rhumb line (loxodrome). 

434 

435 @arg other: The other point (spherical C{LatLon}). 

436 @kwarg radius: Earth radius (C{meter}) or earth model (L{Datum}, 

437 L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}) if 

438 C{B{exact}=True}. 

439 @kwarg exact: If C{True}, use I{Elliptic, Krüger} L{Rhumb} (C{bool}), 

440 default C{False} for backward compatibility. 

441 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the 

442 B{C{other}} point (C{bool}). 

443 

444 @return: Distance (C{meter}, the same units as B{C{radius}} 

445 or C{radians} if B{C{radius}} is C{None}). 

446 

447 @raise TypeError: The B{C{other}} point is incompatible. 

448 

449 @raise ValueError: Invalid B{C{radius}}. 

450 ''' 

451 if exact: # use series, always 

452 r = LatLonBase.rhumbDistanceTo(self, other, exact=False, # Krüger 

453 radius=radius, wrap=wrap) 

454 if radius is None: # angular distance in radians 

455 r = r / self._radius # /= chokes PyChecker 

456 else: 

457 # see <https://www.EdWilliams.org/avform.htm#Rhumb> 

458 r, _, _ = self._rhumbs3(other, wrap, r=True) 

459 if radius is not None: 

460 r *= Radius(radius) 

461 return r 

462 

463 def rhumbIntersecant2(self, circle, point, other, radius=R_M, exact=True, # PYCHOK signature 

464 height=None, wrap=False): 

465 '''Compute the intersections of a circle and a rhumb line given as two 

466 points and as a point and azimuth. 

467 

468 @arg circle: Radius of the circle centered at this location (C{meter}, 

469 same units as B{C{radius}}) or a point on the circle 

470 (this C{LatLon}). 

471 @arg point: The rhumb line's start point (this C{LatLon}). 

472 @arg other: An other point (this I{on} C{LatLon}) or the azimuth I{of} 

473 (compass C{degrees}) the rhumb line. 

474 @kwarg radius: Mean earth radius (C{meter}, conventionally). 

475 @kwarg exact: If C{True} use the I{exact} rhumb methods for azimuth, 

476 destination and distance, if C{False} use the basic 

477 rhumb methods (C{bool}) or if C{None} use the I{great 

478 circle} methods. 

479 @kwarg height: Optional height for the intersection points (C{meter}, 

480 conventionally) or C{None}. 

481 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the points 

482 B{C{circle}}, B{C{point}} and/or B{C{other}} (C{bool}). 

483 

484 @return: 2-Tuple of the intersection points (representing a chord), 

485 each an instance of this class. For a tangent line, both 

486 points are the same instance, wrapped or I{normalized}. 

487 

488 @raise IntersectionError: The circle and line do not intersect. 

489 

490 @raise TypeError: If B{C{point}} is not this C{LatLon} or B{C{circle}} 

491 or B{C{other}} invalid. 

492 

493 @raise UnitError: Invalid B{C{circle}}, B{C{other}}, B{C{radius}}, 

494 B{C{exact}} or B{C{height}}. 

495 ''' 

496 m = LatLonBase.rhumbIntersecant2 if exact else \ 

497 LatLonSphericalBase.intersecant2 

498 return m(self, circle, point, other, radius=radius, exact=exact, 

499 height=height, wrap=wrap) 

500 

501 def rhumbMidpointTo(self, other, height=None, radius=R_M, exact=False, 

502 fraction=_0_5, wrap=False): 

503 '''Return the (loxodromic) midpoint on the rhumb line between 

504 this and an other point. 

505 

506 @arg other: The other point (spherical LatLon). 

507 @kwarg height: Optional height, overriding the mean height (C{meter}). 

508 @kwarg radius: Earth radius (C{meter}) or earth model (L{Datum}, 

509 L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}). 

510 @kwarg exact: If C{True}, use I{Elliptic, Krüger} L{Rhumb} (C{bool}), 

511 default C{False} for backward compatibility. 

512 @kwarg fraction: Midpoint location from this point (C{scalar}), may 

513 be negative if C{B{exact}=True}. 

514 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the B{C{other}} 

515 point (C{bool}). 

516 

517 @return: The (mid)point at the given B{C{fraction}} along the rhumb 

518 line (spherical C{LatLon}). 

519 

520 @raise TypeError: The B{C{other}} point is incompatible. 

521 

522 @raise ValueError: Invalid B{C{height}} or B{C{fraction}} 

523 ''' 

524 if exact: # use series, always 

525 r = LatLonBase.rhumbMidpointTo(self, other, exact=False, # Krüger 

526 radius=radius, height=height, 

527 fraction=fraction, wrap=wrap) 

528 elif fraction is not _0_5: 

529 f = Scalar_(fraction=fraction) # low=_0_0 

530 r, db, dp = self._rhumbs3(other, wrap, r=True) # radians 

531 z = atan2b(db, dp) 

532 h = self._havg(other, f=f, h=height) 

533 r = self.rhumbDestination(r * f, z, radius=None, height=h) 

534 

535 else: # for backward compatibility, unwrapped 

536 # see <https://MathForum.org/library/drmath/view/51822.html> 

537 a1, b1 = self.philam 

538 a2, b2 = self.others(other).philam 

539 

540 if fabs(b2 - b1) > PI: 

541 b1 += PI2 # crossing anti-meridian 

542 

543 a3 = favg(a1, a2) 

544 b3 = favg(b1, b2) 

545 

546 f1 = tanPI_2_2(a1) 

547 if isnon0(f1): 

548 f2 = tanPI_2_2(a2) 

549 f = f2 / f1 

550 if isnon0(f): 

551 f = log(f) 

552 if isnon0(f): 

553 f3 = tanPI_2_2(a3) 

554 b3 = fdot(map1(log, f1, f2, f3), 

555 -b2, b1, b2 - b1) / f 

556 

557 d = self.datum if radius in (None, self._radius) else \ 

558 _spherical_datum(radius, name=self.name, raiser=_radius_) 

559 h = self._havg(other, h=height) 

560 r = self.classof(degrees90(a3), degrees180(b3), datum=d, height=h) 

561 return r 

562 

563 @property_RO 

564 def sphericalLatLon(self): 

565 '''Get this C{LatLon}'s spherical class. 

566 ''' 

567 return type(self) 

568 

569 def toNvector(self, Nvector=NvectorBase, **Nvector_kwds): # PYCHOK signature 

570 '''Convert this point to C{Nvector} components, I{including 

571 height}. 

572 

573 @kwarg Nvector_kwds: Optional, additional B{C{Nvector}} 

574 keyword arguments, ignored if 

575 C{B{Nvector} is None}. 

576 

577 @return: An B{C{Nvector}} or a L{Vector4Tuple}C{(x, y, z, h)} 

578 if B{C{Nvector}} is C{None}. 

579 

580 @raise TypeError: Invalid B{C{Nvector}} or B{C{Nvector_kwds}}. 

581 ''' 

582 return LatLonBase.toNvector(self, Nvector=Nvector, **Nvector_kwds) 

583 

584 

585def _intersecant2(c, r, p, b, radius=R_M, exact=False, height=None, wrap=False): 

586 # (INTERNAL) Intersect a circle and line, see L{intersecant2} 

587 # above, separated to allow callers to embellish any exceptions 

588 

589 if wrap: 

590 p = _unrollon(c, p, wrap=wrap) 

591 nonexact = exact is None 

592 

593 if not isinstanceof(r, c.__class__, p.__class__): 

594 r = Radius_(circle=r) 

595 elif nonexact: 

596 r = c.distanceTo(r, radius=radius, wrap=wrap) 

597 elif isbool(exact): 

598 r = c.rhumbDistanceTo(r, radius=radius, exact=exact, wrap=wrap) 

599 else: 

600 raise _ValueError(exact=exact) 

601 

602 if not isinstanceof(b, c.__class__, p.__class__): 

603 b = Bearing(b) 

604 elif nonexact: 

605 b = p.initialBearingTo(b, wrap=wrap) 

606 else: 

607 b = p.rhumbAzimuthTo(b, radius=radius, exact=exact, wrap=wrap, 

608 b360=True) 

609 

610 d = p.distanceTo(c, radius=radius) if nonexact else \ 

611 p.rhumbDistanceTo(c, radius=radius, exact=exact) 

612 if d > EPS0: 

613 n = _xattr(c, napieradius=0) 

614 a = p.initialBearingTo(c) if nonexact else \ 

615 p.rhumbAzimuthTo(c, radius=radius, exact=exact, b360=True) 

616 s, c = sincos2d(b - a) # Napier's sin(A), cos(A) 

617 if r > n: 

618 # Napier's right spherical triangle rules (R2) and (R1) 

619 # <https://WikiPedia.org/wiki/Spherical_trigonometry> 

620 m = p._mpr(radius=radius, exact=exact) # meter per radian 

621 if fabs(c) > EPS0: 

622 d = d / m # /= chokes PyChecker 

623 a = asin1(sin(d) * fabs(s)) # Napier's a 

624 c = _copysign(cos(a), c) 

625 d = acos1(cos(d) / c) * m 

626 a *= m # meter 

627 else: # point and chord center coincident 

628 a, d = d, 0 

629 c = cos(a / m) 

630 h = (acos1(cos(r / m) / c) * m) if a < r else 0 

631 else: # distance from the chord center to ... 

632 a = fabs(d * s) # ... the cicle center ... 

633 d *= c # ... and to the point 

634 h = sqrt_a(r, a) if a < r else 0 # half chord length 

635 if a > r: 

636 raise IntersectionError(_too_(Fmt.distant(a))) 

637 else: 

638 d, h = 0, r # point and circle center coincident 

639 

640 _intersecant1, kwds = (p.destination, {}) if nonexact else \ 

641 (p.rhumbDestination, dict(exact=exact)) 

642 kwds.update(radius=radius, height=height) 

643 t = (_intersecant1(d + h, b, **kwds),) 

644 if h: 

645 t += (_intersecant1(d - h, b, **kwds),) 

646 else: # same instance twice 

647 t *= 2 

648 return t 

649 

650 

651def _logPI_2_2(a2, a1): 

652 '''(INTERNAL) C{log} of C{tanPI_2_2}'s quotient. 

653 ''' 

654 return log(_over(tanPI_2_2(a2), tanPI_2_2(a1))) 

655 

656 

657def _m2radians(distance, radius, low=EPS): # PYCHOK in .spherical* 

658 '''(INTERNAL) Distance in C{meter} to angular distance in C{radians}. 

659 

660 @raise UnitError: Invalid B{C{distance}} or B{C{radius}}. 

661 ''' 

662 r = float(distance) 

663 if radius: 

664 r = r / Radius_(radius=radius) # /= chokes PyChecker 

665 if low is not None: 

666 # small near0 values from .rhumbDestination not exact OK 

667 r = _0_0 if low < 0 and r < 0 else Radians_(r, low=low) 

668 # _0_0 if low < 0 and low < r < 0 else Radians_(r, low=low) 

669 return r 

670 

671 

672def _radians2m(rad, radius): 

673 '''(INTERNAL) Angular distance in C{radians} to distance in C{meter}. 

674 ''' 

675 if radius is not None: # not in (None, _0_0) 

676 rad *= R_M if radius is R_M else Radius(radius) 

677 return rad 

678 

679 

680def _rads3(rad1, rad2, radius): # in .sphericalTrigonometry 

681 '''(INTERNAL) Convert radii to radians. 

682 ''' 

683 r1 = Radius_(rad1=rad1) 

684 r2 = Radius_(rad2=rad2) 

685 if radius is not None: # convert radii to radians 

686 r1 = _m2radians(r1, radius) 

687 r2 = _m2radians(r2, radius) 

688 

689 x = r1 < r2 

690 if x: 

691 r1, r2 = r2, r1 

692 if r1 > PI: 

693 raise IntersectionError(rad1=rad1, rad2=rad2, 

694 txt=_exceed_PI_radians_) 

695 return r1, r2, x 

696 

697 

698__all__ += _ALL_DOCS(CartesianSphericalBase, LatLonSphericalBase) 

699 

700# **) MIT License 

701# 

702# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved. 

703# 

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

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

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

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

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

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

710# 

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

712# in all copies or substantial portions of the Software. 

713# 

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

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

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

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

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

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

720# OTHER DEALINGS IN THE SOFTWARE.