Coverage for pygeodesy/vector3d.py: 96%

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1 

2# -*- coding: utf-8 -*- 

3 

4u'''Extended 3-D vector class L{Vector3d} and functions. 

5 

6Function L{intersection3d3}, L{intersections2}, L{parse3d}, L{sumOf}, 

7L{trilaterate2d2} and L{trilaterate3d2}. 

8''' 

9 

10from pygeodesy.constants import EPS, EPS0, EPS1, EPS4, INT0, isnear0, \ 

11 _0_0, _1_0 

12from pygeodesy.errors import IntersectionError, _ValueError, VectorError, \ 

13 _xattr, _xError, _xkwds, _xkwds_get, _xkwds_item2 

14from pygeodesy.fmath import euclid, fabs, fdot, hypot, sqrt, fsum1_ 

15# from pygeodesy.fsums import fsum1_ # from .fmath 

16# from pygeodesy.formy import _radical2 # in _intersects2 below 

17from pygeodesy.interns import NN, _COMMA_, _concentric_, _intersection_, \ 

18 _near_, _negative_, _no_, _too_ 

19from pygeodesy.iters import PointsIter, Fmt 

20from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY, _ALL_MODS as _MODS 

21from pygeodesy.named import _xnamed, _xotherError 

22from pygeodesy.namedTuples import Intersection3Tuple, NearestOn2Tuple, \ 

23 NearestOn6Tuple, Vector3Tuple # Vector4Tuple 

24# from pygeodesy.nvectorBase import _nsumOf # _MODS 

25# from pygeodesy.streprs import Fmt # from .iters 

26from pygeodesy.units import _fi_j2, _isDegrees, Radius, Radius_ 

27from pygeodesy.utily import atan2b, sincos2d 

28# from pygeodesy.vector2d import .... # in .... below 

29from pygeodesy.vector3dBase import Vector3dBase 

30 

31# from math import fabs, sqrt # from .fmath 

32 

33__all__ = _ALL_LAZY.vector3d 

34__version__ = '24.02.20' 

35 

36 

37class Vector3d(Vector3dBase): 

38 '''Extended 3-D vector. 

39 

40 In a geodesy context, these may be used to represent: 

41 - n-vector, the normal to a point on the earth's surface 

42 - Earth-Centered, Earth-Fixed (ECEF) cartesian (== spherical n-vector) 

43 - great circle normal to the vector 

44 - motion vector on the earth's surface 

45 - etc. 

46 ''' 

47 

48 def bearing(self, useZ=True): 

49 '''Get this vector's "bearing", the angle off the +Z axis, clockwise. 

50 

51 @kwarg useZ: If C{True}, use the Z component, otherwise ignore the 

52 Z component and consider the +Y as the +Z axis. 

53 

54 @return: Bearing (compass C{degrees}). 

55 ''' 

56 x, y = self.x, self.y 

57 if useZ: 

58 x, y = hypot(x, y), self.z 

59 return atan2b(x, y) 

60 

61 def circin6(self, point2, point3, eps=EPS4): 

62 '''Return the radius and center of the I{inscribed} aka I{In- circle} 

63 of a (3-D) triangle formed by this and two other points. 

64 

65 @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, 

66 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). 

67 @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, 

68 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). 

69 @kwarg eps: Tolerance for function L{pygeodesy.trilaterate3d2} if 

70 C{B{useZ} is True} otherwise L{pygeodesy.trilaterate2d2}. 

71 

72 @return: L{Circin6Tuple}C{(radius, center, deltas, cA, cB, cC)}. The 

73 C{center} and contact points C{cA}, C{cB} and C{cC}, each an 

74 instance of this (sub-)class, are co-planar with this and the 

75 two given points. 

76 

77 @raise ImportError: Package C{numpy} not found, not installed or older 

78 than version 1.10. 

79 

80 @raise IntersectionError: Near-coincident or -colinear points or 

81 a trilateration or C{numpy} issue. 

82 

83 @raise TypeError: Invalid B{C{point2}} or B{C{point3}}. 

84 

85 @see: Function L{pygeodesy.circin6}, U{Incircle 

86 <https://MathWorld.Wolfram.com/Incircle.html>} and U{Contact 

87 Triangle<https://MathWorld.Wolfram.com/ContactTriangle.html>}. 

88 ''' 

89 try: 

90 return _MODS.vector2d._circin6(self, point2, point3, eps=eps, useZ=True) 

91 except (AssertionError, TypeError, ValueError) as x: 

92 raise _xError(x, point=self, point2=point2, point3=point3) 

93 

94 def circum3(self, point2, point3, circum=True, eps=EPS4): 

95 '''Return the radius and center of the smallest circle I{through} or 

96 I{containing} this and two other (3-D) points. 

97 

98 @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} 

99 or C{Vector4Tuple}). 

100 @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} 

101 or C{Vector4Tuple}). 

102 @kwarg circum: If C{True} return the C{circumradius} and C{circumcenter}, 

103 always, ignoring the I{Meeus}' Type I case (C{bool}). 

104 @kwarg eps: Tolerance passed to function L{pygeodesy.trilaterate3d2}. 

105 

106 @return: A L{Circum3Tuple}C{(radius, center, deltas)}. The C{center}, an 

107 instance of this (sub-)class, is co-planar with this and the two 

108 given points. 

109 

110 @raise ImportError: Package C{numpy} not found, not installed or older than 

111 version 1.10. 

112 

113 @raise IntersectionError: Near-concentric, -coincident or -colinear points 

114 or a trilateration or C{numpy} issue. 

115 

116 @raise TypeError: Invalid B{C{point2}} or B{C{point3}}. 

117 

118 @see: Function L{pygeodesy.circum3} and methods L{circum4_} and L{meeus2}. 

119 ''' 

120 try: 

121 return _MODS.vector2d._circum3(self, point2, point3, circum=circum, 

122 eps=eps, useZ=True, clas=self.classof) 

123 except (AssertionError, TypeError, ValueError) as x: 

124 raise _xError(x, point=self, point2=point2, point3=point3, circum=circum) 

125 

126 def circum4_(self, *points): 

127 '''Best-fit a sphere through this and two or more other (3-D) points. 

128 

129 @arg points: Other points (each a C{Cartesian}, L{Vector3d}, C{Vector3Tuple} 

130 or C{Vector4Tuple}). 

131 

132 @return: L{Circum4Tuple}C{(radius, center, rank, residuals)} with C{center} 

133 an instance if this (sub-)class. 

134 

135 @raise ImportError: Package C{numpy} not found, not installed or 

136 older than version 1.10. 

137 

138 @raise NumPyError: Some C{numpy} issue. 

139 

140 @raise PointsError: Too few B{C{points}}. 

141 

142 @raise TypeError: One of the B{C{points}} invalid. 

143 

144 @see: Function L{pygeodesy.circum4_} and methods L{circum3} and L{meeus2}. 

145 ''' 

146 return _MODS.vector2d.circum4_(self, *points, useZ=True, Vector=self.classof) 

147 

148 def iscolinearWith(self, point1, point2, eps=EPS): 

149 '''Check whether this and two other (3-D) points are colinear. 

150 

151 @arg point1: One point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} 

152 or C{Vector4Tuple}). 

153 @arg point2: An other point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} 

154 or C{Vector4Tuple}). 

155 @kwarg eps: Tolerance (C{scalar}), same units as C{x}, 

156 C{y}, and C{z}. 

157 

158 @return: C{True} if this point is colinear with B{C{point1}} and 

159 B{C{point2}}, C{False} otherwise. 

160 

161 @raise TypeError: Invalid B{C{point1}} or B{C{point2}}. 

162 

163 @see: Method L{nearestOn}. 

164 ''' 

165 v = self if self.name else _otherV3d(NN_OK=False, this=self) 

166 return _MODS.vector2d._iscolinearWith(v, point1, point2, eps=eps) 

167 

168 def meeus2(self, point2, point3, circum=False): 

169 '''Return the radius and I{Meeus}' Type of the smallest circle I{through} 

170 or I{containing} this and two other (3-D) points. 

171 

172 @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} 

173 or C{Vector4Tuple}). 

174 @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} 

175 or C{Vector4Tuple}). 

176 @kwarg circum: If C{True} return the C{circumradius} and C{circumcenter} 

177 always, overriding I{Meeus}' Type II case (C{bool}). 

178 

179 @return: L{Meeus2Tuple}C{(radius, Type)}, with C{Type} the C{circumcenter} 

180 iff C{B{circum}=True}. 

181 

182 @raise IntersectionError: Coincident or colinear points, iff C{B{circum}=True}. 

183 

184 @raise TypeError: Invalid B{C{point2}} or B{C{point3}}. 

185 

186 @see: Function L{pygeodesy.meeus2} and methods L{circum3} and L{circum4_}. 

187 ''' 

188 try: 

189 return _MODS.vector2d._meeus2(self, point2, point3, circum, clas=self.classof) 

190 except (TypeError, ValueError) as x: 

191 raise _xError(x, point=self, point2=point2, point3=point3, circum=circum) 

192 

193 def nearestOn(self, point1, point2, within=True): 

194 '''Locate the point between two points closest to this point. 

195 

196 @arg point1: Start point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or 

197 C{Vector4Tuple}). 

198 @arg point2: End point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or 

199 C{Vector4Tuple}). 

200 @kwarg within: If C{True} return the closest point between the given 

201 points, otherwise the closest point on the extended 

202 line through both points (C{bool}). 

203 

204 @return: Closest point, either B{C{point1}} or B{C{point2}} or an instance 

205 of this (sub-)class. 

206 

207 @raise TypeError: Invalid B{C{point1}} or B{C{point2}}. 

208 

209 @see: Method L{sphericalTrigonometry.LatLon.nearestOn3} and U{3-D Point-Line 

210 Distance<https://MathWorld.Wolfram.com/Point-LineDistance3-Dimensional.html>}. 

211 ''' 

212 return _nearestOn2(self, point1, point2, within=within).closest 

213 

214 def nearestOn6(self, points, closed=False, useZ=True): # eps=EPS 

215 '''Locate the point on a path or polygon closest to this point. 

216 

217 The closest point is either on and within the extent of a polygon 

218 edge or the nearest of that edge's end points. 

219 

220 @arg points: The path or polygon points (C{Cartesian}, L{Vector3d}, 

221 C{Vector3Tuple} or C{Vector4Tuple}[]). 

222 @kwarg closed: Optionally, close the path or polygon (C{bool}). 

223 @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}). 

224 

225 @return: A L{NearestOn6Tuple}C{(closest, distance, fi, j, start, end)} 

226 with the C{closest}, the C{start} and the C{end} point each 

227 an instance of this point's (sub-)class. 

228 

229 @raise PointsError: Insufficient number of B{C{points}} 

230 

231 @raise TypeError: Non-cartesian B{C{points}}. 

232 

233 @note: Distances measured with method L{Vector3d.equirectangular}. 

234 

235 @see: Function L{nearestOn6}. 

236 ''' 

237 return nearestOn6(self, points, closed=closed, useZ=useZ) # Vector=self.classof 

238 

239 def parse(self, str3d, sep=_COMMA_, name=NN): 

240 '''Parse an C{"x, y, z"} string to a L{Vector3d} instance. 

241 

242 @arg str3d: X, y and z string (C{str}), see function L{parse3d}. 

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

244 @kwarg name: Optional instance name (C{str}), overriding this name. 

245 

246 @return: The instance (L{Vector3d}). 

247 

248 @raise VectorError: Invalid B{C{str3d}}. 

249 ''' 

250 return parse3d(str3d, sep=sep, Vector=self.classof, name=name or self.name) 

251 

252 def radii11(self, point2, point3): 

253 '''Return the radii of the C{Circum-}, C{In-}, I{Soddy} and C{Tangent} 

254 circles of a (3-D) triangle. 

255 

256 @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, 

257 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). 

258 @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, 

259 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). 

260 

261 @return: L{Radii11Tuple}C{(rA, rB, rC, cR, rIn, riS, roS, a, b, c, s)}. 

262 

263 @raise TriangleError: Near-coincident or -colinear points. 

264 

265 @raise TypeError: Invalid B{C{point2}} or B{C{point3}}. 

266 

267 @see: Function L{pygeodesy.radii11}, U{Incircle 

268 <https://MathWorld.Wolfram.com/Incircle.html>}, U{Soddy Circles 

269 <https://MathWorld.Wolfram.com/SoddyCircles.html>} and U{Tangent 

270 Circles<https://MathWorld.Wolfram.com/TangentCircles.html>}. 

271 ''' 

272 try: 

273 return _MODS.vector2d._radii11ABC(self, point2, point3, useZ=True)[0] 

274 except (TypeError, ValueError) as x: 

275 raise _xError(x, point=self, point2=point2, point3=point3) 

276 

277 def soddy4(self, point2, point3, eps=EPS4): 

278 '''Return the radius and center of the C{inner} I{Soddy} circle of a 

279 (3-D) triangle. 

280 

281 @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, 

282 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). 

283 @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, 

284 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). 

285 @kwarg eps: Tolerance for function L{pygeodesy.trilaterate3d2} if 

286 C{B{useZ} is True} otherwise L{pygeodesy.trilaterate2d2}. 

287 

288 @return: L{Soddy4Tuple}C{(radius, center, deltas, outer)}. The C{center}, 

289 an instance of B{C{point1}}'s (sub-)class, is co-planar with the 

290 three given points. 

291 

292 @raise ImportError: Package C{numpy} not found, not installed or older 

293 than version 1.10. 

294 

295 @raise IntersectionError: Near-coincident or -colinear points or 

296 a trilateration or C{numpy} issue. 

297 

298 @raise TypeError: Invalid B{C{point2}} or B{C{point3}}. 

299 

300 @see: Function L{pygeodesy.soddy4}. 

301 ''' 

302 return _MODS.vector2d.soddy4(self, point2, point3, eps=eps, useZ=True) 

303 

304 def trilaterate2d2(self, radius, center2, radius2, center3, radius3, eps=EPS4, z=INT0): 

305 '''Trilaterate this and two other circles, each given as a (2-D) center 

306 and a radius. 

307 

308 @arg radius: Radius of this circle (same C{units} as this C{x} and C{y}. 

309 @arg center2: Center of the 2nd circle (C{Cartesian}, L{Vector3d}, 

310 C{Vector2Tuple}, C{Vector3Tuple} or C{Vector4Tuple}). 

311 @arg radius2: Radius of this circle (same C{units} as this C{x} and C{y}. 

312 @arg center3: Center of the 3rd circle (C{Cartesian}, L{Vector3d}, 

313 C{Vector2Tuple}, C{Vector3Tuple} or C{Vector4Tuple}). 

314 @arg radius3: Radius of the 3rd circle (same C{units} as this C{x} and C{y}. 

315 @kwarg eps: Tolerance to check the trilaterated point I{delta} on all 

316 3 circles (C{scalar}) or C{None} for no checking. 

317 @kwarg z: Optional Z component of the trilaterated point (C{scalar}). 

318 

319 @return: Trilaterated point, an instance of this (sub-)class with C{z=B{z}}. 

320 

321 @raise IntersectionError: No intersection, near-concentric or -colinear 

322 centers, trilateration failed some other way 

323 or the trilaterated point is off one circle 

324 by more than B{C{eps}}. 

325 

326 @raise TypeError: Invalid B{C{center2}} or B{C{center3}}. 

327 

328 @raise UnitError: Invalid B{C{radius1}}, B{C{radius2}} or B{C{radius3}}. 

329 

330 @see: Function L{pygeodesy.trilaterate2d2}. 

331 ''' 

332 

333 def _xyr3(r, **name_v): 

334 v = _otherV3d(useZ=False, **name_v) 

335 return v.x, v.y, r 

336 

337 try: 

338 return _MODS.vector2d._trilaterate2d2(*(_xyr3(radius, center=self) + 

339 _xyr3(radius2, center2=center2) + 

340 _xyr3(radius3, center3=center3)), 

341 eps=eps, Vector=self.classof, z=z) 

342 except (AssertionError, TypeError, ValueError) as x: 

343 raise _xError(x, center=self, radius=radius, 

344 center2=center2, radius2=radius2, 

345 center3=center3, radius3=radius3) 

346 

347 def trilaterate3d2(self, radius, center2, radius2, center3, radius3, eps=EPS4): 

348 '''Trilaterate this and two other spheres, each given as a (3-D) center 

349 and a radius. 

350 

351 @arg radius: Radius of this sphere (same C{units} as this C{x}, C{y} 

352 and C{z}). 

353 @arg center2: Center of the 2nd sphere (C{Cartesian}, L{Vector3d}, 

354 C{Vector3Tuple} or C{Vector4Tuple}). 

355 @arg radius2: Radius of this sphere (same C{units} as this C{x}, C{y} 

356 and C{z}). 

357 @arg center3: Center of the 3rd sphere (C{Cartesian}, , L{Vector3d}, 

358 C{Vector3Tuple} or C{Vector4Tuple}). 

359 @arg radius3: Radius of the 3rd sphere (same C{units} as this C{x}, C{y} 

360 and C{z}). 

361 @kwarg eps: Pertubation tolerance (C{scalar}), same units as C{x}, C{y} 

362 and C{z} or C{None} for no pertubations. 

363 

364 @return: 2-Tuple with two trilaterated points, each an instance of this 

365 (sub-)class. Both points are the same instance if all three 

366 spheres intersect or abut in a single point. 

367 

368 @raise ImportError: Package C{numpy} not found, not installed or 

369 older than version 1.10. 

370 

371 @raise IntersectionError: Near-concentric, -colinear, too distant or 

372 non-intersecting spheres or C{numpy} issue. 

373 

374 @raise NumPyError: Some C{numpy} issue. 

375 

376 @raise TypeError: Invalid B{C{center2}} or B{C{center3}}. 

377 

378 @raise UnitError: Invalid B{C{radius}}, B{C{radius2}} or B{C{radius3}}. 

379 

380 @note: Package U{numpy<https://PyPI.org/project/numpy>} is required, 

381 version 1.10 or later. 

382 

383 @see: Norrdine, A. U{I{An Algebraic Solution to the Multilateration 

384 Problem}<https://www.ResearchGate.net/publication/275027725>} 

385 and U{I{implementation}<https://www.ResearchGate.net/publication/288825016>}. 

386 ''' 

387 try: 

388 c1 = _otherV3d(center=self, NN_OK=False) 

389 return _MODS.vector2d._trilaterate3d2(c1, Radius_(radius, low=eps), 

390 center2, radius2, 

391 center3, radius3, 

392 eps=eps, clas=self.classof) 

393 except (AssertionError, TypeError, ValueError) as x: 

394 raise _xError(x, center=self, radius=radius, 

395 center2=center2, radius2=radius2, 

396 center3=center3, radius3=radius3) 

397 

398 

399def _intersect3d3(start1, end1, start2, end2, eps=EPS, useZ=False): # MCCABE 16 in .formy.intersection2, .rhumbBase 

400 # (INTERNAL) Intersect two lines, see L{intersection3d3} below, 

401 # separated to allow callers to embellish any exceptions 

402 

403 def _corners2(s1, b1, s2, useZ): 

404 # Get the C{s1'} and C{e1'} corners of a right-angle 

405 # triangle with the hypotenuse thru C{s1} at bearing 

406 # C{b1} and the right angle at C{s2} 

407 dx, dy, d = s2.minus(s1).xyz 

408 if useZ and not isnear0(d): # not supported 

409 raise IntersectionError(useZ=d, bearing=b1) 

410 s, c = sincos2d(b1) 

411 if s and c: 

412 dx *= c / s 

413 dy *= s / c 

414 e1 = Vector3d(s2.x, s1.y + dx, s1.z) 

415 s1 = Vector3d(s1.x + dy, s2.y, s1.z) 

416 else: # orthogonal 

417 d = euclid(dx, dy) # hypot? 

418 e1 = Vector3d(s1.x + s * d, s1.y + c * d, s1.z) 

419 return s1, e1 

420 

421 def _outside(t, d2, o): # -o before start#, +o after end# 

422 return -o if t < 0 else (o if t > d2 else 0) # XXX d2 + eps? 

423 

424 s1 = t = _otherV3d(useZ=useZ, start1=start1) 

425 s2 = _otherV3d(useZ=useZ, start2=start2) 

426 b1 = _isDegrees(end1) 

427 if b1: # bearing, make an e1 

428 s1, e1 = _corners2(s1, end1, s2, useZ) 

429 else: 

430 e1 = _otherV3d(useZ=useZ, end1=end1) 

431 b2 = _isDegrees(end2) 

432 if b2: # bearing, make an e2 

433 s2, e2 = _corners2(s2, end2, t, useZ) 

434 else: 

435 e2 = _otherV3d(useZ=useZ, end2=end2) 

436 

437 a = e1.minus(s1) 

438 b = e2.minus(s2) 

439 c = s2.minus(s1) 

440 

441 ab = a.cross(b) 

442 d = fabs(c.dot(ab)) 

443 e = max(EPS0, eps or _0_0) 

444 if d > EPS0 and ab.length > e: # PYCHOK no cover 

445 d = d / ab.length # /= chokes PyChecker 

446 if d > e: # argonic, skew lines distance 

447 raise IntersectionError(skew_d=d, txt=_no_(_intersection_)) 

448 

449 # co-planar, non-skew lines 

450 ab2 = ab.length2 

451 if ab2 < e: # colinear, parallel or null line(s) 

452 x = a.length2 > b.length2 

453 if x: # make C{a} the shortest 

454 a, b = b, a 

455 s1, s2 = s2, s1 

456 e1, e2 = e2, e1 

457 b1, b2 = b2, b1 

458 if b.length2 < e: # PYCHOK no cover 

459 if c.length < e: 

460 return s1, 0, 0 

461 elif e2.minus(e1).length < e: 

462 return e1, 0, 0 

463 elif a.length2 < e: # null (s1, e1), non-null (s2, e2) 

464 # like _nearestOn2(s1, s2, e2, within=False, eps=e) 

465 t = s1.minus(s2).dot(b) 

466 v = s2.plus(b.times(t / b.length2)) 

467 if s1.minus(v).length < e: 

468 o = 0 if b2 else _outside(t, b.length2, 1 if x else 2) 

469 return (v, o, 0) if x else (v, 0, o) 

470 raise IntersectionError(length2=ab2, txt=_no_(_intersection_)) 

471 

472 cb = c.cross(b) 

473 t = cb.dot(ab) 

474 o1 = 0 if b1 else _outside(t, ab2, 1) 

475 v = s1.plus(a.times(t / ab2)) 

476 o2 = 0 if b2 else _outside(v.minus(s2).dot(b), b.length2, 2) 

477 return v, o1, o2 

478 

479 

480def intersection3d3(start1, end1, start2, end2, eps=EPS, useZ=True, 

481 **Vector_and_kwds): 

482 '''Compute the intersection point of two (2- or 3-D) lines, each defined 

483 by two points or by a point and a bearing. 

484 

485 @arg start1: Start point of the first line (C{Cartesian}, L{Vector3d}, 

486 C{Vector3Tuple} or C{Vector4Tuple}). 

487 @arg end1: End point of the first line (C{Cartesian}, L{Vector3d}, 

488 C{Vector3Tuple} or C{Vector4Tuple}) or the bearing at 

489 B{C{start1}} (compass C{degrees}). 

490 @arg start2: Start point of the second line (C{Cartesian}, L{Vector3d}, 

491 C{Vector3Tuple} or C{Vector4Tuple}). 

492 @arg end2: End point of the second line (C{Cartesian}, L{Vector3d}, 

493 C{Vector3Tuple} or C{Vector4Tuple}) or the bearing at 

494 B{C{start2}} (Ccompass C{degrees}). 

495 @kwarg eps: Tolerance for skew line distance and length (C{EPS}). 

496 @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}). 

497 @kwarg Vector_and_kwds: Optional class C{B{Vector}=None} to return the 

498 intersection points and optional, additional B{C{Vector}} 

499 keyword arguments, otherwise B{C{start1}}'s (sub-)class. 

500 

501 @return: An L{Intersection3Tuple}C{(point, outside1, outside2)} with 

502 C{point} an instance of B{C{Vector}} or B{C{start1}}'s (sub-)class. 

503 

504 @note: The C{outside} values is C{0} for lines specified by point and bearing. 

505 

506 @raise IntersectionError: Invalid, skew, non-co-planar or otherwise 

507 non-intersecting lines. 

508 

509 @see: U{Line-line intersection<https://MathWorld.Wolfram.com/Line-LineIntersection.html>} 

510 and U{line-line distance<https://MathWorld.Wolfram.com/Line-LineDistance.html>}, 

511 U{skew lines<https://MathWorld.Wolfram.com/SkewLines.html>} and U{point-line 

512 distance<https://MathWorld.Wolfram.com/Point-LineDistance3-Dimensional.html>}. 

513 ''' 

514 try: 

515 v, o1, o2 = _intersect3d3(start1, end1, start2, end2, eps=eps, useZ=useZ) 

516 except (TypeError, ValueError) as x: 

517 raise _xError(x, start1=start1, end1=end1, start2=start2, end2=end2) 

518 v = _nVc(v, **_xkwds(Vector_and_kwds, clas=start1.classof, 

519 name=intersection3d3.__name__)) 

520 return Intersection3Tuple(v, o1, o2) 

521 

522 

523def intersections2(center1, radius1, center2, radius2, sphere=True, **Vector_and_kwds): 

524 '''Compute the intersection of two spheres or circles, each defined by a (3-D) 

525 center point and a radius. 

526 

527 @arg center1: Center of the first sphere or circle (C{Cartesian}, L{Vector3d}, 

528 C{Vector3Tuple} or C{Vector4Tuple}). 

529 @arg radius1: Radius of the first sphere or circle (same units as the 

530 B{C{center1}} coordinates). 

531 @arg center2: Center of the second sphere or circle (C{Cartesian}, L{Vector3d}, 

532 C{Vector3Tuple} or C{Vector4Tuple}). 

533 @arg radius2: Radius of the second sphere or circle (same units as the 

534 B{C{center1}} and B{C{center2}} coordinates). 

535 @kwarg sphere: If C{True} compute the center and radius of the intersection of 

536 two spheres. If C{False}, ignore the C{z}-component and compute 

537 the intersection of two circles (C{bool}). 

538 @kwarg Vector_and_kwds: Optional class C{B{Vector}=None} to return the 

539 intersection points and optional, additional B{C{Vector}} 

540 keyword arguments, otherwise B{C{center1}}'s (sub-)class. 

541 

542 @return: If B{C{sphere}} is C{True}, a 2-tuple of the C{center} and C{radius} 

543 of the intersection of the I{spheres}. The C{radius} is C{0.0} for 

544 abutting spheres (and the C{center} is aka the I{radical center}). 

545 

546 If B{C{sphere}} is C{False}, a 2-tuple with the two intersection 

547 points of the I{circles}. For abutting circles, both points are 

548 the same instance, aka the I{radical center}. 

549 

550 @raise IntersectionError: Concentric, invalid or non-intersecting spheres 

551 or circles. 

552 

553 @raise TypeError: Invalid B{C{center1}} or B{C{center2}}. 

554 

555 @raise UnitError: Invalid B{C{radius1}} or B{C{radius2}}. 

556 

557 @see: U{Sphere-Sphere<https://MathWorld.Wolfram.com/Sphere-SphereIntersection.html>} and 

558 U{Circle-Circle<https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>} 

559 Intersection. 

560 ''' 

561 try: 

562 return _intersects2(center1, Radius_(radius1=radius1), 

563 center2, Radius_(radius2=radius2), sphere=sphere, 

564 clas=center1.classof, **Vector_and_kwds) 

565 except (TypeError, ValueError) as x: 

566 raise _xError(x, center1=center1, radius1=radius1, center2=center2, radius2=radius2) 

567 

568 

569def _intersects2(center1, r1, center2, r2, sphere=True, too_d=None, # in CartesianEllipsoidalBase.intersections2, 

570 **clas_Vector_and_kwds): # .ellipsoidalBaseDI._intersections2, .formy.intersections2 

571 # (INTERNAL) Intersect two spheres or circles, see L{intersections2} 

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

573 

574 def _nV3(x, y, z): 

575 v = Vector3d(x, y, z) 

576 n = intersections2.__name__ 

577 return _nVc(v, **_xkwds(clas_Vector_and_kwds, name=n)) 

578 

579 def _xV3(c1, u, x, y): 

580 xy1 = x, y, _1_0 # transform to original space 

581 return _nV3(fdot(xy1, u.x, -u.y, c1.x), 

582 fdot(xy1, u.y, u.x, c1.y), _0_0) 

583 

584 c1 = _otherV3d(useZ=sphere, center1=center1) 

585 c2 = _otherV3d(useZ=sphere, center2=center2) 

586 

587 if r1 < r2: # r1, r2 == R, r 

588 c1, c2 = c2, c1 

589 r1, r2 = r2, r1 

590 

591 m = c2.minus(c1) 

592 d = m.length 

593 if d < max(r2 - r1, EPS): 

594 raise IntersectionError(_near_(_concentric_)) # XXX ConcentricError? 

595 

596 o = fsum1_(-d, r1, r2) # overlap == -(d - (r1 + r2)) 

597 # compute intersections with c1 at (0, 0) and c2 at (d, 0), like 

598 # <https://MathWorld.Wolfram.com/Circle-CircleIntersection.html> 

599 if o > EPS: # overlapping, r1, r2 == R, r 

600 x = _MODS.formy._radical2(d, r1, r2).xline 

601 y = _1_0 - (x / r1)**2 

602 if y > EPS: 

603 y = r1 * sqrt(y) # y == a / 2 

604 elif y < 0: # PYCHOK no cover 

605 raise IntersectionError(_negative_) 

606 else: # abutting 

607 y = _0_0 

608 elif o < 0: # PYCHOK no cover 

609 if too_d is not None: 

610 d = too_d 

611 raise IntersectionError(_too_(Fmt.distant(d))) 

612 else: # abutting 

613 x, y = r1, _0_0 

614 

615 u = m.unit() 

616 if sphere: # sphere center and radius 

617 c = c1 if x < EPS else ( 

618 c2 if x > EPS1 else c1.plus(u.times(x))) 

619 t = _nV3(c.x, c.y, c.z), Radius(y) 

620 

621 elif y > 0: # intersecting circles 

622 t = _xV3(c1, u, x, y), _xV3(c1, u, x, -y) 

623 else: # abutting circles 

624 t = _xV3(c1, u, x, 0) 

625 t = t, t 

626 return t 

627 

628 

629def iscolinearWith(point, point1, point2, eps=EPS, useZ=True): 

630 '''Check whether a point is colinear with two other (2- or 3-D) points. 

631 

632 @arg point: The point (L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). 

633 @arg point1: First point (L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). 

634 @arg point2: Second point (L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). 

635 @kwarg eps: Tolerance (C{scalar}), same units as C{x}, C{y} and C{z}. 

636 @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}). 

637 

638 @return: C{True} if B{C{point}} is colinear B{C{point1}} and B{C{point2}}, 

639 C{False} otherwise. 

640 

641 @raise TypeError: Invalid B{C{point}}, B{C{point1}} or B{C{point2}}. 

642 

643 @see: Function L{nearestOn}. 

644 ''' 

645 p = _otherV3d(useZ=useZ, point=point) 

646 return _MODS.vector2d._iscolinearWith(p, point1, point2, eps=eps, useZ=useZ) 

647 

648 

649def nearestOn(point, point1, point2, within=True, useZ=True, Vector=None, **Vector_kwds): 

650 '''Locate the point between two points closest to a reference (2- or 3-D). 

651 

652 @arg point: Reference point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} 

653 or C{Vector4Tuple}). 

654 @arg point1: Start point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or 

655 C{Vector4Tuple}). 

656 @arg point2: End point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or 

657 C{Vector4Tuple}). 

658 @kwarg within: If C{True} return the closest point between both given 

659 points, otherwise the closest point on the extended line 

660 through both points (C{bool}). 

661 @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}). 

662 @kwarg Vector: Class to return closest point (C{Cartesian}, L{Vector3d} 

663 or C{Vector3Tuple}) or C{None}. 

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

665 ignored if C{B{Vector} is None}. 

666 

667 @return: Closest point, either B{C{point1}} or B{C{point2}} or an instance 

668 of the B{C{point}}'s (sub-)class or B{C{Vector}} if not C{None}. 

669 

670 @raise TypeError: Invalid B{C{point}}, B{C{point1}} or B{C{point2}}. 

671 

672 @see: U{3-D Point-Line Distance<https://MathWorld.Wolfram.com/Point-LineDistance3-Dimensional.html>}, 

673 C{Cartesian} and C{LatLon} methods C{nearestOn}, method L{sphericalTrigonometry.LatLon.nearestOn3} 

674 and function L{sphericalTrigonometry.nearestOn3}. 

675 ''' 

676 p0 = _otherV3d(useZ=useZ, point =point) 

677 p1 = _otherV3d(useZ=useZ, point1=point1) 

678 p2 = _otherV3d(useZ=useZ, point2=point2) 

679 

680 n = nearestOn.__name__ 

681 p, _ = _nearestOn2(p0, p1, p2, within=within) 

682 if Vector is not None: 

683 p = Vector(p.x, p.y, **_xkwds(Vector_kwds, z=p.z, name=n)) 

684 elif p is p1: 

685 p = point1 

686 elif p is p2: 

687 p = point2 

688 else: # ignore Vector_kwds 

689 p = point.classof(p.x, p.y, _xkwds_get(Vector_kwds, z=p.z), name=n) 

690 return p 

691 

692 

693def _nearestOn2(p0, p1, p2, within=True, eps=EPS): 

694 # (INTERNAL) Closest point and fraction, see L{nearestOn} above, 

695 # separated to allow callers to embellish any exceptions 

696 p21 = p2.minus(p1) 

697 d2 = p21.length2 

698 if d2 < eps: # coincident 

699 p = p1 # ~= p2 

700 t = 0 

701 else: # see comments in .points.nearestOn5 

702 t = p0.minus(p1).dot(p21) / d2 

703 if within and t < eps: 

704 p = p1 

705 t = 0 

706 elif within and t > (_1_0 - eps): 

707 p = p2 

708 t = 1 

709 else: 

710 p = p1.plus(p21.times(t)) 

711 return NearestOn2Tuple(p, t) 

712 

713 

714def nearestOn6(point, points, closed=False, useZ=True, **Vector_and_kwds): # eps=EPS 

715 '''Locate the point on a path or polygon closest to a reference point. 

716 

717 The closest point on each polygon edge is either the nearest of that 

718 edge's end points or a point in between. 

719 

720 @arg point: Reference point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or 

721 C{Vector4Tuple}). 

722 @arg points: The path or polygon points (C{Cartesian}, L{Vector3d}, 

723 C{Vector3Tuple} or C{Vector4Tuple}[]). 

724 @kwarg closed: Optionally, close the path or polygon (C{bool}). 

725 @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}). 

726 @kwarg Vector_and_kwds: Optional class C{B{Vector}=None} to return the closest 

727 point and optional, additional B{C{Vector}} keyword 

728 arguments, otherwise B{C{point}}'s (sub-)class. 

729 

730 @return: A L{NearestOn6Tuple}C{(closest, distance, fi, j, start, end)} with the 

731 C{closest}, the C{start} and the C{end} point each an instance of the 

732 B{C{Vector}} keyword argument of if {B{Vector}=None} or not specified, 

733 an instance of the reference B{C{point}}'s (sub-)class. 

734 

735 @raise PointsError: Insufficient number of B{C{points}} 

736 

737 @raise TypeError: Non-cartesian B{C{point}} and B{C{points}}. 

738 

739 @note: Distances measured with method L{Vector3d.equirectangular}. For 

740 geodetic distances use function L{nearestOn5} or one of the 

741 C{LatLon.nearestOn6} methods. 

742 ''' 

743 r = _otherV3d(useZ=useZ, point=point) 

744 D2 = r.equirectangular # distance squared 

745 

746 Ps = PointsIter(points, loop=1, name=nearestOn6.__name__) 

747 p1 = c = s = e = _otherV3d(useZ=useZ, i=0, points=Ps[0]) 

748 c2 = D2(c) # == r.minus(c).length2 

749 

750 f = i = 0 # p1..p2 == points[i]..[j] 

751 for j, p2 in Ps.enumerate(closed=closed): 

752 p2 = _otherV3d(useZ=useZ, i=j, points=p2) 

753 p, t = _nearestOn2(r, p1, p2) # within=True, eps=EPS 

754 d2 = D2(p) # == r.minus(p).length2 

755 if d2 < c2: 

756 c2, c, s, e, f = d2, p, p1, p2, (i + t) 

757 p1, i = p2, j 

758 

759 f, j = _fi_j2(f, len(Ps)) # like .ellipsoidalBaseDI._nearestOn2_ 

760 

761 kwds = _xkwds(Vector_and_kwds, clas=point.classof, name=Ps.name) 

762 v = _nVc(c, **kwds) 

763 s = _nVc(s, **kwds) if s is not c else v 

764 e = _nVc(e, **kwds) if e is not c else v 

765 return NearestOn6Tuple(v, sqrt(c2), f, j, s, e) 

766 

767 

768def _nVc(v, clas=None, name=NN, Vector=None, **Vector_kwds): # in .vector2d 

769 # return a named C{Vector} or C{clas} instance 

770 if Vector is not None: 

771 v = Vector(v.x, v.y, v.z, **Vector_kwds) 

772 elif clas is not None: 

773 v = clas(v.x, v.y, v.z) # ignore Vector_kwds 

774 return _xnamed(v, name) if name else v 

775 

776 

777def _otherV3d(useZ=True, NN_OK=True, i=None, **name_v): 

778 # check named vector instance, return Vector3d 

779 n, v = _xkwds_item2(name_v) 

780 if useZ and isinstance(v, Vector3dBase): 

781 return v if NN_OK or v.name else v.copy(name=Fmt.INDEX(n, i)) 

782 

783 n = Fmt.INDEX(n, i) 

784 try: 

785 return Vector3d(v.x, v.y, (v.z if useZ else INT0), name=n) 

786 except AttributeError: # no .x, .y or .z attr 

787 pass 

788 raise _xotherError(Vector3d(0, 0, 0), v, name=n, up=2) 

789 

790 

791def parse3d(str3d, sep=_COMMA_, Vector=Vector3d, **Vector_kwds): 

792 '''Parse an C{"x, y, z"} string. 

793 

794 @arg str3d: X, y and z values (C{str}). 

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

796 @kwarg Vector: Optional class (L{Vector3d}). 

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

798 ignored if C{B{Vector} is None}. 

799 

800 @return: A B{C{Vector}} instance or if B{C{Vector}} is C{None}, 

801 a named L{Vector3Tuple}C{(x, y, z)}. 

802 

803 @raise VectorError: Invalid B{C{str3d}}. 

804 ''' 

805 try: 

806 v = [float(v.strip()) for v in str3d.split(sep)] 

807 n = len(v) 

808 if n != 3: 

809 raise _ValueError(len=n) 

810 except (TypeError, ValueError) as x: 

811 raise VectorError(str3d=str3d, cause=x) 

812 return _xnamed((Vector3Tuple(v) if Vector is None else # *v 

813 Vector(*v, **Vector_kwds)), parse3d.__name__) 

814 

815 

816def sumOf(vectors, Vector=Vector3d, **Vector_kwds): 

817 '''Compute the I{vectorial} sum of two oe more vectors. 

818 

819 @arg vectors: Vectors to be added (L{Vector3d}[]). 

820 @kwarg Vector: Optional class for the vectorial sum (L{Vector3d}). 

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

822 ignored if C{B{Vector} is None}. 

823 

824 @return: Vectorial sum as B{C{Vector}} or if B{C{Vector}} is 

825 C{None}, a named L{Vector3Tuple}C{(x, y, z)}. 

826 

827 @raise VectorError: No B{C{vectors}}. 

828 ''' 

829 try: 

830 t = _MODS.nvectorBase._nsumOf(vectors, 0, None, {}) # no H 

831 except (TypeError, ValueError) as x: 

832 raise VectorError(vectors=vectors, Vector=Vector, cause=x) 

833 x, y, z = t[:3] 

834 n = sumOf.__name__ 

835 return Vector3Tuple(x, y, z, name=n) if Vector is None else \ 

836 Vector(x, y, z, **_xkwds(Vector_kwds, name=n)) 

837 

838 

839def trilaterate2d2(x1, y1, radius1, x2, y2, radius2, x3, y3, radius3, 

840 eps=None, **Vector_and_kwds): 

841 '''Trilaterate three circles, each given as a (2-D) center and a radius. 

842 

843 @arg x1: Center C{x} coordinate of the 1st circle (C{scalar}). 

844 @arg y1: Center C{y} coordinate of the 1st circle (C{scalar}). 

845 @arg radius1: Radius of the 1st circle (C{scalar}). 

846 @arg x2: Center C{x} coordinate of the 2nd circle (C{scalar}). 

847 @arg y2: Center C{y} coordinate of the 2nd circle (C{scalar}). 

848 @arg radius2: Radius of the 2nd circle (C{scalar}). 

849 @arg x3: Center C{x} coordinate of the 3rd circle (C{scalar}). 

850 @arg y3: Center C{y} coordinate of the 3rd circle (C{scalar}). 

851 @arg radius3: Radius of the 3rd circle (C{scalar}). 

852 @kwarg eps: Tolerance to check the trilaterated point I{delta} on all 

853 3 circles (C{scalar}) or C{None} for no checking. 

854 @kwarg Vector_and_kwds: Optional class C{B{Vector}=None} to return the 

855 trilateration and optional, additional B{C{Vector}} 

856 keyword arguments, otherwise (L{Vector3d}). 

857 

858 @return: Trilaterated point as C{B{Vector}(x, y, **B{Vector_kwds})} 

859 or L{Vector2Tuple}C{(x, y)} if C{B{Vector} is None}.. 

860 

861 @raise IntersectionError: No intersection, near-concentric or -colinear 

862 centers, trilateration failed some other way 

863 or the trilaterated point is off one circle 

864 by more than B{C{eps}}. 

865 

866 @raise UnitError: Invalid B{C{radius1}}, B{C{radius2}} or B{C{radius3}}. 

867 

868 @see: U{Issue #49<https://GitHub.com/mrJean1/PyGeodesy/issues/49>}, 

869 U{Find X location using 3 known (X,Y) location using trilateration 

870 <https://math.StackExchange.com/questions/884807>} and function 

871 L{pygeodesy.trilaterate3d2}. 

872 ''' 

873 return _MODS.vector2d._trilaterate2d2(x1, y1, radius1, 

874 x2, y2, radius2, 

875 x3, y3, radius3, eps=eps, **Vector_and_kwds) 

876 

877 

878def trilaterate3d2(center1, radius1, center2, radius2, center3, radius3, 

879 eps=EPS, **Vector_and_kwds): 

880 '''Trilaterate three spheres, each given as a (3-D) center and a radius. 

881 

882 @arg center1: Center of the 1st sphere (C{Cartesian}, L{Vector3d}, 

883 C{Vector3Tuple} or C{Vector4Tuple}). 

884 @arg radius1: Radius of the 1st sphere (same C{units} as C{x}, C{y} 

885 and C{z}). 

886 @arg center2: Center of the 2nd sphere (C{Cartesian}, L{Vector3d}, 

887 C{Vector3Tuple} or C{Vector4Tuple}). 

888 @arg radius2: Radius of this sphere (same C{units} as C{x}, C{y} 

889 and C{z}). 

890 @arg center3: Center of the 3rd sphere (C{Cartesian}, L{Vector3d}, 

891 C{Vector3Tuple} or C{Vector4Tuple}). 

892 @arg radius3: Radius of the 3rd sphere (same C{units} as C{x}, C{y} 

893 and C{z}). 

894 @kwarg eps: Pertubation tolerance (C{scalar}), same units as C{x}, 

895 C{y} and C{z} or C{None} for no pertubations. 

896 @kwarg Vector_and_kwds: Optional class C{B{Vector}=None} to return the 

897 trilateration and optional, additional B{C{Vector}} 

898 keyword arguments, otherwise B{C{center1}}'s 

899 (sub-)class. 

900 

901 @return: 2-Tuple with two trilaterated points, each a B{C{Vector}} 

902 instance. Both points are the same instance if all three 

903 spheres abut/intersect in a single point. 

904 

905 @raise ImportError: Package C{numpy} not found, not installed or 

906 older than version 1.10. 

907 

908 @raise IntersectionError: Near-concentric, -colinear, too distant or 

909 non-intersecting spheres. 

910 

911 @raise NumPyError: Some C{numpy} issue. 

912 

913 @raise TypeError: Invalid B{C{center1}}, B{C{center2}} or B{C{center3}}. 

914 

915 @raise UnitError: Invalid B{C{radius1}}, B{C{radius2}} or B{C{radius3}}. 

916 

917 @see: Norrdine, A. U{I{An Algebraic Solution to the Multilateration 

918 Problem}<https://www.ResearchGate.net/publication/275027725>}, 

919 the U{I{implementation}<https://www.ResearchGate.net/publication/ 

920 288825016>} and function L{pygeodesy.trilaterate2d2}. 

921 ''' 

922 try: 

923 return _MODS.vector2d._trilaterate3d2(_otherV3d(center1=center1, NN_OK=False), 

924 Radius_(radius1=radius1, low=eps), 

925 center2, radius2, center3, radius3, eps=eps, 

926 clas=center1.classof, **Vector_and_kwds) 

927 except (AssertionError, TypeError, ValueError) as x: 

928 raise _xError(x, center1=center1, radius1=radius1, 

929 center2=center2, radius2=radius2, 

930 center3=center3, radius3=radius3) 

931 

932 

933def _xyzhdn3(xyz, height, datum, ll): # in .cartesianBase, .nvectorBase 

934 '''(INTERNAL) Get a C{(h, d, name)} 3-tuple. 

935 ''' 

936 h = height or _xattr(xyz, height=None) \ 

937 or _xattr(xyz, h=None) \ 

938 or _xattr(ll, height=None) 

939 

940 d = datum or _xattr(xyz, datum=None) \ 

941 or _xattr(ll, datum=None) 

942 

943 return h, d, _xattr(xyz, name=NN) 

944 

945 

946__all__ += _ALL_DOCS(intersections2, sumOf, Vector3dBase) 

947 

948# **) MIT License 

949# 

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

951# 

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

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

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

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

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

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

958# 

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

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

961# 

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

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

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

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

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

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

968# OTHER DEALINGS IN THE SOFTWARE.