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 

10# from pygeodesy.basics import isscalar # from .fmath 

11from pygeodesy.constants import EPS, EPS0, EPS1, EPS4, INT0, isnear0, \ 

12 _0_0, _1_0 

13from pygeodesy.errors import IntersectionError, _ValueError, VectorError, \ 

14 _xattr, _xError, _xkwds_get, _xkwds, _xkwds_popitem 

15from pygeodesy.fmath import euclid, fabs, fdot, hypot, sqrt, \ 

16 fsum1_, isscalar 

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

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

19from pygeodesy.interns import NN, _COMMA_, _concentric_, _intersection_, \ 

20 _near_, _negative_, _no_, _too_ 

21from pygeodesy.iters import PointsIter, Fmt 

22from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY, _ALL_MODS as _MODS 

23from pygeodesy.named import _xnamed, _xotherError 

24from pygeodesy.namedTuples import Intersection3Tuple, NearestOn2Tuple, \ 

25 NearestOn6Tuple, Vector3Tuple # Vector4Tuple 

26# from pygeodesy.nvectorBase import _nsumOf # _MODS 

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

28from pygeodesy.units import _fi_j2, Radius, Radius_ 

29from pygeodesy.utily import atan2b, sincos2d 

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

31from pygeodesy.vector3dBase import Vector3dBase 

32 

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

34 

35__all__ = _ALL_LAZY.vector3d 

36__version__ = '23.06.05' 

37 

38 

39class Vector3d(Vector3dBase): 

40 '''Extended 3-D vector. 

41 

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

43 - earth-centered, earth-fixed cartesian (ECEF) 

44 - n-vector representing a normal to a point on earth's surface 

45 - great circle normal to vector 

46 - motion vector on earth's surface 

47 - etc. 

48 ''' 

49 

50 def bearing(self, useZ=True): 

51 '''Get the "bearing" of this vector. 

52 

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

54 consider the Y as +Z axis. 

55 

56 @return: Bearing (compass C{degrees}), the counter-clockwise 

57 angle off the +Z axis. 

58 ''' 

59 x, y = self.x, self.y 

60 if useZ: 

61 x, y = hypot(x, y), self.z 

62 return atan2b(x, y) 

63 

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

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

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

67 

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

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

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

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

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

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

74 

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

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

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

78 two given points. 

79 

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

81 than version 1.10. 

82 

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

84 a trilateration or C{numpy} issue. 

85 

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

87 

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

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

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

91 ''' 

92 try: 

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

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

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

96 

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

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

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

100 

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

102 or C{Vector4Tuple}). 

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

104 or C{Vector4Tuple}). 

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

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

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

108 

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

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

111 given points. 

112 

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

114 version 1.10. 

115 

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

117 or a trilateration or C{numpy} issue. 

118 

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

120 

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

122 ''' 

123 try: 

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

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

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

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

128 

129 def circum4_(self, *points): 

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

131 

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

133 or C{Vector4Tuple}). 

134 

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

136 an instance if this (sub-)class. 

137 

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

139 older than version 1.10. 

140 

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

142 

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

144 

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

146 

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

148 ''' 

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

150 

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

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

153 

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

155 or C{Vector4Tuple}). 

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

157 or C{Vector4Tuple}). 

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

159 C{y}, and C{z}. 

160 

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

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

163 

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

165 

166 @see: Method L{nearestOn}. 

167 ''' 

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

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

170 

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

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

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

174 

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

176 or C{Vector4Tuple}). 

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

178 or C{Vector4Tuple}). 

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

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

181 

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

183 iff C{B{circum}=True}. 

184 

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

186 

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

188 

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

190 ''' 

191 try: 

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

193 except (TypeError, ValueError) as x: 

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

195 

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

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

198 

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

200 C{Vector4Tuple}). 

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

202 C{Vector4Tuple}). 

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

204 points, otherwise the closest point on the extended 

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

206 

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

208 of this (sub-)class. 

209 

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

211 

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

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

214 ''' 

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

216 

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

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

219 

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

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

222 

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

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

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

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

227 

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

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

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

231 

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

233 

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

235 

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

237 

238 @see: Function L{nearestOn6}. 

239 ''' 

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

241 

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

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

244 

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

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

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

248 

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

250 

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

252 ''' 

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

254 

255 def radii11(self, point2, point3): 

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

257 circles of a (3-D) triangle. 

258 

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

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

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

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

263 

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

265 

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

267 

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

269 

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

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

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

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

274 ''' 

275 try: 

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

277 except (TypeError, ValueError) as x: 

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

279 

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

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

282 (3-D) triangle. 

283 

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

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

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

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

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

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

290 

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

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

293 three given points. 

294 

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

296 than version 1.10. 

297 

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

299 a trilateration or C{numpy} issue. 

300 

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

302 

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

304 ''' 

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

306 

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

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

309 and a radius. 

310 

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

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

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

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

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

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

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

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

319 3 circles (C{scalar}) or C{None} for no checking. 

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

321 

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

323 

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

325 centers, trilateration failed some other way 

326 or the trilaterated point is off one circle 

327 by more than B{C{eps}}. 

328 

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

330 

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

332 

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

334 ''' 

335 

336 def _xyr3(r, **name_v): 

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

338 return v.x, v.y, r 

339 

340 try: 

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

342 _xyr3(radius2, center2=center2) + 

343 _xyr3(radius3, center3=center3)), 

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

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

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

347 center2=center2, radius2=radius2, 

348 center3=center3, radius3=radius3) 

349 

350 def trilaterate3d2(self, radius, center2, radius2, center3, radius3, eps=EPS): 

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

352 and a radius. 

353 

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

355 and C{z}). 

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

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

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

359 and C{z}). 

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

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

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

363 and C{z}). 

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

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

366 

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

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

369 spheres intersect or abut in a single point. 

370 

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

372 older than version 1.10. 

373 

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

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

376 

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

378 

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

380 

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

382 

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

384 version 1.10 or later. 

385 

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

387 Problem}<https://www.ResearchGate.net/publication/ 

388 275027725_An_Algebraic_Solution_to_the_Multilateration_Problem>} 

389 and U{I{implementation}<https://www.ResearchGate.net/publication/ 

390 288825016_Trilateration_Matlab_Code>}. 

391 ''' 

392 try: 

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

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

395 center2, radius2, 

396 center3, radius3, 

397 eps=eps, clas=self.classof) 

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

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

400 center2=center2, radius2=radius2, 

401 center3=center3, radius3=radius3) 

402 

403 

404def _intersect3d3(start1, end1, start2, end2, eps=EPS, useZ=False): # MCCABE 16 in .formy.intersection2, .rhumbx._RhumbLine 

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

406 # separated to allow callers to embellish any exceptions 

407 

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

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

410 

411 def _rightangle2(s1, b1, s2, useZ): 

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

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

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

415 dx, dy, d = s2.minus(s1).xyz 

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

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

418 s, c = sincos2d(b1) 

419 if s and c: 

420 dx *= c / s 

421 dy *= s / c 

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

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

424 else: # orthogonal 

425 d = euclid(dx, dy) # hypot? 

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

427 return s1, e1 

428 

429 s1 = x = _otherV3d(useZ=useZ, start1=start1) 

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

431 b1 = isscalar(end1) 

432 if b1: # bearing, make an e1 

433 s1, e1 = _rightangle2(s1, end1, s2, useZ) 

434 else: 

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

436 b2 = isscalar(end2) 

437 if b2: # bearing, make an e2 

438 s2, e2 = _rightangle2(s2, end2, x, useZ) 

439 else: 

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

441 

442 a = e1.minus(s1) 

443 b = e2.minus(s2) 

444 c = s2.minus(s1) 

445 

446 ab = a.cross(b) 

447 d = fabs(c.dot(ab)) 

448 e = max(EPS0, eps or _0_0) 

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

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

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

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

453 

454 # co-planar, non-skew lines 

455 ab2 = ab.length2 

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

457 x = b.length2 < a.length2 

458 if x: # make C{a} the shortest 

459 a, b = b, a 

460 s1, s2 = s2, s1 

461 e1, e2 = e2, e1 

462 b1, b2 = b2, b1 

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

464 if c.length < e: 

465 return s1, 0, 0 

466 elif e2.minus(e1).length < e: 

467 return e1, 0, 0 

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

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

470 t = s1.minus(s2).dot(b) 

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

472 if s1.minus(v).length < e: 

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

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

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

476 

477 cb = c.cross(b) 

478 t = cb.dot(ab) 

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

480 v = s1.plus(a.times(t / ab2)) 

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

482 return v, o1, o2 

483 

484 

485def intersection3d3(start1, end1, start2, end2, eps=EPS, useZ=True, 

486 **Vector_and_kwds): 

487 '''Compute the intersection point of two lines, each defined by two 

488 points or by a point and a bearing. 

489 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

505 

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

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

508 

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

510 

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

512 non-intersecting lines. 

513 

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

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

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

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

518 ''' 

519 try: 

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

521 except (TypeError, ValueError) as x: 

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

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

524 name=intersection3d3.__name__)) 

525 return Intersection3Tuple(v, o1, o2) 

526 

527 

528def intersections2(center1, radius1, center2, radius2, sphere=True, **Vector_and_kwds): 

529 '''Compute the intersection of two spheres or circles, each defined by a 

530 (3-D) center point and a radius. 

531 

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

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

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

535 B{C{center1}} coordinates). 

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

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

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

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

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

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

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

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

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

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

546 

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

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

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

550 

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

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

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

554 

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

556 or circles. 

557 

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

559 

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

561 

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

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

564 Intersection. 

565 ''' 

566 try: 

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

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

569 clas=center1.classof, **Vector_and_kwds) 

570 except (TypeError, ValueError) as x: 

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

572 

573 

574def _intersects2(center1, r1, center2, r2, sphere=True, too_d=None, # in CartesianEllipsoidalBase.intersections2, 

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

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

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

578 

579 def _nV3(x, y, z): 

580 v = Vector3d(x, y, z) 

581 n = intersections2.__name__ 

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

583 

584 def _xV3(c1, u, x, y): 

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

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

587 fdot(xy1, u.y, u.x, c1.y), _0_0) 

588 

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

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

591 

592 if r1 < r2: # r1, r2 == R, r 

593 c1, c2 = c2, c1 

594 r1, r2 = r2, r1 

595 

596 m = c2.minus(c1) 

597 d = m.length 

598 if d < max(r2 - r1, EPS): 

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

600 

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

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

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

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

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

606 y = _1_0 - (x / r1)**2 

607 if y > EPS: 

608 y = r1 * sqrt(y) # y == a / 2 

609 elif y < 0: # PYCHOK no cover 

610 raise IntersectionError(_negative_) 

611 else: # abutting 

612 y = _0_0 

613 elif o < 0: # PYCHOK no cover 

614 t = d if too_d is None else too_d 

615 raise IntersectionError(_too_(Fmt.distant(t))) 

616 else: # abutting 

617 x, y = r1, _0_0 

618 

619 u = m.unit() 

620 if sphere: # sphere center and radius 

621 c = c1 if x < EPS else ( 

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

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

624 

625 elif y > 0: # intersecting circles 

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

627 else: # abutting circles 

628 t = _xV3(c1, u, x, 0) 

629 t = t, t 

630 return t 

631 

632 

633def iscolinearWith(point, point1, point2, eps=EPS, useZ=True): 

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

635 

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

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

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

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

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

641 

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

643 C{False} otherwise. 

644 

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

646 

647 @see: Function L{nearestOn}. 

648 ''' 

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

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

651 

652 

653def nearestOn(point, point1, point2, within=True, useZ=True, Vector=None, **Vector_kwds): 

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

655 

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

657 or C{Vector4Tuple}). 

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

659 C{Vector4Tuple}). 

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

661 C{Vector4Tuple}). 

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

663 points, otherwise the closest point on the extended line 

664 through both points (C{bool}). 

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

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

667 or C{Vector3Tuple}) or C{None}. 

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

669 ignored if C{B{Vector} is None}. 

670 

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

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

673 

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

675 

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

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

678 and function L{sphericalTrigonometry.nearestOn3}. 

679 ''' 

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

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

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

683 

684 n = nearestOn.__name__ 

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

686 if Vector is not None: 

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

688 elif p is p1: 

689 p = point1 

690 elif p is p2: 

691 p = point2 

692 else: # ignore Vector_kwds 

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

694 return p 

695 

696 

697def _nearestOn2(p0, p1, p2, within=True, eps=EPS): 

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

699 # separated to allow callers to embellish any exceptions 

700 p21 = p2.minus(p1) 

701 d2 = p21.length2 

702 if d2 < eps: # coincident 

703 p = p1 # ~= p2 

704 t = 0 

705 else: # see comments in .points.nearestOn5 

706 t = p0.minus(p1).dot(p21) / d2 

707 if within and t < eps: 

708 p = p1 

709 t = 0 

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

711 p = p2 

712 t = 1 

713 else: 

714 p = p1.plus(p21.times(t)) 

715 return NearestOn2Tuple(p, t) 

716 

717 

718def nearestOn6(point, points, closed=False, useZ=True, **Vector_and_kwds): # eps=EPS 

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

720 

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

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

723 

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

725 C{Vector4Tuple}). 

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

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

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

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

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

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

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

733 

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

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

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

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

738 

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

740 

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

742 

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

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

745 C{LatLon.nearestOn6} methods. 

746 ''' 

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

748 D2 = r.equirectangular # distance squared 

749 

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

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

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

753 

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

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

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

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

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

759 if d2 < c2: 

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

761 p1, i = p2, j 

762 

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

764 

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

766 v = _nVc(c, **kwds) 

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

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

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

770 

771 

772def _nVc(v, clas=None, name=NN, Vector=None, **Vector_kwds): # in .vector2d 

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

774 if Vector is not None: 

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

776 elif clas is not None: 

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

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

779 

780 

781def _otherV3d(useZ=True, NN_OK=True, i=None, **name_v): # in .CartesianEllipsoidalBase.intersections2, 

782 # check named vector instance, return Vector3d .Ellipsoid.height4, .formy.hartzell, .vector2d 

783 def _name_i(name, i): 

784 return name if i is None else Fmt.SQUARE(name, i) 

785 

786 name, v = _xkwds_popitem(name_v) 

787 if useZ and isinstance(v, Vector3dBase): 

788 return v if NN_OK or v.name else v.copy(name=_name_i(name, i)) 

789 try: 

790 return Vector3d(v.x, v.y, (v.z if useZ else INT0), name=_name_i(name, i)) 

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

792 pass 

793 raise _xotherError(Vector3d(0, 0, 0), v, name=_name_i(name, i), up=2) 

794 

795 

796def parse3d(str3d, sep=_COMMA_, Vector=Vector3d, **Vector_kwds): 

797 '''Parse an C{"x, y, z"} string. 

798 

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

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

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

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

803 ignored if C{B{Vector} is None}. 

804 

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

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

807 

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

809 ''' 

810 try: 

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

812 n = len(v) 

813 if n != 3: 

814 raise _ValueError(len=n) 

815 except (TypeError, ValueError) as x: 

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

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

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

819 

820 

821def sumOf(vectors, Vector=Vector3d, **Vector_kwds): 

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

823 

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

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

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

827 ignored if C{B{Vector} is None}. 

828 

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

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

831 

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

833 ''' 

834 try: 

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

836 except (TypeError, ValueError) as x: 

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

838 x, y, z = t[:3] 

839 n = sumOf.__name__ 

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

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

842 

843 

844def trilaterate2d2(x1, y1, radius1, x2, y2, radius2, x3, y3, radius3, 

845 eps=None, **Vector_and_kwds): 

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

847 

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

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

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

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

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

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

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

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

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

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

858 3 circles (C{scalar}) or C{None} for no checking. 

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

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

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

862 

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

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

865 

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

867 centers, trilateration failed some other way 

868 or the trilaterated point is off one circle 

869 by more than B{C{eps}}. 

870 

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

872 

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

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

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

876 L{pygeodesy.trilaterate3d2}. 

877 ''' 

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

879 x2, y2, radius2, 

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

881 

882 

883def trilaterate3d2(center1, radius1, center2, radius2, center3, radius3, 

884 eps=EPS, **Vector_and_kwds): 

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

886 

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

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

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

890 and C{z}). 

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

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

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

894 and C{z}). 

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

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

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

898 and C{z}). 

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

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

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

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

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

904 (sub-)class. 

905 

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

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

908 spheres abut/intersect in a single point. 

909 

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

911 older than version 1.10. 

912 

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

914 non-intersecting spheres. 

915 

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

917 

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

919 

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

921 

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

923 Problem}<https://www.ResearchGate.net/publication/ 

924 275027725_An_Algebraic_Solution_to_the_Multilateration_Problem>}, 

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

926 288825016_Trilateration_Matlab_Code>} and function 

927 L{pygeodesy.trilaterate2d2}. 

928 ''' 

929 try: 

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

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

932 center2, radius2, center3, radius3, eps=eps, 

933 clas=center1.classof, **Vector_and_kwds) 

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

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

936 center2=center2, radius2=radius2, 

937 center3=center3, radius3=radius3) 

938 

939 

940def _xyzhdn3(xyz, height, datum, ll): # in .cartesianBase, .nvectorBase 

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

942 ''' 

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

944 or _xattr(xyz, h=None) \ 

945 or _xattr(ll, height=None) 

946 

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

948 or _xattr(ll, datum=None) 

949 

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

951 

952 

953__all__ += _ALL_DOCS(intersections2, sumOf, Vector3dBase) 

954 

955# **) MIT License 

956# 

957# Copyright (C) 2016-2023 -- mrJean1 at Gmail -- All Rights Reserved. 

958# 

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

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

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

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

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

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

965# 

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

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

968# 

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

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

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

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

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

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

975# OTHER DEALINGS IN THE SOFTWARE.