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__ = '23.08.05' 

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/275027725>} 

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

389 ''' 

390 try: 

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

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

393 center2, radius2, 

394 center3, radius3, 

395 eps=eps, clas=self.classof) 

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

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

398 center2=center2, radius2=radius2, 

399 center3=center3, radius3=radius3) 

400 

401 

402def _intersect3d3(start1, end1, start2, end2, eps=EPS, useZ=False): # MCCABE 16 in .formy.intersection2, .rhumbBase 

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

404 # separated to allow callers to embellish any exceptions 

405 

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

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

408 

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

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

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

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

413 dx, dy, d = s2.minus(s1).xyz 

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

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

416 s, c = sincos2d(b1) 

417 if s and c: 

418 dx *= c / s 

419 dy *= s / c 

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

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

422 else: # orthogonal 

423 d = euclid(dx, dy) # hypot? 

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

425 return s1, e1 

426 

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

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

429 b1 = isscalar(end1) 

430 if b1: # bearing, make an e1 

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

432 else: 

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

434 b2 = isscalar(end2) 

435 if b2: # bearing, make an e2 

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

437 else: 

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

439 

440 a = e1.minus(s1) 

441 b = e2.minus(s2) 

442 c = s2.minus(s1) 

443 

444 ab = a.cross(b) 

445 d = fabs(c.dot(ab)) 

446 e = max(EPS0, eps or _0_0) 

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

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

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

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

451 

452 # co-planar, non-skew lines 

453 ab2 = ab.length2 

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

455 x = b.length2 < a.length2 

456 if x: # make C{a} the shortest 

457 a, b = b, a 

458 s1, s2 = s2, s1 

459 e1, e2 = e2, e1 

460 b1, b2 = b2, b1 

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

462 if c.length < e: 

463 return s1, 0, 0 

464 elif e2.minus(e1).length < e: 

465 return e1, 0, 0 

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

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

468 t = s1.minus(s2).dot(b) 

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

470 if s1.minus(v).length < e: 

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

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

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

474 

475 cb = c.cross(b) 

476 t = cb.dot(ab) 

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

478 v = s1.plus(a.times(t / ab2)) 

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

480 return v, o1, o2 

481 

482 

483def intersection3d3(start1, end1, start2, end2, eps=EPS, useZ=True, 

484 **Vector_and_kwds): 

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

486 points or by a point and a bearing. 

487 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

503 

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

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

506 

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

508 

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

510 non-intersecting lines. 

511 

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

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

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

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

516 ''' 

517 try: 

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

519 except (TypeError, ValueError) as x: 

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

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

522 name=intersection3d3.__name__)) 

523 return Intersection3Tuple(v, o1, o2) 

524 

525 

526def intersections2(center1, radius1, center2, radius2, sphere=True, **Vector_and_kwds): 

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

528 (3-D) center point and a radius. 

529 

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

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

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

533 B{C{center1}} coordinates). 

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

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

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

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

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

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

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

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

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

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

544 

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

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

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

548 

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

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

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

552 

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

554 or circles. 

555 

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

557 

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

559 

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

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

562 Intersection. 

563 ''' 

564 try: 

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

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

567 clas=center1.classof, **Vector_and_kwds) 

568 except (TypeError, ValueError) as x: 

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

570 

571 

572def _intersects2(center1, r1, center2, r2, sphere=True, too_d=None, # in CartesianEllipsoidalBase.intersections2, 

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

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

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

576 

577 def _nV3(x, y, z): 

578 v = Vector3d(x, y, z) 

579 n = intersections2.__name__ 

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

581 

582 def _xV3(c1, u, x, y): 

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

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

585 fdot(xy1, u.y, u.x, c1.y), _0_0) 

586 

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

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

589 

590 if r1 < r2: # r1, r2 == R, r 

591 c1, c2 = c2, c1 

592 r1, r2 = r2, r1 

593 

594 m = c2.minus(c1) 

595 d = m.length 

596 if d < max(r2 - r1, EPS): 

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

598 

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

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

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

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

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

604 y = _1_0 - (x / r1)**2 

605 if y > EPS: 

606 y = r1 * sqrt(y) # y == a / 2 

607 elif y < 0: # PYCHOK no cover 

608 raise IntersectionError(_negative_) 

609 else: # abutting 

610 y = _0_0 

611 elif o < 0: # PYCHOK no cover 

612 t = d if too_d is None else too_d 

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

614 else: # abutting 

615 x, y = r1, _0_0 

616 

617 u = m.unit() 

618 if sphere: # sphere center and radius 

619 c = c1 if x < EPS else ( 

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

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

622 

623 elif y > 0: # intersecting circles 

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

625 else: # abutting circles 

626 t = _xV3(c1, u, x, 0) 

627 t = t, t 

628 return t 

629 

630 

631def iscolinearWith(point, point1, point2, eps=EPS, useZ=True): 

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

633 

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

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

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

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

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

639 

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

641 C{False} otherwise. 

642 

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

644 

645 @see: Function L{nearestOn}. 

646 ''' 

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

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

649 

650 

651def nearestOn(point, point1, point2, within=True, useZ=True, Vector=None, **Vector_kwds): 

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

653 

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

655 or C{Vector4Tuple}). 

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

657 C{Vector4Tuple}). 

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

659 C{Vector4Tuple}). 

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

661 points, otherwise the closest point on the extended line 

662 through both points (C{bool}). 

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

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

665 or C{Vector3Tuple}) or C{None}. 

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

667 ignored if C{B{Vector} is None}. 

668 

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

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

671 

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

673 

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

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

676 and function L{sphericalTrigonometry.nearestOn3}. 

677 ''' 

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

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

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

681 

682 n = nearestOn.__name__ 

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

684 if Vector is not None: 

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

686 elif p is p1: 

687 p = point1 

688 elif p is p2: 

689 p = point2 

690 else: # ignore Vector_kwds 

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

692 return p 

693 

694 

695def _nearestOn2(p0, p1, p2, within=True, eps=EPS): 

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

697 # separated to allow callers to embellish any exceptions 

698 p21 = p2.minus(p1) 

699 d2 = p21.length2 

700 if d2 < eps: # coincident 

701 p = p1 # ~= p2 

702 t = 0 

703 else: # see comments in .points.nearestOn5 

704 t = p0.minus(p1).dot(p21) / d2 

705 if within and t < eps: 

706 p = p1 

707 t = 0 

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

709 p = p2 

710 t = 1 

711 else: 

712 p = p1.plus(p21.times(t)) 

713 return NearestOn2Tuple(p, t) 

714 

715 

716def nearestOn6(point, points, closed=False, useZ=True, **Vector_and_kwds): # eps=EPS 

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

718 

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

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

721 

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

723 C{Vector4Tuple}). 

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

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

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

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

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

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

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

731 

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

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

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

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

736 

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

738 

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

740 

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

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

743 C{LatLon.nearestOn6} methods. 

744 ''' 

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

746 D2 = r.equirectangular # distance squared 

747 

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

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

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

751 

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

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

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

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

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

757 if d2 < c2: 

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

759 p1, i = p2, j 

760 

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

762 

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

764 v = _nVc(c, **kwds) 

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

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

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

768 

769 

770def _nVc(v, clas=None, name=NN, Vector=None, **Vector_kwds): # in .vector2d 

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

772 if Vector is not None: 

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

774 elif clas is not None: 

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

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

777 

778 

779def _otherV3d(useZ=True, NN_OK=True, i=None, **name_v): # in .CartesianEllipsoidalBase.intersections2, 

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

781 def _name_i(name, i): 

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

783 

784 name, v = _xkwds_popitem(name_v) 

785 if useZ and isinstance(v, Vector3dBase): 

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

787 try: 

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

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

790 pass 

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

792 

793 

794def parse3d(str3d, sep=_COMMA_, Vector=Vector3d, **Vector_kwds): 

795 '''Parse an C{"x, y, z"} string. 

796 

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

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

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

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

801 ignored if C{B{Vector} is None}. 

802 

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

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

805 

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

807 ''' 

808 try: 

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

810 n = len(v) 

811 if n != 3: 

812 raise _ValueError(len=n) 

813 except (TypeError, ValueError) as x: 

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

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

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

817 

818 

819def sumOf(vectors, Vector=Vector3d, **Vector_kwds): 

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

821 

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

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

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

825 ignored if C{B{Vector} is None}. 

826 

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

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

829 

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

831 ''' 

832 try: 

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

834 except (TypeError, ValueError) as x: 

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

836 x, y, z = t[:3] 

837 n = sumOf.__name__ 

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

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

840 

841 

842def trilaterate2d2(x1, y1, radius1, x2, y2, radius2, x3, y3, radius3, 

843 eps=None, **Vector_and_kwds): 

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

845 

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

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

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

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

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

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

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

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

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

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

856 3 circles (C{scalar}) or C{None} for no checking. 

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

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

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

860 

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

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

863 

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

865 centers, trilateration failed some other way 

866 or the trilaterated point is off one circle 

867 by more than B{C{eps}}. 

868 

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

870 

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

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

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

874 L{pygeodesy.trilaterate3d2}. 

875 ''' 

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

877 x2, y2, radius2, 

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

879 

880 

881def trilaterate3d2(center1, radius1, center2, radius2, center3, radius3, 

882 eps=EPS, **Vector_and_kwds): 

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

884 

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

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

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

888 and C{z}). 

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

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

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

892 and C{z}). 

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

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

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

896 and C{z}). 

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

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

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

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

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

902 (sub-)class. 

903 

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

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

906 spheres abut/intersect in a single point. 

907 

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

909 older than version 1.10. 

910 

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

912 non-intersecting spheres. 

913 

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

915 

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

917 

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

919 

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

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

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

923 288825016>} and function L{pygeodesy.trilaterate2d2}. 

924 ''' 

925 try: 

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

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

928 center2, radius2, center3, radius3, eps=eps, 

929 clas=center1.classof, **Vector_and_kwds) 

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

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

932 center2=center2, radius2=radius2, 

933 center3=center3, radius3=radius3) 

934 

935 

936def _xyzhdn3(xyz, height, datum, ll): # in .cartesianBase, .nvectorBase 

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

938 ''' 

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

940 or _xattr(xyz, h=None) \ 

941 or _xattr(ll, height=None) 

942 

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

944 or _xattr(ll, datum=None) 

945 

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

947 

948 

949__all__ += _ALL_DOCS(intersections2, sumOf, Vector3dBase) 

950 

951# **) MIT License 

952# 

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

954# 

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

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

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

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

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

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

961# 

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

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

964# 

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

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

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

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

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

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

971# OTHER DEALINGS IN THE SOFTWARE.