Coverage for pygeodesy/vector3d.py: 97%

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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} and 

7L{trilaterate3d2}. 

8''' 

9 

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

11 _0_0, _1_0, typename 

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

13 _xattr, _xError, _xkwds, _xkwds_get, _xkwds_item2 

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

15# from pygeodesy.fsums import fsum1_ # from _MODS 

16# from pygeodesy.formy import _radical2 # _MODS 

17# from pygeodesy.internals import typename # from .constants 

18from pygeodesy.interns import _COMMA_, _concentric_, _intersection_, \ 

19 _near_, _negative_, _no_, _too_ 

20from pygeodesy.iters import PointsIter, Fmt 

21from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY, _ALL_MODS as _MODS 

22from pygeodesy.named import _name__, _name2__, _xnamed, _xotherError 

23from pygeodesy.namedTuples import Intersection3Tuple, NearestOn2Tuple, \ 

24 NearestOn6Tuple, _v2Cls, Vector3Tuple # Vector4Tuple 

25# from pygeodesy.nvectorBase import _nsumOf # _MODS 

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

27from pygeodesy.units import _fi_j2, _isDegrees, Radius, Radius_ 

28from pygeodesy.utily import atan2b, sincos2d 

29# from pygeodesy import vector2d as _vector2d # _MODS.into 

30from pygeodesy.vector3dBase import Vector3dBase 

31 

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

33 

34__all__ = _ALL_LAZY.vector3d 

35__version__ = '25.04.21' 

36 

37_vector2d = _MODS.into(vector2d=__name__) 

38 

39 

40class Vector3d(Vector3dBase): 

41 '''Extended 3-D vector. 

42 

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

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

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

46 - great circle normal to the vector 

47 - motion vector on the earth's surface 

48 - etc. 

49 ''' 

50 

51 def bearing(self, useZ=True): 

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

53 

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

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

56 

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

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 _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 _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 _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 _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 _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): 

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 C{B{name}=NN} (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=self._name__(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 _vector2d._radii11ABC4(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 _vector2d.soddy4(self, point2, point3, eps=eps, useZ=True) 

306 

307 def toCartesian(self, Cartesian, **Cartesian_kwds): 

308 '''Return this C{Vector3d} as a C{Cartesian}. 

309 

310 @arg Cartesian: The C{Cartesian} class to use. 

311 @kwarg Cartesian_kwds: Optional, additional C{Cartesian} 

312 keyword arguments. 

313 

314 @return: The C{B{Cartesian}} instance. 

315 ''' 

316 return _v2Cls(self, Cartesian, Cartesian_kwds) 

317 

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

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

320 and a radius. 

321 

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

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

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

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

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

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

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

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

330 3 circles (C{scalar}) or C{None} for no checking. 

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

332 

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

334 

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

336 centers, trilateration failed some other way 

337 or the trilaterated point is off one circle 

338 by more than B{C{eps}}. 

339 

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

341 

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

343 

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

345 ''' 

346 

347 def _xyr3(r, **name_v): 

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

349 return v.x, v.y, r 

350 

351 try: 

352 return _vector2d._trilaterate2d2(*(_xyr3(radius, center=self) + 

353 _xyr3(radius2, center2=center2) + 

354 _xyr3(radius3, center3=center3)), 

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

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

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

358 center2=center2, radius2=radius2, 

359 center3=center3, radius3=radius3) 

360 

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

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

363 and a radius. 

364 

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

366 and C{z}). 

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

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

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

370 and C{z}). 

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

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

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

374 and C{z}). 

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

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

377 

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

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

380 spheres intersect or abut in a single point. 

381 

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

383 older than version 1.10. 

384 

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

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

387 

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

389 

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

391 

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

393 

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

395 version 1.10 or later. 

396 

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

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

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

400 ''' 

401 try: 

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

403 return _vector2d._trilaterate3d2(c1, Radius_(radius, low=eps), 

404 center2, radius2, 

405 center3, radius3, 

406 eps=eps, clas=self.classof) 

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

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

409 center2=center2, radius2=radius2, 

410 center3=center3, radius3=radius3) 

411 

412 

413def _intersect3d3(start1, end1, start2, end2, eps=EPS, useZ=False): # MCCABE 16 in .formy.intersection2, .rhumbBase 

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

415 # separated to allow callers to embellish any exceptions 

416 

417 def _corners2(s1, b1, s2, useZ): 

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

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

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

421 dx, dy, d = s2.minus(s1).xyz3 

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

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

424 s, c = sincos2d(b1) 

425 if s and c: 

426 dx *= c / s 

427 dy *= s / c 

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

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

430 else: # orthogonal 

431 d = euclid(dx, dy) # hypot? 

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

433 return s1, e1 

434 

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

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

437 

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

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

440 b1 = _isDegrees(end1) 

441 if b1: # bearing, make an e1 

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

443 else: 

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

445 b2 = _isDegrees(end2) 

446 if b2: # bearing, make an e2 

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

448 else: 

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

450 

451 a = e1.minus(s1) 

452 b = e2.minus(s2) 

453 c = s2.minus(s1) 

454 

455 ab = a.cross(b) 

456 d = fabs(c.dot(ab)) 

457 e = max(EPS0, eps or _0_0) 

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

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

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

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

462 

463 # co-planar, non-skew lines 

464 ab2 = ab.length2 

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

466 x = a.length2 > b.length2 

467 if x: # make C{a} the shortest 

468 a, b = b, a 

469 s1, s2 = s2, s1 

470 e1, e2 = e2, e1 

471 b1, b2 = b2, b1 

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

473 if c.length < e: 

474 return s1, 0, 0 

475 elif e2.minus(e1).length < e: 

476 return e1, 0, 0 

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

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

479 t = s1.minus(s2).dot(b) 

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

481 if s1.minus(v).length < e: 

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

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

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

485 

486 cb = c.cross(b) 

487 t = cb.dot(ab) 

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

489 v = s1.plus(a.times(t / ab2)) 

490 t = v.minus(s2).dot(b) 

491 o2 = 0 if b2 else _outside(t, b.length2, 2) 

492 return v, o1, o2 

493 

494 

495def intersection3d3(start1, end1, start2, end2, eps=EPS, useZ=True, 

496 **Vector_and_kwds): 

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

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

499 

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

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

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

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

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

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

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

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

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

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

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

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

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

513 points and optional, additional B{C{Vector}} keyword arguments, 

514 otherwise B{C{start1}}'s (sub-)class. 

515 

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

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

518 

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

520 

521 @raise IntersectionError: Invalid, skew, non-co-planar or otherwise non-intersecting lines. 

522 

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

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

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

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

527 ''' 

528 try: 

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

530 except (TypeError, ValueError) as x: 

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

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

533 name=typename(intersection3d3))) 

534 return Intersection3Tuple(v, o1, o2) 

535 

536 

537def intersections2(center1, radius1, center2, radius2, sphere=True, **Vector_and_kwds): 

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

539 center point and a radius. 

540 

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

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

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

544 B{C{center1}} coordinates). 

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

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

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

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

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

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

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

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

553 points and optionally, additional B{C{Vector}} keyword arguments, 

554 otherwise B{C{center1}}'s (sub-)class. 

555 

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

557 intersection of the I{spheres}. For abutting circles, C{radius} is C{0.0} 

558 and C{center} is the I{radical center}. 

559 

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

561 I{circles}. For abutting circles, both points are the same instance, aka 

562 the I{radical center}. 

563 

564 @raise IntersectionError: Concentric, invalid or non-intersecting spheres or circles. 

565 

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

567 

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

569 

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

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

572 Intersection. 

573 ''' 

574 try: 

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

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

577 clas=center1.classof, **Vector_and_kwds) 

578 except (TypeError, ValueError) as x: 

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

580 

581 

582def _intersects2(center1, r1, center2, r2, sphere=True, too_d=None, # in CartesianEllipsoidalBase.intersections2, 

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

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

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

586 

587 def _nV3(x, y, z): 

588 v = Vector3d(x, y, z) 

589 n = typename(intersections2) 

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

591 

592 def _xV3(c1, u, x, y): 

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

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

595 fdot(xy1, u.y, u.x, c1.y), _0_0) 

596 

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

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

599 

600 if r1 < r2: # r1, r2 == R, r 

601 c1, c2 = c2, c1 

602 r1, r2 = r2, r1 

603 

604 m = c2.minus(c1) 

605 d = m.length 

606 if d < max(r2 - r1, EPS): 

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

608 

609 o = _MODS.fsums.fsum1_(-d, r1, r2) # overlap == -(d - (r1 + r2)) 

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

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

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

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

614 y = _1_0 - (x / r1)**2 

615 if y > EPS: 

616 y = r1 * sqrt(y) # y == a / 2 

617 elif y < 0: # PYCHOK no cover 

618 raise IntersectionError(_negative_) 

619 else: # abutting 

620 y = _0_0 

621 elif o < 0: # PYCHOK no cover 

622 if too_d is not None: 

623 d = too_d 

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

625 else: # abutting 

626 x, y = r1, _0_0 

627 

628 u = m.unit() 

629 if sphere: # sphere center and radius 

630 c = c1 if x < EPS else ( 

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

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

633 

634 elif y > 0: # intersecting circles 

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

636 else: # abutting circles 

637 t = _xV3(c1, u, x, 0) 

638 t = t, t 

639 return t 

640 

641 

642def iscolinearWith(point, point1, point2, eps=EPS, useZ=True): 

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

644 

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

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

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

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

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

650 

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

652 otherwise. 

653 

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

655 

656 @see: Function L{nearestOn}. 

657 ''' 

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

659 return _vector2d._iscolinearWith(p, point1, point2, eps=eps, useZ=useZ) 

660 

661 

662def nearestOn(point, point1, point2, within=True, useZ=True, Vector=None, **Vector_kwds): 

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

664 

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

666 or C{Vector4Tuple}). 

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

668 C{Vector4Tuple}). 

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

670 C{Vector4Tuple}). 

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

672 points, otherwise the closest point on the extended line 

673 through both points (C{bool}). 

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

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

676 C{Vector3Tuple}) or C{None}. 

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

678 ignored if C{B{Vector} is None}. 

679 

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

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

682 

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

684 

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

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

687 and function L{sphericalTrigonometry.nearestOn3}. 

688 ''' 

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

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

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

692 

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

694 if Vector is not None: 

695 p = Vector(p.x, p.y, **_xkwds(Vector_kwds, z=p.z, name__=nearestOn)) 

696 elif p is p1: 

697 p = point1 

698 elif p is p2: 

699 p = point2 

700 else: # ignore Vector_kwds 

701 p = point.classof(p.x, p.y, _xkwds_get(Vector_kwds, z=p.z), name__=nearestOn) 

702 return p 

703 

704 

705def _nearestOn2(p0, p1, p2, within=True, eps=EPS): 

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

707 # separated to allow callers to embellish any exceptions 

708 p21 = p2.minus(p1) 

709 d2 = p21.length2 

710 if d2 < eps: # coincident 

711 p = p1 # ~= p2 

712 t = 0 

713 else: # see comments in .points.nearestOn5 

714 t = p0.minus(p1).dot(p21) / d2 

715 if within and t < eps: 

716 p = p1 

717 t = 0 

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

719 p = p2 

720 t = 1 

721 else: 

722 p = p1.plus(p21.times(t)) 

723 return NearestOn2Tuple(p, t) 

724 

725 

726def nearestOn6(point, points, closed=False, useZ=True, **Vector_and_kwds): # eps=EPS 

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

728 

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

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

731 

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

733 C{Vector4Tuple}). 

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

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

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

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

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

739 point and optionally, additional B{C{Vector}} keyword arguments, 

740 otherwise B{C{point}}'s (sub-)class. 

741 

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

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

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

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

746 

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

748 

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

750 

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

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

753 C{LatLon.nearestOn6} methods. 

754 ''' 

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

756 D2 = r.equirectangular # distance squared 

757 

758 Ps = PointsIter(points, loop=1, name=typename(nearestOn6)) 

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

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

761 

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

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

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

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

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

767 if d2 < c2: 

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

769 p1, i = p2, j 

770 

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

772 

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

774 v = _nVc(c, **kwds) 

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

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

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

778 

779 

780def _nVc(v, clas=None, Vector=None, **name_Vector_kwds): # in .vector2d 

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

782 name, kwds = _name2__(**name_Vector_kwds) 

783 if Vector is not None: 

784 v = Vector(v.x, v.y, v.z, **kwds) 

785 elif clas is not None: 

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

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

788 

789 

790def _otherV3d(useZ=True, NN_OK=True, i=None, **name_vector): 

791 # check named vector instance, return Vector3d 

792 n, v = _xkwds_item2(name_vector) 

793 n = Fmt.INDEX(n, i) 

794 if useZ and isinstance(v, Vector3dBase): 

795 return v if NN_OK or v.name else v.copy(name=n) 

796 try: 

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

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

799 pass 

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

801 

802 

803def parse3d(str3d, sep=_COMMA_, Vector=Vector3d, **Vector_kwds): 

804 '''Parse an C{"x, y, z"} string. 

805 

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

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

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

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

810 ignored if C{B{Vector} is None}. 

811 

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

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

814 

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

816 ''' 

817 try: 

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

819 n = len(v) 

820 if n != 3: 

821 raise _ValueError(len=n) 

822 except (TypeError, ValueError) as x: 

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

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

825 Vector(*v, **Vector_kwds)), name__=parse3d) # .__name__ 

826 

827 

828def sumOf(vectors, Vector=Vector3d, **Vector_kwds): 

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

830 

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

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

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

834 if C{B{Vector} is None}. 

835 

836 @return: Vectorial sum as B{C{Vector}} or if B{C{Vector} is None}, 

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

838 

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

840 ''' 

841 try: 

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

843 except (TypeError, ValueError) as x: 

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

845 x, y, z = t[:3] 

846 return Vector3Tuple(x, y, z, name__=sumOf) if Vector is None else \ 

847 Vector(x, y, z, **_xkwds(Vector_kwds, name__=sumOf)) # .__name__ 

848 

849 

850def trilaterate3d2(center1, radius1, center2, radius2, center3, radius3, 

851 eps=EPS, **Vector_and_kwds): 

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

853 

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

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

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

857 and C{z}). 

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

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

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

861 and C{z}). 

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

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

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

865 and C{z}). 

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

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

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

869 trilateration and optionally, additional B{C{Vector}} 

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

871 

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

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

874 spheres abut/intersect in a single point. 

875 

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

877 than version 1.10. 

878 

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

880 non-intersecting spheres. 

881 

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

883 

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

885 

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

887 

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

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

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

891 288825016>} and function L{pygeodesy.trilaterate2d2}. 

892 ''' 

893 try: 

894 return _vector2d._trilaterate3d2(_otherV3d(center1=center1, NN_OK=False), 

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

896 center2, radius2, center3, radius3, eps=eps, 

897 clas=center1.classof, **Vector_and_kwds) 

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

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

900 center2=center2, radius2=radius2, 

901 center3=center3, radius3=radius3) 

902 

903 

904def _xyzhdlln4(xyz, height, datum, ll=None, **name): # in .cartesianBase, .nvectorBase 

905 '''(INTERNAL) Get a C{(h, d, ll, name)} 4-tuple. 

906 ''' 

907 _x = _xattr 

908 h = height or _x(xyz, height=None) or _x(xyz, h=None) or _x(ll, height=None) 

909 d = datum or _x(xyz, datum=None) or _x(ll, datum=None) 

910 return h, d, ll, _name__(name, _or_nameof=ll) 

911 

912 

913__all__ += _ALL_DOCS(intersections2, sumOf, Vector3dBase) 

914 

915# **) MIT License 

916# 

917# Copyright (C) 2016-2025 -- mrJean1 at Gmail -- All Rights Reserved. 

918# 

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

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

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

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

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

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

925# 

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

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

928# 

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

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

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

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

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

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

935# OTHER DEALINGS IN THE SOFTWARE.