Coverage for pygeodesy/vector3d.py: 97%

235 statements  

« prev     ^ index     » next       coverage.py v7.6.1, created at 2025-01-06 12:20 -0500

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 

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 

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

18 _near_, _negative_, _no_, _too_ 

19from pygeodesy.iters import PointsIter, Fmt 

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

21from pygeodesy.named import _name__, _name2__, _xnamed, _xotherError 

22from pygeodesy.namedTuples import Intersection3Tuple, NearestOn2Tuple, \ 

23 NearestOn6Tuple, _v2Cls, Vector3Tuple # Vector4Tuple 

24# from pygeodesy.nvectorBase import _nsumOf # _MODS 

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

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

27from pygeodesy.utily import atan2b, sincos2d 

28# import pygeodesy.vector2d as _vector2d # _MODS.into 

29from pygeodesy.vector3dBase import Vector3dBase 

30 

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

32 

33__all__ = _ALL_LAZY.vector3d 

34__version__ = '24.11.22' 

35 

36_vector2d = _MODS.into(vector2d=__name__) 

37 

38 

39class Vector3d(Vector3dBase): 

40 '''Extended 3-D vector. 

41 

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

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

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

45 - great circle normal to the vector 

46 - motion vector on the earth's surface 

47 - etc. 

48 ''' 

49 

50 def bearing(self, useZ=True): 

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

52 

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

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

55 

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

57 ''' 

58 x, y = self.x, self.y 

59 if useZ: 

60 x, y = hypot(x, y), self.z 

61 return atan2b(x, y) 

62 

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

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

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

66 

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

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

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

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

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

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

73 

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

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

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

77 two given points. 

78 

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

80 than version 1.10. 

81 

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

83 a trilateration or C{numpy} issue. 

84 

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

86 

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

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

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

90 ''' 

91 try: 

92 return _vector2d._circin6(self, point2, point3, eps=eps, useZ=True) 

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

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

95 

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

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

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

99 

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

101 or C{Vector4Tuple}). 

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

103 or C{Vector4Tuple}). 

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

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

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

107 

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

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

110 given points. 

111 

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

113 version 1.10. 

114 

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

116 or a trilateration or C{numpy} issue. 

117 

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

119 

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

121 ''' 

122 try: 

123 return _vector2d._circum3(self, point2, point3, circum=circum, 

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

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

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

127 

128 def circum4_(self, *points): 

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

130 

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

132 or C{Vector4Tuple}). 

133 

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

135 an instance if this (sub-)class. 

136 

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

138 older than version 1.10. 

139 

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

141 

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

143 

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

145 

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

147 ''' 

148 return _vector2d.circum4_(self, *points, useZ=True, Vector=self.classof) 

149 

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

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

152 

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

154 or C{Vector4Tuple}). 

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

156 or C{Vector4Tuple}). 

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

158 C{y}, and C{z}. 

159 

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

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

162 

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

164 

165 @see: Method L{nearestOn}. 

166 ''' 

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

168 return _vector2d._iscolinearWith(v, point1, point2, eps=eps) 

169 

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

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

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

173 

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

175 or C{Vector4Tuple}). 

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

177 or C{Vector4Tuple}). 

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

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

180 

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

182 iff C{B{circum}=True}. 

183 

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

185 

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

187 

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

189 ''' 

190 try: 

191 return _vector2d._meeus2(self, point2, point3, circum, clas=self.classof) 

192 except (TypeError, ValueError) as x: 

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

194 

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

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

197 

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

199 C{Vector4Tuple}). 

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

201 C{Vector4Tuple}). 

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

203 points, otherwise the closest point on the extended 

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

205 

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

207 of this (sub-)class. 

208 

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

210 

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

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

213 ''' 

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

215 

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

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

218 

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

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

221 

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

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

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

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

226 

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

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

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

230 

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

232 

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

234 

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

236 

237 @see: Function L{nearestOn6}. 

238 ''' 

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

240 

241 def parse(self, str3d, sep=_COMMA_, **name): 

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

243 

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

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

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

247 

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

249 

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

251 ''' 

252 return parse3d(str3d, sep=sep, Vector=self.classof, name=self._name__(name)) 

253 

254 def radii11(self, point2, point3): 

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

256 circles of a (3-D) triangle. 

257 

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

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

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

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

262 

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

264 

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

266 

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

268 

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

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

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

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

273 ''' 

274 try: 

275 return _vector2d._radii11ABC4(self, point2, point3, useZ=True)[0] 

276 except (TypeError, ValueError) as x: 

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

278 

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

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

281 (3-D) triangle. 

282 

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

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

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

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

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

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

289 

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

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

292 three given points. 

293 

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

295 than version 1.10. 

296 

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

298 a trilateration or C{numpy} issue. 

299 

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

301 

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

303 ''' 

304 return _vector2d.soddy4(self, point2, point3, eps=eps, useZ=True) 

305 

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

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

308 

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

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

311 keyword arguments. 

312 

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

314 ''' 

315 return _v2Cls(self, Cartesian, Cartesian_kwds) 

316 

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

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

319 and a radius. 

320 

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

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

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

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

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

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

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

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

329 3 circles (C{scalar}) or C{None} for no checking. 

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

331 

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

333 

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

335 centers, trilateration failed some other way 

336 or the trilaterated point is off one circle 

337 by more than B{C{eps}}. 

338 

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

340 

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

342 

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

344 ''' 

345 

346 def _xyr3(r, **name_v): 

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

348 return v.x, v.y, r 

349 

350 try: 

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

352 _xyr3(radius2, center2=center2) + 

353 _xyr3(radius3, center3=center3)), 

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

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

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

357 center2=center2, radius2=radius2, 

358 center3=center3, radius3=radius3) 

359 

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

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

362 and a radius. 

363 

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

365 and C{z}). 

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

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

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

369 and C{z}). 

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

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

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

373 and C{z}). 

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

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

376 

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

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

379 spheres intersect or abut in a single point. 

380 

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

382 older than version 1.10. 

383 

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

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

386 

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

388 

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

390 

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

392 

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

394 version 1.10 or later. 

395 

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

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

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

399 ''' 

400 try: 

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

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

403 center2, radius2, 

404 center3, radius3, 

405 eps=eps, clas=self.classof) 

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

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

408 center2=center2, radius2=radius2, 

409 center3=center3, radius3=radius3) 

410 

411 

412def _intersect3d3(start1, end1, start2, end2, eps=EPS, useZ=False): # MCCABE 16 in .formy.intersection2, .rhumbBase 

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

414 # separated to allow callers to embellish any exceptions 

415 

416 def _corners2(s1, b1, s2, useZ): 

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

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

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

420 dx, dy, d = s2.minus(s1).xyz3 

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

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

423 s, c = sincos2d(b1) 

424 if s and c: 

425 dx *= c / s 

426 dy *= s / c 

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

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

429 else: # orthogonal 

430 d = euclid(dx, dy) # hypot? 

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

432 return s1, e1 

433 

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

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

436 

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

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

439 b1 = _isDegrees(end1) 

440 if b1: # bearing, make an e1 

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

442 else: 

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

444 b2 = _isDegrees(end2) 

445 if b2: # bearing, make an e2 

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

447 else: 

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

449 

450 a = e1.minus(s1) 

451 b = e2.minus(s2) 

452 c = s2.minus(s1) 

453 

454 ab = a.cross(b) 

455 d = fabs(c.dot(ab)) 

456 e = max(EPS0, eps or _0_0) 

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

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

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

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

461 

462 # co-planar, non-skew lines 

463 ab2 = ab.length2 

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

465 x = a.length2 > b.length2 

466 if x: # make C{a} the shortest 

467 a, b = b, a 

468 s1, s2 = s2, s1 

469 e1, e2 = e2, e1 

470 b1, b2 = b2, b1 

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

472 if c.length < e: 

473 return s1, 0, 0 

474 elif e2.minus(e1).length < e: 

475 return e1, 0, 0 

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

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

478 t = s1.minus(s2).dot(b) 

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

480 if s1.minus(v).length < e: 

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

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

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

484 

485 cb = c.cross(b) 

486 t = cb.dot(ab) 

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

488 v = s1.plus(a.times(t / ab2)) 

489 t = v.minus(s2).dot(b) 

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

491 return v, o1, o2 

492 

493 

494def intersection3d3(start1, end1, start2, end2, eps=EPS, useZ=True, 

495 **Vector_and_kwds): 

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

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

498 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

514 

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

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

517 

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

519 

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

521 

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

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

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

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

526 ''' 

527 try: 

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

529 except (TypeError, ValueError) as x: 

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

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

532 name=intersection3d3.__name__)) 

533 return Intersection3Tuple(v, o1, o2) 

534 

535 

536def intersections2(center1, radius1, center2, radius2, sphere=True, **Vector_and_kwds): 

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

538 center point and a radius. 

539 

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

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

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

543 B{C{center1}} coordinates). 

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

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

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

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

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

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

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

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

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

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

554 

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

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

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

558 

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

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

561 the I{radical center}. 

562 

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

564 

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

566 

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

568 

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

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

571 Intersection. 

572 ''' 

573 try: 

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

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

576 clas=center1.classof, **Vector_and_kwds) 

577 except (TypeError, ValueError) as x: 

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

579 

580 

581def _intersects2(center1, r1, center2, r2, sphere=True, too_d=None, # in CartesianEllipsoidalBase.intersections2, 

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

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

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

585 

586 def _nV3(x, y, z): 

587 v = Vector3d(x, y, z) 

588 n = intersections2.__name__ 

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

590 

591 def _xV3(c1, u, x, y): 

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

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

594 fdot(xy1, u.y, u.x, c1.y), _0_0) 

595 

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

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

598 

599 if r1 < r2: # r1, r2 == R, r 

600 c1, c2 = c2, c1 

601 r1, r2 = r2, r1 

602 

603 m = c2.minus(c1) 

604 d = m.length 

605 if d < max(r2 - r1, EPS): 

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

607 

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

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

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

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

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

613 y = _1_0 - (x / r1)**2 

614 if y > EPS: 

615 y = r1 * sqrt(y) # y == a / 2 

616 elif y < 0: # PYCHOK no cover 

617 raise IntersectionError(_negative_) 

618 else: # abutting 

619 y = _0_0 

620 elif o < 0: # PYCHOK no cover 

621 if too_d is not None: 

622 d = too_d 

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

624 else: # abutting 

625 x, y = r1, _0_0 

626 

627 u = m.unit() 

628 if sphere: # sphere center and radius 

629 c = c1 if x < EPS else ( 

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

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

632 

633 elif y > 0: # intersecting circles 

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

635 else: # abutting circles 

636 t = _xV3(c1, u, x, 0) 

637 t = t, t 

638 return t 

639 

640 

641def iscolinearWith(point, point1, point2, eps=EPS, useZ=True): 

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

643 

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

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

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

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

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

649 

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

651 otherwise. 

652 

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

654 

655 @see: Function L{nearestOn}. 

656 ''' 

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

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

659 

660 

661def nearestOn(point, point1, point2, within=True, useZ=True, Vector=None, **Vector_kwds): 

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

663 

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

665 or C{Vector4Tuple}). 

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

667 C{Vector4Tuple}). 

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

669 C{Vector4Tuple}). 

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

671 points, otherwise the closest point on the extended line 

672 through both points (C{bool}). 

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

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

675 C{Vector3Tuple}) or C{None}. 

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

677 ignored if C{B{Vector} is None}. 

678 

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

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

681 

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

683 

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

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

686 and function L{sphericalTrigonometry.nearestOn3}. 

687 ''' 

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

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

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

691 

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

693 if Vector is not None: 

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

695 elif p is p1: 

696 p = point1 

697 elif p is p2: 

698 p = point2 

699 else: # ignore Vector_kwds 

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

701 return p 

702 

703 

704def _nearestOn2(p0, p1, p2, within=True, eps=EPS): 

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

706 # separated to allow callers to embellish any exceptions 

707 p21 = p2.minus(p1) 

708 d2 = p21.length2 

709 if d2 < eps: # coincident 

710 p = p1 # ~= p2 

711 t = 0 

712 else: # see comments in .points.nearestOn5 

713 t = p0.minus(p1).dot(p21) / d2 

714 if within and t < eps: 

715 p = p1 

716 t = 0 

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

718 p = p2 

719 t = 1 

720 else: 

721 p = p1.plus(p21.times(t)) 

722 return NearestOn2Tuple(p, t) 

723 

724 

725def nearestOn6(point, points, closed=False, useZ=True, **Vector_and_kwds): # eps=EPS 

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

727 

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

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

730 

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

732 C{Vector4Tuple}). 

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

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

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

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

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

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

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

740 

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

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

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

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

745 

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

747 

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

749 

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

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

752 C{LatLon.nearestOn6} methods. 

753 ''' 

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

755 D2 = r.equirectangular # distance squared 

756 

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

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

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

760 

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

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

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

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

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

766 if d2 < c2: 

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

768 p1, i = p2, j 

769 

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

771 

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

773 v = _nVc(c, **kwds) 

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

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

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

777 

778 

779def _nVc(v, clas=None, Vector=None, **name_Vector_kwds): # in .vector2d 

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

781 name, kwds = _name2__(**name_Vector_kwds) 

782 if Vector is not None: 

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

784 elif clas is not None: 

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

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

787 

788 

789def _otherV3d(useZ=True, NN_OK=True, i=None, **name_vector): 

790 # check named vector instance, return Vector3d 

791 n, v = _xkwds_item2(name_vector) 

792 n = Fmt.INDEX(n, i) 

793 if useZ and isinstance(v, Vector3dBase): 

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

795 try: 

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

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

798 pass 

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

800 

801 

802def parse3d(str3d, sep=_COMMA_, Vector=Vector3d, **Vector_kwds): 

803 '''Parse an C{"x, y, z"} string. 

804 

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

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

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

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

809 ignored if C{B{Vector} is None}. 

810 

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

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

813 

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

815 ''' 

816 try: 

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

818 n = len(v) 

819 if n != 3: 

820 raise _ValueError(len=n) 

821 except (TypeError, ValueError) as x: 

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

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

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

825 

826 

827def sumOf(vectors, Vector=Vector3d, **Vector_kwds): 

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

829 

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

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

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

833 if C{B{Vector} is None}. 

834 

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

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

837 

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

839 ''' 

840 try: 

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

842 except (TypeError, ValueError) as x: 

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

844 x, y, z = t[:3] 

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

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

847 

848 

849def trilaterate3d2(center1, radius1, center2, radius2, center3, radius3, 

850 eps=EPS, **Vector_and_kwds): 

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

852 

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

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

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

856 and C{z}). 

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

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

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

860 and C{z}). 

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

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

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

864 and C{z}). 

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

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

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

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

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

870 

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

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

873 spheres abut/intersect in a single point. 

874 

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

876 than version 1.10. 

877 

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

879 non-intersecting spheres. 

880 

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

882 

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

884 

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

886 

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

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

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

890 288825016>} and function L{pygeodesy.trilaterate2d2}. 

891 ''' 

892 try: 

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

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

895 center2, radius2, center3, radius3, eps=eps, 

896 clas=center1.classof, **Vector_and_kwds) 

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

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

899 center2=center2, radius2=radius2, 

900 center3=center3, radius3=radius3) 

901 

902 

903def _xyzhdlln4(xyz, height, datum, ll=None, **name): # in .cartesianBase, .nvectorBase 

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

905 ''' 

906 _x = _xattr 

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

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

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

910 

911 

912__all__ += _ALL_DOCS(intersections2, sumOf, Vector3dBase) 

913 

914# **) MIT License 

915# 

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

917# 

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

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

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

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

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

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

924# 

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

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

927# 

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

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

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

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

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

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

934# OTHER DEALINGS IN THE SOFTWARE.