1

2# -*- coding: utf-8 -*-

3

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

5

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

7 L{trilaterate2d2} and L{trilaterate3d2}.

8'''

9

10 from pygeodesy.basics import isscalar, len2

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

12 _0_0, _1_0

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

14 _xError, _xkwds, _xkwds_popitem

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

16# from pygeodesy.fsums import fsum, fsum1_ # from .fmath

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

18 from pygeodesy.interns import MISSING, NN, _COMMA_, _concentric_, _datum_, \

19 _h_, _height_, _intersection_, _name_, _near_, \

20 _negative_, _no_, _too_, _z_

21 from pygeodesy.iters import Fmt, PointsIter

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

23 from pygeodesy.named import _xnamed, _xotherError

24 from pygeodesy.namedTuples import Intersection3Tuple, NearestOn2Tuple, \

25 NearestOn6Tuple, Vector3Tuple # Vector4Tuple

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

27 from pygeodesy.units import _fi_j2, Radius, Radius_

28 from pygeodesy.utily import atan2b, sincos2d

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

30 from pygeodesy.vector3dBase import Vector3dBase

31

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

33

34__all__ = _ALL_LAZY.vector3d

35__version__ = '23.04.10'

36

37

38 class Vector3d(Vector3dBase):

39 '''Extended 3-D vector.

40

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

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

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

44 - great circle normal to vector

45 - motion vector on earth's surface

46 - etc.

47 '''

48

49 def bearing(self, useZ=True):

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

51

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

53 consider the Y as +Z axis.

54

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

56 angle off the +Z axis.

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 _MODS.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 _MODS.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 _MODS.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 _MODS.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 _MODS.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=NN):

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 name (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=name or self.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 _MODS.vector2d._radii11ABC(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 _MODS.vector2d.soddy4(self, point2, point3, eps=eps, useZ=True)

305

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

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

308 and a radius.

309

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

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

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

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

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

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

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

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

318 3 circles (C{scalar}) or C{None} for no checking.

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

320

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

322

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

324 centers, trilateration failed some other way

325 or the trilaterated point is off one circle

326 by more than B{C{eps}}.

327

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

329

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

331

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

333 '''

334

335 def _xyr3(r, **name_v):

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

337 return v.x, v.y, r

338

339 try:

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

341 _xyr3(radius2, center2=center2) +

342 _xyr3(radius3, center3=center3)),

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

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

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

346 center2=center2, radius2=radius2,

347 center3=center3, radius3=radius3)

348

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

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

351 and a radius.

352

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

354 and C{z}).

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

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

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

358 and C{z}).

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

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

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

362 and C{z}).

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

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

365

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

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

368 spheres intersect or abut in a single point.

369

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

371 older than version 1.10.

372

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

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

375

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

377

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

379

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

381

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

383 version 1.10 or later.

384

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

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

387 275027725_An_Algebraic_Solution_to_the_Multilateration_Problem>}

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

389 288825016_Trilateration_Matlab_Code>}.

390 '''

391 try:

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

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

394 center2, radius2,

395 center3, radius3,

396 eps=eps, clas=self.classof)

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

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

399 center2=center2, radius2=radius2,

400 center3=center3, radius3=radius3)

401

402

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

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

405 # separated to allow callers to embellish any exceptions

406

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

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

409

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

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

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

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

414 dx, dy, d = s2.minus(s1).xyz

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

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

417 s, c = sincos2d(b1)

418 if s and c:

419 dx *= c / s

420 dy *= s / c

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

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

423 else: # orthogonal

424 d = euclid(dx, dy) # hypot?

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

426 return s1, e1

427

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

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

430 b1 = isscalar(end1)

431 if b1: # bearing, make an e1

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

433 else:

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

435 b2 = isscalar(end2)

436 if b2: # bearing, make an e2

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

438 else:

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

440

441 a = e1.minus(s1)

442 b = e2.minus(s2)

443 c = s2.minus(s1)

444

445 ab = a.cross(b)

446 d = fabs(c.dot(ab))

447 e = max(EPS0, eps or _0_0)

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

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

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

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

452

453 # co-planar, non-skew lines

454 ab2 = ab.length2

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

456 x = b.length2 < a.length2

457 if x: # make C{a} the shortest

458 a, b = b, a

459 s1, s2 = s2, s1

460 e1, e2 = e2, e1

461 b1, b2 = b2, b1

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

463 if c.length < e:

464 return s1, 0, 0

465 elif e2.minus(e1).length < e:

466 return e1, 0, 0

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

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

469 t = s1.minus(s2).dot(b)

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

471 if s1.minus(v).length < e:

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

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

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

475

476 cb = c.cross(b)

477 t = cb.dot(ab)

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

479 v = s1.plus(a.times(t / ab2))

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

481 return v, o1, o2

482

483

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

485 **Vector_and_kwds):

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

487 points or by a point and a bearing.

488

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

504

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

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

507

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

509

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

511 non-intersecting lines.

512

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

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

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

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

517 '''

518 try:

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

520 except (TypeError, ValueError) as x:

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

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

523 name=intersection3d3.__name__))

524 return Intersection3Tuple(v, o1, o2)

525

526

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

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

529 (3-D) center point and a radius.

530

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

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

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

534 B{C{center1}} coordinates).

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

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

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

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

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

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

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

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

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

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

545

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

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

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

549

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

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

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

553

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

555 or circles.

556

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

558

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

560

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

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

563 Intersection.

564 '''

565 try:

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

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

568 clas=center1.classof, **Vector_and_kwds)

569 except (TypeError, ValueError) as x:

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

571

572

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

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

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

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

577

578 def _nV3(x, y, z):

579 v = Vector3d(x, y, z)

580 n = intersections2.__name__

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

582

583 def _xV3(c1, u, x, y):

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

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

586 fdot(xy1, u.y, u.x, c1.y), _0_0)

587

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

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

590

591 if r1 < r2: # r1, r2 == R, r

592 c1, c2 = c2, c1

593 r1, r2 = r2, r1

594

595 m = c2.minus(c1)

596 d = m.length

597 if d < max(r2 - r1, EPS):

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

599

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

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

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

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

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

605 y = _1_0 - (x / r1)**2

606 if y > EPS:

607 y = r1 * sqrt(y) # y == a / 2

608 elif y < 0: # PYCHOK no cover

609 raise IntersectionError(_negative_)

610 else: # abutting

611 y = _0_0

612 elif o < 0: # PYCHOK no cover

613 t = d if too_d is None else too_d

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

615 else: # abutting

616 x, y = r1, _0_0

617

618 u = m.unit()

619 if sphere: # sphere center and radius

620 c = c1 if x < EPS else (

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

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

623

624 elif y > 0: # intersecting circles

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

626 else: # abutting circles

627 t = _xV3(c1, u, x, 0)

628 t = t, t

629 return t

630

631

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

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

634

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

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

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

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

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

640

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

642 C{False} otherwise.

643

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

645

646 @see: Function L{nearestOn}.

647 '''

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

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

650

651

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

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

654

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

656 or C{Vector4Tuple}).

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

658 C{Vector4Tuple}).

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

660 C{Vector4Tuple}).

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

662 points, otherwise the closest point on the extended line

663 through both points (C{bool}).

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

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

666 or C{Vector3Tuple}) or C{None}.

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

668 ignored if C{B{Vector} is None}.

669

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

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

672

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

674

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

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

677 and function L{sphericalTrigonometry.nearestOn3}.

678 '''

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

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

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

682

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=nearestOn.__name__))

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, Vector_kwds.get(_z_, p.z), name=nearestOn.__name__)

692 return p

693

694

695 def _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

716 def 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 is either on and within the extent of a polygon edge or

720 the nearest of that edge's end points.

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}.

742

743 @see: Method C{LatLon.nearestOn6} or function L{nearestOn5} for geodetic points.

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

770 def _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

779 def _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

794 def 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

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

820 '''Compute the vectorial sum of several 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 n, vectors = len2(vectors)

833 if n < 1:

834 raise VectorError(vectors=n, txt=MISSING)

835

836 v = Vector3Tuple(fsum(v.x for v in vectors),

837 fsum(v.y for v in vectors),

838 fsum(v.z for v in vectors))

839 return _xnamed((v if Vector is None else

840 Vector(*v, **Vector_kwds)), sumOf.__name__)

841

842

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

844 eps=None, **Vector_and_kwds):

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

846

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

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

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

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

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

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

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

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

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

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

857 3 circles (C{scalar}) or C{None} for no checking.

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

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

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

861

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

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

864

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

866 centers, trilateration failed some other way

867 or the trilaterated point is off one circle

868 by more than B{C{eps}}.

869

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

871

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

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

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

875 L{pygeodesy.trilaterate3d2}.

876 '''

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

878 x2, y2, radius2,

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

880

881

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

883 eps=EPS, **Vector_and_kwds):

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

885

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

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

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

889 and C{z}).

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

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

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

893 and C{z}).

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

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

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

897 and C{z}).

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

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

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

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

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

903 (sub-)class.

904

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

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

907 spheres abut/intersect in a single point.

908

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

910 older than version 1.10.

911

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

913 non-intersecting spheres.

914

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

916

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

918

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

920

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

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

923 275027725_An_Algebraic_Solution_to_the_Multilateration_Problem>},

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

925 288825016_Trilateration_Matlab_Code>} and function

926 L{pygeodesy.trilaterate2d2}.

927 '''

928 try:

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

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

931 center2, radius2, center3, radius3, eps=eps,

932 clas=center1.classof, **Vector_and_kwds)

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

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

935 center2=center2, radius2=radius2,

936 center3=center3, radius3=radius3)

937

938

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

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

941 '''

942 h = height or getattr(xyz, _height_, None) \

943 or getattr(xyz, _h_, None) \

944 or getattr(ll, _height_, None)

945

946 d = datum or getattr(xyz, _datum_, None) \

947 or getattr(ll, _datum_, None)

948

949 return h, d, getattr(xyz, _name_, NN)

950

951

952__all__ += _ALL_DOCS(intersections2, sumOf, Vector3dBase)

953

954# **) MIT License

955#

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

957#

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

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

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

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

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

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

964#

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

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

967#

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

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

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

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

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

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

974# OTHER DEALINGS IN THE SOFTWARE.