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.constants import EPS, EPS0, EPS1, EPS4, INT0, isnear0, \

11 _0_0, _1_0

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

13 _xattr, _xError, _xkwds, _xkwds_get, _xkwds_item2

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

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

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

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

18 _near_, _negative_, _no_, _too_

19 from pygeodesy.iters import PointsIter, Fmt

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

21 from pygeodesy.named import _xnamed, _xotherError

22 from pygeodesy.namedTuples import Intersection3Tuple, NearestOn2Tuple, \

23 NearestOn6Tuple, Vector3Tuple # Vector4Tuple

24# from pygeodesy.nvectorBase import _nsumOf # _MODS

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

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

27 from pygeodesy.utily import atan2b, sincos2d

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

29 from pygeodesy.vector3dBase import Vector3dBase

30

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

32

33__all__ = _ALL_LAZY.vector3d

34__version__ = '24.02.20'

35

36

37 class Vector3d(Vector3dBase):

38 '''Extended 3-D vector.

39

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

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

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

43 - great circle normal to the vector

44 - motion vector on the earth's surface

45 - etc.

46 '''

47

48 def bearing(self, useZ=True):

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

50

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

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

53

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

55 '''

56 x, y = self.x, self.y

57 if useZ:

58 x, y = hypot(x, y), self.z

59 return atan2b(x, y)

60

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

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

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

64

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

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

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

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

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

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

71

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

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

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

75 two given points.

76

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

78 than version 1.10.

79

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

81 a trilateration or C{numpy} issue.

82

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

84

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

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

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

88 '''

89 try:

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

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

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

93

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

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

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

97

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

99 or C{Vector4Tuple}).

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

101 or C{Vector4Tuple}).

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

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

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

105

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

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

108 given points.

109

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

111 version 1.10.

112

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

114 or a trilateration or C{numpy} issue.

115

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

117

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

119 '''

120 try:

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

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

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

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

125

126 def circum4_(self, *points):

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

128

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

130 or C{Vector4Tuple}).

131

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

133 an instance if this (sub-)class.

134

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

136 older than version 1.10.

137

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

139

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

141

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

143

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

145 '''

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

147

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

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

150

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

152 or C{Vector4Tuple}).

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

154 or C{Vector4Tuple}).

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

156 C{y}, and C{z}.

157

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

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

160

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

162

163 @see: Method L{nearestOn}.

164 '''

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

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

167

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

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

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

171

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

173 or C{Vector4Tuple}).

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

175 or C{Vector4Tuple}).

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

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

178

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

180 iff C{B{circum}=True}.

181

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

183

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

185

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

187 '''

188 try:

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

190 except (TypeError, ValueError) as x:

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

192

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

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

195

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

197 C{Vector4Tuple}).

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

199 C{Vector4Tuple}).

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

201 points, otherwise the closest point on the extended

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

203

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

205 of this (sub-)class.

206

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

208

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

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

211 '''

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

213

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

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

216

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

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

219

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

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

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

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

224

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

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

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

228

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

230

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

232

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

234

235 @see: Function L{nearestOn6}.

236 '''

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

238

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

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

241

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

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

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

245

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

247

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

249 '''

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

251

252 def radii11(self, point2, point3):

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

254 circles of a (3-D) triangle.

255

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

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

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

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

260

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

262

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

264

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

266

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

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

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

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

271 '''

272 try:

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

274 except (TypeError, ValueError) as x:

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

276

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

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

279 (3-D) triangle.

280

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

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

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

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

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

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

287

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

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

290 three given points.

291

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

293 than version 1.10.

294

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

296 a trilateration or C{numpy} issue.

297

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

299

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

301 '''

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

303

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

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

306 and a radius.

307

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

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

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

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

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

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

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

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

316 3 circles (C{scalar}) or C{None} for no checking.

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

318

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

320

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

322 centers, trilateration failed some other way

323 or the trilaterated point is off one circle

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

325

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

327

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

329

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

331 '''

332

333 def _xyr3(r, **name_v):

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

335 return v.x, v.y, r

336

337 try:

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

339 _xyr3(radius2, center2=center2) +

340 _xyr3(radius3, center3=center3)),

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

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

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

344 center2=center2, radius2=radius2,

345 center3=center3, radius3=radius3)

346

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

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

349 and a radius.

350

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

352 and C{z}).

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

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

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

356 and C{z}).

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

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

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

360 and C{z}).

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

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

363

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

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

366 spheres intersect or abut in a single point.

367

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

369 older than version 1.10.

370

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

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

373

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

375

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

377

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

379

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

381 version 1.10 or later.

382

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

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

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

386 '''

387 try:

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

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

390 center2, radius2,

391 center3, radius3,

392 eps=eps, clas=self.classof)

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

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

395 center2=center2, radius2=radius2,

396 center3=center3, radius3=radius3)

397

398

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

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

401 # separated to allow callers to embellish any exceptions

402

403 def _corners2(s1, b1, s2, useZ):

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

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

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

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

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

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

410 s, c = sincos2d(b1)

411 if s and c:

412 dx *= c / s

413 dy *= s / c

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

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

416 else: # orthogonal

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

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

419 return s1, e1

420

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

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

423

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

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

426 b1 = _isDegrees(end1)

427 if b1: # bearing, make an e1

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

429 else:

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

431 b2 = _isDegrees(end2)

432 if b2: # bearing, make an e2

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

434 else:

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

436

437 a = e1.minus(s1)

438 b = e2.minus(s2)

439 c = s2.minus(s1)

440

441 ab = a.cross(b)

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

443 e = max(EPS0, eps or _0_0)

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

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

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

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

448

449 # co-planar, non-skew lines

450 ab2 = ab.length2

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

452 x = a.length2 > b.length2

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

454 a, b = b, a

455 s1, s2 = s2, s1

456 e1, e2 = e2, e1

457 b1, b2 = b2, b1

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

459 if c.length < e:

460 return s1, 0, 0

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

462 return e1, 0, 0

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

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

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

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

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

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

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

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

471

472 cb = c.cross(b)

473 t = cb.dot(ab)

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

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

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

477 return v, o1, o2

478

479

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

481 **Vector_and_kwds):

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

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

484

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

500

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

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

503

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

505

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

507 non-intersecting lines.

508

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

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

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

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

513 '''

514 try:

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

516 except (TypeError, ValueError) as x:

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

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

519 name=intersection3d3.__name__))

520 return Intersection3Tuple(v, o1, o2)

521

522

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

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

525 center point and a radius.

526

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

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

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

530 B{C{center1}} coordinates).

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

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

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

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

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

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

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

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

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

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

541

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

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

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

545

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

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

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

549

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

551 or circles.

552

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

554

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

556

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

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

559 Intersection.

560 '''

561 try:

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

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

564 clas=center1.classof, **Vector_and_kwds)

565 except (TypeError, ValueError) as x:

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

567

568

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

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

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

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

573

574 def _nV3(x, y, z):

575 v = Vector3d(x, y, z)

576 n = intersections2.__name__

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

578

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

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

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

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

583

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

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

586

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

588 c1, c2 = c2, c1

589 r1, r2 = r2, r1

590

591 m = c2.minus(c1)

592 d = m.length

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

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

595

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

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

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

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

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

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

602 if y > EPS:

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

604 elif y < 0: # PYCHOK no cover

605 raise IntersectionError(_negative_)

606 else: # abutting

607 y = _0_0

608 elif o < 0: # PYCHOK no cover

609 if too_d is not None:

610 d = too_d

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

612 else: # abutting

613 x, y = r1, _0_0

614

615 u = m.unit()

616 if sphere: # sphere center and radius

617 c = c1 if x < EPS else (

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

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

620

621 elif y > 0: # intersecting circles

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

623 else: # abutting circles

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

625 t = t, t

626 return t

627

628

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

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

631

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

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

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

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

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

637

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

639 C{False} otherwise.

640

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

642

643 @see: Function L{nearestOn}.

644 '''

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

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

647

648

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

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

651

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

653 or C{Vector4Tuple}).

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

655 C{Vector4Tuple}).

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

657 C{Vector4Tuple}).

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

659 points, otherwise the closest point on the extended line

660 through both points (C{bool}).

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

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

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

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

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

666

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

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

669

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

671

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

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

674 and function L{sphericalTrigonometry.nearestOn3}.

675 '''

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

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

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

679

680 n = nearestOn.__name__

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

682 if Vector is not None:

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

684 elif p is p1:

685 p = point1

686 elif p is p2:

687 p = point2

688 else: # ignore Vector_kwds

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

690 return p

691

692

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

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

695 # separated to allow callers to embellish any exceptions

696 p21 = p2.minus(p1)

697 d2 = p21.length2

698 if d2 < eps: # coincident

699 p = p1 # ~= p2

700 t = 0

701 else: # see comments in .points.nearestOn5

702 t = p0.minus(p1).dot(p21) / d2

703 if within and t < eps:

704 p = p1

705 t = 0

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

707 p = p2

708 t = 1

709 else:

710 p = p1.plus(p21.times(t))

711 return NearestOn2Tuple(p, t)

712

713

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

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

716

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

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

719

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

721 C{Vector4Tuple}).

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

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

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

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

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

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

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

729

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

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

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

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

734

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

736

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

738

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

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

741 C{LatLon.nearestOn6} methods.

742 '''

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

744 D2 = r.equirectangular # distance squared

745

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

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

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

749

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

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

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

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

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

755 if d2 < c2:

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

757 p1, i = p2, j

758

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

760

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

762 v = _nVc(c, **kwds)

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

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

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

766

767

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

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

770 if Vector is not None:

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

772 elif clas is not None:

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

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

775

776

777 def _otherV3d(useZ=True, NN_OK=True, i=None, **name_v):

778 # check named vector instance, return Vector3d

779 n, v = _xkwds_item2(name_v)

780 if useZ and isinstance(v, Vector3dBase):

781 return v if NN_OK or v.name else v.copy(name=Fmt.INDEX(n, i))

782

783 n = Fmt.INDEX(n, i)

784 try:

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

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

787 pass

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

789

790

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

792 '''Parse an C{"x, y, z"} string.

793

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

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

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

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

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

799

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

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

802

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

804 '''

805 try:

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

807 n = len(v)

808 if n != 3:

809 raise _ValueError(len=n)

810 except (TypeError, ValueError) as x:

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

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

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

814

815

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

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

818

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

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

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

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

823

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

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

826

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

828 '''

829 try:

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

831 except (TypeError, ValueError) as x:

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

833 x, y, z = t[:3]

834 n = sumOf.__name__

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

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

837

838

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

840 eps=None, **Vector_and_kwds):

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

842

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

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

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

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

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

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

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

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

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

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

853 3 circles (C{scalar}) or C{None} for no checking.

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

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

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

857

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

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

860

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

862 centers, trilateration failed some other way

863 or the trilaterated point is off one circle

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

865

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

867

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

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

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

871 L{pygeodesy.trilaterate3d2}.

872 '''

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

874 x2, y2, radius2,

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

876

877

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

879 eps=EPS, **Vector_and_kwds):

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

881

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

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

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

885 and C{z}).

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

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

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

889 and C{z}).

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

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

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

893 and C{z}).

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

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

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

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

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

899 (sub-)class.

900

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

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

903 spheres abut/intersect in a single point.

904

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

906 older than version 1.10.

907

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

909 non-intersecting spheres.

910

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

912

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

914

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

916

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

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

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

920 288825016>} and function L{pygeodesy.trilaterate2d2}.

921 '''

922 try:

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

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

925 center2, radius2, center3, radius3, eps=eps,

926 clas=center1.classof, **Vector_and_kwds)

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

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

929 center2=center2, radius2=radius2,

930 center3=center3, radius3=radius3)

931

932

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

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

935 '''

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

937 or _xattr(xyz, h=None) \

938 or _xattr(ll, height=None)

939

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

941 or _xattr(ll, datum=None)

942

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

944

945

946__all__ += _ALL_DOCS(intersections2, sumOf, Vector3dBase)

947

948# **) MIT License

949#

950# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved.

951#

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

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

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

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

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

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

958#

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

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

961#

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

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

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

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

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

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

968# OTHER DEALINGS IN THE SOFTWARE.