1

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

3

4 u'''2- or 3-D vectorial functions L{circin6}, L{circum3}, L{circum4_},

5 L{iscolinearWith}, L{meeus2}, L{nearestOn}, L{radii11}, L{soddy4} and

6 L{trilaterate2d2}.

7'''

8

9 from pygeodesy.basics import len2, map2, _xnumpy

10 from pygeodesy.constants import EPS, EPS0, EPS02, EPS4, INF, INT0, \

11 _EPS4e8, isnear0, _0_0, _0_25, _0_5, _N_0_5, \

12 _1_0, _1_0_1T, _N_1_0, _2_0, _N_2_0, _4_0

13 from pygeodesy.errors import _and, _AssertionError, IntersectionError, NumPyError, \

14 PointsError, TriangleError, _xError, _xkwds

15 from pygeodesy.fmath import fabs, fdot, fdot_, hypot, hypot2_, sqrt

16 from pygeodesy.fsums import _Fsumf_, fsumf_, fsum1f_

17 from pygeodesy.interns import NN, _a_, _and_, _b_, _c_, _center_, _coincident_, \

18 _colinear_, _COMMASPACE_, _concentric_, _few_, \

19 _intersection_, _invalid_, _near_, _no_, _of_, \

20 _radius_, _rIn_, _s_, _SPACE_, _too_, _with_

21# from pygeodesy.lazily import _ALL_LAZY # from .named

22 from pygeodesy.named import _ALL_LAZY, _NamedTuple, _Pass, Property_RO

23 from pygeodesy.namedTuples import LatLon3Tuple, Vector2Tuple

24# from pygeodesy.props import Property_RO # from .named

25 from pygeodesy.streprs import Fmt, unstr

26 from pygeodesy.units import Float, Int, Meter, Radius, Radius_

27 from pygeodesy.vector3d import iscolinearWith, nearestOn, _nearestOn2, _nVc, \

28 _otherV3d, trilaterate3d2, Vector3d # PYCHOK unused

29

30 from contextlib import contextmanager

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

32

33__all__ = _ALL_LAZY.vector2d

34__version__ = '24.11.21'

35

36_cA_ = 'cA'

37_cB_ = 'cB'

38_cC_ = 'cC'

39_deltas_ = 'deltas'

40_outer_ = 'outer'

41_raise_ = 'raise' # PYCHOK used!

42_rank_ = 'rank'

43_residuals_ = 'residuals'

44_Type_ = 'Type'

45

46

47 class Circin6Tuple(_NamedTuple):

48 '''6-Tuple C{(radius, center, deltas, cA, cB, cC)} with the C{radius}, the

49 trilaterated C{center} and contact points of the I{inscribed} aka I{In-

50 circle} of a triangle. The C{center} is I{un}ambiguous if C{deltas} is

51 C{None}, otherwise C{center} is the mean and C{deltas} the differences of

52 the L{pygeodesy.trilaterate3d2} results. Contact points C{cA}, C{cB} and

53 C{cC} are the points of tangency, aka the corners of the U{Contact Triangle

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

55 '''

56 _Names_ = (_radius_, _center_, _deltas_, _cA_, _cB_, _cC_)

57 _Units_ = ( Radius, _Pass, _Pass, _Pass, _Pass, _Pass)

58

59

60 class Circum3Tuple(_NamedTuple): # in .latlonBase

61 '''3-Tuple C{(radius, center, deltas)} with the C{circumradius} and trilaterated

62 C{circumcenter} of the C{circumcircle} through 3 points (aka {Meeus}' Type II

63 circle) or the C{radius} and C{center} of the smallest I{Meeus}' Type I circle.

64 The C{center} is I{un}ambiguous if C{deltas} is C{None}, otherwise C{center}

65 is the mean and C{deltas} the differences of the L{pygeodesy.trilaterate3d2}

66 results.

67 '''

68 _Names_ = (_radius_, _center_, _deltas_)

69 _Units_ = ( Radius, _Pass, _Pass)

70

71

72 class Circum4Tuple(_NamedTuple):

73 '''4-Tuple C{(radius, center, rank, residuals)} with C{radius} and C{center}

74 of a sphere I{least-squares} fitted through given points and the C{rank}

75 and C{residuals} -if any- from U{numpy.linalg.lstsq

76 <https://NumPy.org/doc/stable/reference/generated/numpy.linalg.lstsq.html>}.

77 '''

78 _Names_ = (_radius_, _center_, _rank_, _residuals_)

79 _Units_ = ( Radius, _Pass, Int, _Pass)

80

81

82 class Meeus2Tuple(_NamedTuple):

83 '''2-Tuple C{(radius, Type)} with C{radius} and I{Meeus}' C{Type} of the smallest

84 circle I{containing} 3 points. C{Type} is C{None} for a I{Meeus}' Type II

85 C{circumcircle} passing through all 3 points. Otherwise C{Type} is the center

86 of a I{Meeus}' Type I circle with 2 points on (a diameter of) and 1 point

87 inside the circle.

88 '''

89 _Names_ = (_radius_, _Type_)

90 _Units_ = ( Radius, _Pass)

91

92

93 class Radii11Tuple(_NamedTuple):

94 '''11-Tuple C{(rA, rB, rC, cR, rIn, riS, roS, a, b, c, s)} with the C{Tangent}

95 circle radii C{rA}, C{rB} and C{rC}, the C{circumradius} C{cR}, the C{Incircle}

96 radius C{rIn} aka C{inradius}, the inner and outer I{Soddy} circle radii C{riS}

97 and C{roS}, the sides C{a}, C{b} and C{c} and semi-perimeter C{s} of a triangle,

98 all in C{meter} conventionally.

99

100 @note: C{Circumradius} C{cR} and outer I{Soddy} radius C{roS} may be C{INF}.

101 '''

102 _Names_ = ('rA', 'rB', 'rC', 'cR', _rIn_, 'riS', 'roS', _a_, _b_, _c_, _s_)

103 _Units_ = ( Meter,) * len(_Names_)

104

105

106 class Soddy4Tuple(_NamedTuple):

107 '''4-Tuple C{(radius, center, deltas, outer)} with C{radius} and (trilaterated)

108 C{center} of the I{inner} I{Soddy} circle and the radius of the C{outer}

109 I{Soddy} circle. The C{center} is I{un}ambiguous if C{deltas} is C{None},

110 otherwise C{center} is the mean and C{deltas} the differences of the

111 L{pygeodesy.trilaterate3d2} results.

112

113 @note: The outer I{Soddy} radius C{outer} may be C{INF}.

114 '''

115 _Names_ = (_radius_, _center_, _deltas_, _outer_)

116 _Units_ = ( Radius, _Pass, _Pass, Radius)

117

118

119 class Triaxum5Tuple(_NamedTuple):

120 '''5-Tuple C{(a, b, c, rank, residuals)} with the (unordered) triaxial radii

121 C{a}, C{b} and C{c} of an ellipsoid I{least-squares} fitted through given

122 points and the C{rank} and C{residuals} -if any- from U{numpy.linalg.lstsq

123 <https://NumPy.org/doc/stable/reference/generated/numpy.linalg.lstsq.html>}.

124 '''

125 _Names_ = (_a_, _b_, _c_, _rank_, _residuals_)

126 _Units_ = ( Radius, Radius, Radius, Int, _Pass)

127

128

129 def circin6(point1, point2, point3, eps=EPS4, useZ=True):

130 '''Return the radius and center of the I{inscribed} aka I{Incircle} of

131 a (2- or 3-D) triangle.

132

133 @arg point1: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},

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

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

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

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

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

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

140 C{B{useZ} is True} else L{pygeodesy.trilaterate2d2}.

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

142

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

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

145 instance of B{C{point1}}'s (sub-)class, are co-planar with

146 the three given points.

147

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

149 than version 1.10 and C{B{useZ} is True}.

150

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

152 a trilateration or C{numpy} issue.

153

154 @raise TypeError: Invalid B{C{point1}}, B{C{point2}} or B{C{point3}}.

155

156 @see: Functions L{radii11} and L{circum3}, U{Contact Triangle

157 <https://MathWorld.Wolfram.com/ContactTriangle.html>} and

158 U{Incircle<https://MathWorld.Wolfram.com/Incircle.html>}.

159 '''

160 try:

161 return _circin6(point1, point2, point3, eps=eps, useZ=useZ)

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

163 raise _xError(x, point1=point1, point2=point2, point3=point3)

164

165

166 def _circin6(point1, point2, point3, eps=EPS4, useZ=True, dLL3=False, **Vector_kwds):

167 # (INTERNAL) Radius, center, deltas, 3 contact points

168

169 def _fraction(r, a):

170 return (r / a) if a > EPS0 else _0_5

171

172 def _contact2(a, p2, r2, p3, r3, V, V_kwds):

173 c = p2.intermediateTo(p3, _fraction(r2, a)) if r2 > r3 else \

174 p3.intermediateTo(p2, _fraction(r3, a))

175 C = V(c.x, c.y, c.z, **V_kwds)

176 return c, C

177

178 t, p1, p2, p3 = _radii11ABC4(point1, point2, point3, useZ=useZ)

179 V, r1, r2, r3 = point1.classof, t.rA, t.rB, t.rC

180

181 c1, cA = _contact2(t.a, p2, r2, p3, r3, V, _xkwds(Vector_kwds, name=_cA_))

182 c2, cB = _contact2(t.b, p3, r3, p1, r1, V, _xkwds(Vector_kwds, name=_cB_))

183 c3, cC = _contact2(t.c, p1, r1, p2, r2, V, _xkwds(Vector_kwds, name=_cC_))

184

185 r = t.rIn

186 c, d = _tricenter3d2(c1, r, c2, r, c3, r, eps=eps, useZ=useZ, dLL3=dLL3,

187 **_xkwds(Vector_kwds, Vector=V, name=circin6.__name__))

188 return Circin6Tuple(r, c, d, cA, cB, cC)

189

190

191 def circum3(point1, point2, point3, circum=True, eps=EPS4, useZ=True):

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

193 I{containing} three (2- or 3-D) points.

194

195 @arg point1: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or

196 C{Vector4Tuple}).

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

198 C{Vector4Tuple}).

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

200 C{Vector4Tuple}).

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

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

203 @kwarg eps: Tolerance for function L{pygeodesy.trilaterate3d2} if C{B{useZ}

204 is True} else L{pygeodesy.trilaterate2d2}.

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

206

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

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

209 given points.

210

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

212 than version 1.10 and C{B{useZ} is True}.

213

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

215 a trilateration or C{numpy} issue.

216

217 @raise TypeError: Invalid B{C{point1}}, B{C{point2}} or B{C{point3}}.

218

219 @see: Functions L{pygeodesy.circum4_} and L{pygeodesy.meeus2} and Meeus, J.

220 U{I{Astronomical Algorithms}<http://www.Agopax.IT/Libri_astronomia/pdf/

221 Astronomical%20Algorithms.pdf>}, 2nd Ed. 1998, page 127ff, U{circumradius

222 <https://MathWorld.Wolfram.com/Circumradius.html>} and U{circumcircle

223 <https://MathWorld.Wolfram.com/Circumcircle.html>}.

224 '''

225 try:

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

227 return _circum3(p1, point2, point3, circum=circum, eps=eps, useZ=useZ,

228 clas=point1.classof)

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

230 raise _xError(x, point1=point1, point2=point2, point3=point3, circum=circum)

231

232

233 def _circum3(p1, point2, point3, circum=True, eps=EPS4, useZ=True, dLL3=False,

234 clas=Vector3d, **clas_kwds): # in .latlonBase

235 # (INTERNAL) Radius, center, deltas

236 r, d, p2, p3 = _meeus4(p1, point2, point3, circum=circum, useZ=useZ,

237 clas=clas, **clas_kwds)

238 if d is None: # Meeus' Type II or circum=True

239 kwds = _xkwds(clas_kwds, eps=eps, Vector=clas, name=circum3.__name__)

240 c, d = _tricenter3d2(p1, r, p2, r, p3, r, useZ=useZ, dLL3=dLL3, **kwds)

241 else: # Meeus' Type I

242 c, d = d, None

243 return Circum3Tuple(r, c, d)

244

245

246 def circum4(points, useZ=True, **Vector_and_kwds):

247 '''Best-fit a sphere through three or more (3-D) points.

248

249 @arg points: Iterable of points (each a C{Cartesian}, L{Vector3d}, C{Vector3Tuple}

250 or C{Vector4Tuple}).

251 @kwarg useZ: If C{True}, use the points' Z component, otherwise force C{z=INT0}

252 (C{bool}).

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

254 and optionally, additional B{C{Vector}} keyword arguments, otherwise

255 the B{C{points}}' (sub-)class.

256

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

258 instance of C{B{points}[0]}' (sub-)class or B{C{Vector}} if specified.

259

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

261 version 1.10.

262

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

264

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

266

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

268

269 @see: Functions L{pygeodesy.circum3} and L{pygeodesy.meeus2}, I{Charles Jekel}'s

270 U{"Least Squares Sphere Fit"<https://Jekel.me/2015/Least-Squares-Sphere-Fit/>},

271 U{Appendix A<https://hdl.handle.net/10019.1/98627>}, U{numpy.linalg.lstsq

272 <https://NumPy.org/doc/stable/reference/generated/numpy.linalg.lstsq.html>} and U{Eberly

273 6<https://www.sci.Utah.EDU/~balling/FEtools/doc_files/LeastSquaresFitting.pdf>}.

274 '''

275 n, ps = len2(points)

276 if n < 3:

277 raise PointsError(points=n, txt=_too_(_few_))

278

279 A, b = [], []

280 for i, p in enumerate(ps):

281 v = _otherV3d(useZ=useZ, i=i, points=p)

282 A.append(v.times(_2_0).xyz3 + _1_0_1T)

283 b.append(v.length2)

284

285 with _numpy(circum4, n=n) as _np:

286 A = _np.array(A).reshape((n, 4))

287 b = _np.array(b).reshape((n, 1))

288 C, R, rk = _np.least_squares3(A, b)

289

290 c = Vector3d(*C[:3], name__=circum4) # .__name__

291 r = Radius(sqrt(fsumf_(C[3], *c.x2y2z2)), name=c.name)

292

293 c = _nVc(c, **_xkwds(Vector_and_kwds, clas=ps[0].classof, name=c.name))

294 return Circum4Tuple(r, c, rk, R)

295

296

297 def circum4_(*points, **useZ_Vector_and_kwds):

298 '''Best-fit a sphere through three or more (3-D) positional points.

299

300 @arg points: The points (each a C{Cartesian}, L{Vector3d}, C{Vector3Tuple}

301 or C{Vector4Tuple}), all positional.

302 @kwarg useZ_Vector_and_kwds: Keyword arguments C{B{useZ}=True} and

303 C{B{Vector}=None}, see function L{circum4}.

304

305 @see: Function L{circum4} for further details.

306 '''

307 return circum4(points, **useZ_Vector_and_kwds)

308

309

310 def _iscolinearWith(p, point1, point2, eps=EPS, useZ=True):

311 # (INTERNAL) Check colinear, see L{iscolinearWith} above,

312 # separated to allow callers to embellish any exceptions

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

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

315 n, _ = _nearestOn2(p, p1, p2, within=False, eps=eps)

316 return n is p1 or n.minus(p).length2 < eps

317

318

319 def meeus2(point1, point2, point3, circum=False, useZ=True):

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

321 or I{containing} three (2- or 3-D) points.

322

323 @arg point1: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},

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

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

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

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

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

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

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

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

332

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

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

335

336 @raise IntersectionError: Near-coincident or -colinear points, iff C{B{circum}=True}.

337

338 @raise TypeError: Invalid B{C{point1}}, B{C{point2}} or B{C{point3}}.

339

340 @see: Functions L{pygeodesy.circum3} and L{pygeodesy.circum4_} and Meeus, J.

341 U{I{Astronomical Algorithms}<http://www.Agopax.IT/Libri_astronomia/pdf/

342 Astronomical%20Algorithms.pdf>}, 2nd Ed. 1998, page 127ff, U{circumradius

343 <https://MathWorld.Wolfram.com/Circumradius.html>} and U{circumcircle

344 <https://MathWorld.Wolfram.com/Circumcircle.html>}.

345 '''

346 try:

347 A = _otherV3d(useZ=useZ, point1=point1)

348 return _meeus2(A, point2, point3, circum, useZ=useZ, clas=point1.classof)

349 except (TypeError, ValueError) as x:

350 raise _xError(x, point1=point1, point2=point2, point3=point3, circum=circum)

351

352

353 def _meeus2(A, point2, point3, circum, useZ=True, **clas_and_kwds): # in .vector3d

354 # (INTERNAL) Radius and center or Meeus' Type

355 f = _circum3 if circum else _meeus4

356 t = f(A, point2, point3, circum=circum, useZ=useZ, **clas_and_kwds)[:2]

357 return Meeus2Tuple(t)

358

359

360 def _meeus4(A, point2, point3, circum=False, useZ=True, clas=None, **clas_kwds):

361 # (INTERNAL) Radius and Meeus' Type

362 B = p2 = _otherV3d(useZ=useZ, point2=point2)

363 C = p3 = _otherV3d(useZ=useZ, point3=point3)

364

365 a = B.minus(C).length2

366 b = C.minus(A).length2

367 c = A.minus(B).length2

368 if a < b:

369 a, b, A, B = b, a, B, A

370 if a < c:

371 a, c, A, C = c, a, C, A

372

373 if a > EPS02 and (circum or a < (b + c)): # circumradius

374 b = sqrt(b / a)

375 c = sqrt(c / a)

376 R = _Fsumf_(_1_0, b, c) * _Fsumf_(_1_0, b, -c) * \

377 _Fsumf_(_1_0, -b, c) * _Fsumf_(_N_1_0, b, c)

378 r = R.fover(a)

379 if r < EPS02:

380 t = _coincident_ if b < EPS0 or c < EPS0 else (

381 _colinear_ if _iscolinearWith(A, B, C) else _invalid_)

382 raise IntersectionError(t)

383 r = b * c / sqrt(r)

384 t = None # Meeus' Type II

385 else: # obtuse or right angle at A

386 r = sqrt(a * _0_25) if a > EPS02 else INT0

387 t = B.plus(C).times(_0_5) # Meeus' Type I

388 if clas is not None:

389 t = clas(t.x, t.y, t.z, **_xkwds(clas_kwds, name=meeus2.__name__))

390 return r, t, p2, p3

391

392

393 class _numpy(object): # see also .formy._idllmn6, .geodesicw._wargs, .latlonBase._toCartesian3

394 '''(INTERNAL) Partial C{NumPy} wrapper.

395 '''

396 @contextmanager # <https://www.Python.org/dev/peps/pep-0343/> Examples

397 def __call__(self, where, *args, **kwds):

398 '''(INTERNAL) Yield self with any errors raised as L{NumPyError}

399 embellished with all B{C{args}} and B{C{kwds}}.

400 '''

401 np = self.np

402 try: # <https://NumPy.org/doc/stable/reference/generated/numpy.seterr.html>

403 e = np.seterr(all=_raise_) # throw FloatingPointError for numpy errors

404 yield self

405 except Exception as x: # mostly FloatingPointError?

406 t = unstr(where, *args, **kwds)

407 raise NumPyError(t, cause=x) # _xError2?

408 finally: # restore numpy error handling

409 np.seterr(**e)

410

411 @Property_RO

412 def array(self):

413 return self.np.array

414

415 def least_squares3(self, A, b):

416 '''Linear least-squares function.

417 '''

418 C, R, rk, _ = self.np.linalg.lstsq(A, b, rcond=None) # to silence warning

419 C = map2(float, C)

420 R = map2(float, R) # empty if rk < 4 or n <= 4

421 return C, R, int(rk)

422

423 @Property_RO

424 def np(self):

425 '''Import numpy 1.10+ once.

426 '''

427 return _xnumpy(self.__class__, 1, 10)

428

429 def null_space2(self, A, rcond=None):

430 '''Return the C{null_space} and C{rank} of matrix B{C{A}}.

431

432 @see: U{Source<https://docs.SciPy.org/doc/scipy/reference/generated/scipy.linalg.null_space.html>}

433 U{SciPY Cookbook<https://SciPy-Cookbook.ReadTheDocs.io/items/RankNullspace.html>}, U{here

434 <https://NumPy.org/doc/stable/reference/generated/numpy.linalg.svd.html>}, U{here

435 <https://StackOverflow.com/questions/19820921>}, U{here

436 <https://StackOverflow.com/questions/2992947>} and U{here

437 <https://StackOverflow.com/questions/5889142>}.

438 '''

439 def _Error(**kwds):

440 return _AssertionError(txt__=self.null_space2, **kwds)

441

442 np = self.np

443 A = np.array(A)

444 m = max(A.shape)

445 if m != 4: # for this usage

446 raise _Error(shape=m)

447 # if needed, square A, pad with zeros

448 A = np.resize(A, m * m).reshape(m, m)

449# try: # no np.linalg.null_space <https://docs.SciPy.org/doc/>

450# Z = scipy.linalg.null_space(A) # XXX no scipy.linalg?

451# return Z, ...

452# except AttributeError:

453# pass

454 U, S, V = np.linalg.svd(A)

455 s = max(EPS, rcond) if rcond else (EPS * max(U.shape[0], V.shape[1]))

456 t = max(EPS, float(np.max(S) * s)) # abs_tol, rel_tol * largest singular

457 r = int(np.sum(S > t)) # rank

458 if r == 3: # get null_space

459 Z = np.transpose(V[r:])

460 s = map2(int, Z.shape)

461 if s != (m, 1): # bad null_space shape

462 raise _Error(shape=s, m=m)

463 D = A.dot(Z) # near-zeros-vector

464 n = float(np.linalg.norm(D, INF)) # INF = max(fabs(D)), 2 = hypot_(*D)

465 if n > t: # largest exceed tol

466 raise _Error(dot=tuple(D.ravel()), norm=n, tol=t)

467 else: # coincident, colinear, concentric centers, ambiguous, etc.

468 Z = None

469 # del A, S, U, V # release numpy

470 return Z, r

471

472 @Property_RO

473 def pseudo_inverse(self):

474 '''Moore-Penrose pseudo-inverse function.

475 '''

476 return self.np.linalg.pinv

477

478 def real_roots(self, *coeffs):

479 '''Compute the real, non-complex roots of a polynomial.

480 '''

481 np = self.np

482 rs = np.polynomial.polynomial.polyroots(coeffs)

483 return tuple(float(r) for r in rs if not np.iscomplex(r))

484

485_numpy = _numpy() # PYCHOK singleton

486

487

488 def radii11(point1, point2, point3, useZ=True):

489 '''Return the radii of the C{In-}, I{Soddy} and C{Tangent} circles of a

490 (2- or 3-D) triangle.

491

492 @arg point1: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},

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

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

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

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

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

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

499

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

501

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

503

504 @raise TypeError: Invalid B{C{point1}}, B{C{point2}} or B{C{point3}}.

505

506 @see: U{Circumradius<https://MathWorld.Wolfram.com/Circumradius.html>},

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

508 Circles<https://MathWorld.Wolfram.com/SoddyCircles.html>} and

509 U{Tangent Circles<https://MathWorld.Wolfram.com/TangentCircles.html>}.

510 '''

511 try:

512 return _radii11ABC4(point1, point2, point3, useZ=useZ)[0]

513 except (TypeError, ValueError) as x:

514 raise _xError(x, point1=point1, point2=point2, point3=point3)

515

516

517 def _radii11ABC4(point1, point2, point3, useZ=True):

518 # (INTERNAL) Tangent, Circum, Incircle, Soddy radii, sides and semi-perimeter

519 A = _otherV3d(useZ=useZ, point1=point1, NN_OK=False)

520 B = _otherV3d(useZ=useZ, point2=point2, NN_OK=False)

521 C = _otherV3d(useZ=useZ, point3=point3, NN_OK=False)

522

523 a = B.minus(C).length

524 b = C.minus(A).length

525 c = A.minus(B).length

526

527 S = _Fsumf_(a, b, c) * _0_5

528 s = float(S) # semi-perimeter

529 if s > EPS0:

530 rs = float(S - a), float(S - b), float(S - c)

531 r3, r2, r1 = sorted(rs) # r3 <= r2 <= r1

532 if r3 > EPS0: # and r2 > EPS0 and r1 > EPS0

533 r3_r1 = r3 / r1

534 r3_r2 = r3 / r2

535 # t = r1 * r2 * r3 * (r1 + r2 + r3)

536 # = r1**2 * r2 * r3 * (1 + r2 / r1 + r3 / r1)

537 # = (r1 * r2)**2 * (r3 / r2) * (1 + r2 / r1 + r3 / r1)

538 t = r3_r2 * fsum1f_(_1_0, r2 / r1, r3_r1) # * (r1 * r2)**2

539 if t > EPS02:

540 t = sqrt(t) * _2_0 # * r1 * r2

541 # d = r1 * r2 + r2 * r3 + r3 * r1

542 # = r1 * (r2 + r2 * r3 / r1 + r3)

543 # = r1 * r2 * (1 + r3 / r1 + r3 / r2)

544 d = fsum1f_(_1_0, r3_r1, r3_r2) # * r1 * r2

545 # si/o = r1 * r2 * r3 / (r1 * r2 * (d +/- t))

546 # = r3 / (d +/- t)

547 si = r3 / (d + t)

548 so = (r3 / (d - t)) if d > t else INF

549 # ci = sqrt(r1 * r2 * r3 / s)

550 # = r1 * sqrt(r2 * r3 / r1 / s)

551 ci = r1 * sqrt(r2 * r3_r1 / s)

552 # co = a * b * c / (4 * ci * s)

553 t = ci * s * _4_0

554 co = (a * b * c / t) if t > EPS0 else INF

555 r1, r2, r3 = rs # original order

556 t = Radii11Tuple(r1, r2, r3, co, ci, si, so, a, b, c, s)

557 return t, A, B, C

558

559 raise TriangleError(_near_(_coincident_) if min(a, b, c) < EPS0 else (

560 _colinear_ if _iscolinearWith(A, B, C) else _invalid_))

561

562

563 def soddy4(point1, point2, point3, eps=EPS4, useZ=True):

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

565 (2- or 3-D) triangle.

566

567 @arg point1: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},

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

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

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

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

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

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

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

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

576

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

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

579 three given points. The C{outer} I{Soddy} radius may be C{INF}.

580

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

582 than version 1.10 and C{B{useZ} is True}.

583

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

585 a trilateration or C{numpy} issue.

586

587 @raise TypeError: Invalid B{C{point1}}, B{C{point2}} or B{C{point3}}.

588

589 @see: Functions L{radii11} and L{circum3} and U{Soddy Circles

590 <https://MathWorld.Wolfram.com/SoddyCircles.html>}.

591 '''

592 t, p1, p2, p3 = _radii11ABC4(point1, point2, point3, useZ=useZ)

593

594 r = t.riS

595 c, d = _tricenter3d2(p1, t.rA + r,

596 p2, t.rB + r,

597 p3, t.rC + r, eps=eps, useZ=useZ,

598 Vector=point1.classof, name=soddy4.__name__)

599 return Soddy4Tuple(r, c, d, t.roS)

600

601

602 def triaxum5(points, useZ=True):

603 '''Best-fit a triaxial ellipsoid through three or more (3-D) points.

604

605 @arg points: Iterable of points (each a C{Cartesian}, L{Vector3d}, C{Vector3Tuple}

606 or C{Vector4Tuple}).

607 @kwarg useZ: If C{True}, use the points' Z component, otherwise force C{z=INT0}

608 (C{bool}).

609

610 @return: L{Triaxum5Tuple}C{(a, b, c, rank, residuals)} with the unordered triaxial

611 radii C{a}, C{b} and C{c} in C{meter}, same units as the points' coordinates.

612

613 @raise ImportError: Package C{numpy} not found, not installed or older than version 1.10.

614

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

616

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

618

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

620

621 @see: I{Charles Jekel}'s U{"Least Squares Ellipsoid Fit"<https://Jekel.me/2020/Least-Squares-Ellipsoid-Fit/>}

622 and U{numpy.linalg.lstsq<https://NumPy.org/doc/stable/reference/generated/numpy.linalg.lstsq.html>}.

623 '''

624 n, ps = len2(points)

625 if n < 3:

626 raise PointsError(points=n, txt=_too_(_few_))

627

628 A = []

629 for i, p in enumerate(ps):

630 v = _otherV3d(useZ=useZ, i=i, points=p)

631 A.append(v.x2y2z2)

632

633 with _numpy(triaxum5, n=n) as _np:

634 A = _np.array(A)

635 b = _1_0_1T * n

636 T, R, rk = _np.least_squares3(A, b)

637

638 def _1_sqrt(x):

639 return sqrt(_1_0 / x) if x else _0_0 # INF

640

641 a, b, c = map(_1_sqrt, T)

642 return Triaxum5Tuple(a, b, c, rk, R)

643

644

645 def _tricenter3d2(p1, r1, p2, r2, p3, r3, eps=EPS4, useZ=True, dLL3=False, **kwds):

646 # (INTERNAL) Trilaterate and disambiguate the 3-D center

647 d, kwds = None, _xkwds(kwds, eps=eps, coin=True)

648 if useZ and p1.z != p2.z != p3.z: # ignore z if all match

649 a, b = _trilaterate3d2(p1, r1, p2, r2, p3, r3, **kwds)

650 if a is b: # no unambiguity

651 c = a # == b

652 else:

653 c = a.plus(b).times(_0_5) # mean

654 if not a.isconjugateTo(b, minum=0, eps=eps):

655 if dLL3: # deltas as (lat, lon, height)

656 a = a.toLatLon()

657 b = b.toLatLon()

658 d = LatLon3Tuple(b.lat - a.lat,

659 b.lon - a.lon,

660 b.height - a.height, name=_deltas_)

661 else:

662 d = b.minus(a) # vectorial deltas

663 else:

664 if useZ: # pass z to Vector if given

665 kwds = _xkwds(kwds, z=p1.z)

666 c = _trilaterate2d2(p1.x, p1.y, r1,

667 p2.x, p2.y, r2,

668 p3.x, p3.y, r3, **kwds)

669 return c, d

670

671

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

673 eps=None, **Vector_and_kwds):

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

675

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

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

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

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

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

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

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

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

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

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

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

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

688 the trilateration and optionally, additional B{C{Vector}}

689 keyword arguments).

690

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

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

693

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

695 centers, trilateration failed some other way

696 or the trilaterated point is off one circle

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

698

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

700

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

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

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

704 L{pygeodesy.trilaterate3d2}.

705 '''

706 return _trilaterate2d2(x1, y1, radius1,

707 x2, y2, radius2,

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

709

710

711 def _trilaterate2d2(x1, y1, radius1, x2, y2, radius2, x3, y3, radius3,

712 coin=False, eps=None,

713 Vector=None, **Vector_kwds):

714 # (INTERNAL) Trilaterate three circles, see L{pygeodesy.trilaterate2d2}

715

716 def _abct4(x1, y1, r1, x2, y2, r2):

717 a = x2 - x1

718 b = y2 - y1

719 t = _tri3near2far(r1, r2, hypot(a, b), coin)

720 c = _0_0 if t else (hypot2_(r1, x2, y2) - hypot2_(r2, x1, y1))

721 return a, b, c, t

722

723 def _astr(**kwds): # kwds as (name=value, ...) strings

724 return Fmt.PAREN(_COMMASPACE_(*(Fmt.EQUALg(*t) for t in kwds.items())))

725

726 r1 = Radius_(radius1=radius1)

727 r2 = Radius_(radius2=radius2)

728 r3 = Radius_(radius3=radius3)

729

730 a, b, c, t = _abct4(x1, y1, r1, x2, y2, r2)

731 if t:

732 raise IntersectionError(_and(_astr(x1=x1, y1=y1, radius1=r1),

733 _astr(x2=x2, y2=y2, radius2=r2)), txt=t)

734

735 d, e, f, t = _abct4(x2, y2, r2, x3, y3, r3)

736 if t:

737 raise IntersectionError(_and(_astr(x2=x2, y2=y2, radius2=r2),

738 _astr(x3=x3, y3=y3, radius3=r3)), txt=t)

739

740 _, _, _, t = _abct4(x3, y3, r3, x1, y1, r1)

741 if t:

742 raise IntersectionError(_and(_astr(x3=x3, y3=y3, radius3=r3),

743 _astr(x1=x1, y1=y1, radius1=r1)), txt=t)

744

745 q = fdot_(a, e, -b, d) * _2_0

746 if isnear0(q):

747 t = _no_(_intersection_)

748 raise IntersectionError(_and(_astr(x1=x1, y1=y1, radius1=r1),

749 _astr(x2=x2, y2=y2, radius2=r2),

750 _astr(x3=x3, y3=y3, radius3=r3)), txt=t)

751 t = Vector2Tuple(fdot_(c, e, -b, f) / q,

752 fdot_(a, f, -c, d) / q, name=trilaterate2d2.__name__)

753

754 if eps and eps > 0: # check distances to center vs radius

755 for x, y, r in ((x1, y1, r1), (x2, y2, r2), (x3, y3, r3)):

756 d = hypot(x - t.x, y - t.y)

757 e = fabs(d - r)

758 if e > eps:

759 t = _and(Float(delta=e).toRepr(), r.toRepr(),

760 Float(distance=d).toRepr(), t.toRepr())

761 raise IntersectionError(t, txt=Fmt.exceeds_eps(eps))

762

763 if Vector is not None:

764 t = Vector(t.x, t.y, **_xkwds(Vector_kwds, name=t.name))

765 return t

766

767

768 def _trilaterate3d2(c1, r1, c2, r2, c3, r3, eps=EPS4, coin=False, # MCCABE 13

769 **clas_Vector_and_kwds):

770 # (INTERNAL) Intersect three spheres or circles, see function

771 # L{pygeodesy.trilaterate3d2}, separated to allow callers to

772 # embellish exceptions, like C{FloatingPointError}s from C{numpy}

773

774 def _Arow4(c):

775 # make a row for matrix A (1, -2x, -2y, -2z)

776 return _1_0_1T + c.times(_N_2_0).xyz3

777

778 def _F4d3(F):

779 # map numpy 4-vector to floats and xyz3

780 T = map2(float, F)

781 t = T[1:]

782 return T, t, Vector3d(*t)

783

784 def _N3(t01, x, z):

785 # compute x, y and z and return as B{C{clas}} or B{C{Vector}}

786 v = x.plus(z.times(t01))

787 n = trilaterate3d2.__name__

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

789

790 c2 = _otherV3d(center2=c2, NN_OK=False)

791 c3 = _otherV3d(center3=c3, NN_OK=False)

792 rs = (r1, Radius_(radius2=r2, low=EPS),

793 Radius_(radius3=r3, low=EPS))

794

795 # get matrix A[3 x 4], its null_space Z and pseudo-inverse

796 A = [_Arow4(c) for c in (c1, c2, c3)]

797 with _numpy(trilaterate3d2, A=A, eps=eps) as _np:

798 Z, _ = _np.null_space2(A, eps)

799 if Z is not None:

800 Z, _, z = _F4d3(Z) # [4 x 1]

801 z2 = z.length2

802 A = _np.pseudo_inverse(A) # [4 x 3]

803 bs = [c.length2 for c in (c1, c2, c3)]

804 # perturb radii slightly by eps and eps * 4

805 for p in _tri5perturbs(eps, min(rs)):

806 b = [((r + p)**2 - b) for r, b in zip(rs, bs)] # [3 x 1]

807 X, t, x = _F4d3(A.dot(b)) # [4 * 1]

808 # quadratic polynomial, coefficients order (^0, ^1, ^2)

809 t = _np.real_roots(fdot(X, _N_1_0, *t),

810 fdot(Z, _N_0_5, *t) * _2_0, z2)

811 if t:

812 v = _N3(t[0], x, z)

813 if len(t) < 2: # one intersection

814 t = v, v

815 elif fabs(t[0] - t[1]) < eps: # abutting

816 t = v, v

817 else: # "lowest" intersection first (to avoid test failures)

818 u = _N3(t[1], x, z)

819 t = (u, v) if u.x < v.x else (v, u)

820 return t

821

822 def _no_itersection(coin, Z):

823 t = _no_(_intersection_)

824 if coin:

825 def _reprs(*crs):

826 return _and(*map(repr, crs))

827

828 r = repr(r1) if r1 == r2 == r3 else _reprs(r1, r2, r3)

829 t = _SPACE_(t, _of_, _reprs(c1, c2, c3), _with_, _radius_, r)

830 elif Z is None:

831 t = _COMMASPACE_(t, _no_(_numpy.null_space2.__name__))

832 return t

833

834 # coincident, concentric, colinear, too distant, no intersection:

835 # create the explanation and and throw an IntersectionError

836 t = _tri4near2far(c1, r1, c2, r2, coin) or \

837 _tri4near2far(c1, r1, c3, r3, coin) or \

838 _tri4near2far(c2, r2, c3, r3, coin) or (

839 _colinear_ if _iscolinearWith(c1, c2, c3, eps=eps) else

840 _no_itersection(coin, Z))

841 raise IntersectionError(t, txt=None)

842

843

844 def _tri3near2far(r1, r2, h, coin):

845 # check for near-coincident/-concentric or too distant spheres/circles

846 return _too_(Fmt.distant(h)) if h > (r1 + r2) else (_near_(

847 _coincident_ if coin else _concentric_) if h < fabs(r1 - r2) else NN)

848

849

850 def _tri4near2far(c1, r1, c2, r2, coin):

851 # check for near-coincident/-concentric or too distant spheres/circles

852 t = _tri3near2far(r1, r2, c1.minus(c2).length, coin)

853 return _SPACE_(c1.name, _and_, c2.name, t) if t else NN

854

855

856 def _tri5perturbs(eps, r):

857 # perturb the radii to handle this corner case

858 # <https://GitHub.com/mrJean1/PyGeodesy/issues/49>

859 yield _0_0

860 if eps and eps > 0:

861 p = max(eps, EPS)

862 yield p

863 m = min(p, r)

864 yield -m

865 q = max(eps * _4_0, _EPS4e8)

866 if q > p:

867 yield q

868 q = min(q, r)

869 if q > m:

870 yield -q

871

872# **) MIT License

873#

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

875#

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

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

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

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

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

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

882#

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

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

885#

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

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

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

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

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

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

892# OTHER DEALINGS IN THE SOFTWARE.