Coverage for pygeodesy/vector2d.py: 98%

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1 

2# -*- coding: utf-8 -*- 

3 

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

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

6''' 

7 

8from pygeodesy.basics import len2, map2, _xnumpy 

9from pygeodesy.constants import EPS, EPS0, EPS02, EPS4, INF, INT0, \ 

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

11 _1_0, _1_0_1T, _N_1_0, _2_0, _N_2_0, _4_0 

12from pygeodesy.errors import _and, _AssertionError, IntersectionError, NumPyError, \ 

13 PointsError, TriangleError, _xError, _xkwds 

14from pygeodesy.fmath import fabs, fdot, hypot, hypot2_, sqrt 

15from pygeodesy.fsums import _Fsumf_, fsumf_, fsum1f_ 

16from pygeodesy.interns import NN, _a_, _and_, _b_, _c_, _center_, _coincident_, \ 

17 _colinear_, _COMMASPACE_, _concentric_, _few_, \ 

18 _intersection_, _invalid_, _near_, _no_, _of_, \ 

19 _radius_, _rIn_, _s_, _SPACE_, _too_, _with_ 

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

21from pygeodesy.named import _ALL_LAZY, _NamedTuple, _Pass, Property_RO 

22from pygeodesy.namedTuples import LatLon3Tuple, Vector2Tuple 

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

24from pygeodesy.streprs import Fmt, unstr 

25from pygeodesy.units import Float, Int, Meter, Radius, Radius_ 

26from pygeodesy.vector3d import iscolinearWith, nearestOn, _nearestOn2, _nVc, _otherV3d, \ 

27 trilaterate2d2, trilaterate3d2, Vector3d # PYCHOK unused 

28 

29from contextlib import contextmanager 

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

31 

32__all__ = _ALL_LAZY.vector2d 

33__version__ = '24.05.04' 

34 

35_cA_ = 'cA' 

36_cB_ = 'cB' 

37_cC_ = 'cC' 

38_deltas_ = 'deltas' 

39_outer_ = 'outer' 

40_raise_ = 'raise' # PYCHOK used! 

41_rank_ = 'rank' 

42_residuals_ = 'residuals' 

43_Type_ = 'Type' 

44 

45 

46class Circin6Tuple(_NamedTuple): 

47 '''6-Tuple C{(radius, center, deltas, cA, cB, cC)} with the C{radius}, the 

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

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

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

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

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

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

54 ''' 

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

56 _Units_ = ( Radius, _Pass, _Pass, _Pass, _Pass, _Pass) 

57 

58 

59class Circum3Tuple(_NamedTuple): # in .latlonBase 

60 '''3-Tuple C{(radius, center, deltas)} with the C{circumradius} and trilaterated 

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

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

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

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

65 results. 

66 ''' 

67 _Names_ = (_radius_, _center_, _deltas_) 

68 _Units_ = ( Radius, _Pass, _Pass) 

69 

70 

71class Circum4Tuple(_NamedTuple): 

72 '''4-Tuple C{(radius, center, rank, residuals)} with C{radius} and C{center} 

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

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

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

76 ''' 

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

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

79 

80 

81class Meeus2Tuple(_NamedTuple): 

82 '''2-Tuple C{(radius, Type)} with C{radius} and I{Meeus}' C{Type} of the smallest 

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

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

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

86 inside the circle. 

87 ''' 

88 _Names_ = (_radius_, _Type_) 

89 _Units_ = ( Radius, _Pass) 

90 

91 

92class Radii11Tuple(_NamedTuple): 

93 '''11-Tuple C{(rA, rB, rC, cR, rIn, riS, roS, a, b, c, s)} with the C{Tangent} 

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

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

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

97 all in C{meter} conventionally. 

98 

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

100 ''' 

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

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

103 

104 

105class Soddy4Tuple(_NamedTuple): 

106 '''4-Tuple C{(radius, center, deltas, outer)} with C{radius} and (trilaterated) 

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

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

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

110 L{pygeodesy.trilaterate3d2} results. 

111 

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

113 ''' 

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

115 _Units_ = ( Radius, _Pass, _Pass, Radius) 

116 

117 

118def circin6(point1, point2, point3, eps=EPS4, useZ=True): 

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

120 a (2- or 3-D) triangle. 

121 

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

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

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

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

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

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

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

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

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

131 

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

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

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

135 the three given points. 

136 

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

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

139 

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

141 a trilateration or C{numpy} issue. 

142 

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

144 

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

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

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

148 ''' 

149 try: 

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

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

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

153 

154 

155def _circin6(point1, point2, point3, eps=EPS4, useZ=True, dLL3=False, **Vector_kwds): 

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

157 

158 def _fraction(r, a): 

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

160 

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

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

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

164 C = V(c.x, c.y, c.z, **V_kwds) 

165 return c, C 

166 

167 t, p1, p2, p3 = _radii11ABC(point1, point2, point3, useZ=useZ) 

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

169 

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

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

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

173 

174 r = t.rIn 

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

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

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

178 

179 

180def circum3(point1, point2, point3, circum=True, eps=EPS4, useZ=True): 

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

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

183 

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

185 C{Vector4Tuple}). 

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

187 C{Vector4Tuple}). 

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

189 C{Vector4Tuple}). 

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

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

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

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

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

195 

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

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

198 given points. 

199 

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

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

202 

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

204 a trilateration or C{numpy} issue. 

205 

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

207 

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

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

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

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

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

213 ''' 

214 try: 

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

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

217 clas=point1.classof) 

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

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

220 

221 

222def _circum3(p1, point2, point3, circum=True, eps=EPS4, useZ=True, dLL3=False, 

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

224 # (INTERNAL) Radius, center, deltas 

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

226 clas=clas, **clas_kwds) 

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

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

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

230 else: # Meeus' Type I 

231 c, d = d, None 

232 return Circum3Tuple(r, c, d) 

233 

234 

235def circum4_(*points, **useZ_Vector_and_kwds): 

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

237 

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

239 or C{Vector4Tuple}). 

240 @kwarg useZ_Vector_and_kwds: Keyword arguments C{B{useZ}=True} (C{bool}) 

241 to use the Z components, otherwise force all C{z=INT0}, class 

242 C{B{Vector}=None} to return the center point with optionally, 

243 additional nB{C{Vector}} keyword arguments, otherwise the 

244 first B{C{points}}' (sub-)class is used. 

245 

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

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

248 

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

250 version 1.10. 

251 

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

253 

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

255 

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

257 

258 @see: Functions L{pygeodesy.circum3} and L{pygeodesy.meeus2}, Jekel, Charles F. U{I{Least 

259 Squares Sphere Fit}<https://Jekel.me/2015/Least-Squares-Sphere-Fit/>} Sep 13, 2015, 

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

261 NumPy.org/doc/stable/reference/generated/numpy.linalg.lstsq.html>} and U{Eberly 6 

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

263 ''' 

264 def _useZ_kwds(useZ=True, **kwds): 

265 return useZ, kwds 

266 

267 n, ps = len2(points) 

268 if n < 3: 

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

270 useZ, kwds = _useZ_kwds(**useZ_Vector_and_kwds) 

271 

272 A, b = [], [] 

273 for i, p in enumerate(ps): 

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

275 A.append(v.times(_2_0).xyz + _1_0_1T) 

276 b.append(v.length2) 

277 

278 with _numpy(circum4_, n=n) as _np: 

279 A = _np.array(A).reshape((n, 4)) 

280 b = _np.array(b).reshape((n, 1)) 

281 C, R, rk, _ = _np.least_squares4(A, b, rcond=None) # to silence warning 

282 C = map2(float, C) 

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

284 

285 n = circum4_.__name__ 

286 c = Vector3d(*C[:3], name=n) 

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

288 

289 c = _nVc(c, **_xkwds(kwds, clas=ps[0].classof, name=n)) 

290 return Circum4Tuple(r, c, rk, R) 

291 

292 

293def _iscolinearWith(p, point1, point2, eps=EPS, useZ=True): 

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

295 # separated to allow callers to embellish any exceptions 

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

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

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

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

300 

301 

302def meeus2(point1, point2, point3, circum=False, useZ=True): 

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

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

305 

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

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

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

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

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

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

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

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

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

315 

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

317 iff C{B{circum}=True}. 

318 

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

320 

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

322 

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

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

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

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

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

328 ''' 

329 try: 

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

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

332 except (TypeError, ValueError) as x: 

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

334 

335 

336def _meeus2(A, point2, point3, circum, useZ=True, **clas_and_kwds): # in .vector3d 

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

338 f = _circum3 if circum else _meeus4 

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

340 return Meeus2Tuple(t) 

341 

342 

343def _meeus4(A, point2, point3, circum=False, useZ=True, clas=None, **clas_kwds): 

344 # (INTERNAL) Radius and Meeus' Type 

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

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

347 

348 a = B.minus(C).length2 

349 b = C.minus(A).length2 

350 c = A.minus(B).length2 

351 if a < b: 

352 a, b, A, B = b, a, B, A 

353 if a < c: 

354 a, c, A, C = c, a, C, A 

355 

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

357 b = sqrt(b / a) 

358 c = sqrt(c / a) 

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

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

361 r = R.fover(a) 

362 if r < EPS02: 

363 raise IntersectionError(_coincident_ if b < EPS0 or c < EPS0 else ( 

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

365 r = b * c / sqrt(r) 

366 t = None # Meeus' Type II 

367 else: # obtuse or right angle at A 

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

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

370 if clas is not None: 

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

372 return r, t, p2, p3 

373 

374 

375class _numpy(object): # see also .formy._idllmn6, .geodesicw._wargs, .latlonBase._toCartesian3 

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

377 ''' 

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

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

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

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

382 ''' 

383 np = self.np 

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

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

386 yield self 

387 except Exception as x: # mostly FloatingPointError? 

388 t = unstr(where, *args, **kwds) 

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

390 finally: # restore numpy error handling 

391 np.seterr(**e) 

392 

393 @Property_RO 

394 def array(self): 

395 return self.np.array 

396 

397 @Property_RO 

398 def least_squares4(self): 

399 '''Linear least-squares function. 

400 ''' 

401 return self.np.linalg.lstsq 

402 

403 @Property_RO 

404 def np(self): 

405 '''Import numpy 1.10+ once. 

406 ''' 

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

408 

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

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

411 

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

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

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

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

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

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

418 ''' 

419 def _Error(txt=self.null_space2.__name__, **kwds): 

420 return _AssertionError(txt=txt, **kwds) 

421 

422 np = self.np 

423 A = np.array(A) 

424 m = max(A.shape) 

425 if m != 4: # for this usage 

426 raise _Error(shape=m) 

427 # if needed, square A, pad with zeros 

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

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

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

431# return Z, ... 

432# except AttributeError: 

433# pass 

434 U, S, V = np.linalg.svd(A) 

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

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

437 r = int(np.sum(S > t)) # rank 

438 if r == 3: # get null_space 

439 Z = np.transpose(V[r:]) 

440 s = map2(int, Z.shape) 

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

442 raise _Error(shape=s, m=m) 

443 D = A.dot(Z) # near-zeros-vector 

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

445 if n > t: # largest exceed tol 

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

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

448 Z = None 

449 # del A, S, U, V # release numpy 

450 return Z, r 

451 

452 @Property_RO 

453 def pseudo_inverse(self): 

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

455 ''' 

456 return self.np.linalg.pinv 

457 

458 def real_roots(self, *coeffs): 

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

460 ''' 

461 np = self.np 

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

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

464 

465_numpy = _numpy() # PYCHOK singleton 

466 

467 

468def radii11(point1, point2, point3, useZ=True): 

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

470 (2- or 3-D) triangle. 

471 

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

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

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

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

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

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

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

479 

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

481 

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

483 

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

485 

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

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

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

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

490 ''' 

491 try: 

492 return _radii11ABC(point1, point2, point3, useZ=useZ)[0] 

493 except (TypeError, ValueError) as x: 

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

495 

496 

497def _radii11ABC(point1, point2, point3, useZ=True): 

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

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

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

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

502 

503 a = B.minus(C).length 

504 b = C.minus(A).length 

505 c = A.minus(B).length 

506 

507 S = _Fsumf_(a, b, c) * _0_5 

508 s = float(S) # semi-perimeter 

509 if s > EPS0: 

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

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

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

513 r3_r1 = r3 / r1 

514 r3_r2 = r3 / r2 

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

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

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

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

519 if t > EPS02: 

520 t = sqrt(t) * _2_0 # * r1 * r2 

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

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

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

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

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

526 # = r3 / (d +/- t) 

527 si = r3 / (d + t) 

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

529 # ci = sqrt(r1 * r2 * r3 / s) 

530 # = r1 * sqrt(r2 * r3 / r1 / s) 

531 ci = r1 * sqrt(r2 * r3_r1 / s) 

532 # co = a * b * c / (4 * ci * s) 

533 t = ci * s * _4_0 

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

535 r1, r2, r3 = rs # original order 

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

537 return t, A, B, C 

538 

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

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

541 

542 

543def soddy4(point1, point2, point3, eps=EPS4, useZ=True): 

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

545 (2- or 3-D) triangle. 

546 

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

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

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

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

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

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

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

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

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

556 

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

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

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

560 

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

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

563 

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

565 a trilateration or C{numpy} issue. 

566 

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

568 

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

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

571 ''' 

572 t, p1, p2, p3 = _radii11ABC(point1, point2, point3, useZ=useZ) 

573 

574 r = t.riS 

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

576 p2, t.rB + r, 

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

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

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

580 

581 

582def _tricenter3d2(p1, r1, p2, r2, p3, r3, eps=EPS4, useZ=True, dLL3=False, **kwds): 

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

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

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

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

587 if a is b: # no unambiguity 

588 c = a # == b 

589 else: 

590 c = a.plus(b).times(_0_5) # mean 

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

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

593 a = a.toLatLon() 

594 b = b.toLatLon() 

595 d = LatLon3Tuple(b.lat - a.lat, 

596 b.lon - a.lon, 

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

598 else: 

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

600 else: 

601 if useZ: # pass z to Vector if given 

602 kwds = _xkwds(kwds, z=p1.z) 

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

604 p2.x, p2.y, r2, 

605 p3.x, p3.y, r3, **kwds) 

606 return c, d 

607 

608 

609def _trilaterate2d2(x1, y1, radius1, x2, y2, radius2, x3, y3, radius3, 

610 coin=False, eps=None, 

611 Vector=None, **Vector_kwds): 

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

613 

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

615 a = x2 - x1 

616 b = y2 - y1 

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

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

619 return a, b, c, t 

620 

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

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

623 

624 r1 = Radius_(radius1=radius1) 

625 r2 = Radius_(radius2=radius2) 

626 r3 = Radius_(radius3=radius3) 

627 

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

629 if t: 

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

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

632 

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

634 if t: 

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

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

637 

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

639 if t: 

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

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

642 

643 q = (a * e - b * d) * _2_0 

644 if isnear0(q): 

645 t = _no_(_intersection_) 

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

647 _astr(x2=x2, y2=y2, radius2=r2), 

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

649 t = Vector2Tuple((c * e - b * f) / q, 

650 (a * f - c * d) / q, name=trilaterate2d2.__name__) 

651 

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

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

654 d = hypot(x - t.x, y - t.y) 

655 e = fabs(d - r) 

656 if e > eps: 

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

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

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

660 

661 if Vector is not None: 

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

663 return t 

664 

665 

666def _trilaterate3d2(c1, r1, c2, r2, c3, r3, eps=EPS4, coin=False, 

667 **clas_Vector_and_kwds): 

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

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

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

671 

672 def _F3d2(F): 

673 # map numpy 4-vector to floats tuple and Vector3d 

674 T = map2(float, F) 

675 return T, Vector3d(*T[1:]) 

676 

677 def _N3(t01, x, z): 

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

679 v = x.plus(z.times(t01)) 

680 n = trilaterate3d2.__name__ 

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

682 

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

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

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

686 Radius_(radius3=r3, low=EPS)) 

687 

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

689 A = [(_1_0_1T + c.times(_N_2_0).xyz) for c in (c1, c2, c3)] 

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

691 Z, _ = _np.null_space2(A, eps) 

692 if Z is not None: 

693 Z, z = _F3d2(Z) # [4 x 1] 

694 z2 = z.length2 

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

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

697 # perturbe radii and vector b slightly by eps and eps * 4 

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

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

700 X, x = _F3d2(A.dot(b)) 

701 # quadratic polynomial, coefficients ordered (^0, ^1, ^2) 

702 t = _np.real_roots(fdot(X, _N_1_0, *x.xyz), 

703 fdot(Z, _N_0_5, *x.xyz) * _2_0, z2) 

704 if t: 

705 v = _N3(t[0], x, z) 

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

707 t = v, v 

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

709 t = v, v 

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

711 u = _N3(t[1], x, z) 

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

713 return t 

714 

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

716 # create the explanation and and throw an IntersectionError 

717 

718 def _no_intersection(coin): 

719 t = _no_(_intersection_) 

720 if coin: 

721 def _reprs(*crs): 

722 return _and(*map(repr, crs)) 

723 

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

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

726 elif Z is None: 

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

728 return t 

729 

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

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

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

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

734 _no_intersection(coin)) 

735 raise IntersectionError(t, txt=None) 

736 

737 

738def _tri3near2far(r1, r2, h, coin): 

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

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

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

742 

743 

744def _tri4near2far(c1, r1, c2, r2, coin): 

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

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

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

748 

749 

750def _tri5perturbs(eps, r): 

751 # perturb the radii to handle this corner case 

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

753 yield _0_0 

754 if eps and eps > 0: 

755 p = max(eps, EPS) 

756 yield p 

757 m = min(p, r) 

758 yield -m 

759 q = max(eps * _4_0, _EPS4e8) 

760 if q > p: 

761 yield q 

762 q = min(q, r) 

763 if q > m: 

764 yield -q 

765 

766# **) MIT License 

767# 

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

769# 

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

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

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

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

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

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

776# 

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

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

779# 

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

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

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

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

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

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

786# OTHER DEALINGS IN THE SOFTWARE.