Coverage for pygeodesy/vector2d.py: 98%

340 statements  

« prev     ^ index     » next       coverage.py v7.6.1, created at 2025-05-04 12:01 -0400

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}, L{soddy4}, 

6L{triaxum5} and L{trilaterate2d2}. 

7 

8@note: Functions L{circin6}, L{circum3}, L{circum4_}, L{soddy4} and 

9 L{triaxum5} require U{numpyhttps://PyPI.org/project/numpy>} 

10 version 1.10 or newer to be installed. 

11''' 

12 

13from pygeodesy.basics import len2, map2, _xnumpy, typename 

14from pygeodesy.constants import EPS, EPS0, EPS02, EPS4, INF, INT0, \ 

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

16 _1_0, _1_0_1T, _N_1_0, _2_0, _N_2_0, _4_0 

17from pygeodesy.errors import _and, _AssertionError, IntersectionError, NumPyError, \ 

18 PointsError, TriangleError, _xError, _xkwds 

19from pygeodesy.fmath import fabs, fdot, Fdot_, fdot_, hypot, hypot2_, sqrt 

20from pygeodesy.fsums import _Fsumf_, fsumf_, fsum1f_ 

21# from pygeodesy.internals import typename # from .basics 

22from pygeodesy.interns import NN, _a_, _and_, _b_, _c_, _center_, _coincident_, \ 

23 _colinear_, _COMMASPACE_, _concentric_, _few_, \ 

24 _intersection_, _invalid_, _near_, _no_, _of_, \ 

25 _radius_, _rIn_, _s_, _SPACE_, _too_, _with_ 

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

27from pygeodesy.named import _ALL_LAZY, _NamedTuple, _Pass, Property_RO 

28from pygeodesy.namedTuples import LatLon3Tuple, Vector2Tuple 

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

30from pygeodesy.streprs import Fmt, unstr 

31from pygeodesy.units import Float, Int, Meter, Radius, Radius_ 

32from pygeodesy.vector3d import iscolinearWith, nearestOn, _nearestOn2, _nVc, \ 

33 _otherV3d, trilaterate3d2, Vector3d # PYCHOK unused 

34 

35from contextlib import contextmanager 

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

37 

38__all__ = _ALL_LAZY.vector2d 

39__version__ = '25.04.30' 

40 

41_cA_ = 'cA' 

42_cB_ = 'cB' 

43_cC_ = 'cC' 

44_deltas_ = 'deltas' 

45_outer_ = 'outer' 

46_raise_ = 'raise' # PYCHOK used! 

47_rank_ = 'rank' 

48_residuals_ = 'residuals' 

49_Type_ = 'Type' 

50 

51 

52class Circin6Tuple(_NamedTuple): 

53 '''6-Tuple C{(radius, center, deltas, cA, cB, cC)} with the C{radius}, the 

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

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

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

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

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

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

60 ''' 

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

62 _Units_ = ( Radius, _Pass, _Pass, _Pass, _Pass, _Pass) 

63 

64 

65class Circum3Tuple(_NamedTuple): # in .latlonBase 

66 '''3-Tuple C{(radius, center, deltas)} with the C{circumradius} and trilaterated 

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

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

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

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

71 results. 

72 ''' 

73 _Names_ = (_radius_, _center_, _deltas_) 

74 _Units_ = ( Radius, _Pass, _Pass) 

75 

76 

77class Circum4Tuple(_NamedTuple): 

78 '''4-Tuple C{(radius, center, rank, residuals)} with C{radius} and C{center} 

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

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

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

82 ''' 

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

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

85 

86 

87class Meeus2Tuple(_NamedTuple): 

88 '''2-Tuple C{(radius, Type)} with C{radius} and I{Meeus}' C{Type} of the smallest 

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

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

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

92 inside the circle. 

93 ''' 

94 _Names_ = (_radius_, _Type_) 

95 _Units_ = ( Radius, _Pass) 

96 

97 

98class Radii11Tuple(_NamedTuple): 

99 '''11-Tuple C{(rA, rB, rC, cR, rIn, riS, roS, a, b, c, s)} with the C{Tangent} 

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

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

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

103 all in C{meter} conventionally. 

104 

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

106 ''' 

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

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

109 

110 

111class Soddy4Tuple(_NamedTuple): 

112 '''4-Tuple C{(radius, center, deltas, outer)} with C{radius} and (trilaterated) 

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

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

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

116 L{pygeodesy.trilaterate3d2} results. 

117 

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

119 ''' 

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

121 _Units_ = ( Radius, _Pass, _Pass, Radius) 

122 

123 

124class Triaxum5Tuple(_NamedTuple): 

125 '''5-Tuple C{(a, b, c, rank, residuals)} with the (unordered) triaxial radii 

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

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

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

129 ''' 

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

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

132 

133 

134def circin6(point1, point2, point3, eps=EPS4, useZ=True): 

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

136 a (2- or 3-D) triangle. 

137 

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

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

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

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

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

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

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

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

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

147 

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

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

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

151 the three given points. 

152 

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

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

155 

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

157 a trilateration or C{numpy} issue. 

158 

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

160 

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

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

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

164 ''' 

165 try: 

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

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

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

169 

170 

171def _circin6(point1, point2, point3, eps=EPS4, useZ=True, dLL3=False, **Vector_kwds): 

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

173 

174 def _fraction(r, a): 

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

176 

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

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

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

180 C = V(c.x, c.y, c.z, **V_kwds) 

181 return c, C 

182 

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

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

185 

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

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

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

189 

190 r = t.rIn 

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

192 **_xkwds(Vector_kwds, Vector=V, name=typename(circin6))) 

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

194 

195 

196def circum3(point1, point2, point3, circum=True, eps=EPS4, useZ=True): 

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

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

199 

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

201 C{Vector4Tuple}). 

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

203 C{Vector4Tuple}). 

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

205 C{Vector4Tuple}). 

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

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

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

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

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

211 

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

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

214 given points. 

215 

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

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

218 

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

220 a trilateration or C{numpy} issue. 

221 

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

223 

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

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

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

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

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

229 ''' 

230 try: 

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

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

233 clas=point1.classof) 

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

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

236 

237 

238def _circum3(p1, point2, point3, circum=True, eps=EPS4, useZ=True, dLL3=False, 

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

240 # (INTERNAL) Radius, center, deltas 

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

242 clas=clas, **clas_kwds) 

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

244 kwds = _xkwds(clas_kwds, eps=eps, Vector=clas, name=typename(circum3)) 

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

246 else: # Meeus' Type I 

247 c, d = d, None 

248 return Circum3Tuple(r, c, d) 

249 

250 

251def circum4(points, useZ=True, **Vector_and_kwds): 

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

253 

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

255 or C{Vector4Tuple}). 

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

257 (C{bool}). 

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

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

260 the B{C{points}}' (sub-)class. 

261 

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

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

264 

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

266 version 1.10. 

267 

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

269 

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

271 

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

273 

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

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

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

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

278 6<https://www.sci.Utah.EDU/~balling/FEtools/doc_files/LeastSquaresFitting.pdf>}. 

279 ''' 

280 n, ps = len2(points) 

281 if n < 3: 

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

283 

284 A, b = [], [] 

285 for i, p in enumerate(ps): 

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

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

288 b.append(v.length2) 

289 

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

291 A = _np.array(A).reshape((n, 4)) 

292 b = _np.array(b).reshape((n, 1)) 

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

294 

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

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

297 

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

299 return Circum4Tuple(r, c, rk, R) 

300 

301 

302def circum4_(*points, **useZ_Vector_and_kwds): 

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

304 

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

306 or C{Vector4Tuple}), all positional. 

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

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

309 

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

311 ''' 

312 return circum4(points, **useZ_Vector_and_kwds) 

313 

314 

315def _iscolinearWith(p, point1, point2, eps=EPS, useZ=True): 

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

317 # separated to allow callers to embellish any exceptions 

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

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

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

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

322 

323 

324def meeus2(point1, point2, point3, circum=False, useZ=True): 

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

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

327 

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

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

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

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

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

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

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

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

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

337 

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

339 iff C{B{circum}=True}. 

340 

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

342 

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

344 

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

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

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

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

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

350 ''' 

351 try: 

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

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

354 except (TypeError, ValueError) as x: 

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

356 

357 

358def _meeus2(A, point2, point3, circum, useZ=True, **clas_and_kwds): # in .vector3d 

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

360 f = _circum3 if circum else _meeus4 

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

362 return Meeus2Tuple(t) 

363 

364 

365def _meeus4(A, point2, point3, circum=False, useZ=True, clas=None, **clas_kwds): 

366 # (INTERNAL) Radius and Meeus' Type 

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

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

369 

370 a = B.minus(C).length2 

371 b = C.minus(A).length2 

372 c = A.minus(B).length2 

373 if a < b: 

374 a, b, A, B = b, a, B, A 

375 if a < c: 

376 a, c, A, C = c, a, C, A 

377 

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

379 b = sqrt(b / a) 

380 c = sqrt(c / a) 

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

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

383 r = R.fover(a) 

384 if r < EPS02: 

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

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

387 raise IntersectionError(t) 

388 r = b * c / sqrt(r) 

389 t = None # Meeus' Type II 

390 else: # obtuse or right angle at A 

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

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

393 if clas is not None: 

394 t = clas(t.x, t.y, t.z, **_xkwds(clas_kwds, name=typename(meeus2))) 

395 return r, t, p2, p3 

396 

397 

398class _numpy(object): # see also .formy._idllmn6, .geodesicw._wargs, .latlonBase._toCartesian3 

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

400 ''' 

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

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

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

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

405 ''' 

406 np = self.np 

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

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

409 yield self 

410 except Exception as x: # mostly FloatingPointError? 

411 t = unstr(where, *args, **kwds) 

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

413 finally: # restore numpy error handling 

414 np.seterr(**e) 

415 

416 @Property_RO 

417 def array(self): 

418 '''Get U{numpy.array<https://NumPy.org/doc/2.2/reference/generated/numpy.array.html#numpy.array>}. 

419 ''' 

420 return self.np.array 

421 

422 def least_squares3(self, A, b): 

423 '''Linear least-squares function. 

424 ''' 

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

426 C = map2(float, C) 

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

428 return C, R, int(rk) 

429 

430 @Property_RO 

431 def np(self): 

432 '''Import numpy 1.10+ once. 

433 ''' 

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

435 

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

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

438 

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

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

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

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

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

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

445 ''' 

446 def _Error(**kwds): 

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

448 

449 np = self.np 

450 A = np.array(A) 

451 m = max(A.shape) 

452 if m != 4: # for this usage 

453 raise _Error(shape=m) 

454 # if needed, square A, pad with zeros 

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

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

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

458# return Z, ... 

459# except AttributeError: 

460# pass 

461 U, S, V = np.linalg.svd(A) 

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

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

464 r = int(np.sum(S > t)) # rank 

465 if r == 3: # get null_space 

466 Z = np.transpose(V[r:]) 

467 s = map2(int, Z.shape) 

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

469 raise _Error(shape=s, m=m) 

470 D = A.dot(Z) # near-zeros-vector 

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

472 if n > t: # largest exceed tol 

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

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

475 Z = None 

476 # del A, S, U, V # release numpy 

477 return Z, r 

478 

479 @Property_RO 

480 def pseudo_inverse(self): 

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

482 ''' 

483 return self.np.linalg.pinv 

484 

485 def real_roots(self, *coeffs): 

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

487 ''' 

488 np = self.np 

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

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

491 

492_numpy = _numpy() # PYCHOK singleton 

493 

494 

495def radii11(point1, point2, point3, useZ=True): 

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

497 (2- or 3-D) triangle. 

498 

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

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

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

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

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

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

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

506 

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

508 

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

510 

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

512 

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

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

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

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

517 ''' 

518 try: 

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

520 except (TypeError, ValueError) as x: 

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

522 

523 

524def _radii11ABC4(point1, point2, point3, useZ=True): 

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

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

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

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

529 

530 a = B.minus(C).length 

531 b = C.minus(A).length 

532 c = A.minus(B).length 

533 

534 S = _Fsumf_(a, b, c) * _0_5 

535 s = float(S) # semi-perimeter 

536 if s > EPS0: 

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

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

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

540 r3_r1 = r3 / r1 

541 r3_r2 = r3 / r2 

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

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

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

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

546 if t > EPS02: 

547 t = sqrt(t) * _2_0 # * r1 * r2 

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

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

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

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

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

553 # = r3 / (d +/- t) 

554 si = r3 / (d + t) 

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

556 # ci = sqrt(r1 * r2 * r3 / s) 

557 # = r1 * sqrt(r2 * r3 / r1 / s) 

558 ci = r1 * sqrt(r2 * r3_r1 / s) 

559 # co = a * b * c / (4 * ci * s) 

560 t = ci * s * _4_0 

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

562 r1, r2, r3 = rs # original order 

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

564 return t, A, B, C 

565 

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

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

568 

569 

570def soddy4(point1, point2, point3, eps=EPS4, useZ=True): 

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

572 (2- or 3-D) triangle. 

573 

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

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

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

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

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

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

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

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

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

583 

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

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

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

587 

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

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

590 

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

592 a trilateration or C{numpy} issue. 

593 

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

595 

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

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

598 ''' 

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

600 

601 r = t.riS 

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

603 p2, t.rB + r, 

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

605 Vector=point1.classof, name=typename(soddy4)) 

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

607 

608 

609def triaxum5(points, useZ=True): 

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

611 

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

613 or C{Vector4Tuple}). 

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

615 (C{bool}). 

616 

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

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

619 

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

621 

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

623 

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

625 

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

627 

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

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

630 ''' 

631 n, ps = len2(points) 

632 if n < 3: 

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

634 

635 A = [] 

636 for i, p in enumerate(ps): 

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

638 A.append(v.x2y2z2) 

639 

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

641 A = _np.array(A) 

642 b = _1_0_1T * n 

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

644 

645 def _1_sqrt(x): 

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

647 

648 a, b, c = map(_1_sqrt, T) 

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

650 

651 

652def _tricenter3d2(p1, r1, p2, r2, p3, r3, eps=EPS4, useZ=True, dLL3=False, **kwds): 

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

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

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

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

657 if a is b: # no unambiguity 

658 c = a # == b 

659 else: 

660 c = a.plus(b).times(_0_5) # mean 

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

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

663 a = a.toLatLon() 

664 b = b.toLatLon() 

665 d = LatLon3Tuple(b.lat - a.lat, 

666 b.lon - a.lon, 

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

668 else: 

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

670 else: 

671 if useZ: # pass z to Vector if given 

672 kwds = _xkwds(kwds, z=p1.z) 

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

674 p2.x, p2.y, r2, 

675 p3.x, p3.y, r3, **kwds) 

676 return c, d 

677 

678 

679def trilaterate2d2(x1, y1, radius1, x2, y2, radius2, x3, y3, radius3, 

680 eps=None, **Vector_and_kwds): 

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

682 

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

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

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

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

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

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

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

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

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

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

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

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

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

696 keyword arguments). 

697 

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

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

700 

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

702 centers, trilateration failed some other way 

703 or the trilaterated point is off one circle 

704 by more than B{C{eps}}. 

705 

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

707 

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

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

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

711 L{pygeodesy.trilaterate3d2}. 

712 ''' 

713 return _trilaterate2d2(x1, y1, radius1, 

714 x2, y2, radius2, 

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

716 

717 

718def _trilaterate2d2(x1, y1, radius1, x2, y2, radius2, x3, y3, radius3, 

719 coin=False, eps=None, 

720 Vector=None, **Vector_kwds): 

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

722 

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

724 a = x2 - x1 

725 b = y2 - y1 

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

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

728 return a, b, c, t 

729 

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

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

732 

733 r1 = Radius_(radius1=radius1) 

734 r2 = Radius_(radius2=radius2) 

735 r3 = Radius_(radius3=radius3) 

736 

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

738 if t: 

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

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

741 

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

743 if t: 

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

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

746 

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

748 if t: 

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

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

751 

752 q = fdot_(a, e, -b, d) * _2_0 

753 if isnear0(q): 

754 t = _no_(_intersection_) 

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

756 _astr(x2=x2, y2=y2, radius2=r2), 

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

758 t = Vector2Tuple(Fdot_(c, e, -b, f).fover(q), 

759 Fdot_(a, f, -c, d).fover(q), name=typename(trilaterate2d2)) 

760 

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

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

763 d = hypot(x - t.x, y - t.y) 

764 e = fabs(d - r) 

765 if e > eps: 

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

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

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

769 

770 if Vector is not None: 

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

772 return t 

773 

774 

775def _trilaterate3d2(c1, r1, c2, r2, c3, r3, eps=EPS4, coin=False, # MCCABE 13 

776 **clas_Vector_and_kwds): 

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

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

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

780 

781 def _Arow4(c): 

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

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

784 

785 def _F4d3(F): 

786 # map numpy 4-vector to floats and xyz3 

787 T = map2(float, F) 

788 t = T[1:] 

789 return T, t, Vector3d(*t) 

790 

791 def _N3(t01, x, z): 

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

793 v = x.plus(z.times(t01)) 

794 n = typename(trilaterate3d2) 

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

796 

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

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

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

800 Radius_(radius3=r3, low=EPS)) 

801 

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

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

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

805 Z, _ = _np.null_space2(A, eps) 

806 if Z is not None: 

807 Z, _, z = _F4d3(Z) # [4 x 1] 

808 z2 = z.length2 

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

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

811 # perturb radii slightly by eps and eps * 4 

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

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

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

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

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

817 fdot(Z, _N_0_5, *t) * _2_0, z2) 

818 if t: 

819 v = _N3(t[0], x, z) 

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

821 t = v, v 

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

823 t = v, v 

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

825 u = _N3(t[1], x, z) 

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

827 return t 

828 

829 def _no_itersection(coin, Z): 

830 t = _no_(_intersection_) 

831 if coin: 

832 def _reprs(*crs): 

833 return _and(*map(repr, crs)) 

834 

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

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

837 elif Z is None: 

838 t = _COMMASPACE_(t, _no_(typename(_numpy.null_space2))) 

839 return t 

840 

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

842 # create the explanation and and throw an IntersectionError 

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

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

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

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

847 _no_itersection(coin, Z)) 

848 raise IntersectionError(t, txt=None) 

849 

850 

851def _tri3near2far(r1, r2, h, coin): 

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

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

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

855 

856 

857def _tri4near2far(c1, r1, c2, r2, coin): 

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

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

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

861 

862 

863def _tri5perturbs(eps, r): 

864 # perturb the radii to handle this corner case 

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

866 yield _0_0 

867 if eps and eps > 0: 

868 p = max(eps, EPS) 

869 yield p 

870 m = min(p, r) 

871 yield -m 

872 q = max(eps * _4_0, _EPS4e8) 

873 if q > p: 

874 yield q 

875 q = min(q, r) 

876 if q > m: 

877 yield -q 

878 

879# **) MIT License 

880# 

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

882# 

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

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

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

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

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

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

889# 

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

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

892# 

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

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

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

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

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

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

899# OTHER DEALINGS IN THE SOFTWARE.