1

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

3

4 u'''3-Point resection functions L{cassini}, L{collins5}, L{pierlot}, L{pierlotx} and

5 L{tienstra7}, survey functions L{snellius3} and L{wildberger3} and triangle functions

6 L{triAngle}, L{triAngle5}, L{triSide}, L{triSide2} and L{triSide4}.

7

8@note: Functions L{pierlot} and L{pierlotx} are transcoded to Python with permission from

9 U{triangulationPierlot<http://www.Telecom.ULg.ac.BE/triangulation/doc/total_8c.html>} and

10 U{Pierlot<http://www.Telecom.ULg.ac.BE/publi/publications/pierlot/Pierlot2014ANewThree>}.

11'''

12# make sure int/int division yields float quotient

13 from __future__ import division as _; del _ # PYCHOK semicolon

14

15 from pygeodesy.basics import map1, map2, _zip, _ALL_LAZY

16 from pygeodesy.constants import EPS, EPS0, EPS02, INT0, PI, PI2, PI_2, PI_4, \

17 _0_0, _0_5, _1_0, _N_1_0, _2_0, _N_2_0, _4_0, \

18 _16_0, _180_0, _360_0, isnear0, _over, _umod_360

19 from pygeodesy.errors import _and, _or, TriangleError, _ValueError, _xcallable, \

20 _xkwds, _xkwds_pop2

21 from pygeodesy.fmath import favg, Fdot, fidw, fmean, hypot, hypot2_

22 from pygeodesy.fsums import _Fsumf_, fsumf_, fsum1, fsum1f_

23 from pygeodesy.interns import _a_, _A_, _area_, _b_, _B_, _c_, _C_, _coincident_, \

24 _colinear_, _d_, _invalid_, _negative_, _not_, \

25 _rIn_, _SPACE_

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

27 from pygeodesy.named import _NamedTuple, _Pass, Fmt

28# from pygeodesy.streprs import Fmt # from .named

29 from pygeodesy.units import Degrees, Distance, Radians

30 from pygeodesy.utily import acos1, asin1, atan2, sincos2, sincos2_, \

31 sincos2d, sincos2d_

32 from pygeodesy.vector3d import _otherV3d, Vector3d

33

34 from math import cos, degrees, fabs, radians, sin, sqrt

35

36__all__ = _ALL_LAZY.resections

37__version__ = '24.11.27'

38

39_concyclic_ = 'concyclic'

40_PA_ = 'PA'

41_PB_ = 'PB'

42_PC_ = 'PC'

43_pointH_ = 'pointH'

44_pointP_ = 'pointP'

45_positive_ = 'positive'

46_radA_ = 'radA'

47_radB_ = 'radB'

48_radC_ = 'radC'

49

50

51 class Collins5Tuple(_NamedTuple):

52 '''5-Tuple C{(pointP, pointH, a, b, c)} with survey C{pointP}, auxiliary

53 C{pointH}, each an instance of B{C{pointA}}'s (sub-)class and triangle

54 sides C{a}, C{b} and C{c} in C{meter}, conventionally.

55 '''

56 _Names_ = (_pointP_, _pointH_, _a_, _b_, _c_)

57 _Units_ = (_Pass, _Pass, Distance, Distance, Distance)

58

59

60 class ResectionError(_ValueError):

61 '''Error raised for issues in L{pygeodesy.resections}.

62 '''

63 pass

64

65

66 class Survey3Tuple(_NamedTuple):

67 '''3-Tuple C{(PA, PB, PC)} with distance from survey point C{P} to each of

68 the triangle corners C{A}, C{B} and C{C} in C{meter}, conventionally.

69 '''

70 _Names_ = (_PA_, _PB_, _PC_)

71 _Units_ = ( Distance, Distance, Distance)

72

73

74 class Tienstra7Tuple(_NamedTuple):

75 '''7-Tuple C{(pointP, A, B, C, a, b, c)} with survey C{pointP}, interior

76 triangle angles C{A}, C{B} and C{C} in C{degrees} and triangle sides

77 C{a}, C{b} and C{c} in C{meter}, conventionally.

78 '''

79 _Names_ = (_pointP_, _A_, _B_, _C_, _a_, _b_, _c_)

80 _Units_ = (_Pass, Degrees, Degrees, Degrees, Distance, Distance, Distance)

81

82

83 class TriAngle5Tuple(_NamedTuple):

84 '''5-Tuple C{(radA, radB, radC, rIn, area)} with the interior angles at

85 triangle corners C{A}, C{B} and C{C} in C{radians}, the C{InCircle}

86 radius C{rIn} aka C{inradius} in C{meter} and the triangle C{area}

87 in C{meter} I{squared}, conventionally.

88 '''

89 _Names_ = (_radA_, _radB_, _radC_, _rIn_, _area_)

90 _Units_ = ( Radians, Radians, Radians, Distance, _Pass)

91

92

93 class TriSide2Tuple(_NamedTuple):

94 '''2-Tuple C{(a, radA)} with triangle side C{a} in C{meter}, conventionally

95 and angle C{radA} at the opposite triangle corner in C{radians}.

96 '''

97 _Names_ = (_a_, _radA_)

98 _Units_ = ( Distance, Radians)

99

100

101 class TriSide4Tuple(_NamedTuple):

102 '''4-Tuple C{(a, b, radC, d)} with interior angle C{radC} at triangle corner

103 C{C} in C{radians}with the length of triangle sides C{a} and C{b} and

104 with triangle height C{d} perpendicular to triangle side C{c}, in the

105 same units as triangle sides C{a} and C{b}.

106 '''

107 _Names_ = (_a_, _b_, _radC_, _d_)

108 _Units_ = ( Distance, Distance, Radians, Distance)

109

110

111 def _ABC3(useZ, pointA, pointB, pointC):

112 '''(INTERNAL) Helper for L{cassini} and L{tienstra7}.

113 '''

114 return (_otherV3d(useZ=useZ, pointA=pointA),

115 _otherV3d(useZ=useZ, pointB=pointB),

116 _otherV3d(useZ=useZ, pointC=pointC))

117

118

119 def _B3(useZ, point1, point2, point3):

120 '''(INTERNAL) Helper for L{pierlot} and L{pierlotx}.

121 '''

122 return (_otherV3d(useZ=useZ, point1=point1),

123 _otherV3d(useZ=useZ, point2=point2),

124 _otherV3d(useZ=useZ, point3=point3))

125

126

127 def cassini(pointA, pointB, pointC, alpha, beta, useZ=False, **Clas_and_kwds):

128 '''3-Point resection using U{Cassini<https://NL.WikiPedia.org/wiki/Achterwaartse_insnijding>}'s method.

129

130 @arg pointA: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},

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

132 @arg pointB: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},

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

134 @arg pointC: Center point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},

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

136 @arg alpha: Angle subtended by triangle side B{C{pointA}} to B{C{pointC}}

137 (C{degrees}, non-negative).

138 @arg beta: Angle subtended by triangle side B{C{pointB}} to B{C{pointC}}

139 (C{degrees}, non-negative).

140 @kwarg useZ: If C{True}, use and interpolate the Z component, otherwise

141 force C{z=INT0} (C{bool}).

142 @kwarg Clas_and_kwds: Optional class C{B{Clas}=B{pointA}.classof} to

143 return the survey point with optionally other B{C{Clas}}

144 keyword arguments to instantiate the survey point.

145

146 @note: Typically, B{C{pointC}} is between B{C{pointA}} and B{C{pointB}}.

147

148 @return: The survey point, an instance of B{C{Clas}} or B{C{pointA}}'s

149 (sub-)class.

150

151 @raise ResectionError: Near-coincident, -colinear or -concyclic points

152 or negative or invalid B{C{alpha}} or B{C{beta}}.

153

154 @raise TypeError: Invalid B{C{pointA}}, B{C{pointB}} or B{C{pointM}}.

155

156 @see: U{Three Point Resection Problem<https://Dokumen.tips/documents/

157 three-point-resection-problem-introduction-kaestner-burkhardt-method.html>}

158 and functions L{collins5}, L{pierlot}, L{pierlotx} and L{tienstra7}.

159 '''

160

161 def _H(A, C, sa):

162 s, c = sincos2d(sa)

163 if isnear0(s):

164 raise ValueError(_or(_coincident_, _colinear_))

165 t = s, c, c

166 x = Fdot(t, A.x, C.y, -A.y).fover(s)

167 y = Fdot(t, A.y, -C.x, A.x).fover(s)

168 return x, y

169

170 A, B, C = _ABC3(useZ, pointA, pointB, pointC)

171 try:

172 sa, sb = map1(float, alpha, beta)

173 if min(sa, sb) < 0:

174 raise ValueError(_negative_)

175 if fsumf_(_360_0, -sa, -sb) < EPS0:

176 raise ValueError()

177

178 x1, y1 = _H(A, C, sa)

179 x2, y2 = _H(B, C, -sb)

180

181 x = x1 - x2

182 y = y1 - y2

183 if isnear0(x) or isnear0(y):

184 raise ValueError(_SPACE_(_concyclic_, (x, y)))

185

186 m = y / x

187 n = x / y

188 N = n + m

189 if isnear0(N):

190 raise ValueError(_SPACE_(_concyclic_, (m, n, N)))

191

192 t = n, m, _1_0, _N_1_0

193 x = Fdot(t, C.x, x1, C.y, y1).fover(N)

194 y = Fdot(t, y1, C.y, C.x, x1).fover(N)

195 z = _zidw(x, y, useZ, A, B, C)

196 return _Clas(cassini, pointA, Clas_and_kwds, x, y, z)

197

198 except (TypeError, ValueError) as x:

199 raise ResectionError(pointA=pointA, pointB=pointB, pointC=pointC,

200 alpha=alpha, beta=beta, cause=x)

201

202

203 def _Clas(which, point, Clas_and_kwds, *args):

204 '''(INTERNAL) Return a C{B{Clas}=point.classof} survey point.

205 '''

206 Clas, kwds = _xkwds_pop2(Clas_and_kwds, Clas=point.classof)

207 return Clas(*args, **_xkwds(kwds, name=which.__name__))

208

209

210 def collins5(pointA, pointB, pointC, alpha, beta, useZ=False, **Clas_and_kwds):

211 '''3-Point resection using U{Collins<https://Dokumen.tips/documents/

212 three-point-resection-problem-introduction-kaestner-burkhardt-method.html>}' method.

213

214 @arg pointA: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},

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

216 @arg pointB: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},

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

218 @arg pointC: Center point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},

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

220 @arg alpha: Angle subtended by triangle side C{b} from B{C{pointA}} to

221 B{C{pointC}} (C{degrees}, non-negative).

222 @arg beta: Angle subtended by triangle side C{a} from B{C{pointB}} to

223 B{C{pointC}} (C{degrees}, non-negative).

224 @kwarg useZ: If C{True}, use and interpolate the Z component, otherwise

225 force C{z=INT0} (C{bool}).

226 @kwarg Clas_and_kwds: Optional class C{B{Clas}=B{pointA}.classof} to

227 return the survey point with optionally other B{C{Clas}}

228 keyword arguments to instantiate the survey point.

229

230 @note: Typically, B{C{pointC}} is between B{C{pointA}} and B{C{pointB}}.

231

232 @return: L{Collins5Tuple}C{(pointP, pointH, a, b, c)} with survey C{pointP},

233 auxiliary C{pointH}, each an instance of B{C{Clas}} or B{C{pointA}}'s

234 (sub-)class and triangle sides C{a}, C{b} and C{c} in C{meter},

235 conventionally.

236

237 @raise ResectionError: Near-coincident, -colinear or -concyclic points

238 or negative or invalid B{C{alpha}} or B{C{beta}}.

239

240 @raise TypeError: Invalid B{C{pointA}}, B{C{pointB}} or B{C{pointM}}.

241

242 @see: U{Collins' methode<https://NL.WikiPedia.org/wiki/Achterwaartse_insnijding>}

243 and functions L{cassini}, L{pierlot}, L{pierlotx} and L{tienstra7}.

244 '''

245

246 def _azi_len2(A, B, pi2=PI2):

247 v = B.minus(A)

248 r = atan2(v.x, v.y)

249 if r < 0 and pi2:

250 r += pi2

251 return r, v.length

252

253 def _xyz(d, r, A, B, C, useZ):

254 s, c = sincos2(r)

255 x = d * s + A.x # fma(d, s, A.x)

256 y = d * c + A.y # fma(d, c, A.y)

257 z = _zidw(x, y, useZ, A, B, C)

258 return x, y, z

259

260 A, B, C = _ABC3(useZ, pointA, pointB, pointC)

261 try:

262 ra, rb = radians(alpha), radians(beta)

263 if min(ra, rb) < 0:

264 raise ValueError(_negative_)

265

266 sra, srH = sin(ra), sin(ra + rb - PI) # rH = PI - ((PI - ra) + (PI - rb))

267 if isnear0(sra) or isnear0(srH):

268 raise ValueError(_or(_coincident_, _colinear_, _concyclic_))

269

270# za, a = _azi_len2(C, B)

271 zb, b = _azi_len2(C, A)

272 zc, c = _azi_len2(A, B, 0)

273

274# d = c * sin(PI - rb) / srH # B.minus(H).length

275 d = c * sin(PI - ra) / srH # A.minus(H).length

276 r = zc + PI - rb # zh = zc + (PI - rb)

277 H = _xyz(d, r, A, B, C, useZ)

278

279 zh, _ = _azi_len2(C, Vector3d(*H))

280

281# d = a * sin(za - zh) / sin(rb) # B.minus(P).length

282 d = b * sin(zb - zh) / sra # A.minus(P).length

283 r = zh - ra # zb - PI + (PI - ra - (zb - zh))

284 P = _xyz(d, r, A, B, C, useZ)

285

286 P = _Clas(collins5, pointA, Clas_and_kwds, *P)

287 H = _Clas(collins5, pointA, Clas_and_kwds, *H)

288 a = B.minus(C).length

289 return Collins5Tuple(P, H, a, b, c, name=collins5.__name__)

290

291 except (TypeError, ValueError) as x:

292 raise ResectionError(pointA=pointA, pointB=pointB, pointC=pointC,

293 alpha=alpha, beta=beta, cause=x)

294

295

296 def pierlot(point1, point2, point3, alpha12, alpha23, useZ=False, eps=EPS,

297 **Clas_and_kwds):

298 '''3-Point resection using U{Pierlot<http://www.Telecom.ULg.ac.BE/publi/publications/

299 pierlot/Pierlot2014ANewThree>}'s method C{ToTal} with I{approximate} limits for

300 the (pseudo-)singularities.

301

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

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

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

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

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

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

308 @arg alpha12: Angle subtended from B{C{point1}} to B{C{point2}} or

309 B{C{alpha2 - alpha1}} (C{degrees}).

310 @arg alpha23: Angle subtended from B{C{point2}} to B{C{point3}} or

311 B{C{alpha3 - alpha2}}(C{degrees}).

312 @kwarg useZ: If C{True}, interpolate the survey point's Z component,

313 otherwise use C{z=INT0} (C{bool}).

314 @kwarg eps: Tolerance for C{cot}angent (pseudo-)singularities (C{float}).

315 @kwarg Clas_and_kwds: Optional class C{B{Clas}=B{point1}.classof} to

316 return the survey point with optionally other B{C{Clas}}

317 keyword arguments to instantiate the survey point.

318

319 @note: Typically, B{C{point1}}, B{C{point2}} and B{C{point3}} are ordered

320 by angle, modulo 360, counter-clockwise.

321

322 @return: The survey (or robot) point, an instance of B{C{Clas}} or B{C{point1}}'s

323 (sub-)class.

324

325 @raise ResectionError: Near-coincident, -colinear or -concyclic points

326 or invalid B{C{alpha12}} or B{C{alpha23}} or

327 non-positive B{C{eps}}.

328

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

330

331 @see: I{Pierlot}'s C function U{triangulationPierlot<http://www.Telecom.ULg.ac.BE/

332 triangulation/doc/total_8c_source.html>}, U{V. Pierlot, M. Van Droogenbroeck,

333 "A New Three Object Triangulation Algorithm for Mobile Robot Positioning"

334 <https://ORBi.ULiege.BE/bitstream/2268/157469/1/Pierlot2014ANewThree.pdf>},

335 U{Vincent Pierlot, Marc Van Droogenbroeck, "18 Triangulation Algorithms for 2D

336 Positioning (also known as the Resection Problem)"<http://www.Telecom.ULg.ac.BE/

337 triangulation>} and functions L{pierlotx}, L{cassini}, L{collins5} and L{tienstra7}.

338 '''

339

340 def _cot(s, c): # -eps < I{approximate} cotangent < eps

341 if eps > 0:

342 return c / (min(s, -eps) if s < 0 else max(s, eps))

343 t = Fmt.PARENSPACED(eps=eps)

344 raise ValueError(_SPACE_(t, _not_, _positive_))

345

346 B1, B2, B3 = _B3(useZ, point1, point2, point3)

347 try:

348 xyz = _pierlot3(B1, B2, B3, alpha12, alpha23, useZ, _cot)

349 return _Clas(pierlot, point1, Clas_and_kwds, *xyz)

350

351 except (TypeError, ValueError) as x:

352 raise ResectionError(point1=point1, point2=point2, point3=point3,

353 alpha12=alpha12, alpha23=alpha23, eps=eps, cause=x)

354

355

356 def _pierlot3(B1, B2, B3, a12, a23, useZ, _cot):

357 '''(INTERNAL) Shared L{pierlot} and L{pierlotx}.

358 '''

359 x1_, y1_, _ = B1.minus(B2).xyz3

360 x3_, y3_, _ = B3.minus(B2).xyz3

361

362 s12, c12, s23, c23 = sincos2d_(a12, a23)

363 # cot31 = (1 - cot12 * cot23) / (cot12 + cot32)

364 # = (1 - c12 / s12 * c23 / s23) / (c12 / s12 + c23 / s23)

365 # = (1 - (c12 * c23) / (s12 * s23)) / (c12 * s23 + s12 * c23) / (s12 * s23)

366 # = (s12 * s23 - c12 * c23) / (c12 * s23 + s12 * c23)

367 # = c31 / s31

368 cot31 = _cot(fsum1f_(c12 * s23, s12 * c23), # s31

369 fsum1f_(s12 * s23, -c12 * c23)) # c31

370

371 K = _Fsumf_(x3_ * x1_, cot31 * (y3_ * x1_),

372 y3_ * y1_, -cot31 * (x3_ * y1_))

373 if K:

374 cot12 = _cot(s12, c12)

375 cot23 = _cot(s23, c23)

376

377 # x12 = x1_ + cot12 * y1_

378 # y12 = y1_ - cot12 * x1_

379

380 # x23 = x3_ - cot23 * y3_

381 # y23 = y3_ + cot23 * x3_

382

383 # x31 = x3_ + x1_ + cot31 * (y3_ - y1_)

384 # y31 = y3_ + y1_ - cot31 * (x3_ - x1_)

385

386 # x12 - x23 = x1_ + cot12 * y1_ - x3_ + cot23 * y3_

387 X12_23 = _Fsumf_(x1_, cot12 * y1_, -x3_, cot23 * y3_)

388 # y12 - y23 = y1_ - cot12 * x1_ - y3_ - cot23 * x3_

389 Y12_23 = _Fsumf_(y1_, -cot12 * x1_, -y3_, -cot23 * x3_)

390

391 # x31 - x23 = x3_ + x1_ + cot31 * (y3_ - y1_) - x3_ + cot23 * y3_

392 # = x1_ + cot31 * y3_ - cot31 * y1_ + cot23 * y3_

393 X31_23 = _Fsumf_(x1_, -cot31 * y1_, cot31 * y3_, cot23 * y3_)

394 # y31 - y23 = y3_ + y1_ - cot31 * (x3_ - x1_) - y3_ - cot23 * x3_

395 # = y1_ - cot31 * x3_ + cot31 * x1_ - cot23 * x3_

396 Y31_23 = _Fsumf_(y1_, cot31 * x1_, -cot31 * x3_, -cot23 * x3_)

397

398 # d = (x12 - x23) * (y23 - y31) + (x31 - x23) * (y12 - y23)

399 # = (x31 - x23) * (y12 - y23) - (x12 - x23) * (y31 - y23)

400 # x = (d * B2.x + K * Y12_23).fover(d)

401 # y = (d * B2.y - K * X12_23).fover(d)

402 x, y = _pierlotxy2(B2, -K, Y12_23, X12_23, (X31_23 * Y12_23 -

403 X12_23 * Y31_23))

404 else:

405 x, y, _ = B2.xyz3

406 return x, y, _zidw(x, y, useZ, B1, B2, B3)

407

408

409 def pierlotx(point1, point2, point3, alpha1, alpha2, alpha3, useZ=False,

410 **Clas_and_kwds):

411 '''3-Point resection using U{Pierlot<http://www.Telecom.ULg.ac.BE/publi/

412 publications/pierlot/Pierlot2014ANewThree>}'s method C{ToTal} with

413 I{exact} limits for the (pseudo-)singularities.

414

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

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

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

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

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

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

421 @arg alpha1: Angle at B{C{point1}} (C{degrees}, counter-clockwise).

422 @arg alpha2: Angle at B{C{point2}} (C{degrees}, counter-clockwise).

423 @arg alpha3: Angle at B{C{point3}} (C{degrees}, counter-clockwise).

424 @kwarg useZ: If C{True}, interpolate the survey point's Z component,

425 otherwise use C{z=INT0} (C{bool}).

426 @kwarg Clas_and_kwds: Optional class C{B{Clas}=B{point1}.classof} to

427 return the survey point with optionally other B{C{Clas}}

428 keyword arguments to instantiate the survey point.

429

430 @return: The survey (or robot) point, an instance of B{C{Clas}} or

431 B{C{point1}}'s (sub-)class.

432

433 @raise ResectionError: Near-coincident, -colinear or -concyclic points or

434 invalid B{C{alpha1}}, B{C{alpha2}} or B{C{alpha3}}.

435

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

437

438 @see: I{Pierlot}'s C function U{triangulationPierlot2<http://www.Telecom.ULg.ac.BE/

439 triangulation/doc/total_8c_source.html>} and function L{pierlot}, L{cassini},

440 L{collins5} and L{tienstra7}.

441 '''

442

443 def _a_z_Bs(Bs, *alphas):

444 ds = map2(_umod_360, alphas) # 0 <= alphas < 360

445 ds, Bs = zip(*sorted(_zip(ds, Bs))) # unzip

446 for p, d, B in _zip(ds, _rotate(ds), Bs):

447 d -= p # a12 = a2 - a1, ...

448 z = isnear0(fabs(d) % _180_0)

449 yield d, z, B

450

451 def _cot(s, c): # I{exact} cotangent

452 try:

453 return (c / s) # if c else _copysign_0_0(s)

454 except ZeroDivisionError:

455 raise ValueError(_or(_coincident_, _colinear_))

456

457 Bs = _B3(useZ, point1, point2, point3)

458 try:

459 Cs = [0] # pseudo-global, passing the exception Case

460 xyz = _pierlotx3(_a_z_Bs(Bs, alpha1, alpha2, alpha3),

461 useZ, _cot, Cs.append)

462 return _Clas(pierlotx, point1, Clas_and_kwds, *xyz)

463

464 except (TypeError, ValueError) as x:

465 raise ResectionError(point1=point1, point2=point2, point3=point3, C=Cs.pop(),

466 alpha1=alpha1, alpha2=alpha2, alpha3=alpha3, cause=x)

467

468

469 def _pierlotx3(a_z_Bs, useZ, _cot, Cs):

470 '''(INTERNAL) Core of L{pierlotx}.

471 '''

472 (a12, z12, B1), \

473 (a23, z23, B2), \

474 (a31, z31, B3) = a_z_Bs

475 if z12 and not z23:

476 Cs(1)

477 elif z23 and not z31:

478 Cs(2)

479 a23, B1, B2, B3 = a31, B2, B3, B1

480 elif z31 and not z12:

481 Cs(3)

482 a23, B2, B3 = a12, B3, B2

483 else:

484 Cs(4)

485 return _pierlot3(B1, B2, B3, a12, a23, useZ, _cot)

486

487 x1_, y1_, _ = B1.minus(B3).xyz3

488 x2_, y2_, _ = B2.minus(B3).xyz3

489

490 K = _Fsumf_(y1_ * x2_, -x1_ * y2_)

491 if K:

492 cot23 = _cot(*sincos2d(a23))

493

494 # x23 = x2_ + cot23 * y2_

495 # y23 = y2_ - cot23 * x2_

496

497 # x31 = x1_ + cot23 * y1_

498 # y31 = y1_ - cot23 * x1_

499

500 # x31 - x23 = x1_ + cot23 * y1_ - x2_ - cot23 * y2_

501 X31_23 = _Fsumf_(x1_, cot23 * y1_, -x2_, -cot23 * y2_)

502 # y31 - y23 = y1_ - cot23 * x1_ - y2_ + cot23 * x2_

503 Y31_23 = _Fsumf_(y1_, -cot23 * x1_, -y2_, cot23 * x2_)

504

505 # d = (x31 - x23) * (x2_ - x1_) + (y31 - y23) * (y2_ - y1_)

506 # x = (D * B3.x - K * Y31_23).fover(d)

507 # y = (D * B3.y + K * X31_23).fover(d)

508 x, y = _pierlotxy2(B3, K, Y31_23, X31_23, (X31_23 * _Fsumf_(x2_, -x1_) +

509 Y31_23 * _Fsumf_(y2_, -y1_)))

510 else:

511 x, y, _ = B3.xyz3

512 return x, y, _zidw(x, y, useZ, B1, B2, B3)

513

514

515 def _pierlotxy2(B, K, X, Y, D):

516 '''(INTERNAL) Helper for C{_pierlot3} and C{_pierlotx3}.

517 '''

518 d = float(D)

519 if isnear0(d):

520 raise ValueError(_or(_coincident_, _colinear_, _concyclic_))

521 x = (D * B.x - K * X).fover(d)

522 y = (D * B.y + K * Y).fover(d)

523 return x, y

524

525

526 def _rotate(xs, n=1):

527 '''Rotate list or tuple C{xs} by C{n} items, right if C{n > 0} else left.

528 '''

529 return xs[n:] + xs[:n]

530

531

532 def snellius3(a, b, degC, alpha, beta):

533 '''Snellius' surveying using U{Snellius Pothenot<https://WikiPedia.org/wiki/Snellius–Pothenot_problem>}.

534

535 @arg a: Length of the triangle side between corners C{B} and C{C} and opposite of

536 triangle corner C{A} (C{scalar}, non-negative C{meter}, conventionally).

537 @arg b: Length of the triangle side between corners C{C} and C{A} and opposite of

538 triangle corner C{B} (C{scalar}, non-negative C{meter}, conventionally).

539 @arg degC: Angle at triangle corner C{C}, opposite triangle side C{c} (non-negative C{degrees}).

540 @arg alpha: Angle subtended by triangle side B{C{b}} (non-negative C{degrees}).

541 @arg beta: Angle subtended by triangle side B{C{a}} (non-negative C{degrees}).

542

543 @return: L{Survey3Tuple}C{(PA, PB, PC)} with distance from survey point C{P} to

544 each of the triangle corners C{A}, C{B} and C{C}, same units as triangle

545 sides B{C{a}}, B{C{b}} and B{C{c}}.

546

547 @raise TriangleError: Invalid B{C{a}}, B{C{b}} or B{C{degC}} or negative B{C{alpha}}

548 or B{C{beta}}.

549

550 @see: Function L{wildberger3}.

551 '''

552 try:

553 a, b, degC, alpha, beta = t = map1(float, a, b, degC, alpha, beta)

554 if min(t) < 0:

555 raise ValueError(_negative_)

556 ra, rb, rC = map1(radians, alpha, beta, degC)

557

558 r = fsum1f_(ra, rb, rC) * _0_5

559 k = PI - r

560 if min(k, r) < 0:

561 raise ValueError(_or(_coincident_, _colinear_))

562

563 sa, sb = map1(sin, ra, rb)

564 p = atan2(sa * a, sb * b)

565 sp, cp, sr, cr = sincos2_(PI_4 - p, r)

566 p = atan2(sp * sr, cp * cr)

567 pa = k + p

568 pb = k - p

569

570 if fabs(sb) > fabs(sa):

571 pc = fabs(a * sin(pb) / sb)

572 elif sa:

573 pc = fabs(b * sin(pa) / sa)

574 else:

575 raise ValueError(_or(_colinear_, _coincident_))

576

577 pa = _triSide(b, pc, fsumf_(PI, -ra, -pa))

578 pb = _triSide(a, pc, fsumf_(PI, -rb, -pb))

579 return Survey3Tuple(pa, pb, pc, name=snellius3.__name__)

580

581 except (TypeError, ValueError) as x:

582 raise TriangleError(a=a, b=b, degC=degC, alpha=alpha, beta=beta, cause=x)

583

584

585 def tienstra7(pointA, pointB, pointC, alpha, beta=None, gamma=None,

586 useZ=False, **Clas_and_kwds):

587 '''3-Point resection using U{Tienstra<https://WikiPedia.org/wiki/Tienstra_formula>}'s formula.

588

589 @arg pointA: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or

590 C{Vector2Tuple} if C{B{useZ}=False}).

591 @arg pointB: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or

592 C{Vector2Tuple} if C{B{useZ}=False}).

593 @arg pointC: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or

594 C{Vector2Tuple} if C{B{useZ}=False}).

595 @arg alpha: Angle subtended by triangle side C{a} from B{C{pointB}} to B{C{pointC}}

596 (C{degrees}, non-negative).

597 @kwarg beta: Angle subtended by triangle side C{b} from B{C{pointA}} to B{C{pointC}}

598 (C{degrees}, non-negative) or C{None} if C{B{gamma} is not None}.

599 @kwarg gamma: Angle subtended by triangle side C{c} from B{C{pointA}} to B{C{pointB}}

600 (C{degrees}, non-negative) or C{None} if C{B{beta} is not None}.

601 @kwarg useZ: If C{True}, use and interpolate the Z component, otherwise force C{z=INT0}

602 (C{bool}).

603 @kwarg Clas_and_kwds: Optional class C{B{Clas}=B{pointA}.classof} to return the survey

604 point with optionally other B{C{Clas}} keyword arguments to instantiate

605 the survey point.

606

607 @note: Points B{C{pointA}}, B{C{pointB}} and B{C{pointC}} are ordered clockwise.

608

609 @return: L{Tienstra7Tuple}C{(pointP, A, B, C, a, b, c)} with survey C{pointP}, an

610 instance of B{C{Clas}} or B{C{pointA}}'s (sub-)class, with triangle angles C{A}

611 at B{C{pointA}}, C{B} at B{C{pointB}} and C{C} at B{C{pointC}} in C{degrees}

612 and with triangle sides C{a}, C{b} and C{c} in C{meter}, conventionally.

613

614 @raise ResectionError: Near-coincident, -colinear or -concyclic points or sum of

615 B{C{alpha}}, B{C{beta}} and B{C{gamma}} not C{360} or negative

616 B{C{alpha}}, B{C{beta}} or B{C{gamma}}.

617

618 @raise TypeError: Invalid B{C{pointA}}, B{C{pointB}} or B{C{pointC}}.

619

620 @see: U{3-Point Resection Solver<http://MesaMike.org/geocache/GC1B0Q9/tienstra/>},

621 U{V. Pierlot, M. Van Droogenbroeck, "A New Three Object Triangulation..."

622 <http://www.Telecom.ULg.ac.BE/publi/publications/pierlot/Pierlot2014ANewThree/>},

623 U{18 Triangulation Algorithms...<http://www.Telecom.ULg.ac.BE/triangulation/>} and

624 functions L{cassini}, L{collins5}, L{pierlot} and L{pierlotx}.

625 '''

626

627 def _deg_ks(r, s, ks, N):

628 if isnear0(fsumf_(PI, r, -s)): # r + (PI2 - s) == PI

629 raise ValueError(Fmt.PARENSPACED(concyclic=N))

630 # k = 1 / (cot(r) - cot(s))

631 # = 1 / (cos(r) / sin(r) - cos(s) / sin(s))

632 # = 1 / (cos(r) * sin(s) - cos(s) * sin(r)) / (sin(r) * sin(s))

633 # = sin(r) * sin(s) / (cos(r) * sin(s) - cos(s) * sin(r))

634 sr, cr, ss, cs = sincos2_(r, s)

635 c = fsum1f_(cr * ss, -cs * sr)

636 if isnear0(c):

637 raise ValueError(Fmt.PARENSPACED(cotan=N))

638 ks.append(sr * ss / c)

639 return Degrees(degrees(r), name=N) # C degrees

640

641 A, B, C = _ABC3(useZ, pointA, pointB, pointC)

642 try:

643 sa, sb, sc = map1(radians, alpha, (beta or 0), (gamma or 0))

644 if beta is None:

645 if gamma is None:

646 raise ValueError(_and(Fmt.EQUAL(beta=beta), Fmt.EQUAL(gamma=gamma)))

647 sb = fsumf_(PI2, -sa, -sc)

648 elif gamma is None:

649 sc = fsumf_(PI2, -sa, -sb)

650 else: # subtended angles must add to 360 degrees

651 r = fsum1f_(sa, sb, sc)

652 if fabs(r - PI2) > EPS:

653 raise ValueError(Fmt.EQUAL(sum=degrees(r)))

654 if min(sa, sb, sc) < 0:

655 raise ValueError(_negative_)

656

657 # triangle sides

658 a = B.minus(C).length

659 b = A.minus(C).length

660 c = A.minus(B).length

661

662 ks = [] # 3 Ks and triangle angles

663 dA = _deg_ks(_triAngle(b, c, a), sa, ks, _A_)

664 dB = _deg_ks(_triAngle(a, c, b), sb, ks, _B_)

665 dC = _deg_ks(_triAngle(a, b, c), sc, ks, _C_)

666

667 k = fsum1(ks)

668 if isnear0(k):

669 raise ValueError(Fmt.EQUAL(K=k))

670 x = Fdot(ks, A.x, B.x, C.x).fover(k)

671 y = Fdot(ks, A.y, B.y, C.y).fover(k)

672 z = _zidw(x, y, useZ, A, B, C)

673

674 P = _Clas(tienstra7, pointA, Clas_and_kwds, x, y, z)

675 return Tienstra7Tuple(P, dA, dB, dC, a, b, c, name=tienstra7.__name__)

676

677 except (TypeError, ValueError) as x:

678 raise ResectionError(pointA=pointA, pointB=pointB, pointC=pointC,

679 alpha=alpha, beta=beta, gamma=gamma, cause=x)

680

681

682 def triAngle(a, b, c):

683 '''Compute one angle of a triangle.

684

685 @arg a: Adjacent triangle side length (C{scalar}, non-negative

686 C{meter}, conventionally).

687 @arg b: Adjacent triangle side length (C{scalar}, non-negative

688 C{meter}, conventionally).

689 @arg c: Opposite triangle side length (C{scalar}, non-negative

690 C{meter}, conventionally).

691

692 @return: Angle in C{radians} at triangle corner C{C}, opposite

693 triangle side B{C{c}}.

694

695 @raise TriangleError: Invalid or negative B{C{a}}, B{C{b}} or B{C{c}}.

696

697 @see: Functions L{triAngle5} and L{triSide}.

698 '''

699 try:

700 return _triAngle(a, b, c)

701 except (TypeError, ValueError) as x:

702 raise TriangleError(a=a, b=b, c=c, cause=x)

703

704

705 def _triAngle(a, b, c):

706 # (INTERNAL) To allow callers to embellish errors

707 a, b, c = map1(float, a, b, c)

708 if a < b:

709 a, b = b, a

710 if b < 0 or c < 0:

711 raise ValueError(_negative_)

712 if a < EPS0:

713 raise ValueError(_coincident_)

714 b_a = b / a

715 if b_a < EPS0:

716 raise ValueError(_coincident_)

717 t = fsumf_(_1_0, b_a**2, -(c / a)**2) / (b_a * _2_0)

718 return acos1(t)

719

720

721 def triAngle5(a, b, c):

722 '''Compute the angles of a triangle.

723

724 @arg a: Length of the triangle side opposite of triangle corner C{A}

725 (C{scalar}, non-negative C{meter}, conventionally).

726 @arg b: Length of the triangle side opposite of triangle corner C{B}

727 (C{scalar}, non-negative C{meter}, conventionally).

728 @arg c: Length of the triangle side opposite of triangle corner C{C}

729 (C{scalar}, non-negative C{meter}, conventionally).

730

731 @return: L{TriAngle5Tuple}C{(radA, radB, radC, rIn, area)} with angles

732 C{radA}, C{radB} and C{radC} at triangle corners C{A}, C{B}

733 and C{C}, all in C{radians}, the C{InCircle} radius C{rIn}

734 aka C{inradius}, same units as triangle sides B{C{a}},

735 B{C{b}} and B{C{c}} and the triangle C{area} in those same

736 units I{squared}.

737

738 @raise TriangleError: Invalid or negative B{C{a}}, B{C{b}} or B{C{c}}.

739

740 @see: Functions L{triAngle} and L{triArea}.

741 '''

742 try:

743 x, y, z = map1(float, a, b, c)

744 ab = x < y

745 if ab:

746 x, y = y, x

747 bc = y < z

748 if bc:

749 y, z = z, y

750

751 if z > EPS0: # z = min(a, b, c)

752 s = fsum1f_(z, y, x) * _0_5

753 sa, sb, r = (s - x), (s - y), (s - z)

754 r *= _over(sa * sb, s)

755 if r < EPS02:

756 raise ValueError(_coincident_)

757 r = sqrt(r)

758 rA = atan2(r, sa) * _2_0

759 rB = atan2(r, sb) * _2_0

760 rC = fsumf_(PI, -rA, -rB)

761 if min(rA, rB, rC) < 0:

762 raise ValueError(_colinear_)

763 s *= r # Heron's area

764 elif z < 0:

765 raise ValueError(_negative_)

766 else: # 0 <= c <= EPS0

767 rA = rB = PI_2

768 rC = r = s = _0_0

769

770 if bc:

771 rB, rC = rC, rB

772 if ab:

773 rA, rB = rB, rA

774 return TriAngle5Tuple(rA, rB, rC, r, s, name=triAngle5.__name__)

775

776 except (TypeError, ValueError) as x:

777 raise TriangleError(a=a, b=b, c=c, cause=x)

778

779

780 def triArea(a, b, c):

781 '''Compute the area of a triangle using U{Heron's<https://

782 WikiPedia.org/wiki/Heron%27s_formula>} C{stable} formula.

783

784 @arg a: Length of the triangle side opposite of triangle corner C{A}

785 (C{scalar}, non-negative C{meter}, conventionally).

786 @arg b: Length of the triangle side opposite of triangle corner C{B}

787 (C{scalar}, non-negative C{meter}, conventionally).

788 @arg c: Length of the triangle side opposite of triangle corner C{C}

789 (C{scalar}, non-negative C{meter}, conventionally).

790

791 @return: The triangle area (C{float}, conventionally C{meter} or

792 same units as B{C{a}}, B{C{b}} and B{C{c}} I{squared}).

793

794 @raise TriangleError: Invalid or negative B{C{a}}, B{C{b}} or B{C{c}}.

795 '''

796 try:

797 r, y, x = sorted(map1(float, a, b, c))

798 if r > 0: # r = min(a, b, c)

799 ab = x - y

800 bc = y - r

801 y += r

802 r = (x + y) * (r - ab) * (r + ab) * (x + bc)

803 if r:

804 r = sqrt(r / _16_0)

805 elif r < 0:

806 raise ValueError(_negative_)

807 return r

808

809 except (TypeError, ValueError) as x:

810 raise TriangleError(a=a, b=b, c=c, cause=x)

811

812

813 def triSide(a, b, radC):

814 '''Compute one side of a triangle.

815

816 @arg a: Adjacent triangle side length (C{scalar},

817 non-negative C{meter}, conventionally).

818 @arg b: Adjacent triangle side length (C{scalar},

819 non-negative C{meter}, conventionally).

820 @arg radC: Angle included by sides B{C{a}} and B{C{b}},

821 opposite triangle side C{c} (C{radians}).

822

823 @return: Length of triangle side C{c}, opposite triangle

824 corner C{C} and angle B{C{radC}}, same units as

825 B{C{a}} and B{C{b}}.

826

827 @raise TriangleError: Invalid B{C{a}}, B{C{b}} or B{C{radC}}.

828

829 @see: Functions L{sqrt_a}, L{triAngle}, L{triSide2} and L{triSide4}.

830 '''

831 try:

832 return _triSide(a, b, radC)

833 except (TypeError, ValueError) as x:

834 raise TriangleError(a=a, b=b, radC=radC, cause=x)

835

836

837 def _triSide(a, b, radC):

838 # (INTERNAL) To allow callers to embellish errors

839 a, b, r = t = map1(float, a, b, radC)

840 if min(t) < 0:

841 raise ValueError(_negative_)

842

843 if a < b:

844 a, b = b, a

845 if a > EPS0:

846 ba = b / a

847 c2 = fsumf_(_1_0, ba**2, _N_2_0 * ba * cos(r))

848 if c2 > EPS02:

849 return a * sqrt(c2)

850 elif c2 < 0:

851 raise ValueError(_invalid_)

852 return hypot(a, b)

853

854

855 def triSide2(b, c, radB):

856 '''Compute a side and its opposite angle of a triangle.

857

858 @arg b: Adjacent triangle side length (C{scalar},

859 non-negative C{meter}, conventionally).

860 @arg c: Adjacent triangle side length (C{scalar},

861 non-negative C{meter}, conventionally).

862 @arg radB: Angle included by sides B{C{a}} and B{C{c}},

863 opposite triangle side C{b} (C{radians}).

864

865 @return: L{TriSide2Tuple}C{(a, radA)} with triangle angle

866 C{radA} in C{radians} and length of the opposite

867 triangle side C{a}, same units as B{C{b}} and B{C{c}}.

868

869 @raise TriangleError: Invalid B{C{b}} or B{C{c}} or either

870 B{C{b}} or B{C{radB}} near zero.

871

872 @see: Functions L{sqrt_a}, L{triSide} and L{triSide4}.

873 '''

874 try:

875 return _triSide2(b, c, radB)

876 except (TypeError, ValueError) as x:

877 raise TriangleError(b=b, c=c, radB=radB, cause=x)

878

879

880 def _triSide2(b, c, radB):

881 # (INTERNAL) To allow callers to embellish errors

882 b, c, rB = map1(float, b, c, radB)

883 if min(b, c, rB) < 0:

884 raise ValueError(_negative_)

885 sB, cB = sincos2(rB)

886 if isnear0(sB):

887 if not isnear0(b):

888 raise ValueError(_invalid_)

889 a, rA = ((b + c), PI) if cB < 0 else (fabs(b - c), _0_0)

890 elif isnear0(b):

891 raise ValueError(_invalid_)

892 else:

893 rA = fsumf_(PI, -rB, -asin1(c * sB / b))

894 a = sin(rA) * b / sB

895 return TriSide2Tuple(a, rA, name=triSide2.__name__)

896

897

898 def triSide4(radA, radB, c):

899 '''Compute two sides and the height of a triangle.

900

901 @arg radA: Angle at triangle corner C{A}, opposite triangle side C{a}

902 (non-negative C{radians}).

903 @arg radB: Angle at triangle corner C{B}, opposite triangle side C{b}

904 (non-negative C{radians}).

905 @arg c: Length of triangle side between triangle corners C{A} and C{B},

906 (C{scalar}, non-negative C{meter}, conventionally).

907

908 @return: L{TriSide4Tuple}C{(a, b, radC, d)} with triangle sides C{a} and

909 C{b} and triangle height C{d} perpendicular to triangle side

910 B{C{c}}, all in the same units as B{C{c}} and interior angle

911 C{radC} in C{radians} at triangle corner C{C}, opposite

912 triangle side B{C{c}}.

913

914 @raise TriangleError: Invalid or negative B{C{radA}}, B{C{radB}} or B{C{c}}.

915

916 @see: U{Triangulation, Surveying<https://WikiPedia.org/wiki/Triangulation_(surveying)>}

917 and functions L{sqrt_a}, L{triSide} and L{triSide2}.

918 '''

919 try:

920 rA, rB, c = map1(float, radA, radB, c)

921 rC = fsumf_(PI, -rA, -rB)

922 if min(rC, rA, rB, c) < 0:

923 raise ValueError(_negative_)

924 sa, ca, sb, cb = sincos2_(rA, rB)

925 sc = fsum1f_(sa * cb, sb * ca)

926 if sc < EPS0 or min(sa, sb) < 0:

927 raise ValueError(_invalid_)

928 sc = c / sc

929 return TriSide4Tuple((sa * sc), (sb * sc), rC, (sa * sb * sc),

930 name=triSide4.__name__)

931

932 except (TypeError, ValueError) as x:

933 raise TriangleError(radA=radA, radB=radB, c=c, cause=x)

934

935

936 def wildberger3(a, b, c, alpha, beta, R3=min):

937 '''Snellius' surveying using U{Rational Trigonometry

938 <https://WikiPedia.org/wiki/Snellius–Pothenot_problem>}.

939

940 @arg a: Length of the triangle side between corners C{B} and C{C} and opposite of

941 triangle corner C{A} (C{scalar}, non-negative C{meter}, conventionally).

942 @arg b: Length of the triangle side between corners C{C} and C{A} and opposite of

943 triangle corner C{B} (C{scalar}, non-negative C{meter}, conventionally).

944 @arg c: Length of the triangle side between corners C{A} and C{B} and opposite of

945 triangle corner C{C} (C{scalar}, non-negative C{meter}, conventionally).

946 @arg alpha: Angle subtended by triangle side B{C{b}} (C{degrees}, non-negative).

947 @arg beta: Angle subtended by triangle side B{C{a}} (C{degrees}, non-negative).

948 @kwarg R3: Callable to determine C{R3} from C{(R3 - C)**2 = D}, typically standard

949 Python function C{min} or C{max}, invoked with 2 arguments.

950

951 @return: L{Survey3Tuple}C{(PA, PB, PC)} with distance from survey point C{P} to

952 each of the triangle corners C{A}, C{B} and C{C}, same units as B{C{a}},

953 B{C{b}} and B{C{c}}.

954

955 @raise TriangleError: Invalid B{C{a}}, B{C{b}} or B{C{c}} or negative B{C{alpha}} or

956 B{C{beta}} or B{C{R3}} not C{callable}.

957

958 @see: U{Wildberger, Norman J.<https://Math.Sc.Chula.ac.TH/cjm/content/

959 survey-article-greek-geometry-rational-trigonometry-and-snellius-–-pothenot-surveying>},

960 U{Devine Proportions, page 252<http://www.MS.LT/derlius/WildbergerDivineProportions.pdf>}

961 and function L{snellius3}.

962 '''

963 def _s(x):

964 return sin(x)**2

965

966 def _vpa(r3, q2, q3, s2, s3):

967 r1 = s2 * q3 / s3

968 r = r1 * r3 * _4_0

969 n = (r - _Fsumf_(r1, r3, -q2)**2).fover(s3)

970 if n < 0 or r < EPS0:

971 raise ValueError(_coincident_)

972 return sqrt((n / r) * q3) if n else _0_0

973

974 try:

975 a, b, c, da, db = q = map1(float, a, b, c, alpha, beta)

976 if min(q) < 0:

977 raise ValueError(_negative_)

978

979 q1, q2, q3 = q = a**2, b**2, c**2

980 if min(q) < EPS02:

981 raise ValueError(_coincident_)

982

983 ra, rb = map1(radians, da, db)

984 s1, s2, s3 = s = map1(_s, rb, ra, ra + rb) # rb, ra!

985 if min(s) < EPS02:

986 raise ValueError(_or(_coincident_, _colinear_))

987

988 q4 = hypot2_(*q) * _2_0 # a**4 + ...

989 Qs = _Fsumf_(*q) # == hypot2_(a, b, c)

990 d0 = (Qs**2 - q4).fmul(s1 * s2).fover(s3)

991 if d0 < 0:

992 raise ValueError(_negative_)

993 s += _Fsumf_(*s), # == fsum1(s),

994 C0 = Fdot(s, q1, q2, q3, -Qs * _0_5)

995 r3 = C0.fover(-s3) # C0 /= -s3

996 if d0 > EPS02: # > c0

997 _xcallable(R3=R3)

998 d0 = sqrt(d0)

999 r3 = R3(float(C0 + d0), float(C0 - d0)) # XXX min or max

1000

1001 pa = _vpa(r3, q2, q3, s2, s3)

1002 pb = _vpa(r3, q1, q3, s1, s3)

1003 pc = favg(_triSide2(b, pa, ra).a,

1004 _triSide2(a, pb, rb).a)

1005 return Survey3Tuple(pa, pb, pc, name=wildberger3.__name__)

1006

1007 except (TypeError, ValueError) as x:

1008 raise TriangleError(a=a, b=b, c=c, alpha=alpha, beta=beta, R3=R3, cause=x)

1009

1010

1011 def _zidw(x, y, useZ, *ABC):

1012 if useZ: # interpolate z or coplanar with A, B and C?

1013 t = tuple(_.z for _ in ABC)

1014 v = Vector3d(x, y, fmean(t))

1015 z = fidw(t, (v.minus(T).length for T in ABC))

1016 else:

1017 z = INT0

1018 return z

1019

1020# **) MIT License

1021#

1022# Copyright (C) 2016-2025 -- mrJean1 at Gmail -- All Rights Reserved.

1023#

1024# Permission is hereby granted, free of charge, to any person obtaining a

1025# copy of this software and associated documentation files (the "Software"),

1026# to deal in the Software without restriction, including without limitation

1027# the rights to use, copy, modify, merge, publish, distribute, sublicense,

1028# and/or sell copies of the Software, and to permit persons to whom the

1029# Software is furnished to do so, subject to the following conditions:

1030#

1031# The above copyright notice and this permission notice shall be included

1032# in all copies or substantial portions of the Software.

1033#

1034# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS

1035# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

1036# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL

1037# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR

1038# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,

1039# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR

1040# OTHER DEALINGS IN THE SOFTWARE.