Coverage for pygeodesy/resections.py: 97%

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1 

2# -*- coding: utf-8 -*- 

3 

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

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

6L{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 

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

14 

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

16from 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, _copysign_0_0, isnear0, \ 

19 _over, _umod_360 

20from pygeodesy.errors import _and, _or, TriangleError, _ValueError, _xcallable, \ 

21 _xkwds, _xkwds_pop2 

22from pygeodesy.fmath import favg, Fdot, fidw, fmean, hypot, hypot2_ 

23from pygeodesy.fsums import Fsum, fsumf_, fsum1, fsum1f_ 

24from pygeodesy.interns import _a_, _A_, _area_, _b_, _B_, _c_, _C_, _coincident_, \ 

25 _colinear_, _d_, _eps_, _invalid_, _negative_, _not_, \ 

26 _rIn_, _SPACE_ 

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

28from pygeodesy.named import _NamedTuple, _Pass, Fmt 

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

30from pygeodesy.units import Degrees, Distance, Radians 

31from pygeodesy.utily import acos1, asin1, sincos2, sincos2_, sincos2d, sincos2d_ 

32from pygeodesy.vector3d import _otherV3d, Vector3d 

33 

34from math import cos, atan2, degrees, fabs, radians, sin, sqrt 

35 

36__all__ = _ALL_LAZY.resections 

37__version__ = '24.04.04' 

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 

51class 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 

60def _F1(*xs): # class 

61 '''(INTERNAL) An L{Fsum}, 1-primed. 

62 ''' 

63 F = Fsum(_1_0, *xs) 

64 F += _N_1_0 

65 return F 

66 

67 

68class ResectionError(_ValueError): 

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

70 ''' 

71 pass 

72 

73 

74class Survey3Tuple(_NamedTuple): 

75 '''3-Tuple C{(PA, PB, PC)} with distance from survey point C{P} to each of 

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

77 ''' 

78 _Names_ = (_PA_, _PB_, _PC_) 

79 _Units_ = ( Distance, Distance, Distance) 

80 

81 

82class Tienstra7Tuple(_NamedTuple): 

83 '''7-Tuple C{(pointP, A, B, C, a, b, c)} with survey C{pointP}, interior 

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

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

86 ''' 

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

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

89 

90 

91class TriAngle5Tuple(_NamedTuple): 

92 '''5-Tuple C{(radA, radB, radC, rIn, area)} with the interior angles at 

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

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

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

96 ''' 

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

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

99 

100 

101class TriSide2Tuple(_NamedTuple): 

102 '''2-Tuple C{(a, radA)} with triangle side C{a} in C{meter}, conventionally 

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

104 ''' 

105 _Names_ = (_a_, _radA_) 

106 _Units_ = ( Distance, Radians) 

107 

108 

109class TriSide4Tuple(_NamedTuple): 

110 '''4-Tuple C{(a, b, radC, d)} with interior angle C{radC} at triangle corner 

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

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

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

114 ''' 

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

116 _Units_ = ( Distance, Distance, Radians, Distance) 

117 

118 

119def _ABC3(useZ, pointA, pointB, pointC): 

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

121 ''' 

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

123 _otherV3d(useZ=useZ, pointB=pointB), 

124 _otherV3d(useZ=useZ, pointC=pointC)) 

125 

126 

127def _B3(useZ, point1, point2, point3): 

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

129 ''' 

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

131 _otherV3d(useZ=useZ, point2=point2), 

132 _otherV3d(useZ=useZ, point3=point3)) 

133 

134 

135def cassini(pointA, pointB, pointC, alpha, beta, useZ=False, **Clas_and_kwds): 

136 '''3-Point resection using U{Cassini<https://NL.WikiPedia.org/wiki/Achterwaartse_insnijding>}'s method. 

137 

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

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

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

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

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

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

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

145 (C{degrees}, non-negative). 

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

147 (C{degrees}, non-negative). 

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

149 force C{z=INT0} (C{bool}). 

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

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

152 keyword arguments to instantiate the survey point. 

153 

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

155 

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

157 (sub-)class. 

158 

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

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

161 

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

163 

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

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

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

167 ''' 

168 

169 def _H(A, C, sa): 

170 s, c = sincos2d(sa) 

171 if isnear0(s): 

172 raise ValueError(_or(_coincident_, _colinear_)) 

173 t = s, c, c 

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

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

176 return x, y 

177 

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

179 try: 

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

181 if min(sa, sb) < 0: 

182 raise ValueError(_negative_) 

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

184 raise ValueError() 

185 

186 x1, y1 = _H(A, C, sa) 

187 x2, y2 = _H(B, C, -sb) 

188 

189 x = x1 - x2 

190 y = y1 - y2 

191 if isnear0(x) or isnear0(y): 

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

193 

194 m = y / x 

195 n = x / y 

196 N = n + m 

197 if isnear0(N): 

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

199 

200 t = n, m, _1_0, _N_1_0 

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

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

203 z = _zidw(x, y, useZ, A, B, C) 

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

205 

206 except (TypeError, ValueError) as x: 

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

208 alpha=alpha, beta=beta, cause=x) 

209 

210 

211def _Clas(where, point, Clas_and_kwds, *args): 

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

213 ''' 

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

215 return Clas(*args, **_xkwds(kwds, name=where.__name__)) 

216 

217 

218def collins5(pointA, pointB, pointC, alpha, beta, useZ=False, **Clas_and_kwds): 

219 '''3-Point resection using U{Collins<https://Dokumen.tips/documents/ 

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

221 

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

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

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

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

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

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

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

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

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

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

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

233 force C{z=INT0} (C{bool}). 

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

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

236 keyword arguments to instantiate the survey point. 

237 

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

239 

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

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

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

243 conventionally. 

244 

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

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

247 

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

249 

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

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

252 ''' 

253 

254 def _azi_len2(A, B, pi2=PI2): 

255 v = B.minus(A) 

256 r = atan2(v.x, v.y) 

257 if r < 0 and pi2: 

258 r += pi2 

259 return r, v.length 

260 

261 def _xyz(d, r, A, B, C, useZ): 

262 s, c = sincos2(r) 

263 x = A.x + d * s 

264 y = A.y + d * c 

265 z = _zidw(x, y, useZ, A, B, C) 

266 return x, y, z 

267 

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

269 try: 

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

271 if min(ra, rb) < 0: 

272 raise ValueError(_negative_) 

273 

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

275 if isnear0(sra) or isnear0(srH): 

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

277 

278# za, a = _azi_len2(C, B) 

279 zb, b = _azi_len2(C, A) 

280 zc, c = _azi_len2(A, B, 0) 

281 

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

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

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

285 H = _xyz(d, r, A, B, C, useZ) 

286 

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

288 

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

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

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

292 P = _xyz(d, r, A, B, C, useZ) 

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

294 

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

296 a = B.minus(C).length 

297 

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

299 

300 except (TypeError, ValueError) as x: 

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

302 alpha=alpha, beta=beta, cause=x) 

303 

304 

305def pierlot(point1, point2, point3, alpha12, alpha23, useZ=False, eps=EPS, 

306 **Clas_and_kwds): 

307 '''3-Point resection using U{Pierlot<http://www.Telecom.ULg.ac.BE/publi/publications/ 

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

309 the (pseudo-)singularities. 

310 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

326 keyword arguments to instantiate the survey point. 

327 

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

329 by angle, modulo 360, counter-clockwise. 

330 

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

332 (sub-)class. 

333 

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

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

336 non-positive B{C{eps}}. 

337 

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

339 

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

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

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

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

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

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

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

347 ''' 

348 

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

350 if eps > 0: 

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

352 raise ValueError(_SPACE_(_eps_, _not_, _positive_)) 

353 

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

355 try: 

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

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

358 

359 except (TypeError, ValueError) as x: 

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

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

362 

363 

364def _pierlot3(B1, B2, B3, a12, a23, useZ, cot): 

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

366 ''' 

367 x1_, y1_, _ = B1.minus(B2).xyz 

368 x3_, y3_, _ = B3.minus(B2).xyz 

369 

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

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

372 # = (1 - c12 / s12 * c23 / s23) / (c12 / s12 + c23 / s23) 

373 # = (1 - (c12 * c23) / (s12 * s23)) / (c12 * s23 + s12 * c23) / (s12 * s23) 

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

375 # = c31 / s31 

376 cot31 = cot(fsum1f_(c12 * s23, s12 * c23), # s31 

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

378 

379 K = _F1(x3_ * x1_, cot31 * (y3_ * x1_), 

380 y3_ * y1_, -cot31 * (x3_ * y1_)) 

381 if K: 

382 cot12 = cot(s12, c12) 

383 cot23 = cot(s23, c23) 

384 

385 # x12 = x1_ + cot12 * y1_ 

386 # y12 = y1_ - cot12 * x1_ 

387 

388 # x23 = x3_ - cot23 * y3_ 

389 # y23 = y3_ + cot23 * x3_ 

390 

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

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

393 

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

395 X12_23 = _F1(x1_, cot12 * y1_, -x3_, cot23 * y3_) 

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

397 Y12_23 = _F1(y1_, -cot12 * x1_, -y3_, -cot23 * x3_) 

398 

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

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

401 X31_23 = _F1(x1_, -cot31 * y1_, cot31 * y3_, cot23 * y3_) 

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

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

404 Y31_23 = _F1(y1_, cot31 * x1_, -cot31 * x3_, -cot23 * x3_) 

405 

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

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

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

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

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

411 X12_23 * Y31_23)) 

412 else: 

413 x, y, _ = B2.xyz 

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

415 

416 

417def pierlotx(point1, point2, point3, alpha1, alpha2, alpha3, useZ=False, 

418 **Clas_and_kwds): 

419 '''3-Point resection using U{Pierlot<http://www.Telecom.ULg.ac.BE/publi/ 

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

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

422 

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

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

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

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

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

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

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

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

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

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

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

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

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

436 keyword arguments to instantiate the survey point. 

437 

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

439 (sub-)class. 

440 

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

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

443 

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

445 

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

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

448 L{collins5} and L{tienstra7}. 

449 ''' 

450 

451 def _a_z_Bs(Bs, *alphas): 

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

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

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

455 d -= p # a12 = a2 - a1, ... 

456 z = isnear0(fabs(d) % _180_0) 

457 yield d, z, B 

458 

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

460 try: 

461 return (c / s) if c else _copysign_0_0(s) 

462 except ZeroDivisionError: 

463 raise ValueError(_or(_coincident_, _colinear_)) 

464 

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

466 try: 

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

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

469 useZ, _cot, Cs.append) 

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

471 

472 except (TypeError, ValueError) as x: 

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

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

475 

476 

477def _pierlotx3(a_z_Bs, useZ, cot, Cs): 

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

479 ''' 

480 (a12, z12, B1), \ 

481 (a23, z23, B2), \ 

482 (a31, z31, B3) = a_z_Bs 

483 if z12 and not z23: 

484 Cs(1) 

485 elif z23 and not z31: 

486 Cs(2) 

487 a23, B1, B2, B3 = a31, B2, B3, B1 

488 elif z31 and not z12: 

489 Cs(3) 

490 a23, B2, B3 = a12, B3, B2 

491 else: 

492 Cs(4) 

493 return _pierlot3(B1, B2, B3, a12, a23, useZ, cot) 

494 

495 x1_, y1_, _ = B1.minus(B3).xyz 

496 x2_, y2_, _ = B2.minus(B3).xyz 

497 

498 K = _F1(y1_ * x2_, -x1_ * y2_) 

499 if K: 

500 cot23 = cot(*sincos2d(a23)) 

501 

502 # x23 = x2_ + cot23 * y2_ 

503 # y23 = y2_ - cot23 * x2_ 

504 

505 # x31 = x1_ + cot23 * y1_ 

506 # y31 = y1_ - cot23 * x1_ 

507 

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

509 X31_23 = _F1(x1_, cot23 * y1_, -x2_, -cot23 * y2_) 

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

511 Y31_23 = _F1(y1_, -cot23 * x1_, -y2_, cot23 * x2_) 

512 

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

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

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

516 x, y = _pierlotxy2(B3, K, Y31_23, X31_23, (X31_23 * _F1(x2_, -x1_) + 

517 Y31_23 * _F1(y2_, -y1_))) 

518 else: 

519 x, y, _ = B3.xyz 

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

521 

522 

523def _pierlotxy2(B, K, X, Y, D): 

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

525 ''' 

526 d = float(D) 

527 if isnear0(d): 

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

529 x = (D * B.x - K * X).fover(d) 

530 y = (D * B.y + K * Y).fover(d) 

531 return x, y 

532 

533 

534def _rotate(xs, n=1): 

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

536 ''' 

537 return xs[n:] + xs[:n] 

538 

539 

540def snellius3(a, b, degC, alpha, beta): 

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

542 

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

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

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

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

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

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

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

550 

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

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

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

554 

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

556 or B{C{beta}}. 

557 

558 @see: Function L{wildberger3}. 

559 ''' 

560 try: 

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

562 if min(t) < 0: 

563 raise ValueError(_negative_) 

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

565 

566 r = fsum1f_(ra, rb, rC) * _0_5 

567 k = PI - r 

568 if min(k, r) < 0: 

569 raise ValueError(_or(_coincident_, _colinear_)) 

570 

571 sa, sb = map1(sin, ra, rb) 

572 p = atan2(sa * a, sb * b) 

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

574 p = atan2(sp * sr, cp * cr) 

575 pa = k + p 

576 pb = k - p 

577 

578 if fabs(sb) > fabs(sa): 

579 pc = fabs(a * sin(pb) / sb) 

580 elif sa: 

581 pc = fabs(b * sin(pa) / sa) 

582 else: 

583 raise ValueError(_or(_colinear_, _coincident_)) 

584 

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

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

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

588 

589 except (TypeError, ValueError) as x: 

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

591 

592 

593def tienstra7(pointA, pointB, pointC, alpha, beta=None, gamma=None, 

594 useZ=False, **Clas_and_kwds): 

595 '''3-Point resection using U{Tienstra<https://WikiPedia.org/wiki/Tienstra_formula>}'s formula. 

596 

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

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

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

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

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

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

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

604 (C{degrees}, non-negative). 

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

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

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

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

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

610 (C{bool}). 

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

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

613 the survey point. 

614 

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

616 

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

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

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

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

621 

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

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

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

625 

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

627 

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

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

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

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

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

633 ''' 

634 

635 def _deg_ks(r, s, ks, N): 

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

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

638 # k = 1 / (cot(r) - cot(s)) 

639 # = 1 / (cos(r) / sin(r) - cos(s) / sin(s)) 

640 # = 1 / (cos(r) * sin(s) - cos(s) * sin(r)) / (sin(r) * sin(s)) 

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

642 sr, cr, ss, cs = sincos2_(r, s) 

643 c = fsum1f_(cr * ss, -cs * sr) 

644 if isnear0(c): 

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

646 ks.append(sr * ss / c) 

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

648 

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

650 try: 

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

652 if beta is None: 

653 if gamma is None: 

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

655 sb = fsumf_(PI2, -sa, -sc) 

656 elif gamma is None: 

657 sc = fsumf_(PI2, -sa, -sb) 

658 else: # subtended angles must add to 360 degrees 

659 r = fsum1f_(sa, sb, sc) 

660 if fabs(r - PI2) > EPS: 

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

662 if min(sa, sb, sc) < 0: 

663 raise ValueError(_negative_) 

664 

665 # triangle sides 

666 a = B.minus(C).length 

667 b = A.minus(C).length 

668 c = A.minus(B).length 

669 

670 ks = [] # 3 Ks and triangle angles 

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

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

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

674 

675 k = fsum1(ks, floats=True) 

676 if isnear0(k): 

677 raise ValueError(Fmt.EQUAL(K=k)) 

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

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

680 z = _zidw(x, y, useZ, A, B, C) 

681 

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

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

684 

685 except (TypeError, ValueError) as x: 

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

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

688 

689 

690def triAngle(a, b, c): 

691 '''Compute one angle of a triangle. 

692 

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

694 C{meter}, conventionally). 

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

696 C{meter}, conventionally). 

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

698 C{meter}, conventionally). 

699 

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

701 triangle side B{C{c}}. 

702 

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

704 

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

706 ''' 

707 try: 

708 return _triAngle(a, b, c) 

709 except (TypeError, ValueError) as x: 

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

711 

712 

713def _triAngle(a, b, c): 

714 # (INTERNAL) To allow callers to embellish errors 

715 a, b, c = map1(float, a, b, c) 

716 if a < b: 

717 a, b = b, a 

718 if b < 0 or c < 0: 

719 raise ValueError(_negative_) 

720 if a < EPS0: 

721 raise ValueError(_coincident_) 

722 b_a = b / a 

723 if b_a < EPS0: 

724 raise ValueError(_coincident_) 

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

726 return acos1(t) 

727 

728 

729def triAngle5(a, b, c): 

730 '''Compute the angles of a triangle. 

731 

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

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

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

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

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

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

738 

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

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

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

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

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

744 units I{squared}. 

745 

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

747 

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

749 ''' 

750 try: 

751 x, y, z = map1(float, a, b, c) 

752 ab = x < y 

753 if ab: 

754 x, y = y, x 

755 bc = y < z 

756 if bc: 

757 y, z = z, y 

758 

759 if z > EPS0: # z = min(a, b, c) 

760 s = fsum1f_(z, y, x) * _0_5 

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

762 r *= _over(sa * sb, s) 

763 if r < EPS02: 

764 raise ValueError(_coincident_) 

765 r = sqrt(r) 

766 rA = atan2(r, sa) * _2_0 

767 rB = atan2(r, sb) * _2_0 

768 rC = fsumf_(PI, -rA, -rB) 

769 if min(rA, rB, rC) < 0: 

770 raise ValueError(_colinear_) 

771 s *= r # Heron's area 

772 elif z < 0: 

773 raise ValueError(_negative_) 

774 else: # 0 <= c <= EPS0 

775 rA = rB = PI_2 

776 rC = r = s = _0_0 

777 

778 if bc: 

779 rB, rC = rC, rB 

780 if ab: 

781 rA, rB = rB, rA 

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

783 

784 except (TypeError, ValueError) as x: 

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

786 

787 

788def triArea(a, b, c): 

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

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

791 

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

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

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

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

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

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

798 

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

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

801 

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

803 ''' 

804 try: 

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

806 if r > 0: # r = min(a, b, c) 

807 ab = x - y 

808 bc = y - r 

809 y += r 

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

811 if r: 

812 r = sqrt(r / _16_0) 

813 elif r < 0: 

814 raise ValueError(_negative_) 

815 return r 

816 

817 except (TypeError, ValueError) as x: 

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

819 

820 

821def triSide(a, b, radC): 

822 '''Compute one side of a triangle. 

823 

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

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

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

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

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

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

830 

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

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

833 B{C{a}} and B{C{b}}. 

834 

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

836 

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

838 ''' 

839 try: 

840 return _triSide(a, b, radC) 

841 except (TypeError, ValueError) as x: 

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

843 

844 

845def _triSide(a, b, radC): 

846 # (INTERNAL) To allow callers to embellish errors 

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

848 if min(t) < 0: 

849 raise ValueError(_negative_) 

850 

851 if a < b: 

852 a, b = b, a 

853 if a > EPS0: 

854 ba = b / a 

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

856 if c2 > EPS02: 

857 return a * sqrt(c2) 

858 elif c2 < 0: 

859 raise ValueError(_invalid_) 

860 return hypot(a, b) 

861 

862 

863def triSide2(b, c, radB): 

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

865 

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

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

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

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

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

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

872 

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

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

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

876 

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

878 B{C{b}} or B{C{radB}} near zero. 

879 

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

881 ''' 

882 try: 

883 return _triSide2(b, c, radB) 

884 except (TypeError, ValueError) as x: 

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

886 

887 

888def _triSide2(b, c, radB): 

889 # (INTERNAL) To allow callers to embellish errors 

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

891 if min(b, c, rB) < 0: 

892 raise ValueError(_negative_) 

893 sB, cB = sincos2(rB) 

894 if isnear0(sB): 

895 if not isnear0(b): 

896 raise ValueError(_invalid_) 

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

898 elif isnear0(b): 

899 raise ValueError(_invalid_) 

900 else: 

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

902 a = sin(rA) * b / sB 

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

904 

905 

906def triSide4(radA, radB, c): 

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

908 

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

910 (non-negative C{radians}). 

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

912 (non-negative C{radians}). 

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

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

915 

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

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

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

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

920 triangle side B{C{c}}. 

921 

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

923 

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

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

926 ''' 

927 try: 

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

929 rC = fsumf_(PI, -rA, -rB) 

930 if min(rC, rA, rB, c) < 0: 

931 raise ValueError(_negative_) 

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

933 sc = fsum1f_(sa * cb, sb * ca) 

934 if sc < EPS0 or min(sa, sb) < 0: 

935 raise ValueError(_invalid_) 

936 sc = c / sc 

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

938 name=triSide4.__name__) 

939 

940 except (TypeError, ValueError) as x: 

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

942 

943 

944def wildberger3(a, b, c, alpha, beta, R3=min): 

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

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

947 

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

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

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

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

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

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

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

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

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

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

958 

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

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

961 B{C{b}} and B{C{c}}. 

962 

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

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

965 

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

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

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

969 and function L{snellius3}. 

970 ''' 

971 def _s(x): 

972 return sin(x)**2 

973 

974 def _vpa(r3, q2, q3, s2, s3): 

975 r1 = s2 * q3 / s3 

976 r = r1 * r3 * _4_0 

977 n = (r - _F1(r1, r3, -q2)**2).fover(s3) 

978 if n < 0 or r < EPS0: 

979 raise ValueError(_coincident_) 

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

981 

982 try: 

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

984 if min(q) < 0: 

985 raise ValueError(_negative_) 

986 

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

988 if min(q) < EPS02: 

989 raise ValueError(_coincident_) 

990 

991 ra, rb = map1(radians, da, db) 

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

993 if min(s) < EPS02: 

994 raise ValueError(_or(_coincident_, _colinear_)) 

995 

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

997 Qs = _F1(*q) # == hypot2_(a, b, c) 

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

999 if d0 < 0: 

1000 raise ValueError(_negative_) 

1001 s += _F1(*s), # == fsum1(s), 

1002 C0 = Fdot(s, q1, q2, q3, -Qs * _0_5) 

1003 r3 = C0.fover(-s3) # C0 /= -s3 

1004 if d0 > EPS02: # > c0 

1005 _xcallable(R3=R3) 

1006 d0 = sqrt(d0) 

1007 r3 = R3(float(C0 + d0), float(C0 - d0)) # XXX min or max 

1008 

1009 pa = _vpa(r3, q2, q3, s2, s3) 

1010 pb = _vpa(r3, q1, q3, s1, s3) 

1011 pc = favg(_triSide2(b, pa, ra).a, 

1012 _triSide2(a, pb, rb).a) 

1013 return Survey3Tuple(pa, pb, pc, name=wildberger3.__name__) 

1014 

1015 except (TypeError, ValueError) as x: 

1016 raise TriangleError(a=a, b=b, c=c, alpha=alpha, beta=beta, R3=R3, cause=x) 

1017 

1018 

1019def _zidw(x, y, useZ, *ABC): 

1020 if useZ: # interpolate z or coplanar with A, B and C? 

1021 t = tuple(_.z for _ in ABC) 

1022 v = Vector3d(x, y, fmean(t)) 

1023 z = fidw(t, (v.minus(T).length for T in ABC)) 

1024 else: 

1025 z = INT0 

1026 return z 

1027 

1028# **) MIT License 

1029# 

1030# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved. 

1031# 

1032# Permission is hereby granted, free of charge, to any person obtaining a 

1033# copy of this software and associated documentation files (the "Software"), 

1034# to deal in the Software without restriction, including without limitation 

1035# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1036# and/or sell copies of the Software, and to permit persons to whom the 

1037# Software is furnished to do so, subject to the following conditions: 

1038# 

1039# The above copyright notice and this permission notice shall be included 

1040# in all copies or substantial portions of the Software. 

1041# 

1042# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

1043# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1044# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

1045# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

1046# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

1047# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

1048# OTHER DEALINGS IN THE SOFTWARE.