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, typename 

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, isnear0, _over, _umod_360 

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

20 _xkwds, _xkwds_pop2 

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

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

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

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

25 _colinear_, _d_, _invalid_, _negative_, _rIn_, _SPACE_ 

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

27from pygeodesy.named import _NamedTuple, _Pass, Fmt 

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

29from pygeodesy.units import Degrees, Distance, Radians 

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

31 sincos2d, sincos2d_ 

32from pygeodesy.vector3d import _otherV3d, Vector3d 

33 

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

35 

36__all__ = _ALL_LAZY.resections 

37__version__ = '25.05.04' 

38 

39_concyclic_ = 'concyclic' 

40_PA_ = 'PA' 

41_PB_ = 'PB' 

42_PC_ = 'PC' 

43_pointH_ = 'pointH' 

44_pointP_ = 'pointP' 

45_radA_ = 'radA' 

46_radB_ = 'radB' 

47_radC_ = 'radC' 

48 

49 

50class Collins5Tuple(_NamedTuple): 

51 '''5-Tuple C{(pointP, pointH, a, b, c)} with survey C{pointP}, auxiliary 

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

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

54 ''' 

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

56 _Units_ = (_Pass, _Pass, Distance, Distance, Distance) 

57 

58 

59class ResectionError(_ValueError): 

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

61 ''' 

62 pass 

63 

64 

65class Survey3Tuple(_NamedTuple): 

66 '''3-Tuple C{(PA, PB, PC)} with distance from survey point C{P} to each of 

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

68 ''' 

69 _Names_ = (_PA_, _PB_, _PC_) 

70 _Units_ = ( Distance, Distance, Distance) 

71 

72 

73class Tienstra7Tuple(_NamedTuple): 

74 '''7-Tuple C{(pointP, A, B, C, a, b, c)} with survey C{pointP}, interior 

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

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

77 ''' 

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

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

80 

81 

82class TriAngle5Tuple(_NamedTuple): 

83 '''5-Tuple C{(radA, radB, radC, rIn, area)} with the interior angles at 

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

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

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

87 ''' 

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

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

90 

91 

92class TriSide2Tuple(_NamedTuple): 

93 '''2-Tuple C{(a, radA)} with triangle side C{a} in C{meter}, conventionally 

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

95 ''' 

96 _Names_ = (_a_, _radA_) 

97 _Units_ = ( Distance, Radians) 

98 

99 

100class TriSide4Tuple(_NamedTuple): 

101 '''4-Tuple C{(a, b, radC, d)} with interior angle C{radC} at triangle corner 

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

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

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

105 ''' 

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

107 _Units_ = ( Distance, Distance, Radians, Distance) 

108 

109 

110def _ABC3(useZ, pointA, pointB, pointC): 

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

112 ''' 

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

114 _otherV3d(useZ=useZ, pointB=pointB), 

115 _otherV3d(useZ=useZ, pointC=pointC)) 

116 

117 

118def _B3(useZ, point1, point2, point3): 

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

120 ''' 

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

122 _otherV3d(useZ=useZ, point2=point2), 

123 _otherV3d(useZ=useZ, point3=point3)) 

124 

125 

126def cassini(pointA, pointB, pointC, alpha, beta, useZ=False, **Clas_and_kwds): 

127 '''3-Point resection using U{Cassini<https://NL.WikiPedia.org/wiki/Achterwaartse_insnijding>}'s method. 

128 

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

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

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

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

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

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

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

136 (C{degrees}, non-negative). 

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

138 (C{degrees}, non-negative). 

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

140 force C{z=INT0} (C{bool}). 

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

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

143 keyword arguments to instantiate the survey point. 

144 

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

146 

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

148 (sub-)class. 

149 

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

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

152 

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

154 

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

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

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

158 ''' 

159 

160 def _H(A, C, sa): 

161 s, c = sincos2d(sa) 

162 if isnear0(s): 

163 raise ValueError(_or(_coincident_, _colinear_)) 

164 t = s, c, c 

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

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

167 return x, y 

168 

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

170 try: 

171 sa, sb = _noneg(alpha, beta) 

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

173 raise ValueError(_colinear_) 

174 

175 x1, y1 = _H(A, C, sa) 

176 x2, y2 = _H(B, C, -sb) 

177 

178 x = x1 - x2 

179 y = y1 - y2 

180 if isnear0(x) or isnear0(y): 

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

182 

183 m = y / x 

184 n = x / y 

185 N = n + m 

186 if isnear0(N): 

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

188 

189 t = n, m, _1_0, _N_1_0 

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

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

192 z = _zidw(x, y, useZ, A, B, C) 

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

194 

195 except (TypeError, ValueError) as x: 

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

197 alpha=alpha, beta=beta, cause=x) 

198 

199 

200def _Clas(which, point, Clas_and_kwds, *args): 

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

202 ''' 

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

204 return Clas(*args, **_xkwds(kwds, name=typename(which))) 

205 

206 

207def collins5(pointA, pointB, pointC, alpha, beta, useZ=False, **Clas_and_kwds): 

208 '''3-Point resection using U{Collins<https://Dokumen.tips/documents/ 

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

210 

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

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

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

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

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

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

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

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

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

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

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

222 force C{z=INT0} (C{bool}). 

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

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

225 keyword arguments to instantiate the survey point. 

226 

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

228 

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

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

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

232 conventionally. 

233 

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

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

236 

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

238 

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

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

241 ''' 

242 

243 def _azi_len2(A, B, pi2=PI2): 

244 v = B.minus(A) 

245 r = atan2(v.x, v.y) 

246 if r < 0 and pi2: 

247 r += pi2 

248 return r, v.length 

249 

250 def _xyz(d, r, A, B, C, useZ): 

251 s, c = sincos2(r) 

252 x = _fma(d, s, A.x) 

253 y = _fma(d, c, A.y) 

254 z = _zidw(x, y, useZ, A, B, C) 

255 return x, y, z 

256 

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

258 try: 

259 ra, rb = t = radians(alpha), radians(beta) 

260 if min(t) < 0: 

261 raise ValueError(_negative_) 

262 

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

264 if isnear0(sra) or isnear0(srH): 

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

266 

267# za, a = _azi_len2(C, B) 

268 zb, b = _azi_len2(C, A) 

269 zc, c = _azi_len2(A, B, 0) 

270 

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

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

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

274 H = _xyz(d, r, A, B, C, useZ) 

275 

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

277 

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

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

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

281 P = _xyz(d, r, A, B, C, useZ) 

282 

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

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

285 a = B.minus(C).length 

286 return Collins5Tuple(P, H, a, b, c, name=typename(collins5)) 

287 

288 except (TypeError, ValueError) as x: 

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

290 alpha=alpha, beta=beta, cause=x) 

291 

292 

293def _noneg(*xs): 

294 '''(INTERNAL) Return non-negative C{float}s. 

295 ''' 

296 xs = tuple(map(float, xs)) 

297 if min(xs) < 0: 

298 raise ValueError(_negative_) 

299 return xs 

300 

301 

302def pierlot(point1, point2, point3, alpha12, alpha23, useZ=False, eps=EPS, 

303 **Clas_and_kwds): 

304 '''3-Point resection using U{Pierlot<http://www.Telecom.ULg.ac.BE/publi/publications/ 

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

306 the (pseudo-)singularities. 

307 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

323 keyword arguments to instantiate the survey point. 

324 

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

326 by angle, modulo 360, counter-clockwise. 

327 

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

329 (sub-)class. 

330 

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

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

333 non-positive B{C{eps}}. 

334 

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

336 

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

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

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

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

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

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

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

344 ''' 

345 

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

347 if eps > 0: 

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

349 t = Fmt.PARENSPACED(eps=eps) 

350 raise ValueError(_SPACE_(t, _invalid_)) 

351 

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

353 try: 

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

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

356 

357 except (TypeError, ValueError) as x: 

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

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

360 

361 

362def _pierlot3(B1, B2, B3, a12, a23, useZ, _cot): 

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

364 ''' 

365 x1_, y1_, _ = B1.minus(B2).xyz3 

366 x3_, y3_, _ = B3.minus(B2).xyz3 

367 

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

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

370 # = (1 - c12 / s12 * c23 / s23) / (c12 / s12 + c23 / s23) 

371 # = (1 - (c12 * c23) / (s12 * s23)) / (c12 * s23 + s12 * c23) / (s12 * s23) 

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

373 # = c31 / s31 

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

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

376 

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

378 y3_ * y1_, -cot31 * (x3_ * y1_)) 

379 if K: 

380 cot12 = _cot(s12, c12) 

381 cot23 = _cot(s23, c23) 

382 

383 # x12 = x1_ + cot12 * y1_ 

384 # y12 = y1_ - cot12 * x1_ 

385 

386 # x23 = x3_ - cot23 * y3_ 

387 # y23 = y3_ + cot23 * x3_ 

388 

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

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

391 

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

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

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

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

396 

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

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

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

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

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

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

403 

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

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

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

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

408 x, y = _pierlotxy2(B2, -K, Y12_23, X12_23, Fdot_(X31_23, Y12_23, 

409 -X12_23, Y31_23)) 

410 else: 

411 x, y, _ = B2.xyz3 

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

413 

414 

415def pierlotx(point1, point2, point3, alpha1, alpha2, alpha3, useZ=False, 

416 **Clas_and_kwds): 

417 '''3-Point resection using U{Pierlot<http://www.Telecom.ULg.ac.BE/publi/ 

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

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

420 

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

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

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

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

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

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

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

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

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

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

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

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

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

434 keyword arguments to instantiate the survey point. 

435 

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

437 B{C{point1}}'s (sub-)class. 

438 

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

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

441 

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

443 

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

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

446 L{collins5} and L{tienstra7}. 

447 ''' 

448 

449 def _a_z_Bs(Bs, *alphas): 

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

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

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

453 d -= p # a12 = a2 - a1, ... 

454 z = isnear0(fabs(d) % _180_0) 

455 yield d, z, B 

456 

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

458 try: 

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

460 except ZeroDivisionError: 

461 raise ValueError(_or(_coincident_, _colinear_)) 

462 

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

464 try: 

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

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

467 useZ, _cot, Cs.append) 

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

469 

470 except (TypeError, ValueError) as x: 

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

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

473 

474 

475def _pierlotx3(a_z_Bs, useZ, _cot, Cs): 

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

477 ''' 

478 (a12, z12, B1), \ 

479 (a23, z23, B2), \ 

480 (a31, z31, B3) = a_z_Bs 

481 if z12 and not z23: 

482 Cs(1) 

483 elif z23 and not z31: 

484 Cs(2) 

485 a23, B1, B2, B3 = a31, B2, B3, B1 

486 elif z31 and not z12: 

487 Cs(3) 

488 a23, B2, B3 = a12, B3, B2 

489 else: 

490 Cs(4) 

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

492 

493 x1_, y1_, _ = B1.minus(B3).xyz3 

494 x2_, y2_, _ = B2.minus(B3).xyz3 

495 

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

497 if K: 

498 cot23 = _cot(*sincos2d(a23)) 

499 

500 # x23 = x2_ + cot23 * y2_ # _fma( cot23, y2_, x2_) 

501 # y23 = y2_ - cot23 * x2_ # _fma(-cot23, x2_, y2_) 

502 

503 # x31 = x1_ + cot23 * y1_ # _fma( cot23, y1_, x1_) 

504 # y31 = y1_ - cot23 * x1_ # _fma(-cot23, x1_, y1_) 

505 

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

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

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

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

510 

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

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

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

514 x, y = _pierlotxy2(B3, K, Y31_23, X31_23, Fdot_(X31_23, _Fsumf_(x2_, -x1_), 

515 Y31_23, _Fsumf_(y2_, -y1_))) 

516 else: 

517 x, y, _ = B3.xyz3 

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

519 

520 

521def _pierlotxy2(B, K, X, Y, D): 

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

523 ''' 

524 d = float(D) 

525 if isnear0(d): 

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

527 x = Fdot_(D, B.x, -K, X).fover(d) 

528 y = Fdot_(D, B.y, K, Y).fover(d) 

529 return x, y 

530 

531 

532def _rotate(xs, n=1): 

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

534 ''' 

535 return xs[n:] + xs[:n] 

536 

537 

538def snellius3(a, b, degC, alpha, beta): 

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

540 

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

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

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

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

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

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

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

548 

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

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

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

552 

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

554 or B{C{beta}}. 

555 

556 @see: Function L{wildberger3}. 

557 ''' 

558 try: 

559 a, b, degC, alpha, beta = _noneg(a, b, degC, alpha, beta) 

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

561 

562 r = fsum1f_(ra, rb, rC) * _0_5 

563 k = PI - r 

564 if min(k, r) < 0: 

565 raise ValueError(_or(_coincident_, _colinear_)) 

566 

567 sa, sb = map1(sin, ra, rb) 

568 p = atan2(sa * a, sb * b) 

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

570 p = atan2(sp * sr, cp * cr) 

571 pa = k + p 

572 pb = k - p 

573 

574 if fabs(sb) > fabs(sa): 

575 pc = fabs(a * sin(pb) / sb) 

576 elif sa: 

577 pc = fabs(b * sin(pa) / sa) 

578 else: 

579 raise ValueError(_or(_colinear_, _coincident_)) 

580 

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

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

583 return Survey3Tuple(pa, pb, pc, name=typename(snellius3)) 

584 

585 except (TypeError, ValueError) as x: 

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

587 

588 

589def tienstra7(pointA, pointB, pointC, alpha, beta=None, gamma=None, 

590 useZ=False, **Clas_and_kwds): 

591 '''3-Point resection using U{Tienstra<https://WikiPedia.org/wiki/Tienstra_formula>}'s formula. 

592 

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

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

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

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

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

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

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

600 (C{degrees}, non-negative). 

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

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

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

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

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

606 (C{bool}). 

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

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

609 the survey point. 

610 

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

612 

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

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

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

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

617 

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

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

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

621 

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

623 

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

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

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

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

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

629 ''' 

630 

631 def _deg_ks(r, s, ks, N): 

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

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

634 # k = 1 / (cot(r) - cot(s)) 

635 # = 1 / (cos(r) / sin(r) - cos(s) / sin(s)) 

636 # = 1 / (cos(r) * sin(s) - cos(s) * sin(r)) / (sin(r) * sin(s)) 

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

638 sr, cr, ss, cs = sincos2_(r, s) 

639 c = fsum1f_(cr * ss, -cs * sr) 

640 if isnear0(c): 

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

642 ks.append(sr * ss / c) 

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

644 

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

646 try: 

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

648 if beta is None: 

649 if gamma is None: 

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

651 sb = fsumf_(PI2, -sa, -sc) 

652 elif gamma is None: 

653 sc = fsumf_(PI2, -sa, -sb) 

654 else: # subtended angles must add to 360 degrees 

655 r = fsum1f_(sa, sb, sc) 

656 if fabs(r - PI2) > EPS: 

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

658 if min(sa, sb, sc) < 0: 

659 raise ValueError(_negative_) 

660 

661 # triangle sides 

662 a = B.minus(C).length 

663 b = A.minus(C).length 

664 c = A.minus(B).length 

665 

666 ks = [] # 3 Ks and triangle angles 

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

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

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

670 

671 k = fsum1(ks) 

672 if isnear0(k): 

673 raise ValueError(Fmt.EQUAL(K=k)) 

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

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

676 z = _zidw(x, y, useZ, A, B, C) 

677 

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

679 return Tienstra7Tuple(P, dA, dB, dC, a, b, c, name=typename(tienstra7)) 

680 

681 except (TypeError, ValueError) as x: 

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

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

684 

685 

686def triAngle(a, b, c): 

687 '''Compute one angle of a triangle. 

688 

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

690 C{meter}, conventionally). 

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

692 C{meter}, conventionally). 

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

694 C{meter}, conventionally). 

695 

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

697 triangle side B{C{c}}. 

698 

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

700 

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

702 ''' 

703 try: 

704 return _triAngle(a, b, c) 

705 except (TypeError, ValueError) as x: 

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

707 

708 

709def _triAngle(a, b, c): 

710 # (INTERNAL) To allow callers to embellish errors 

711 a, b, c = _noneg(a, b, c) 

712 if b > a: 

713 a, b = b, a 

714 if a < EPS0: 

715 raise ValueError(_coincident_) 

716 b_a = b / a 

717 if b_a < EPS0: 

718 raise ValueError(_coincident_) 

719 t = _Fsumf_(_1_0, b_a**2, -(c / a)**2).fover(b_a * _2_0) 

720 return acos1(t) 

721 

722 

723def triAngle5(a, b, c): 

724 '''Compute the angles of a triangle. 

725 

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

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

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

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

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

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

732 

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

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

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

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

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

738 units I{squared}. 

739 

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

741 

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

743 ''' 

744 try: 

745 x, y, z = map1(float, a, b, c) 

746 ab = x < y 

747 if ab: 

748 x, y = y, x 

749 bc = y < z 

750 if bc: 

751 y, z = z, y 

752 

753 if z > EPS0: # z = min(a, b, c) 

754 s = fsum1f_(z, y, x) * _0_5 

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

756 r *= _over(sa * sb, s) 

757 if r < EPS02: 

758 raise ValueError(_coincident_) 

759 r = sqrt(r) 

760 rA = atan2(r, sa) * _2_0 

761 rB = atan2(r, sb) * _2_0 

762 rC = fsumf_(PI, -rA, -rB) 

763 if min(rA, rB, rC) < 0: 

764 raise ValueError(_colinear_) 

765 s *= r # Heron's area 

766 elif z < 0: 

767 raise ValueError(_negative_) 

768 else: # 0 <= c <= EPS0 

769 rA = rB = PI_2 

770 rC = r = s = _0_0 

771 

772 if bc: 

773 rB, rC = rC, rB 

774 if ab: 

775 rA, rB = rB, rA 

776 return TriAngle5Tuple(rA, rB, rC, r, s, name=typename(triAngle5)) 

777 

778 except (TypeError, ValueError) as x: 

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

780 

781 

782def triArea(a, b, c): 

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

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

785 

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

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

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

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

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

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

792 

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

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

795 

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

797 ''' 

798 try: 

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

800 if r > 0: # r = min(a, b, c) 

801 z = r 

802 d = x - y 

803 r = (z + d) * (z - d) 

804 if r: 

805 x += y 

806 r *= (x + z) * (x - z) 

807 if r < 0: 

808 raise ValueError(_negative_) 

809 return sqrt(r / _16_0) if r else _0_0 

810 

811 except (TypeError, ValueError) as x: 

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

813 

814 

815def triSide(a, b, radC): 

816 '''Compute one side of a triangle. 

817 

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

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

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

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

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

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

824 

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

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

827 B{C{a}} and B{C{b}}. 

828 

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

830 

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

832 ''' 

833 try: 

834 return _triSide(a, b, radC) 

835 except (TypeError, ValueError) as x: 

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

837 

838 

839def _triSide(a, b, radC): 

840 # (INTERNAL) To allow callers to embellish errors 

841 a, b, r = _noneg(a, b, radC) 

842 if b < a: 

843 a, b = b, a 

844 if a > EPS0: 

845 ba = b / a 

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

847 if c2 > EPS02: 

848 return a * sqrt(c2) 

849 elif c2 < 0: 

850 raise ValueError(_invalid_) 

851 return hypot(a, b) 

852 

853 

854def triSide2(b, c, radB): 

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

856 

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

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

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

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

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

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

863 

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

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

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

867 

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

869 B{C{b}} or B{C{radB}} near zero. 

870 

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

872 ''' 

873 try: 

874 return _triSide2(b, c, radB) 

875 except (TypeError, ValueError) as x: 

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

877 

878 

879def _triSide2(b, c, radB): 

880 # (INTERNAL) To allow callers to embellish errors 

881 b, c, rB = _noneg(b, c, radB) 

882 sB, cB = sincos2(rB) 

883 if isnear0(b) or isnear0(sB): 

884 if isnear0(b) and isnear0(sB): 

885 if cB < 0: 

886 rA = PI 

887 a = b + c 

888 else: 

889 rA = _0_0 

890 a = fabs(b - c) 

891 else: 

892 raise ValueError(_invalid_) 

893 else: 

894 rC = asin1(c * sB / b) 

895 rA = max(fsumf_(PI, -rB, -rC), _0_0) 

896 a = sin(rA) * b / sB 

897 return TriSide2Tuple(a, rA, name=typename(triSide2)) 

898 

899 

900def triSide4(radA, radB, c): 

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

902 

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

904 (non-negative C{radians}). 

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

906 (non-negative C{radians}). 

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

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

909 

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

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

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

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

914 triangle side B{C{c}}. 

915 

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

917 

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

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

920 ''' 

921 try: 

922 rA, rB, c = _noneg(radA, radB, c) 

923 rC = fsumf_(PI, -rA, -rB) 

924 if rC < 0: 

925 raise ValueError(_negative_) 

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

927 sc = fsum1f_(sa * cb, sb * ca) 

928 if sc < EPS0 or min(sa, sb) < 0: 

929 raise ValueError(_invalid_) 

930 sc = c / sc 

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

932 name=typename(triSide4)) 

933 

934 except (TypeError, ValueError) as x: 

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

936 

937 

938def wildberger3(a, b, c, alpha, beta, R3=min): 

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

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

941 

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

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

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

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

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

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

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

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

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

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

952 

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

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

955 B{C{b}} and B{C{c}}. 

956 

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

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

959 

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

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

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

963 and function L{snellius3}. 

964 ''' 

965 def _s(x): 

966 return sin(x)**2 

967 

968 def _vpa(r3, q2, q3, s2, s3): 

969 r1 = s2 * q3 / s3 

970 r = r1 * r3 * _4_0 

971 R = _Fsumf_(r1, r3, -q2) 

972 R *= R # -(R**2 ... 

973 R -= r # ... - r) / s3 

974 n = -R.fover(s3) 

975 if n < 0 or r < EPS0: 

976 raise ValueError(_coincident_) 

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

978 

979 try: 

980 a, b, c, da, db = _noneg(a, b, c, alpha, beta) 

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

982 if min(q) < EPS02: 

983 raise ValueError(_coincident_) 

984 

985 ra, rb = map1(radians, da, db) 

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

987 if min(s) < EPS02: 

988 raise ValueError(_or(_coincident_, _colinear_)) 

989 

990 Q = _Fsumf_(*q) # == a**2 + b**2 + ... 

991 s += _Fsumf_(*s), # == fsum1(s), 

992 C0 = Fdot(s, q1, q2, q3, -Q * _0_5) 

993 r3 = C0.fover(-s3) # C0 /= -s3 

994 Q *= Q # Q**2 - 2 * (a**4 + b**4 ... 

995 Q -= hypot2_(*q) *_2_0 # ... + c**4) 

996 d0 = Q.fmul(s1 * s2).fover(s3) 

997 if d0 > EPS02: # > c0 

998 _xcallable(R3=R3) 

999 d0 = sqrt(d0) 

1000 r3 = R3(float(C0 + d0), float(C0 - d0)) # XXX min or max 

1001 elif d0 < (-EPS02): 

1002 raise ValueError(_negative_) 

1003 

1004 pa = _vpa(r3, q2, q3, s2, s3) 

1005 pb = _vpa(r3, q1, q3, s1, s3) 

1006 pc = favg(_triSide2(b, pa, ra).a, 

1007 _triSide2(a, pb, rb).a) 

1008 return Survey3Tuple(pa, pb, pc, name=typename(wildberger3)) 

1009 

1010 except (TypeError, ValueError) as x: 

1011 raise TriangleError(a=a, b=b, c=c, alpha=alpha, beta=beta, R3=R3, cause=x) 

1012 

1013 

1014def _zidw(x, y, useZ, *ABC): 

1015 if useZ: # interpolate z or coplanar with A, B and C? 

1016 t = tuple(_.z for _ in ABC) 

1017 v = Vector3d(x, y, fmean(t)) 

1018 z = fidw(t, (v.minus(T).length for T in ABC)) 

1019 else: 

1020 z = INT0 

1021 return z 

1022 

1023# **) MIT License 

1024# 

1025# Copyright (C) 2016-2025 -- mrJean1 at Gmail -- All Rights Reserved. 

1026# 

1027# Permission is hereby granted, free of charge, to any person obtaining a 

1028# copy of this software and associated documentation files (the "Software"), 

1029# to deal in the Software without restriction, including without limitation 

1030# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1031# and/or sell copies of the Software, and to permit persons to whom the 

1032# Software is furnished to do so, subject to the following conditions: 

1033# 

1034# The above copyright notice and this permission notice shall be included 

1035# in all copies or substantial portions of the Software. 

1036# 

1037# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

1038# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1039# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

1040# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

1041# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

1042# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

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