Coverage for pygeodesy/resections.py: 99%

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1 

2# -*- coding: utf-8 -*- 

3 

4u'''3-Point resection functions L{cassini}, L{collins5}, L{pierlot} and L{tienstra7}, 

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

6L{triAngle4}, L{triSide}, L{triSide2} and L{triSide4}. 

7 

8@note: Function L{pierlot} is transcoded with permission from U{triangulationPierlot 

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

10 <http://www.Telecom.ULg.ac.Be/publi/publications/pierlot/Pierlot2014ANewThree>}. 

11''' 

12 

13from pygeodesy.basics import map1, _ALL_LAZY 

14from pygeodesy.constants import EPS, EPS0, EPS02, INT0, PI, PI2, PI_2, PI_4, isnear0, \ 

15 _0_0, _0_5, _1_0, _N_1_0, _2_0, _N_2_0, _4_0, _360_0 

16from pygeodesy.errors import _and, _or, TriangleError, _ValueError, _xkwds 

17from pygeodesy.fmath import favg, Fdot, fidw, fmean, hypot, hypot2_ 

18from pygeodesy.fsums import Fsum, fsum_, fsum1, fsum1_ 

19from pygeodesy.interns import _a_, _A_, _b_, _B_, _c_, _C_, _coincident_, _colinear_, \ 

20 _d_, _invalid_, _negative_, _not_, _rIn_, _SPACE_ 

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

22from pygeodesy.named import Fmt, _NamedTuple, _Pass 

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

24from pygeodesy.units import Degrees, Distance, Radians 

25from pygeodesy.utily import acos1, asin1, sincos2, sincos2_, sincos2d, sincos2d_ 

26from pygeodesy.vector3d import _otherV3d, Vector3d 

27 

28from math import cos, atan2, degrees, fabs, radians, sin, sqrt 

29 

30__all__ = _ALL_LAZY.resections 

31__version__ = '23.04.02' 

32 

33_concyclic_ = 'concyclic' 

34_PA_ = 'PA' 

35_PB_ = 'PB' 

36_PC_ = 'PC' 

37_pointH_ = 'pointH' 

38_pointP_ = 'pointP' 

39_R3__ = 'R3 ' 

40_radA_ = 'radA' 

41_radB_ = 'radB' 

42_radC_ = 'radC' 

43 

44 

45class Collins5Tuple(_NamedTuple): 

46 '''5-Tuple C{(pointP, pointH, a, b, c)} with survey C{pointP}, auxiliary 

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

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

49 ''' 

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

51 _Units_ = (_Pass, _Pass, Distance, Distance, Distance) 

52 

53 

54class ResectionError(_ValueError): 

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

56 ''' 

57 pass 

58 

59 

60class Survey3Tuple(_NamedTuple): 

61 '''3-Tuple C{(PA, PB, PC)} with distance from survey point C{P} to each of 

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

63 ''' 

64 _Names_ = (_PA_, _PB_, _PC_) 

65 _Units_ = ( Distance, Distance, Distance) 

66 

67 

68class Tienstra7Tuple(_NamedTuple): 

69 '''7-Tuple C{(pointP, A, B, C, a, b, c)} with survey C{pointP}, interior 

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

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

72 ''' 

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

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

75 

76 

77class TriAngle4Tuple(_NamedTuple): 

78 '''4-Tuple C{(radA, radB, radC, rIn)} with the interior angles at triangle 

79 corners C{A}, C{B} and C{C} in C{radians} and the C{InCircle} radius 

80 C{rIn} aka C{inradius} in C{meter}, conventionally. 

81 ''' 

82 _Names_ = (_radA_, _radB_, _radC_, _rIn_) 

83 _Units_ = ( Radians, Radians, Radians, Distance) 

84 

85 

86class TriSide2Tuple(_NamedTuple): 

87 '''2-Tuple C{(a, radA)} with triangle side C{a} in C{meter}, conventionally 

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

89 ''' 

90 _Names_ = (_a_, _radA_) 

91 _Units_ = ( Distance, Radians) 

92 

93 

94class TriSide4Tuple(_NamedTuple): 

95 '''4-Tuple C{(a, b, radC, d)} with interior angle C{radC} at triangle corner 

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

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

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

99 ''' 

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

101 _Units_ = ( Distance, Distance, Radians, Distance) 

102 

103 

104def cassini(pointA, pointB, pointC, alpha, beta, useZ=False, Clas=None, **Clas_kwds): 

105 '''3-Point resection using U{Cassini<https://NL.WikiPedia.org/wiki/Achterwaartse_insnijding>}'s method. 

106 

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

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

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

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

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

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

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

114 (C{degrees}, non-negative). 

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

116 (C{degrees}, non-negative). 

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

118 force C{z=INT0} (C{bool}). 

119 @kwarg Clas: Optional class to return the survey point or C{None} for 

120 B{C{pointA}}'s (sub-)class. 

121 @kwarg Clas_kwds: Optional additional keyword argument for the survey 

122 point instance. 

123 

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

125 

126 @return: The survey point, an instance of B{C{Clas}} or if C{B{Clas} is 

127 None} of B{C{pointA}}'s (sub-)class. 

128 

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

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

131 

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

133 

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

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

136 and functions L{pygeodesy.collins5}, L{pygeodesy.pierlot} and 

137 L{pygeodesy.tienstra7}. 

138 ''' 

139 

140 def _H(A, C, sa): 

141 s, c = sincos2d(sa) 

142 if isnear0(s): 

143 raise ValueError(_or(_coincident_, _colinear_)) 

144 t = s, c, c 

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

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

147 return Vector3d(x, y, 0) 

148 

149 A = _otherV3d(useZ=useZ, pointA=pointA) 

150 B = _otherV3d(useZ=useZ, pointB=pointB) 

151 C = _otherV3d(useZ=useZ, pointC=pointC) 

152 

153 try: 

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

155 if min(sa, sb) < 0: 

156 raise ValueError(_negative_) 

157 if fsum_(_360_0, -sa, -sb) < EPS0: 

158 raise ValueError() 

159 

160 H1 = _H(A, C, sa) 

161 H2 = _H(B, C, -sb) 

162 

163 x = H1.x - H2.x 

164 y = H1.y - H2.y 

165 # x, y, _ = H1.minus(H2).xyz 

166 if isnear0(x) or isnear0(y): 

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

168 

169 m = y / x 

170 n = x / y 

171 N = n + m 

172 if isnear0(N): 

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

174 

175 t = n, m, _1_0, _N_1_0 

176 x = Fdot(t, C.x, H1.x, C.y, H1.y).fover(N) 

177 y = Fdot(t, H1.y, C.y, C.x, H1.x).fover(N) 

178 z = _zidw(A, B, C, x, y) if useZ else INT0 

179 

180 clas = Clas or pointA.classof 

181 return clas(x, y, z, **_xkwds(Clas_kwds, name=cassini.__name__)) 

182 

183 except (TypeError, ValueError) as x: 

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

185 alpha=alpha, beta=beta, cause=x) 

186 

187 

188def collins5(pointA, pointB, pointC, alpha, beta, useZ=False, Clas=None, **Clas_kwds): 

189 '''3-Point resection using U{Collins<https://Dokumen.tips/documents/ 

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

191 

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

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

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

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

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

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

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

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

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

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

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

203 force C{z=INT0} (C{bool}). 

204 @kwarg Clas: Optional class to return the survey and auxiliary point 

205 or C{None} for B{C{pointA}}'s (sub-)class. 

206 @kwarg Clas_kwds: Optional additional keyword argument for the survey 

207 and auxiliary point instance. 

208 

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

210 

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

212 auxiliary C{pointH}, each an instance of B{C{Clas}} or if C{B{Clas} 

213 is None} of B{C{pointA}}'s (sub-)class and triangle sides C{a}, 

214 C{b} and C{c} in C{meter}, conventionally. 

215 

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

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

218 

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

220 

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

222 and functions L{pygeodesy.cassini}, L{pygeodesy.pierlot} and 

223 L{pygeodesy.tienstra7}. 

224 ''' 

225 

226 def _azi_len2(A, B, pi2): 

227 v = B.minus(A) 

228 r = atan2(v.x, v.y) 

229 if pi2 and r < 0: 

230 r += pi2 

231 return r, v.length 

232 

233 def _cV3(d, r, A, B, C, useZ, V3, **kwds): 

234 s, c = sincos2(r) 

235 x = A.x + d * s 

236 y = A.y + d * c 

237 z = _zidw(A, B, C, x, y) if useZ else INT0 

238 return V3(x, y, z, **kwds) 

239 

240 A = _otherV3d(useZ=useZ, pointA=pointA) 

241 B = _otherV3d(useZ=useZ, pointB=pointB) 

242 C = _otherV3d(useZ=useZ, pointC=pointC) 

243 

244 try: 

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

246 if min(ra, rb) < 0: 

247 raise ValueError(_negative_) 

248 

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

250 if isnear0(sra) or isnear0(srH): 

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

252 

253 clas = Clas or pointA.classof 

254 kwds = _xkwds(Clas_kwds, name=collins5.__name__) 

255 

256# za, a = _azi_len2(C, B, PI2) 

257 zb, b = _azi_len2(C, A, PI2) 

258 zc, c = _azi_len2(A, B, 0) 

259 

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

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

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

263 H = _cV3(d, r, A, B, C, useZ, Vector3d) 

264 

265 zh, _ = _azi_len2(C, H, PI2) 

266 

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

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

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

270 P = _cV3(d, r, A, B, C, useZ, clas, **kwds) 

271 

272 H = clas(H.x, H.y, H.z, **kwds) 

273 a = B.minus(C).length 

274 

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

276 

277 except (TypeError, ValueError) as x: 

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

279 alpha=alpha, beta=beta, cause=x) 

280 

281 

282def pierlot(point1, point2, point3, alpha12, alpha23, useZ=False, Clas=None, **Clas_kwds): 

283 '''3-Point resection using U{Pierlot<http://www.Telecom.ULg.ac.Be/publi/publications/ 

284 pierlot/Pierlot2014ANewThree>}'s method C{ToTal}. 

285 

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

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

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

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

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

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

292 @arg alpha12: Angle subtended from B{C{point1}} to B{C{point2}} (C{degrees}). 

293 @arg alpha23: Angle subtended from B{C{point2}} to B{C{point3}} (C{degrees}). 

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

295 (C{bool}). 

296 @kwarg Clas: Optional class to return the survey point or C{None} for 

297 B{C{point1}}'s (sub-)class. 

298 @kwarg Clas_kwds: Optional additional keyword arguments for the survey 

299 point instance. 

300 

301 @note: Points B{C{point1}}, B{C{point2}} and B{C{point3}} are ordered 

302 counter-clockwise, typically. 

303 

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

305 C{B{Clas} is None} of B{C{point1}}'s (sub-)class. 

306 

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

308 or invalid B{C{alpha12}} or B{C{alpha23}}. 

309 

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

311 

312 @see: U{V. Pierlot, M. Van Droogenbroeck, "A New Three Object Triangulation 

313 Algorithm for Mobile Robot Positioning"<https://ORBi.ULiege.Be/ 

314 bitstream/2268/157469/1/Pierlot2014ANewThree.pdf>}, U{Vincent Pierlot, 

315 Marc Van Droogenbroeck, "18 Triangulation Algorithms for 2D Positioning 

316 (also known as the Resection Problem)"<http://www.Telecom.ULg.ac.Be/ 

317 triangulation>} and functions L{pygeodesy.cassini}, L{pygeodesy.collins5} 

318 and L{pygeodesy.tienstra7}. 

319 ''' 

320 B1 = _otherV3d(useZ=useZ, point1=point1) 

321 B2 = _otherV3d(useZ=useZ, point2=point2) 

322 B3 = _otherV3d(useZ=useZ, point3=point3) 

323 

324 try: # (INTERNAL) Raises error for (pseudo-)singularities 

325 s12, c12, s23, c23 = sincos2d_(alpha12, alpha23) 

326 if isnear0(s12) or isnear0(s23): 

327 raise ValueError(_or(_coincident_, _colinear_)) 

328 cot12 = c12 / s12 

329 cot23 = c23 / s23 

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

331 d = fsum1_(c12 * s23, s12 * c23) 

332 if isnear0(d): 

333 raise ValueError(_or(_colinear_, _coincident_)) 

334 cot31 = Fsum(_1_0, s12 * s23, -c12 * c23, _N_1_0).fover(d) 

335 

336 x1_, y1_, _ = B1.minus(B2).xyz 

337 x3_, y3_, _ = B3.minus(B2).xyz 

338 

339# x23 = x3_ - cot23 * y3_ 

340# y23 = y3_ + cot23 * x3_ 

341 

342 X12_23 = Fsum(x1_, cot12 * y1_, -x3_, cot23 * y3_) 

343 Y12_23 = Fsum(y1_, -cot12 * x1_, -y3_, -cot23 * x3_) 

344 

345 X31_23 = Fsum(x1_, -cot31 * y1_, cot31 * y3_, cot23 * y3_) 

346 Y31_23 = Fsum(y1_, cot31 * x1_, -cot31 * x3_, -cot23 * x3_) 

347 

348 d = float(X31_23 * Y12_23 - X12_23 * Y31_23) 

349 if isnear0(d): 

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

351 K = Fsum(x3_ * x1_, cot31 * (y3_ * x1_), 

352 y3_ * y1_, -cot31 * (x3_ * y1_)) 

353 

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

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

356 z = _zidw(B1, B2, B3, x, y) if useZ else INT0 

357 

358 clas = Clas or point1.classof 

359 return clas(x, y, z, **_xkwds(Clas_kwds, name=pierlot.__name__)) 

360 

361 except (TypeError, ValueError) as x: 

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

363 alpha12=alpha12, alpha23=alpha23, cause=x) 

364 

365 

366def snellius3(a, b, degC, alpha, beta): 

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

368 

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

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

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

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

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

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

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

376 

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

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

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

380 

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

382 or B{C{beta}}. 

383 

384 @see: Function L{pygeodesy.wildberger3}. 

385 ''' 

386 try: 

387 a, b, degC, alpha, beta = map1(float, a, b, degC, alpha, beta) 

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

389 if min(ra, rb, rC, a, b) < 0: 

390 raise ValueError(_negative_) 

391 

392 r = fsum_(ra, rb, rC) * _0_5 

393 k = PI - r 

394 if min(k, r) < 0: 

395 raise ValueError(_or(_coincident_, _colinear_)) 

396 

397 sa, sb = sin(ra), sin(rb) 

398 p = atan2(a * sa, b * sb) 

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

400 w = atan2(sp * sr, cp * cr) 

401 x = k + w 

402 y = k - w 

403 

404 s = fabs(sa) 

405 if fabs(sb) > s: 

406 pc = fabs(a * sin(y) / sb) 

407 elif s: 

408 pc = fabs(b * sin(x) / sa) 

409 else: 

410 raise ValueError(_or(_colinear_, _coincident_)) 

411 

412 pa = _triSide(b, pc, fsum_(PI, -ra, -x)) 

413 pb = _triSide(a, pc, fsum_(PI, -rb, -y)) 

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

415 

416 except (TypeError, ValueError) as x: 

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

418 

419 

420def tienstra7(pointA, pointB, pointC, alpha, beta=None, gamma=None, 

421 useZ=False, Clas=None, **Clas_kwds): 

422 '''3-Point resection using U{Tienstra<https://WikiPedia.org/wiki/Tienstra_formula>}'s formula. 

423 

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

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

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

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

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

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

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

431 (C{degrees}, non-negative). 

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

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

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

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

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

437 (C{bool}). 

438 @kwarg Clas: Optional class to return the survey point or C{None} for B{C{pointA}}'s 

439 (sub-)class. 

440 @kwarg Clas_kwds: Optional additional keyword arguments for the survey point instance. 

441 

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

443 

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

445 instance of B{C{Clas}} or if C{B{Clas} is None} of B{C{pointA}}'s (sub-)class, 

446 with triangle angles C{A} at B{C{pointA}}, C{B} at B{C{pointB}} and C{C} at 

447 B{C{pointC}} in C{degrees} and with triangle sides C{a}, C{b} and C{c} in 

448 C{meter}, conventionally. 

449 

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

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

452 negative B{C{alpha}}, B{C{beta}} or B{C{gamma}}. 

453 

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

455 

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

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

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

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

460 functions L{pygeodesy.cassini}, L{pygeodesy.collins5} and L{pygeodesy.pierlot}. 

461 ''' 

462 

463 def _deg_ks(r, s, ks, N): 

464 if isnear0(fsum_(PI, r, -s)): # r + (PI2 - s) == PI 

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

466 # k = 1 / (cot(r) - cot(s)) 

467 # = 1 / (cos(r) / sin(r) - cos(s) / sin(s)) 

468 # = 1 / (cos(r) * sin(s) - cos(s) * sin(r)) / (sin(r) * sin(s)) 

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

470 sr, cr, ss, cs = sincos2_(r, s) 

471 c = cr * ss - cs * sr 

472 if isnear0(c): 

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

474 ks.append(sr * ss / c) 

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

476 

477 A = _otherV3d(useZ=useZ, pointA=pointA) 

478 B = _otherV3d(useZ=useZ, pointB=pointB) 

479 C = _otherV3d(useZ=useZ, pointC=pointC) 

480 

481 try: 

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

483 if beta is None: 

484 if gamma is None: 

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

486 sb = fsum_(PI2, -sa, -sc) 

487 elif gamma is None: 

488 sc = fsum_(PI2, -sa, -sb) 

489 else: # subtended angles must add to 360 degrees 

490 r = fsum1_(sa, sb, sc) 

491 if fabs(r - PI2) > EPS: 

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

493 if min(sa, sb, sc) < 0: 

494 raise ValueError(_negative_) 

495 

496 # triangle sides 

497 a = B.minus(C).length 

498 b = A.minus(C).length 

499 c = A.minus(B).length 

500 

501 ks = [] # 3 Ks and triangle angles 

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

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

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

505 

506 k = fsum1(ks, floats=True) 

507 if isnear0(k): 

508 raise ValueError(Fmt.EQUAL(K=k)) 

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

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

511 z = _zidw(A, B, C, x, y) if useZ else INT0 

512 

513 n = tienstra7.__name__ 

514 clas = Clas or pointA.classof 

515 P = clas(x, y, z, **_xkwds(Clas_kwds, name=n)) 

516 return Tienstra7Tuple(P, dA, dB, dC, a, b, c, name=n) 

517 

518 except (TypeError, ValueError) as x: 

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

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

521 

522 

523def triAngle(a, b, c): 

524 '''Compute one angle of a triangle. 

525 

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

527 C{meter}, conventionally). 

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

529 C{meter}, conventionally). 

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

531 C{meter}, conventionally). 

532 

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

534 triangle side B{C{c}}. 

535 

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

537 

538 @see: Functions L{pygeodesy.triAngle4} and L{pygeodesy.triSide}. 

539 ''' 

540 try: 

541 return _triAngle(a, b, c) 

542 except (TypeError, ValueError) as x: 

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

544 

545 

546def _triAngle(a, b, c): 

547 # (INTERNAL) To allow callers to embellish errors 

548 a, b, c = map1(float, a, b, c) 

549 if a < b: 

550 a, b = b, a 

551 if b < 0 or c < 0: 

552 raise ValueError(_negative_) 

553 if a < EPS0: 

554 raise ValueError(_coincident_) 

555 b_a = b / a 

556 if b_a < EPS0: 

557 raise ValueError(_coincident_) 

558 return acos1(fsum_(_1_0, b_a**2, -(c / a)**2) / (b_a * _2_0)) 

559 

560 

561def triAngle4(a, b, c): 

562 '''Compute the angles of a triangle. 

563 

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

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

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

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

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

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

570 

571 @return: L{TriAngle4Tuple}C{(radA, radB, radC, rIn)} with angles C{radA}, 

572 C{radB} and C{radC} at triangle corners C{A}, C{B} and C{C}, all 

573 in C{radians} and the C{InCircle} radius C{rIn} aka C{inradius}, 

574 same units as triangle sides B{C{a}}, B{C{b}} and B{C{c}}. 

575 

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

577 

578 @see: Function L{pygeodesy.triAngle}. 

579 ''' 

580 try: 

581 a, b, c = map1(float, a, b, c) 

582 ab = a < b 

583 if ab: 

584 a, b = b, a 

585 bc = b < c 

586 if bc: 

587 b, c = c, b 

588 

589 if c > EPS0: # c = min(a, b, c) 

590 s = float(Fsum(a, b, c) * _0_5) 

591 if s < EPS0: 

592 raise ValueError(_negative_) 

593 sa, sb, sc = (s - a), (s - b), (s - c) 

594 r = sa * sb * sc / s 

595 if r < EPS02: 

596 raise ValueError(_coincident_) 

597 r = sqrt(r) 

598 rA = atan2(r, sa) * _2_0 

599 rB = atan2(r, sb) * _2_0 

600 rC = fsum_(PI, -rA, -rB) 

601 if min(rA, rB, rC) < 0: 

602 raise ValueError(_colinear_) 

603 elif c < 0: 

604 raise ValueError(_negative_) 

605 else: # 0 <= c <= EPS0 

606 rA = rB = PI_2 

607 rC = r = _0_0 

608 

609 if bc: 

610 rB, rC = rC, rB 

611 if ab: 

612 rA, rB = rB, rA 

613 return TriAngle4Tuple(rA, rB, rC, r, name=triAngle4.__name__) 

614 

615 except (TypeError, ValueError) as x: 

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

617 

618 

619def triSide(a, b, radC): 

620 '''Compute one side of a triangle. 

621 

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

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

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

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

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

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

628 

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

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

631 B{C{a}} and B{C{b}}. 

632 

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

634 

635 @see: Functions L{pygeodesy.sqrt_a}, L{pygeodesy.triAngle}, 

636 L{pygeodesy.triSide2} and L{pygeodesy.triSide4}. 

637 ''' 

638 try: 

639 return _triSide(a, b, radC) 

640 except (TypeError, ValueError) as x: 

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

642 

643 

644def _triSide(a, b, radC): 

645 # (INTERNAL) To allow callers to embellish errors 

646 a, b, r = map1(float, a, b, radC) 

647 if min(a, b, r) < 0: 

648 raise ValueError(_negative_) 

649 

650 if a < b: 

651 a, b = b, a 

652 if a > EPS0: 

653 ba = b / a 

654 c2 = fsum_(_1_0, ba**2, _N_2_0 * ba * cos(r)) 

655 if c2 > EPS02: 

656 return a * sqrt(c2) 

657 elif c2 < 0: 

658 raise ValueError(_invalid_) 

659 return hypot(a, b) 

660 

661 

662def triSide2(b, c, radB): 

663 '''Compute one side and the opposite angle of a triangle. 

664 

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

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

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

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

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

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

671 

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

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

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

675 

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

677 B{C{b}} or B{C{radB}} near zero. 

678 

679 @see: Functions L{pygeodesy.sqrt_a}, L{pygeodesy.triSide} 

680 and L{pygeodesy.triSide4}. 

681 ''' 

682 try: 

683 return _triSide2(b, c, radB) 

684 except (TypeError, ValueError) as x: 

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

686 

687 

688def _triSide2(b, c, radB): 

689 # (INTERNAL) To allow callers to embellish errors 

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

691 if min(b, c, rB) < 0: 

692 raise ValueError(_negative_) 

693 sB, cB = sincos2(rB) 

694 if isnear0(sB): 

695 if not isnear0(b): 

696 raise ValueError(_invalid_) 

697 if cB < 0: 

698 a, rA = (b + c), PI 

699 else: 

700 a, rA = fabs(b - c), _0_0 

701 elif isnear0(b): 

702 raise ValueError(_invalid_) 

703 else: 

704 rA = fsum_(PI, -rB, -asin1(c * sB / b)) 

705 a = sin(rA) * b / sB 

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

707 

708 

709def triSide4(radA, radB, c): 

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

711 

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

713 (non-negative C{radians}). 

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

715 (non-negative C{radians}). 

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

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

718 

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

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

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

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

723 triangle side B{C{c}}. 

724 

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

726 

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

728 and functions L{pygeodesy.sqrt_a}, L{pygeodesy.triSide} and L{pygeodesy.triSide2}. 

729 ''' 

730 try: 

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

732 rC = fsum_(PI, -rA, -rB) 

733 if min(rC, rA, rB, c) < 0: 

734 raise ValueError(_negative_) 

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

736 sc = fsum1_(sa * cb, sb * ca) 

737 if sc < EPS0 or min(sa, sb) < 0: 

738 raise ValueError(_invalid_) 

739 sc = c / sc 

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

741 name=triSide4.__name__) 

742 

743 except (TypeError, ValueError) as x: 

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

745 

746 

747def wildberger3(a, b, c, alpha, beta, R3=min): 

748 '''Snellius' surveying using U{Rational Trigonometry<https://WikiPedia.org/wiki/Snellius–Pothenot_problem>}. 

749 

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

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

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

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

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

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

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

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

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

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

760 

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

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

763 B{C{b}} and B{C{c}}. 

764 

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

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

767 

768 @see: U{Wildberger, Norman J.<https://math.sc.Chula.ac.TH/cjm/content/ 

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

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

771 and function L{pygeodesy.snellius3}. 

772 ''' 

773 def _s(x): 

774 return sin(x)**2 

775 

776 def _vpa(r1, r3, q2, q3, s3): 

777 r = r1 * r3 * _4_0 

778 n = (r - Fsum(r1, r3, -q2).fpow(2)).fover(s3) 

779 if n < 0 or isnear0(r): 

780 raise ValueError(_coincident_) 

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

782 

783 try: 

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

785 if min(a, b, c, da, db) < 0: 

786 raise ValueError(_negative_) 

787 

788 ra, rb = radians(da), radians(db) 

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

790 if min(s) < EPS02: 

791 raise ValueError(_or(_coincident_, _colinear_)) 

792 

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

794 if min(q) < EPS02: 

795 raise ValueError(_coincident_) 

796 

797 r1 = s2 * q3 / s3 # s2! 

798 r2 = s1 * q3 / s3 # s1! 

799 Qs = Fsum(*q) # == hypot2_(a, b, c) 

800 Ss = Fsum(*s) # == fsum1(s, floats=True) 

801 s += Qs * _0_5, # tuple! 

802 C0 = Fdot(s, q1, q2, q3, -Ss) 

803 r3 = C0.fover(-s3) 

804 d0 = Qs.fpow(2).fsub_(hypot2_(*q) * _2_0).fmul(s1 * s2).fover(s3) 

805 if d0 > EPS02: # > c0 

806 d0 = sqrt(d0) 

807 if not callable(R3): 

808 raise ValueError(_R3__ + _not_(callable.__name__)) 

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

810 elif d0 < 0: 

811 raise ValueError(_negative_) 

812 

813 pa = _vpa(r1, r3, q2, q3, s3) 

814 pb = _vpa(r2, r3, q1, q3, s3) 

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

816 _triSide2(a, pb, rb).a) 

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

818 

819 except (TypeError, ValueError) as x: 

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

821 

822 

823def _zidw(A, B, C, x, y): 

824 # interpolate z or coplanar with A, B and C? 

825 t = A.z, B.z, C.z 

826 m = Vector3d(x, y, fmean(t)).minus 

827 return fidw(t, (m(T).length for T in (A, B, C))) 

828 

829# **) MIT License 

830# 

831# Copyright (C) 2016-2023 -- mrJean1 at Gmail -- All Rights Reserved. 

832# 

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

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

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

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

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

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

839# 

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

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

842# 

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

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

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

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

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

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

849# OTHER DEALINGS IN THE SOFTWARE.