1

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

3

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

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

6 L{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# make sure int/int division yields float quotient

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

14

15 from pygeodesy.basics import map1, _ALL_LAZY

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

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

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

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

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

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

22 _d_, _invalid_, _negative_, _not_, _rIn_, _SPACE_

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

24 from pygeodesy.named import Fmt, _NamedTuple, _Pass

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

26 from pygeodesy.units import Degrees, Distance, Radians

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

28 from pygeodesy.vector3d import _otherV3d, Vector3d

29

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

31

32__all__ = _ALL_LAZY.resections

33__version__ = '23.05.15'

34

35_concyclic_ = 'concyclic'

36_PA_ = 'PA'

37_PB_ = 'PB'

38_PC_ = 'PC'

39_pointH_ = 'pointH'

40_pointP_ = 'pointP'

41_R3__ = 'R3 '

42_radA_ = 'radA'

43_radB_ = 'radB'

44_radC_ = 'radC'

45

46

47 class Collins5Tuple(_NamedTuple):

48 '''5-Tuple C{(pointP, pointH, a, b, c)} with survey C{pointP}, auxiliary

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

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

51 '''

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

53 _Units_ = (_Pass, _Pass, Distance, Distance, Distance)

54

55

56 class ResectionError(_ValueError):

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

58 '''

59 pass

60

61

62 class Survey3Tuple(_NamedTuple):

63 '''3-Tuple C{(PA, PB, PC)} with distance from survey point C{P} to each of

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

65 '''

66 _Names_ = (_PA_, _PB_, _PC_)

67 _Units_ = ( Distance, Distance, Distance)

68

69

70 class Tienstra7Tuple(_NamedTuple):

71 '''7-Tuple C{(pointP, A, B, C, a, b, c)} with survey C{pointP}, interior

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

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

74 '''

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

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

77

78

79 class TriAngle4Tuple(_NamedTuple):

80 '''4-Tuple C{(radA, radB, radC, rIn)} with the interior angles at triangle

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

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

83 '''

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

85 _Units_ = ( Radians, Radians, Radians, Distance)

86

87

88 class TriSide2Tuple(_NamedTuple):

89 '''2-Tuple C{(a, radA)} with triangle side C{a} in C{meter}, conventionally

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

91 '''

92 _Names_ = (_a_, _radA_)

93 _Units_ = ( Distance, Radians)

94

95

96 class TriSide4Tuple(_NamedTuple):

97 '''4-Tuple C{(a, b, radC, d)} with interior angle C{radC} at triangle corner

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

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

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

101 '''

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

103 _Units_ = ( Distance, Distance, Radians, Distance)

104

105

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

107 '''3-Point resection using U{Cassini<https://NL.WikiPedia.org/wiki/Achterwaartse_insnijding>}'s method.

108

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

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

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

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

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

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

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

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

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

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

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

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

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

122 B{C{pointA}}'s (sub-)class.

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

124 point instance.

125

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

127

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

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

130

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

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

133

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

135

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

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

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

139 L{pygeodesy.tienstra7}.

140 '''

141

142 def _H(A, C, sa):

143 s, c = sincos2d(sa)

144 if isnear0(s):

145 raise ValueError(_or(_coincident_, _colinear_))

146 t = s, c, c

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

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

149 return Vector3d(x, y, 0)

150

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

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

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

154

155 try:

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

157 if min(sa, sb) < 0:

158 raise ValueError(_negative_)

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

160 raise ValueError()

161

162 H1 = _H(A, C, sa)

163 H2 = _H(B, C, -sb)

164

165 x = H1.x - H2.x

166 y = H1.y - H2.y

167 # x, y, _ = H1.minus(H2).xyz

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

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

170

171 m = y / x

172 n = x / y

173 N = n + m

174 if isnear0(N):

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

176

177 t = n, m, _1_0, _N_1_0

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

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

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

181

182 clas = Clas or pointA.classof

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

184

185 except (TypeError, ValueError) as x:

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

187 alpha=alpha, beta=beta, cause=x)

188

189

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

191 '''3-Point resection using U{Collins<https://Dokumen.tips/documents/

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

193

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

209 and auxiliary point instance.

210

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

212

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

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

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

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

217

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

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

220

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

222

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

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

225 L{pygeodesy.tienstra7}.

226 '''

227

228 def _azi_len2(A, B, pi2):

229 v = B.minus(A)

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

231 if pi2 and r < 0:

232 r += pi2

233 return r, v.length

234

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

236 s, c = sincos2(r)

237 x = A.x + d * s

238 y = A.y + d * c

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

240 return V3(x, y, z, **kwds)

241

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

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

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

245

246 try:

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

248 if min(ra, rb) < 0:

249 raise ValueError(_negative_)

250

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

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

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

254

255 clas = Clas or pointA.classof

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

257

258# za, a = _azi_len2(C, B, PI2)

259 zb, b = _azi_len2(C, A, PI2)

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

261

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

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

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

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

266

267 zh, _ = _azi_len2(C, H, PI2)

268

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

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

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

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

273

274 H = clas(H.x, H.y, H.z, **kwds)

275 a = B.minus(C).length

276

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

278

279 except (TypeError, ValueError) as x:

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

281 alpha=alpha, beta=beta, cause=x)

282

283

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

285 '''3-Point resection using U{Pierlot<http://www.Telecom.ULg.ac.Be/publi/publications/

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

287

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

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

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

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

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

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

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

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

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

297 (C{bool}).

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

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

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

301 point instance.

302

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

304 counter-clockwise, typically.

305

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

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

308

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

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

311

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

313

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

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

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

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

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

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

320 and L{pygeodesy.tienstra7}.

321 '''

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

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

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

325

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

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

328 if isnear0(s12) or isnear0(s23):

329 raise ValueError(_or(_coincident_, _colinear_))

330 cot12 = c12 / s12

331 cot23 = c23 / s23

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

333 d = fsum1f_(c12 * s23, s12 * c23)

334 if isnear0(d):

335 raise ValueError(_or(_colinear_, _coincident_))

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

337

338 x1_, y1_, _ = B1.minus(B2).xyz

339 x3_, y3_, _ = B3.minus(B2).xyz

340

341# x23 = x3_ - cot23 * y3_

342# y23 = y3_ + cot23 * x3_

343

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

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

346

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

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

349

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

351 if isnear0(d):

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

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

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

355

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

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

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

359

360 clas = Clas or point1.classof

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

362

363 except (TypeError, ValueError) as x:

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

365 alpha12=alpha12, alpha23=alpha23, cause=x)

366

367

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

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

370

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

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

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

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

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

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

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

378

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

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

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

382

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

384 or B{C{beta}}.

385

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

387 '''

388 try:

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

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

391 if min(ra, rb, rC, a, b) < 0:

392 raise ValueError(_negative_)

393

394 r = fsumf_(ra, rb, rC) * _0_5

395 k = PI - r

396 if min(k, r) < 0:

397 raise ValueError(_or(_coincident_, _colinear_))

398

399 sa, sb = sin(ra), sin(rb)

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

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

402 w = atan2(sp * sr, cp * cr)

403 x = k + w

404 y = k - w

405

406 s = fabs(sa)

407 if fabs(sb) > s:

408 pc = fabs(a * sin(y) / sb)

409 elif s:

410 pc = fabs(b * sin(x) / sa)

411 else:

412 raise ValueError(_or(_colinear_, _coincident_))

413

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

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

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

417

418 except (TypeError, ValueError) as x:

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

420

421

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

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

424 '''3-Point resection using U{Tienstra<https://WikiPedia.org/wiki/Tienstra_formula>}'s formula.

425

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

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

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

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

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

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

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

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

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

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

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

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

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

439 (C{bool}).

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

441 (sub-)class.

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

443

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

445

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

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

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

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

450 C{meter}, conventionally.

451

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

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

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

455

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

457

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

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

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

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

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

463 '''

464

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

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

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

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

469 # = 1 / (cos(r) / sin(r) - cos(s) / sin(s))

470 # = 1 / (cos(r) * sin(s) - cos(s) * sin(r)) / (sin(r) * sin(s))

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

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

473 c = cr * ss - cs * sr

474 if isnear0(c):

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

476 ks.append(sr * ss / c)

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

478

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

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

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

482

483 try:

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

485 if beta is None:

486 if gamma is None:

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

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

489 elif gamma is None:

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

491 else: # subtended angles must add to 360 degrees

492 r = fsum1f_(sa, sb, sc)

493 if fabs(r - PI2) > EPS:

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

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

496 raise ValueError(_negative_)

497

498 # triangle sides

499 a = B.minus(C).length

500 b = A.minus(C).length

501 c = A.minus(B).length

502

503 ks = [] # 3 Ks and triangle angles

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

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

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

507

508 k = fsum1(ks, floats=True)

509 if isnear0(k):

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

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

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

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

514

515 n = tienstra7.__name__

516 clas = Clas or pointA.classof

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

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

519

520 except (TypeError, ValueError) as x:

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

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

523

524

525 def triAngle(a, b, c):

526 '''Compute one angle of a triangle.

527

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

529 C{meter}, conventionally).

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

531 C{meter}, conventionally).

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

533 C{meter}, conventionally).

534

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

536 triangle side B{C{c}}.

537

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

539

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

541 '''

542 try:

543 return _triAngle(a, b, c)

544 except (TypeError, ValueError) as x:

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

546

547

548 def _triAngle(a, b, c):

549 # (INTERNAL) To allow callers to embellish errors

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

551 if a < b:

552 a, b = b, a

553 if b < 0 or c < 0:

554 raise ValueError(_negative_)

555 if a < EPS0:

556 raise ValueError(_coincident_)

557 b_a = b / a

558 if b_a < EPS0:

559 raise ValueError(_coincident_)

560 return acos1(fsumf_(_1_0, b_a**2, -(c / a)**2) / (b_a * _2_0))

561

562

563 def triAngle4(a, b, c):

564 '''Compute the angles of a triangle.

565

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

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

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

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

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

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

572

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

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

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

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

577

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

579

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

581 '''

582 try:

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

584 ab = a < b

585 if ab:

586 a, b = b, a

587 bc = b < c

588 if bc:

589 b, c = c, b

590

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

592 s = float(Fsum(a, b, c) * _0_5)

593 if s < EPS0:

594 raise ValueError(_negative_)

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

596 r = sa * sb * sc / s

597 if r < EPS02:

598 raise ValueError(_coincident_)

599 r = sqrt(r)

600 rA = atan2(r, sa) * _2_0

601 rB = atan2(r, sb) * _2_0

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

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

604 raise ValueError(_colinear_)

605 elif c < 0:

606 raise ValueError(_negative_)

607 else: # 0 <= c <= EPS0

608 rA = rB = PI_2

609 rC = r = _0_0

610

611 if bc:

612 rB, rC = rC, rB

613 if ab:

614 rA, rB = rB, rA

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

616

617 except (TypeError, ValueError) as x:

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

619

620

621 def triSide(a, b, radC):

622 '''Compute one side of a triangle.

623

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

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

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

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

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

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

630

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

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

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

634

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

636

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

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

639 '''

640 try:

641 return _triSide(a, b, radC)

642 except (TypeError, ValueError) as x:

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

644

645

646 def _triSide(a, b, radC):

647 # (INTERNAL) To allow callers to embellish errors

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

649 if min(a, b, r) < 0:

650 raise ValueError(_negative_)

651

652 if a < b:

653 a, b = b, a

654 if a > EPS0:

655 ba = b / a

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

657 if c2 > EPS02:

658 return a * sqrt(c2)

659 elif c2 < 0:

660 raise ValueError(_invalid_)

661 return hypot(a, b)

662

663

664 def triSide2(b, c, radB):

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

666

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

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

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

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

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

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

673

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

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

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

677

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

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

680

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

682 and L{pygeodesy.triSide4}.

683 '''

684 try:

685 return _triSide2(b, c, radB)

686 except (TypeError, ValueError) as x:

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

688

689

690 def _triSide2(b, c, radB):

691 # (INTERNAL) To allow callers to embellish errors

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

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

694 raise ValueError(_negative_)

695 sB, cB = sincos2(rB)

696 if isnear0(sB):

697 if not isnear0(b):

698 raise ValueError(_invalid_)

699 if cB < 0:

700 a, rA = (b + c), PI

701 else:

702 a, rA = fabs(b - c), _0_0

703 elif isnear0(b):

704 raise ValueError(_invalid_)

705 else:

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

707 a = sin(rA) * b / sB

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

709

710

711 def triSide4(radA, radB, c):

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

713

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

715 (non-negative C{radians}).

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

717 (non-negative C{radians}).

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

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

720

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

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

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

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

725 triangle side B{C{c}}.

726

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

728

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

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

731 '''

732 try:

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

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

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

736 raise ValueError(_negative_)

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

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

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

740 raise ValueError(_invalid_)

741 sc = c / sc

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

743 name=triSide4.__name__)

744

745 except (TypeError, ValueError) as x:

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

747

748

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

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

751

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

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

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

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

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

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

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

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

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

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

762

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

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

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

766

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

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

769

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

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

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

773 and function L{pygeodesy.snellius3}.

774 '''

775 def _s(x):

776 return sin(x)**2

777

778 def _vpa(r1, r3, q2, q3, s3):

779 r = r1 * r3 * _4_0

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

781 if n < 0 or isnear0(r):

782 raise ValueError(_coincident_)

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

784

785 try:

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

787 if min(a, b, c, da, db) < 0:

788 raise ValueError(_negative_)

789

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

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

792 if min(s) < EPS02:

793 raise ValueError(_or(_coincident_, _colinear_))

794

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

796 if min(q) < EPS02:

797 raise ValueError(_coincident_)

798

799 r1 = s2 * q3 / s3 # s2!

800 r2 = s1 * q3 / s3 # s1!

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

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

803 s += Qs * _0_5, # tuple!

804 C0 = Fdot(s, q1, q2, q3, -Ss)

805 r3 = C0.fover(-s3)

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

807 if d0 > EPS02: # > c0

808 d0 = sqrt(d0)

809 if not callable(R3):

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

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

812 elif d0 < 0:

813 raise ValueError(_negative_)

814

815 pa = _vpa(r1, r3, q2, q3, s3)

816 pb = _vpa(r2, r3, q1, q3, s3)

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

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

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

820

821 except (TypeError, ValueError) as x:

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

823

824

825 def _zidw(A, B, C, x, y):

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

827 t = A.z, B.z, C.z

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

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

830

831# **) MIT License

832#

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

834#

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

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

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

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

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

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

841#

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

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

844#

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

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

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

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

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

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

851# OTHER DEALINGS IN THE SOFTWARE.