1

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

3

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

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

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

7

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

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

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

11'''

12# make sure int/int division yields float quotient

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

14

15 from pygeodesy.basics import map1, _ALL_LAZY

16 from pygeodesy.constants import EPS, EPS0, EPS02, INT0, NEG0, PI, PI2, PI_2, PI_4, \

17 isnear0, _umod_360, _0_0, _0_5, _1_0, _N_1_0, _2_0, \

18 _N_2_0, _4_0, _180_0, _360_0

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

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

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

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

23 _d_, _eps_, _invalid_, _negative_, _not_, _rIn_, _SPACE_

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

25 from pygeodesy.named import Fmt, _NamedTuple, _Pass

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

27 from pygeodesy.units import Degrees, Distance, Radians

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

29 from pygeodesy.vector3d import _otherV3d, Vector3d

30

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

32

33__all__ = _ALL_LAZY.resections

34__version__ = '23.06.07'

35

36_concyclic_ = 'concyclic'

37_PA_ = 'PA'

38_PB_ = 'PB'

39_PC_ = 'PC'

40_pointH_ = 'pointH'

41_pointP_ = 'pointP'

42_positive_ = 'positive'

43_R3__ = 'R3 '

44_radA_ = 'radA'

45_radB_ = 'radB'

46_radC_ = 'radC'

47

48

49 class Collins5Tuple(_NamedTuple):

50 '''5-Tuple C{(pointP, pointH, a, b, c)} with survey C{pointP}, auxiliary

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

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

53 '''

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

55 _Units_ = (_Pass, _Pass, Distance, Distance, Distance)

56

57

58 class ResectionError(_ValueError):

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

60 '''

61 pass

62

63

64 class Survey3Tuple(_NamedTuple):

65 '''3-Tuple C{(PA, PB, PC)} with distance from survey point C{P} to each of

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

67 '''

68 _Names_ = (_PA_, _PB_, _PC_)

69 _Units_ = ( Distance, Distance, Distance)

70

71

72 class Tienstra7Tuple(_NamedTuple):

73 '''7-Tuple C{(pointP, A, B, C, a, b, c)} with survey C{pointP}, interior

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

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

76 '''

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

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

79

80

81 class TriAngle4Tuple(_NamedTuple):

82 '''4-Tuple C{(radA, radB, radC, rIn)} with the interior angles at triangle

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

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

85 '''

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

87 _Units_ = ( Radians, Radians, Radians, Distance)

88

89

90 class TriSide2Tuple(_NamedTuple):

91 '''2-Tuple C{(a, radA)} with triangle side C{a} in C{meter}, conventionally

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

93 '''

94 _Names_ = (_a_, _radA_)

95 _Units_ = ( Distance, Radians)

96

97

98 class TriSide4Tuple(_NamedTuple):

99 '''4-Tuple C{(a, b, radC, d)} with interior angle C{radC} at triangle corner

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

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

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

103 '''

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

105 _Units_ = ( Distance, Distance, Radians, Distance)

106

107

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

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

110 '''

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

112 _otherV3d(useZ=useZ, pointB=pointB),

113 _otherV3d(useZ=useZ, pointC=pointC))

114

115

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

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

118 '''

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

120 _otherV3d(useZ=useZ, point2=point2),

121 _otherV3d(useZ=useZ, point3=point3))

122

123

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

125 '''3-Point resection using U{Cassini<https://NL.WikiPedia.org/wiki/Achterwaartse_insnijding>}'s method.

126

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

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

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

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

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

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

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

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

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

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

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

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

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

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

141 @kwarg Clas_kwds: Optional, additional keyword argument for the survey

142 point instance.

143

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

145

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

147 None} an instance of B{C{pointA}}'s (sub-)class.

148

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

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

151

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

153

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

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

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

157 L{pygeodesy.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 = map1(float, alpha, beta)

172 if min(sa, sb) < 0:

173 raise ValueError(_negative_)

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

175 raise ValueError()

176

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

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

179

180 x = x1 - x2

181 y = y1 - y2

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

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

184

185 m = y / x

186 n = x / y

187 N = n + m

188 if isnear0(N):

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

190

191 t = n, m, _1_0, _N_1_0

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

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

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

195

196 clas = Clas or pointA.classof

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

198

199 except (TypeError, ValueError) as x:

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

201 alpha=alpha, beta=beta, cause=x)

202

203

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

205 '''3-Point resection using U{Collins<https://Dokumen.tips/documents/

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

207

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

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

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

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

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

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

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

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

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

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

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

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

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

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

222 @kwarg Clas_kwds: Optional, additional keyword argument for the survey

223 and auxiliary point instance.

224

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

226

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

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

229 is None} an instance of B{C{pointA}}'s (sub-)class and triangle

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

231

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

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

234

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

236

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

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

239 L{pygeodesy.tienstra7}.

240 '''

241

242 def _azi_len2(A, B, pi2):

243 v = B.minus(A)

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

245 if pi2 and r < 0:

246 r += pi2

247 return r, v.length

248

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

250 s, c = sincos2(r)

251 x = A.x + d * s

252 y = A.y + d * c

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

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

255

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

257 try:

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

259 if min(ra, rb) < 0:

260 raise ValueError(_negative_)

261

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

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

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

265

266 clas = Clas or pointA.classof

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

268

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

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

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

272

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

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

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

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

277

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

279

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

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

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

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

284

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

286 a = B.minus(C).length

287

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

289

290 except (TypeError, ValueError) as x:

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

292 alpha=alpha, beta=beta, cause=x)

293

294

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

296 Clas=None, **Clas_kwds):

297 '''3-Point resection using U{Pierlot<http://www.Telecom.ULg.ac.BE/publi/publications/

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

299 the (pseudo-)singularities.

300

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

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

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

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

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

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

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

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

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

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

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

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

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

314 @kwarg Clas: Optional class to return the survey point, if C{None} use

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

316 @kwarg Clas_kwds: Optional, additional keyword arguments for the survey

317 point instance.

318

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

320 by angle, modulo 360, counter-clockwise.

321

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

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

324

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

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

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

328

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

330

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

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

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

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

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

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

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

338 L{pygeodesy.collins5} and L{pygeodesy.tienstra7}.

339 '''

340

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

342 if eps > 0:

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

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

345

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

347 try:

348 x, y, z = _pierlot3(B1, B2, B3, alpha12, alpha23, useZ, _cot)

349 clas = Clas or point1.classof

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

351

352 except (TypeError, ValueError) as x:

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

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

355

356

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

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

359 '''

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

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

362

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

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

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

366 # = (1 - (c12 * c23) / (s12 * s23)) / (c12 * s23 + s12 * c23) / (s12 * s23)

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

368 # = c31 / s31

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

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

371

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

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

374 if K:

375 cot12 = cot(s12, c12)

376 cot23 = cot(s23, c23)

377

378 # x12 = x1_ + cot12 * y1_

379 # y12 = y1_ - cot12 * x1_

380

381 # x23 = x3_ - cot23 * y3_

382 # y23 = y3_ + cot23 * x3_

383

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

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

386

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

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

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

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

391

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

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

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

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

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

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

398

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

400 # = (x31 - x23) * (y12 - y23) - (x12 - x23) * (y12 - y23)

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

402 if isnear0(d):

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

404

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

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

407 else:

408 x, y, _ = B2.xyz

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

410

411

412 def _pierlotx3(a_z_Bs, useZ, cot, C):

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

414 '''

415 (a12, z12, B1), \

416 (a23, z23, B2), \

417 (a31, z31, B3) = a_z_Bs

418 if z12 and not z23:

419 C(1)

420 elif z23 and not z31:

421 C(2)

422 a23, B1, B2, B3 = a31, B2, B3, B1

423 elif z31 and not z12:

424 C(3)

425 a23, B2, B3 = a12, B3, B2

426 else:

427 C(4)

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

429

430 x1_, y1_, _ = B1.minus(B3).xyz

431 x2_, y2_, _ = B2.minus(B3).xyz

432

433 K = Fsum(_1_0, y1_ * x2_, -x1_ * y2_, _N_1_0) # 1-primed

434 if K:

435 cot23 = cot(*sincos2d(a23))

436

437 # x23 = x2_ + cot23 * y2_

438 # y23 = y2_ - cot23 * x2_

439

440 # x31 = x1_ + cot23 * y1_

441 # y31 = y1_ - cot23 * x1_

442

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

444 X31_23 = Fsum(x1_, cot23 * y1_, -x2_, -cot23 * y2_)

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

446 Y31_23 = Fsum(y1_, -cot23 * x1_, -y2_, cot23 * x2_)

447

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

449 d = float(X31_23 * x2_ - X31_23 * x1_ +

450 Y31_23 * y2_ - Y31_23 * y1_)

451 if isnear0(d):

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

453

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

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

456 else:

457 x, y, _ = B3.xyz

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

459

460

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

462 Clas=None, **Clas_kwds):

463 '''3-Point resection using U{Pierlot<http://www.Telecom.ULg.ac.BE/publi/

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

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

466

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

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

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

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

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

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

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

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

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

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

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

478 @kwarg Clas: Optional class to return the survey point, if C{None} use

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

480 @kwarg Clas_kwds: Optional, additional keyword arguments for the survey

481 point instance.

482

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

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

485

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

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

488

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

490

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

492 triangulation/doc/total_8c_source.html>} and function L{pygeodesy.pierlot}.

493 '''

494

495 def _a_z_Bs(Bs, *alphas):

496 a3 = map(_umod_360, alphas) # 0 <= alphas < 360

497 (a1, a2, a3), (B1, B2, B3) = zip(*sorted(zip(a3, Bs)))

498 for a, d, B in ((a1, a2, B1), (a2, a3, B2), (a3, a1, B3)):

499 d -= a # a12 = a2 - a1, ...

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

501 yield d, z, B

502

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

504 try:

505 return (c / s) if c else (NEG0 if s < 0 else _0_0)

506 except ZeroDivisionError:

507 raise ValueError(_or(_coincident_, _colinear_))

508

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

510 try:

511 C = [0] # pseudo-global, passing the exception Case

512 x, y, z = _pierlotx3(_a_z_Bs(Bs, alpha1, alpha2, alpha3),

513 useZ, _cot, C.append)

514 clas = Clas or point1.classof

515 return clas(x, y, z, **_xkwds(Clas_kwds, name=pierlotx.__name__))

516

517 except (TypeError, ValueError) as x:

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

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

520

521

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

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

524

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

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

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

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

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

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

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

532

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

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

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

536

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

538 or B{C{beta}}.

539

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

541 '''

542 try:

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

544 if min(t) < 0:

545 raise ValueError(_negative_)

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

547

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

549 k = PI - r

550 if min(k, r) < 0:

551 raise ValueError(_or(_coincident_, _colinear_))

552

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

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

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

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

557 x = k + w

558 y = k - w

559

560 s = fabs(sa)

561 if fabs(sb) > s:

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

563 elif s:

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

565 else:

566 raise ValueError(_or(_colinear_, _coincident_))

567

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

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

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

571

572 except (TypeError, ValueError) as x:

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

574

575

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

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

578 '''3-Point resection using U{Tienstra<https://WikiPedia.org/wiki/Tienstra_formula>}'s formula.

579

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

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

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

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

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

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

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

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

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

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

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

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

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

593 (C{bool}).

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

595 (sub-)class.

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

597

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

599

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

601 instance of B{C{Clas}} or if C{B{Clas} is None} an instance of B{C{pointA}}'s

602 (sub-)class, with triangle angles C{A} at B{C{pointA}}, C{B} at B{C{pointB}}

603 and C{C} at B{C{pointC}} in C{degrees} and with triangle sides C{a}, C{b} and

604 C{c} in C{meter}, conventionally.

605

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

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

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

609

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

611

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

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

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

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

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

617 '''

618

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

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

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

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

623 # = 1 / (cos(r) / sin(r) - cos(s) / sin(s))

624 # = 1 / (cos(r) * sin(s) - cos(s) * sin(r)) / (sin(r) * sin(s))

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

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

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

628 if isnear0(c):

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

630 ks.append(sr * ss / c)

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

632

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

634 try:

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

636 if beta is None:

637 if gamma is None:

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

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

640 elif gamma is None:

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

642 else: # subtended angles must add to 360 degrees

643 r = fsum1f_(sa, sb, sc)

644 if fabs(r - PI2) > EPS:

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

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

647 raise ValueError(_negative_)

648

649 # triangle sides

650 a = B.minus(C).length

651 b = A.minus(C).length

652 c = A.minus(B).length

653

654 ks = [] # 3 Ks and triangle angles

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

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

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

658

659 k = fsum1(ks, floats=True)

660 if isnear0(k):

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

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

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

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

665

666 n = tienstra7.__name__

667 clas = Clas or pointA.classof

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

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

670

671 except (TypeError, ValueError) as x:

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

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

674

675

676 def triAngle(a, b, c):

677 '''Compute one angle of a triangle.

678

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

680 C{meter}, conventionally).

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

682 C{meter}, conventionally).

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

684 C{meter}, conventionally).

685

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

687 triangle side B{C{c}}.

688

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

690

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

692 '''

693 try:

694 return _triAngle(a, b, c)

695 except (TypeError, ValueError) as x:

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

697

698

699 def _triAngle(a, b, c):

700 # (INTERNAL) To allow callers to embellish errors

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

702 if a < b:

703 a, b = b, a

704 if b < 0 or c < 0:

705 raise ValueError(_negative_)

706 if a < EPS0:

707 raise ValueError(_coincident_)

708 b_a = b / a

709 if b_a < EPS0:

710 raise ValueError(_coincident_)

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

712 return acos1(t)

713

714

715 def triAngle4(a, b, c):

716 '''Compute the angles of a triangle.

717

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

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

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

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

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

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

724

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

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

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

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

729

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

731

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

733 '''

734 try:

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

736 ab = a < b

737 if ab:

738 a, b = b, a

739 bc = b < c

740 if bc:

741 b, c = c, b

742

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

744 s = fsumf_(a, b, c) * _0_5

745 if s < EPS0:

746 raise ValueError(_negative_)

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

748 r = sa * sb * sc / s

749 if r < EPS02:

750 raise ValueError(_coincident_)

751 r = sqrt(r)

752 rA = atan2(r, sa) * _2_0

753 rB = atan2(r, sb) * _2_0

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

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

756 raise ValueError(_colinear_)

757 elif c < 0:

758 raise ValueError(_negative_)

759 else: # 0 <= c <= EPS0

760 rA = rB = PI_2

761 rC = r = _0_0

762

763 if bc:

764 rB, rC = rC, rB

765 if ab:

766 rA, rB = rB, rA

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

768

769 except (TypeError, ValueError) as x:

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

771

772

773 def triSide(a, b, radC):

774 '''Compute one side of a triangle.

775

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

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

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

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

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

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

782

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

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

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

786

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

788

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

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

791 '''

792 try:

793 return _triSide(a, b, radC)

794 except (TypeError, ValueError) as x:

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

796

797

798 def _triSide(a, b, radC):

799 # (INTERNAL) To allow callers to embellish errors

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

801 if min(t) < 0:

802 raise ValueError(_negative_)

803

804 if a < b:

805 a, b = b, a

806 if a > EPS0:

807 ba = b / a

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

809 if c2 > EPS02:

810 return a * sqrt(c2)

811 elif c2 < 0:

812 raise ValueError(_invalid_)

813 return hypot(a, b)

814

815

816 def triSide2(b, c, radB):

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

818

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

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

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

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

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

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

825

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

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

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

829

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

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

832

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

834 and L{pygeodesy.triSide4}.

835 '''

836 try:

837 return _triSide2(b, c, radB)

838 except (TypeError, ValueError) as x:

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

840

841

842 def _triSide2(b, c, radB):

843 # (INTERNAL) To allow callers to embellish errors

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

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

846 raise ValueError(_negative_)

847 sB, cB = sincos2(rB)

848 if isnear0(sB):

849 if not isnear0(b):

850 raise ValueError(_invalid_)

851 if cB < 0:

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

853 else:

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

855 elif isnear0(b):

856 raise ValueError(_invalid_)

857 else:

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

859 a = sin(rA) * b / sB

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

861

862

863 def triSide4(radA, radB, c):

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

865

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

867 (non-negative C{radians}).

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

869 (non-negative C{radians}).

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

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

872

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

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

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

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

877 triangle side B{C{c}}.

878

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

880

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

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

883 '''

884 try:

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

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

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

888 raise ValueError(_negative_)

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

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

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

892 raise ValueError(_invalid_)

893 sc = c / sc

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

895 name=triSide4.__name__)

896

897 except (TypeError, ValueError) as x:

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

899

900

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

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

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

904

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

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

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

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

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

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

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

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

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

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

915

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

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

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

919

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

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

922

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

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

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

926 and function L{pygeodesy.snellius3}.

927 '''

928 def _s(x):

929 return sin(x)**2

930

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

932 r = r1 * r3 * _4_0

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

934 if n < 0 or isnear0(r):

935 raise ValueError(_coincident_)

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

937

938 try:

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

940 if min(t) < 0:

941 raise ValueError(_negative_)

942

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

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

945 if min(s) < EPS02:

946 raise ValueError(_or(_coincident_, _colinear_))

947

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

949 if min(q) < EPS02:

950 raise ValueError(_coincident_)

951

952 r1 = s2 * q3 / s3 # s2!

953 r2 = s1 * q3 / s3 # s1!

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

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

956 s += Qs * _0_5, # tuple!

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

958 r3 = C0.fover(-s3)

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

960 if d0 > EPS02: # > c0

961 d0 = sqrt(d0)

962 if not callable(R3):

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

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

965 elif d0 < 0:

966 raise ValueError(_negative_)

967

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

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

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

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

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

973

974 except (TypeError, ValueError) as x:

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

976

977

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

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

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

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

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

983 else:

984 z = INT0

985 return z

986

987# **) MIT License

988#

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

990#

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

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

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

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

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

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

997#

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

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

1000#

1001# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS

1002# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

1003# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL

1004# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR

1005# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,

1006# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR

1007# OTHER DEALINGS IN THE SOFTWARE.