Coverage for pygeodesy/resections.py: 97%
370 statements
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2# -*- coding: utf-8 -*-
4u'''3-Point resection functions L{cassini}, L{collins5}, L{pierlot}, L{pierlotx} and
5L{tienstra7}, survey functions L{snellius3} and L{wildberger3} and triangle functions
6L{triAngle}, L{triAngle5}, L{triSide}, L{triSide2} and L{triSide4}.
8@note: Functions L{pierlot} and L{pierlotx} are transcoded to Python with permission from
9 U{triangulationPierlot<http://www.Telecom.ULg.ac.BE/triangulation/doc/total_8c.html>} and
10 U{Pierlot<http://www.Telecom.ULg.ac.BE/publi/publications/pierlot/Pierlot2014ANewThree>}.
11'''
12# make sure int/int division yields float quotient
13from __future__ import division as _; del _ # PYCHOK semicolon
15from pygeodesy.basics import map1, map2, _zip, _ALL_LAZY
16from pygeodesy.constants import EPS, EPS0, EPS02, INT0, PI, PI2, PI_2, PI_4, \
17 _0_0, _0_5, _1_0, _N_1_0, _2_0, _N_2_0, _4_0, \
18 _16_0, _180_0, _360_0, _copysign_0_0, isnear0, \
19 _over, _umod_360
20from pygeodesy.errors import _and, _or, TriangleError, _ValueError, _xcallable, \
21 _xkwds, _xkwds_pop2
22from pygeodesy.fmath import favg, Fdot, fidw, fmean, hypot, hypot2_
23from pygeodesy.fsums import Fsum, fsumf_, fsum1, fsum1f_
24from pygeodesy.interns import _a_, _A_, _area_, _b_, _B_, _c_, _C_, _coincident_, \
25 _colinear_, _d_, _eps_, _invalid_, _negative_, _not_, \
26 _rIn_, _SPACE_
27# from pygeodesy.lazily import _ALL_LAZY # from .basics
28from pygeodesy.named import _NamedTuple, _Pass, Fmt
29# from pygeodesy.streprs import Fmt # from .named
30from pygeodesy.units import Degrees, Distance, Radians
31from pygeodesy.utily import acos1, asin1, sincos2, sincos2_, sincos2d, sincos2d_
32from pygeodesy.vector3d import _otherV3d, Vector3d
34from math import cos, atan2, degrees, fabs, radians, sin, sqrt
36__all__ = _ALL_LAZY.resections
37__version__ = '24.04.04'
39_concyclic_ = 'concyclic'
40_PA_ = 'PA'
41_PB_ = 'PB'
42_PC_ = 'PC'
43_pointH_ = 'pointH'
44_pointP_ = 'pointP'
45_positive_ = 'positive'
46_radA_ = 'radA'
47_radB_ = 'radB'
48_radC_ = 'radC'
51class Collins5Tuple(_NamedTuple):
52 '''5-Tuple C{(pointP, pointH, a, b, c)} with survey C{pointP}, auxiliary
53 C{pointH}, each an instance of B{C{pointA}}'s (sub-)class and triangle
54 sides C{a}, C{b} and C{c} in C{meter}, conventionally.
55 '''
56 _Names_ = (_pointP_, _pointH_, _a_, _b_, _c_)
57 _Units_ = (_Pass, _Pass, Distance, Distance, Distance)
60def _F1(*xs): # class
61 '''(INTERNAL) An L{Fsum}, 1-primed.
62 '''
63 F = Fsum(_1_0, *xs)
64 F += _N_1_0
65 return F
68class ResectionError(_ValueError):
69 '''Error raised for issues in L{pygeodesy.resections}.
70 '''
71 pass
74class Survey3Tuple(_NamedTuple):
75 '''3-Tuple C{(PA, PB, PC)} with distance from survey point C{P} to each of
76 the triangle corners C{A}, C{B} and C{C} in C{meter}, conventionally.
77 '''
78 _Names_ = (_PA_, _PB_, _PC_)
79 _Units_ = ( Distance, Distance, Distance)
82class Tienstra7Tuple(_NamedTuple):
83 '''7-Tuple C{(pointP, A, B, C, a, b, c)} with survey C{pointP}, interior
84 triangle angles C{A}, C{B} and C{C} in C{degrees} and triangle sides
85 C{a}, C{b} and C{c} in C{meter}, conventionally.
86 '''
87 _Names_ = (_pointP_, _A_, _B_, _C_, _a_, _b_, _c_)
88 _Units_ = (_Pass, Degrees, Degrees, Degrees, Distance, Distance, Distance)
91class TriAngle5Tuple(_NamedTuple):
92 '''5-Tuple C{(radA, radB, radC, rIn, area)} with the interior angles at
93 triangle corners C{A}, C{B} and C{C} in C{radians}, the C{InCircle}
94 radius C{rIn} aka C{inradius} in C{meter} and the triangle C{area}
95 in C{meter} I{squared}, conventionally.
96 '''
97 _Names_ = (_radA_, _radB_, _radC_, _rIn_, _area_)
98 _Units_ = ( Radians, Radians, Radians, Distance, _Pass)
101class TriSide2Tuple(_NamedTuple):
102 '''2-Tuple C{(a, radA)} with triangle side C{a} in C{meter}, conventionally
103 and angle C{radA} at the opposite triangle corner in C{radians}.
104 '''
105 _Names_ = (_a_, _radA_)
106 _Units_ = ( Distance, Radians)
109class TriSide4Tuple(_NamedTuple):
110 '''4-Tuple C{(a, b, radC, d)} with interior angle C{radC} at triangle corner
111 C{C} in C{radians}with the length of triangle sides C{a} and C{b} and
112 with triangle height C{d} perpendicular to triangle side C{c}, in the
113 same units as triangle sides C{a} and C{b}.
114 '''
115 _Names_ = (_a_, _b_, _radC_, _d_)
116 _Units_ = ( Distance, Distance, Radians, Distance)
119def _ABC3(useZ, pointA, pointB, pointC):
120 '''(INTERNAL) Helper for L{cassini} and L{tienstra7}.
121 '''
122 return (_otherV3d(useZ=useZ, pointA=pointA),
123 _otherV3d(useZ=useZ, pointB=pointB),
124 _otherV3d(useZ=useZ, pointC=pointC))
127def _B3(useZ, point1, point2, point3):
128 '''(INTERNAL) Helper for L{pierlot} and L{pierlotx}.
129 '''
130 return (_otherV3d(useZ=useZ, point1=point1),
131 _otherV3d(useZ=useZ, point2=point2),
132 _otherV3d(useZ=useZ, point3=point3))
135def cassini(pointA, pointB, pointC, alpha, beta, useZ=False, **Clas_and_kwds):
136 '''3-Point resection using U{Cassini<https://NL.WikiPedia.org/wiki/Achterwaartse_insnijding>}'s method.
138 @arg pointA: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
139 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
140 @arg pointB: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
141 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
142 @arg pointC: Center point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
143 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
144 @arg alpha: Angle subtended by triangle side B{C{pointA}} to B{C{pointC}}
145 (C{degrees}, non-negative).
146 @arg beta: Angle subtended by triangle side B{C{pointB}} to B{C{pointC}}
147 (C{degrees}, non-negative).
148 @kwarg useZ: If C{True}, use and interpolate the Z component, otherwise
149 force C{z=INT0} (C{bool}).
150 @kwarg Clas_and_kwds: Optional class C{B{Clas}=B{pointA}.classof} to
151 return the survey point with optionally other B{C{Clas}}
152 keyword arguments to instantiate the survey point.
154 @note: Typically, B{C{pointC}} is between B{C{pointA}} and B{C{pointB}}.
156 @return: The survey point, an instance of B{C{Clas}} or B{C{pointA}}'s
157 (sub-)class.
159 @raise ResectionError: Near-coincident, -colinear or -concyclic points
160 or negative or invalid B{C{alpha}} or B{C{beta}}.
162 @raise TypeError: Invalid B{C{pointA}}, B{C{pointB}} or B{C{pointM}}.
164 @see: U{Three Point Resection Problem<https://Dokumen.tips/documents/
165 three-point-resection-problem-introduction-kaestner-burkhardt-method.html>}
166 and functions L{collins5}, L{pierlot}, L{pierlotx} and L{tienstra7}.
167 '''
169 def _H(A, C, sa):
170 s, c = sincos2d(sa)
171 if isnear0(s):
172 raise ValueError(_or(_coincident_, _colinear_))
173 t = s, c, c
174 x = Fdot(t, A.x, C.y, -A.y).fover(s)
175 y = Fdot(t, A.y, -C.x, A.x).fover(s)
176 return x, y
178 A, B, C = _ABC3(useZ, pointA, pointB, pointC)
179 try:
180 sa, sb = map1(float, alpha, beta)
181 if min(sa, sb) < 0:
182 raise ValueError(_negative_)
183 if fsumf_(_360_0, -sa, -sb) < EPS0:
184 raise ValueError()
186 x1, y1 = _H(A, C, sa)
187 x2, y2 = _H(B, C, -sb)
189 x = x1 - x2
190 y = y1 - y2
191 if isnear0(x) or isnear0(y):
192 raise ValueError(_SPACE_(_concyclic_, (x, y)))
194 m = y / x
195 n = x / y
196 N = n + m
197 if isnear0(N):
198 raise ValueError(_SPACE_(_concyclic_, (m, n, N)))
200 t = n, m, _1_0, _N_1_0
201 x = Fdot(t, C.x, x1, C.y, y1).fover(N)
202 y = Fdot(t, y1, C.y, C.x, x1).fover(N)
203 z = _zidw(x, y, useZ, A, B, C)
204 return _Clas(cassini, pointA, Clas_and_kwds, x, y, z)
206 except (TypeError, ValueError) as x:
207 raise ResectionError(pointA=pointA, pointB=pointB, pointC=pointC,
208 alpha=alpha, beta=beta, cause=x)
211def _Clas(where, point, Clas_and_kwds, *args):
212 '''(INTERNAL) Return a C{B{Clas}=point.classof} survey point.
213 '''
214 Clas, kwds = _xkwds_pop2(Clas_and_kwds, Clas=point.classof)
215 return Clas(*args, **_xkwds(kwds, name=where.__name__))
218def collins5(pointA, pointB, pointC, alpha, beta, useZ=False, **Clas_and_kwds):
219 '''3-Point resection using U{Collins<https://Dokumen.tips/documents/
220 three-point-resection-problem-introduction-kaestner-burkhardt-method.html>}' method.
222 @arg pointA: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
223 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
224 @arg pointB: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
225 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
226 @arg pointC: Center point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
227 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
228 @arg alpha: Angle subtended by triangle side C{b} from B{C{pointA}} to
229 B{C{pointC}} (C{degrees}, non-negative).
230 @arg beta: Angle subtended by triangle side C{a} from B{C{pointB}} to
231 B{C{pointC}} (C{degrees}, non-negative).
232 @kwarg useZ: If C{True}, use and interpolate the Z component, otherwise
233 force C{z=INT0} (C{bool}).
234 @kwarg Clas_and_kwds: Optional class C{B{Clas}=B{pointA}.classof} to
235 return the survey point with optionally other B{C{Clas}}
236 keyword arguments to instantiate the survey point.
238 @note: Typically, B{C{pointC}} is between B{C{pointA}} and B{C{pointB}}.
240 @return: L{Collins5Tuple}C{(pointP, pointH, a, b, c)} with survey C{pointP},
241 auxiliary C{pointH}, each an instance of B{C{Clas}} or B{C{pointA}}'s
242 (sub-)class and triangle sides C{a}, C{b} and C{c} in C{meter},
243 conventionally.
245 @raise ResectionError: Near-coincident, -colinear or -concyclic points
246 or negative or invalid B{C{alpha}} or B{C{beta}}.
248 @raise TypeError: Invalid B{C{pointA}}, B{C{pointB}} or B{C{pointM}}.
250 @see: U{Collins' methode<https://NL.WikiPedia.org/wiki/Achterwaartse_insnijding>}
251 and functions L{cassini}, L{pierlot}, L{pierlotx} and L{tienstra7}.
252 '''
254 def _azi_len2(A, B, pi2=PI2):
255 v = B.minus(A)
256 r = atan2(v.x, v.y)
257 if r < 0 and pi2:
258 r += pi2
259 return r, v.length
261 def _xyz(d, r, A, B, C, useZ):
262 s, c = sincos2(r)
263 x = A.x + d * s
264 y = A.y + d * c
265 z = _zidw(x, y, useZ, A, B, C)
266 return x, y, z
268 A, B, C = _ABC3(useZ, pointA, pointB, pointC)
269 try:
270 ra, rb = radians(alpha), radians(beta)
271 if min(ra, rb) < 0:
272 raise ValueError(_negative_)
274 sra, srH = sin(ra), sin(ra + rb - PI) # rH = PI - ((PI - ra) + (PI - rb))
275 if isnear0(sra) or isnear0(srH):
276 raise ValueError(_or(_coincident_, _colinear_, _concyclic_))
278# za, a = _azi_len2(C, B)
279 zb, b = _azi_len2(C, A)
280 zc, c = _azi_len2(A, B, 0)
282# d = c * sin(PI - rb) / srH # B.minus(H).length
283 d = c * sin(PI - ra) / srH # A.minus(H).length
284 r = zc + PI - rb # zh = zc + (PI - rb)
285 H = _xyz(d, r, A, B, C, useZ)
287 zh, _ = _azi_len2(C, Vector3d(*H))
289# d = a * sin(za - zh) / sin(rb) # B.minus(P).length
290 d = b * sin(zb - zh) / sra # A.minus(P).length
291 r = zh - ra # zb - PI + (PI - ra - (zb - zh))
292 P = _xyz(d, r, A, B, C, useZ)
293 P = _Clas(collins5, pointA, Clas_and_kwds, *P)
295 H = _Clas(collins5, pointA, Clas_and_kwds, *H)
296 a = B.minus(C).length
298 return Collins5Tuple(P, H, a, b, c, name=collins5.__name__)
300 except (TypeError, ValueError) as x:
301 raise ResectionError(pointA=pointA, pointB=pointB, pointC=pointC,
302 alpha=alpha, beta=beta, cause=x)
305def pierlot(point1, point2, point3, alpha12, alpha23, useZ=False, eps=EPS,
306 **Clas_and_kwds):
307 '''3-Point resection using U{Pierlot<http://www.Telecom.ULg.ac.BE/publi/publications/
308 pierlot/Pierlot2014ANewThree>}'s method C{ToTal} with I{approximate} limits for
309 the (pseudo-)singularities.
311 @arg point1: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
312 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
313 @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
314 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
315 @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
316 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
317 @arg alpha12: Angle subtended from B{C{point1}} to B{C{point2}} or
318 B{C{alpha2 - alpha1}} (C{degrees}).
319 @arg alpha23: Angle subtended from B{C{point2}} to B{C{point3}} or
320 B{C{alpha3 - alpha2}}(C{degrees}).
321 @kwarg useZ: If C{True}, interpolate the survey point's Z component,
322 otherwise use C{z=INT0} (C{bool}).
323 @kwarg eps: Tolerance for C{cot} (pseudo-)singularities (C{float}).
324 @kwarg Clas_and_kwds: Optional class C{B{Clas}=B{point1}.classof} to
325 return the survey point with optionally other B{C{Clas}}
326 keyword arguments to instantiate the survey point.
328 @note: Typically, B{C{point1}}, B{C{point2}} and B{C{point3}} are ordered
329 by angle, modulo 360, counter-clockwise.
331 @return: The survey (or robot) point, an instance of B{C{Clas}} or B{C{point1}}'s
332 (sub-)class.
334 @raise ResectionError: Near-coincident, -colinear or -concyclic points
335 or invalid B{C{alpha12}} or B{C{alpha23}} or
336 non-positive B{C{eps}}.
338 @raise TypeError: Invalid B{C{point1}}, B{C{point2}} or B{C{point3}}.
340 @see: I{Pierlot}'s C function U{triangulationPierlot<http://www.Telecom.ULg.ac.BE/
341 triangulation/doc/total_8c_source.html>}, U{V. Pierlot, M. Van Droogenbroeck,
342 "A New Three Object Triangulation Algorithm for Mobile Robot Positioning"
343 <https://ORBi.ULiege.BE/bitstream/2268/157469/1/Pierlot2014ANewThree.pdf>},
344 U{Vincent Pierlot, Marc Van Droogenbroeck, "18 Triangulation Algorithms for 2D
345 Positioning (also known as the Resection Problem)"<http://www.Telecom.ULg.ac.BE/
346 triangulation>} and functions L{pierlotx}, L{cassini}, L{collins5} and L{tienstra7}.
347 '''
349 def _cot(s, c): # -eps < I{approximate} cotangent < eps
350 if eps > 0:
351 return c / (min(s, -eps) if s < 0 else max(s, eps))
352 raise ValueError(_SPACE_(_eps_, _not_, _positive_))
354 B1, B2, B3 = _B3(useZ, point1, point2, point3)
355 try:
356 xyz = _pierlot3(B1, B2, B3, alpha12, alpha23, useZ, _cot)
357 return _Clas(pierlot, point1, Clas_and_kwds, *xyz)
359 except (TypeError, ValueError) as x:
360 raise ResectionError(point1=point1, point2=point2, point3=point3,
361 alpha12=alpha12, alpha23=alpha23, eps=eps, cause=x)
364def _pierlot3(B1, B2, B3, a12, a23, useZ, cot):
365 '''(INTERNAL) Shared L{pierlot} and L{pierlotx}.
366 '''
367 x1_, y1_, _ = B1.minus(B2).xyz
368 x3_, y3_, _ = B3.minus(B2).xyz
370 s12, c12, s23, c23 = sincos2d_(a12, a23)
371 # cot31 = (1 - cot12 * cot23) / (cot12 + cot32)
372 # = (1 - c12 / s12 * c23 / s23) / (c12 / s12 + c23 / s23)
373 # = (1 - (c12 * c23) / (s12 * s23)) / (c12 * s23 + s12 * c23) / (s12 * s23)
374 # = (s12 * s23 - c12 * c23) / (c12 * s23 + s12 * c23)
375 # = c31 / s31
376 cot31 = cot(fsum1f_(c12 * s23, s12 * c23), # s31
377 fsum1f_(s12 * s23, -c12 * c23)) # c31
379 K = _F1(x3_ * x1_, cot31 * (y3_ * x1_),
380 y3_ * y1_, -cot31 * (x3_ * y1_))
381 if K:
382 cot12 = cot(s12, c12)
383 cot23 = cot(s23, c23)
385 # x12 = x1_ + cot12 * y1_
386 # y12 = y1_ - cot12 * x1_
388 # x23 = x3_ - cot23 * y3_
389 # y23 = y3_ + cot23 * x3_
391 # x31 = x3_ + x1_ + cot31 * (y3_ - y1_)
392 # y31 = y3_ + y1_ - cot31 * (x3_ - x1_)
394 # x12 - x23 = x1_ + cot12 * y1_ - x3_ + cot23 * y3_
395 X12_23 = _F1(x1_, cot12 * y1_, -x3_, cot23 * y3_)
396 # y12 - y23 = y1_ - cot12 * x1_ - y3_ - cot23 * x3_
397 Y12_23 = _F1(y1_, -cot12 * x1_, -y3_, -cot23 * x3_)
399 # x31 - x23 = x3_ + x1_ + cot31 * (y3_ - y1_) - x3_ + cot23 * y3_
400 # = x1_ + cot31 * y3_ - cot31 * y1_ + cot23 * y3_
401 X31_23 = _F1(x1_, -cot31 * y1_, cot31 * y3_, cot23 * y3_)
402 # y31 - y23 = y3_ + y1_ - cot31 * (x3_ - x1_) - y3_ - cot23 * x3_
403 # = y1_ - cot31 * x3_ + cot31 * x1_ - cot23 * x3_
404 Y31_23 = _F1(y1_, cot31 * x1_, -cot31 * x3_, -cot23 * x3_)
406 # d = (x12 - x23) * (y23 - y31) + (x31 - x23) * (y12 - y23)
407 # = (x31 - x23) * (y12 - y23) - (x12 - x23) * (y31 - y23)
408 # x = (d * B2.x + K * Y12_23).fover(d)
409 # y = (d * B2.y - K * X12_23).fover(d)
410 x, y = _pierlotxy2(B2, -K, Y12_23, X12_23, (X31_23 * Y12_23 -
411 X12_23 * Y31_23))
412 else:
413 x, y, _ = B2.xyz
414 return x, y, _zidw(x, y, useZ, B1, B2, B3)
417def pierlotx(point1, point2, point3, alpha1, alpha2, alpha3, useZ=False,
418 **Clas_and_kwds):
419 '''3-Point resection using U{Pierlot<http://www.Telecom.ULg.ac.BE/publi/
420 publications/pierlot/Pierlot2014ANewThree>}'s method C{ToTal} with
421 I{exact} limits for the (pseudo-)singularities.
423 @arg point1: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
424 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
425 @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
426 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
427 @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
428 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
429 @arg alpha1: Angle at B{C{point1}} (C{degrees}, counter-clockwise).
430 @arg alpha2: Angle at B{C{point2}} (C{degrees}, counter-clockwise).
431 @arg alpha3: Angle at B{C{point3}} (C{degrees}, counter-clockwise).
432 @kwarg useZ: If C{True}, interpolate the survey point's Z component,
433 otherwise use C{z=INT0} (C{bool}).
434 @kwarg Clas_and_kwds: Optional class C{B{Clas}=B{point1}.classof} to
435 return the survey point with optionally other B{C{Clas}}
436 keyword arguments to instantiate the survey point.
438 @return: The survey (or robot) point, an instance of B{C{Clas}} or B{C{point1}}'s
439 (sub-)class.
441 @raise ResectionError: Near-coincident, -colinear or -concyclic points or
442 invalid B{C{alpha1}}, B{C{alpha2}} or B{C{alpha3}}.
444 @raise TypeError: Invalid B{C{point1}}, B{C{point2}} or B{C{point3}}.
446 @see: I{Pierlot}'s C function U{triangulationPierlot2<http://www.Telecom.ULg.ac.BE/
447 triangulation/doc/total_8c_source.html>} and function L{pierlot}, L{cassini},
448 L{collins5} and L{tienstra7}.
449 '''
451 def _a_z_Bs(Bs, *alphas):
452 ds = map2(_umod_360, alphas) # 0 <= alphas < 360
453 ds, Bs = zip(*sorted(_zip(ds, Bs))) # unzip
454 for p, d, B in _zip(ds, _rotate(ds), Bs):
455 d -= p # a12 = a2 - a1, ...
456 z = isnear0(fabs(d) % _180_0)
457 yield d, z, B
459 def _cot(s, c): # I{exact} cotangent
460 try:
461 return (c / s) if c else _copysign_0_0(s)
462 except ZeroDivisionError:
463 raise ValueError(_or(_coincident_, _colinear_))
465 Bs = _B3(useZ, point1, point2, point3)
466 try:
467 Cs = [0] # pseudo-global, passing the exception Case
468 xyz = _pierlotx3(_a_z_Bs(Bs, alpha1, alpha2, alpha3),
469 useZ, _cot, Cs.append)
470 return _Clas(pierlotx, point1, Clas_and_kwds, *xyz)
472 except (TypeError, ValueError) as x:
473 raise ResectionError(point1=point1, point2=point2, point3=point3, C=Cs.pop(),
474 alpha1=alpha1, alpha2=alpha2, alpha3=alpha3, cause=x)
477def _pierlotx3(a_z_Bs, useZ, cot, Cs):
478 '''(INTERNAL) Core of L{pierlotx}.
479 '''
480 (a12, z12, B1), \
481 (a23, z23, B2), \
482 (a31, z31, B3) = a_z_Bs
483 if z12 and not z23:
484 Cs(1)
485 elif z23 and not z31:
486 Cs(2)
487 a23, B1, B2, B3 = a31, B2, B3, B1
488 elif z31 and not z12:
489 Cs(3)
490 a23, B2, B3 = a12, B3, B2
491 else:
492 Cs(4)
493 return _pierlot3(B1, B2, B3, a12, a23, useZ, cot)
495 x1_, y1_, _ = B1.minus(B3).xyz
496 x2_, y2_, _ = B2.minus(B3).xyz
498 K = _F1(y1_ * x2_, -x1_ * y2_)
499 if K:
500 cot23 = cot(*sincos2d(a23))
502 # x23 = x2_ + cot23 * y2_
503 # y23 = y2_ - cot23 * x2_
505 # x31 = x1_ + cot23 * y1_
506 # y31 = y1_ - cot23 * x1_
508 # x31 - x23 = x1_ + cot23 * y1_ - x2_ - cot23 * y2_
509 X31_23 = _F1(x1_, cot23 * y1_, -x2_, -cot23 * y2_)
510 # y31 - y23 = y1_ - cot23 * x1_ - y2_ + cot23 * x2_
511 Y31_23 = _F1(y1_, -cot23 * x1_, -y2_, cot23 * x2_)
513 # d = (x31 - x23) * (x2_ - x1_) + (y31 - y23) * (y2_ - y1_)
514 # x = (D * B3.x - K * Y31_23).fover(d)
515 # y = (D * B3.y + K * X31_23).fover(d)
516 x, y = _pierlotxy2(B3, K, Y31_23, X31_23, (X31_23 * _F1(x2_, -x1_) +
517 Y31_23 * _F1(y2_, -y1_)))
518 else:
519 x, y, _ = B3.xyz
520 return x, y, _zidw(x, y, useZ, B1, B2, B3)
523def _pierlotxy2(B, K, X, Y, D):
524 '''(INTERNAL) Helper for C{_pierlot3} and C{_pierlotx3}.
525 '''
526 d = float(D)
527 if isnear0(d):
528 raise ValueError(_or(_coincident_, _colinear_, _concyclic_))
529 x = (D * B.x - K * X).fover(d)
530 y = (D * B.y + K * Y).fover(d)
531 return x, y
534def _rotate(xs, n=1):
535 '''Rotate list or tuple C{xs} by C{n} items, right if C{n > 0} else left.
536 '''
537 return xs[n:] + xs[:n]
540def snellius3(a, b, degC, alpha, beta):
541 '''Snellius' surveying using U{Snellius Pothenot<https://WikiPedia.org/wiki/Snellius–Pothenot_problem>}.
543 @arg a: Length of the triangle side between corners C{B} and C{C} and opposite of
544 triangle corner C{A} (C{scalar}, non-negative C{meter}, conventionally).
545 @arg b: Length of the triangle side between corners C{C} and C{A} and opposite of
546 triangle corner C{B} (C{scalar}, non-negative C{meter}, conventionally).
547 @arg degC: Angle at triangle corner C{C}, opposite triangle side C{c} (non-negative C{degrees}).
548 @arg alpha: Angle subtended by triangle side B{C{b}} (non-negative C{degrees}).
549 @arg beta: Angle subtended by triangle side B{C{a}} (non-negative C{degrees}).
551 @return: L{Survey3Tuple}C{(PA, PB, PC)} with distance from survey point C{P} to
552 each of the triangle corners C{A}, C{B} and C{C}, same units as triangle
553 sides B{C{a}}, B{C{b}} and B{C{c}}.
555 @raise TriangleError: Invalid B{C{a}}, B{C{b}} or B{C{degC}} or negative B{C{alpha}}
556 or B{C{beta}}.
558 @see: Function L{wildberger3}.
559 '''
560 try:
561 a, b, degC, alpha, beta = t = map1(float, a, b, degC, alpha, beta)
562 if min(t) < 0:
563 raise ValueError(_negative_)
564 ra, rb, rC = map1(radians, alpha, beta, degC)
566 r = fsum1f_(ra, rb, rC) * _0_5
567 k = PI - r
568 if min(k, r) < 0:
569 raise ValueError(_or(_coincident_, _colinear_))
571 sa, sb = map1(sin, ra, rb)
572 p = atan2(sa * a, sb * b)
573 sp, cp, sr, cr = sincos2_(PI_4 - p, r)
574 p = atan2(sp * sr, cp * cr)
575 pa = k + p
576 pb = k - p
578 if fabs(sb) > fabs(sa):
579 pc = fabs(a * sin(pb) / sb)
580 elif sa:
581 pc = fabs(b * sin(pa) / sa)
582 else:
583 raise ValueError(_or(_colinear_, _coincident_))
585 pa = _triSide(b, pc, fsumf_(PI, -ra, -pa))
586 pb = _triSide(a, pc, fsumf_(PI, -rb, -pb))
587 return Survey3Tuple(pa, pb, pc, name=snellius3.__name__)
589 except (TypeError, ValueError) as x:
590 raise TriangleError(a=a, b=b, degC=degC, alpha=alpha, beta=beta, cause=x)
593def tienstra7(pointA, pointB, pointC, alpha, beta=None, gamma=None,
594 useZ=False, **Clas_and_kwds):
595 '''3-Point resection using U{Tienstra<https://WikiPedia.org/wiki/Tienstra_formula>}'s formula.
597 @arg pointA: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or
598 C{Vector2Tuple} if C{B{useZ}=False}).
599 @arg pointB: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or
600 C{Vector2Tuple} if C{B{useZ}=False}).
601 @arg pointC: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or
602 C{Vector2Tuple} if C{B{useZ}=False}).
603 @arg alpha: Angle subtended by triangle side C{a} from B{C{pointB}} to B{C{pointC}}
604 (C{degrees}, non-negative).
605 @kwarg beta: Angle subtended by triangle side C{b} from B{C{pointA}} to B{C{pointC}}
606 (C{degrees}, non-negative) or C{None} if C{B{gamma} is not None}.
607 @kwarg gamma: Angle subtended by triangle side C{c} from B{C{pointA}} to B{C{pointB}}
608 (C{degrees}, non-negative) or C{None} if C{B{beta} is not None}.
609 @kwarg useZ: If C{True}, use and interpolate the Z component, otherwise force C{z=INT0}
610 (C{bool}).
611 @kwarg Clas_and_kwds: Optional class C{B{Clas}=B{pointA}.classof} to return the survey
612 point with optionally other B{C{Clas}} keyword arguments to instantiate
613 the survey point.
615 @note: Points B{C{pointA}}, B{C{pointB}} and B{C{pointC}} are ordered clockwise.
617 @return: L{Tienstra7Tuple}C{(pointP, A, B, C, a, b, c)} with survey C{pointP}, an
618 instance of B{C{Clas}} or B{C{pointA}}'s (sub-)class, with triangle angles C{A}
619 at B{C{pointA}}, C{B} at B{C{pointB}} and C{C} at B{C{pointC}} in C{degrees}
620 and with triangle sides C{a}, C{b} and C{c} in C{meter}, conventionally.
622 @raise ResectionError: Near-coincident, -colinear or -concyclic points or sum of
623 B{C{alpha}}, B{C{beta}} and B{C{gamma}} not C{360} or negative
624 B{C{alpha}}, B{C{beta}} or B{C{gamma}}.
626 @raise TypeError: Invalid B{C{pointA}}, B{C{pointB}} or B{C{pointC}}.
628 @see: U{3-Point Resection Solver<http://MesaMike.org/geocache/GC1B0Q9/tienstra/>},
629 U{V. Pierlot, M. Van Droogenbroeck, "A New Three Object Triangulation..."
630 <http://www.Telecom.ULg.ac.BE/publi/publications/pierlot/Pierlot2014ANewThree/>},
631 U{18 Triangulation Algorithms...<http://www.Telecom.ULg.ac.BE/triangulation/>} and
632 functions L{cassini}, L{collins5}, L{pierlot} and L{pierlotx}.
633 '''
635 def _deg_ks(r, s, ks, N):
636 if isnear0(fsumf_(PI, r, -s)): # r + (PI2 - s) == PI
637 raise ValueError(Fmt.PARENSPACED(concyclic=N))
638 # k = 1 / (cot(r) - cot(s))
639 # = 1 / (cos(r) / sin(r) - cos(s) / sin(s))
640 # = 1 / (cos(r) * sin(s) - cos(s) * sin(r)) / (sin(r) * sin(s))
641 # = sin(r) * sin(s) / (cos(r) * sin(s) - cos(s) * sin(r))
642 sr, cr, ss, cs = sincos2_(r, s)
643 c = fsum1f_(cr * ss, -cs * sr)
644 if isnear0(c):
645 raise ValueError(Fmt.PARENSPACED(cotan=N))
646 ks.append(sr * ss / c)
647 return Degrees(degrees(r), name=N) # C degrees
649 A, B, C = _ABC3(useZ, pointA, pointB, pointC)
650 try:
651 sa, sb, sc = map1(radians, alpha, (beta or 0), (gamma or 0))
652 if beta is None:
653 if gamma is None:
654 raise ValueError(_and(Fmt.EQUAL(beta=beta), Fmt.EQUAL(gamma=gamma)))
655 sb = fsumf_(PI2, -sa, -sc)
656 elif gamma is None:
657 sc = fsumf_(PI2, -sa, -sb)
658 else: # subtended angles must add to 360 degrees
659 r = fsum1f_(sa, sb, sc)
660 if fabs(r - PI2) > EPS:
661 raise ValueError(Fmt.EQUAL(sum=degrees(r)))
662 if min(sa, sb, sc) < 0:
663 raise ValueError(_negative_)
665 # triangle sides
666 a = B.minus(C).length
667 b = A.minus(C).length
668 c = A.minus(B).length
670 ks = [] # 3 Ks and triangle angles
671 dA = _deg_ks(_triAngle(b, c, a), sa, ks, _A_)
672 dB = _deg_ks(_triAngle(a, c, b), sb, ks, _B_)
673 dC = _deg_ks(_triAngle(a, b, c), sc, ks, _C_)
675 k = fsum1(ks, floats=True)
676 if isnear0(k):
677 raise ValueError(Fmt.EQUAL(K=k))
678 x = Fdot(ks, A.x, B.x, C.x).fover(k)
679 y = Fdot(ks, A.y, B.y, C.y).fover(k)
680 z = _zidw(x, y, useZ, A, B, C)
682 P = _Clas(tienstra7, pointA, Clas_and_kwds, x, y, z)
683 return Tienstra7Tuple(P, dA, dB, dC, a, b, c, name=tienstra7.__name__)
685 except (TypeError, ValueError) as x:
686 raise ResectionError(pointA=pointA, pointB=pointB, pointC=pointC,
687 alpha=alpha, beta=beta, gamma=gamma, cause=x)
690def triAngle(a, b, c):
691 '''Compute one angle of a triangle.
693 @arg a: Adjacent triangle side length (C{scalar}, non-negative
694 C{meter}, conventionally).
695 @arg b: Adjacent triangle side length (C{scalar}, non-negative
696 C{meter}, conventionally).
697 @arg c: Opposite triangle side length (C{scalar}, non-negative
698 C{meter}, conventionally).
700 @return: Angle in C{radians} at triangle corner C{C}, opposite
701 triangle side B{C{c}}.
703 @raise TriangleError: Invalid or negative B{C{a}}, B{C{b}} or B{C{c}}.
705 @see: Functions L{triAngle5} and L{triSide}.
706 '''
707 try:
708 return _triAngle(a, b, c)
709 except (TypeError, ValueError) as x:
710 raise TriangleError(a=a, b=b, c=c, cause=x)
713def _triAngle(a, b, c):
714 # (INTERNAL) To allow callers to embellish errors
715 a, b, c = map1(float, a, b, c)
716 if a < b:
717 a, b = b, a
718 if b < 0 or c < 0:
719 raise ValueError(_negative_)
720 if a < EPS0:
721 raise ValueError(_coincident_)
722 b_a = b / a
723 if b_a < EPS0:
724 raise ValueError(_coincident_)
725 t = fsumf_(_1_0, b_a**2, -(c / a)**2) / (b_a * _2_0)
726 return acos1(t)
729def triAngle5(a, b, c):
730 '''Compute the angles of a triangle.
732 @arg a: Length of the triangle side opposite of triangle corner C{A}
733 (C{scalar}, non-negative C{meter}, conventionally).
734 @arg b: Length of the triangle side opposite of triangle corner C{B}
735 (C{scalar}, non-negative C{meter}, conventionally).
736 @arg c: Length of the triangle side opposite of triangle corner C{C}
737 (C{scalar}, non-negative C{meter}, conventionally).
739 @return: L{TriAngle5Tuple}C{(radA, radB, radC, rIn, area)} with angles
740 C{radA}, C{radB} and C{radC} at triangle corners C{A}, C{B}
741 and C{C}, all in C{radians}, the C{InCircle} radius C{rIn}
742 aka C{inradius}, same units as triangle sides B{C{a}},
743 B{C{b}} and B{C{c}} and the triangle C{area} in those same
744 units I{squared}.
746 @raise TriangleError: Invalid or negative B{C{a}}, B{C{b}} or B{C{c}}.
748 @see: Functions L{triAngle} and L{triArea}.
749 '''
750 try:
751 x, y, z = map1(float, a, b, c)
752 ab = x < y
753 if ab:
754 x, y = y, x
755 bc = y < z
756 if bc:
757 y, z = z, y
759 if z > EPS0: # z = min(a, b, c)
760 s = fsum1f_(z, y, x) * _0_5
761 sa, sb, r = (s - x), (s - y), (s - z)
762 r *= _over(sa * sb, s)
763 if r < EPS02:
764 raise ValueError(_coincident_)
765 r = sqrt(r)
766 rA = atan2(r, sa) * _2_0
767 rB = atan2(r, sb) * _2_0
768 rC = fsumf_(PI, -rA, -rB)
769 if min(rA, rB, rC) < 0:
770 raise ValueError(_colinear_)
771 s *= r # Heron's area
772 elif z < 0:
773 raise ValueError(_negative_)
774 else: # 0 <= c <= EPS0
775 rA = rB = PI_2
776 rC = r = s = _0_0
778 if bc:
779 rB, rC = rC, rB
780 if ab:
781 rA, rB = rB, rA
782 return TriAngle5Tuple(rA, rB, rC, r, s, name=triAngle5.__name__)
784 except (TypeError, ValueError) as x:
785 raise TriangleError(a=a, b=b, c=c, cause=x)
788def triArea(a, b, c):
789 '''Compute the area of a triangle using U{Heron's<https://
790 WikiPedia.org/wiki/Heron%27s_formula>} C{stable} formula.
792 @arg a: Length of the triangle side opposite of triangle corner C{A}
793 (C{scalar}, non-negative C{meter}, conventionally).
794 @arg b: Length of the triangle side opposite of triangle corner C{B}
795 (C{scalar}, non-negative C{meter}, conventionally).
796 @arg c: Length of the triangle side opposite of triangle corner C{C}
797 (C{scalar}, non-negative C{meter}, conventionally).
799 @return: The triangle area (C{float}, conventionally C{meter} or
800 same units as B{C{a}}, B{C{b}} and B{C{c}} I{squared}).
802 @raise TriangleError: Invalid or negative B{C{a}}, B{C{b}} or B{C{c}}.
803 '''
804 try:
805 r, y, x = sorted(map1(float, a, b, c))
806 if r > 0: # r = min(a, b, c)
807 ab = x - y
808 bc = y - r
809 y += r
810 r = (x + y) * (r - ab) * (r + ab) * (x + bc)
811 if r:
812 r = sqrt(r / _16_0)
813 elif r < 0:
814 raise ValueError(_negative_)
815 return r
817 except (TypeError, ValueError) as x:
818 raise TriangleError(a=a, b=b, c=c, cause=x)
821def triSide(a, b, radC):
822 '''Compute one side of a triangle.
824 @arg a: Adjacent triangle side length (C{scalar},
825 non-negative C{meter}, conventionally).
826 @arg b: Adjacent triangle side length (C{scalar},
827 non-negative C{meter}, conventionally).
828 @arg radC: Angle included by sides B{C{a}} and B{C{b}},
829 opposite triangle side C{c} (C{radians}).
831 @return: Length of triangle side C{c}, opposite triangle
832 corner C{C} and angle B{C{radC}}, same units as
833 B{C{a}} and B{C{b}}.
835 @raise TriangleError: Invalid B{C{a}}, B{C{b}} or B{C{radC}}.
837 @see: Functions L{sqrt_a}, L{triAngle}, L{triSide2} and L{triSide4}.
838 '''
839 try:
840 return _triSide(a, b, radC)
841 except (TypeError, ValueError) as x:
842 raise TriangleError(a=a, b=b, radC=radC, cause=x)
845def _triSide(a, b, radC):
846 # (INTERNAL) To allow callers to embellish errors
847 a, b, r = t = map1(float, a, b, radC)
848 if min(t) < 0:
849 raise ValueError(_negative_)
851 if a < b:
852 a, b = b, a
853 if a > EPS0:
854 ba = b / a
855 c2 = fsumf_(_1_0, ba**2, _N_2_0 * ba * cos(r))
856 if c2 > EPS02:
857 return a * sqrt(c2)
858 elif c2 < 0:
859 raise ValueError(_invalid_)
860 return hypot(a, b)
863def triSide2(b, c, radB):
864 '''Compute a side and its opposite angle of a triangle.
866 @arg b: Adjacent triangle side length (C{scalar},
867 non-negative C{meter}, conventionally).
868 @arg c: Adjacent triangle side length (C{scalar},
869 non-negative C{meter}, conventionally).
870 @arg radB: Angle included by sides B{C{a}} and B{C{c}},
871 opposite triangle side C{b} (C{radians}).
873 @return: L{TriSide2Tuple}C{(a, radA)} with triangle angle
874 C{radA} in C{radians} and length of the opposite
875 triangle side C{a}, same units as B{C{b}} and B{C{c}}.
877 @raise TriangleError: Invalid B{C{b}} or B{C{c}} or either
878 B{C{b}} or B{C{radB}} near zero.
880 @see: Functions L{sqrt_a}, L{triSide} and L{triSide4}.
881 '''
882 try:
883 return _triSide2(b, c, radB)
884 except (TypeError, ValueError) as x:
885 raise TriangleError(b=b, c=c, radB=radB, cause=x)
888def _triSide2(b, c, radB):
889 # (INTERNAL) To allow callers to embellish errors
890 b, c, rB = map1(float, b, c, radB)
891 if min(b, c, rB) < 0:
892 raise ValueError(_negative_)
893 sB, cB = sincos2(rB)
894 if isnear0(sB):
895 if not isnear0(b):
896 raise ValueError(_invalid_)
897 a, rA = ((b + c), PI) if cB < 0 else (fabs(b - c), _0_0)
898 elif isnear0(b):
899 raise ValueError(_invalid_)
900 else:
901 rA = fsumf_(PI, -rB, -asin1(c * sB / b))
902 a = sin(rA) * b / sB
903 return TriSide2Tuple(a, rA, name=triSide2.__name__)
906def triSide4(radA, radB, c):
907 '''Compute two sides and the height of a triangle.
909 @arg radA: Angle at triangle corner C{A}, opposite triangle side C{a}
910 (non-negative C{radians}).
911 @arg radB: Angle at triangle corner C{B}, opposite triangle side C{b}
912 (non-negative C{radians}).
913 @arg c: Length of triangle side between triangle corners C{A} and C{B},
914 (C{scalar}, non-negative C{meter}, conventionally).
916 @return: L{TriSide4Tuple}C{(a, b, radC, d)} with triangle sides C{a} and
917 C{b} and triangle height C{d} perpendicular to triangle side
918 B{C{c}}, all in the same units as B{C{c}} and interior angle
919 C{radC} in C{radians} at triangle corner C{C}, opposite
920 triangle side B{C{c}}.
922 @raise TriangleError: Invalid or negative B{C{radA}}, B{C{radB}} or B{C{c}}.
924 @see: U{Triangulation, Surveying<https://WikiPedia.org/wiki/Triangulation_(surveying)>}
925 and functions L{sqrt_a}, L{triSide} and L{triSide2}.
926 '''
927 try:
928 rA, rB, c = map1(float, radA, radB, c)
929 rC = fsumf_(PI, -rA, -rB)
930 if min(rC, rA, rB, c) < 0:
931 raise ValueError(_negative_)
932 sa, ca, sb, cb = sincos2_(rA, rB)
933 sc = fsum1f_(sa * cb, sb * ca)
934 if sc < EPS0 or min(sa, sb) < 0:
935 raise ValueError(_invalid_)
936 sc = c / sc
937 return TriSide4Tuple((sa * sc), (sb * sc), rC, (sa * sb * sc),
938 name=triSide4.__name__)
940 except (TypeError, ValueError) as x:
941 raise TriangleError(radA=radA, radB=radB, c=c, cause=x)
944def wildberger3(a, b, c, alpha, beta, R3=min):
945 '''Snellius' surveying using U{Rational Trigonometry
946 <https://WikiPedia.org/wiki/Snellius–Pothenot_problem>}.
948 @arg a: Length of the triangle side between corners C{B} and C{C} and opposite of
949 triangle corner C{A} (C{scalar}, non-negative C{meter}, conventionally).
950 @arg b: Length of the triangle side between corners C{C} and C{A} and opposite of
951 triangle corner C{B} (C{scalar}, non-negative C{meter}, conventionally).
952 @arg c: Length of the triangle side between corners C{A} and C{B} and opposite of
953 triangle corner C{C} (C{scalar}, non-negative C{meter}, conventionally).
954 @arg alpha: Angle subtended by triangle side B{C{b}} (C{degrees}, non-negative).
955 @arg beta: Angle subtended by triangle side B{C{a}} (C{degrees}, non-negative).
956 @kwarg R3: Callable to determine C{R3} from C{(R3 - C)**2 = D}, typically standard
957 Python function C{min} or C{max}, invoked with 2 arguments.
959 @return: L{Survey3Tuple}C{(PA, PB, PC)} with distance from survey point C{P} to
960 each of the triangle corners C{A}, C{B} and C{C}, same units as B{C{a}},
961 B{C{b}} and B{C{c}}.
963 @raise TriangleError: Invalid B{C{a}}, B{C{b}} or B{C{c}} or negative B{C{alpha}} or
964 B{C{beta}} or B{C{R3}} not C{callable}.
966 @see: U{Wildberger, Norman J.<https://Math.Sc.Chula.ac.TH/cjm/content/
967 survey-article-greek-geometry-rational-trigonometry-and-snellius-–-pothenot-surveying>},
968 U{Devine Proportions, page 252<http://www.MS.LT/derlius/WildbergerDivineProportions.pdf>}
969 and function L{snellius3}.
970 '''
971 def _s(x):
972 return sin(x)**2
974 def _vpa(r3, q2, q3, s2, s3):
975 r1 = s2 * q3 / s3
976 r = r1 * r3 * _4_0
977 n = (r - _F1(r1, r3, -q2)**2).fover(s3)
978 if n < 0 or r < EPS0:
979 raise ValueError(_coincident_)
980 return sqrt((n / r) * q3) if n else _0_0
982 try:
983 a, b, c, da, db = q = map1(float, a, b, c, alpha, beta)
984 if min(q) < 0:
985 raise ValueError(_negative_)
987 q1, q2, q3 = q = a**2, b**2, c**2
988 if min(q) < EPS02:
989 raise ValueError(_coincident_)
991 ra, rb = map1(radians, da, db)
992 s1, s2, s3 = s = map1(_s, rb, ra, ra + rb) # rb, ra!
993 if min(s) < EPS02:
994 raise ValueError(_or(_coincident_, _colinear_))
996 q4 = hypot2_(*q) * _2_0 # a**4 + ...
997 Qs = _F1(*q) # == hypot2_(a, b, c)
998 d0 = (Qs**2 - q4).fmul(s1 * s2).fover(s3)
999 if d0 < 0:
1000 raise ValueError(_negative_)
1001 s += _F1(*s), # == fsum1(s),
1002 C0 = Fdot(s, q1, q2, q3, -Qs * _0_5)
1003 r3 = C0.fover(-s3) # C0 /= -s3
1004 if d0 > EPS02: # > c0
1005 _xcallable(R3=R3)
1006 d0 = sqrt(d0)
1007 r3 = R3(float(C0 + d0), float(C0 - d0)) # XXX min or max
1009 pa = _vpa(r3, q2, q3, s2, s3)
1010 pb = _vpa(r3, q1, q3, s1, s3)
1011 pc = favg(_triSide2(b, pa, ra).a,
1012 _triSide2(a, pb, rb).a)
1013 return Survey3Tuple(pa, pb, pc, name=wildberger3.__name__)
1015 except (TypeError, ValueError) as x:
1016 raise TriangleError(a=a, b=b, c=c, alpha=alpha, beta=beta, R3=R3, cause=x)
1019def _zidw(x, y, useZ, *ABC):
1020 if useZ: # interpolate z or coplanar with A, B and C?
1021 t = tuple(_.z for _ in ABC)
1022 v = Vector3d(x, y, fmean(t))
1023 z = fidw(t, (v.minus(T).length for T in ABC))
1024 else:
1025 z = INT0
1026 return z
1028# **) MIT License
1029#
1030# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved.
1031#
1032# Permission is hereby granted, free of charge, to any person obtaining a
1033# copy of this software and associated documentation files (the "Software"),
1034# to deal in the Software without restriction, including without limitation
1035# the rights to use, copy, modify, merge, publish, distribute, sublicense,
1036# and/or sell copies of the Software, and to permit persons to whom the
1037# Software is furnished to do so, subject to the following conditions:
1038#
1039# The above copyright notice and this permission notice shall be included
1040# in all copies or substantial portions of the Software.
1041#
1042# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
1043# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
1044# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
1045# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
1046# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
1047# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
1048# OTHER DEALINGS IN THE SOFTWARE.