Coverage for pygeodesy/resections.py: 97%
366 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, typename
16from pygeodesy.constants import EPS, EPS0, EPS02, INT0, PI, PI2, PI_2, PI_4, \
17 _0_0, _0_5, _1_0, _N_1_0, _2_0, _N_2_0, _4_0, \
18 _16_0, _180_0, _360_0, isnear0, _over, _umod_360
19from pygeodesy.errors import _and, _or, TriangleError, _ValueError, _xcallable, \
20 _xkwds, _xkwds_pop2
21from pygeodesy.fmath import favg, Fdot, fidw, fmean, hypot, hypot2_
22from pygeodesy.fsums import _Fsumf_, fsumf_, fsum1, fsum1f_
23# from pygeodesy.internals import typename # from .basics
24from pygeodesy.interns import _a_, _A_, _area_, _b_, _B_, _c_, _C_, _coincident_, \
25 _colinear_, _d_, _invalid_, _negative_, _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, atan2, sincos2, sincos2_, \
32 sincos2d, sincos2d_
33from pygeodesy.vector3d import _otherV3d, Vector3d
35from math import cos, degrees, fabs, radians, sin, sqrt
37__all__ = _ALL_LAZY.resections
38__version__ = '25.04.14'
40_concyclic_ = 'concyclic'
41_PA_ = 'PA'
42_PB_ = 'PB'
43_PC_ = 'PC'
44_pointH_ = 'pointH'
45_pointP_ = 'pointP'
46_positive_ = 'positive'
47_radA_ = 'radA'
48_radB_ = 'radB'
49_radC_ = 'radC'
52class Collins5Tuple(_NamedTuple):
53 '''5-Tuple C{(pointP, pointH, a, b, c)} with survey C{pointP}, auxiliary
54 C{pointH}, each an instance of B{C{pointA}}'s (sub-)class and triangle
55 sides C{a}, C{b} and C{c} in C{meter}, conventionally.
56 '''
57 _Names_ = (_pointP_, _pointH_, _a_, _b_, _c_)
58 _Units_ = (_Pass, _Pass, Distance, Distance, Distance)
61class ResectionError(_ValueError):
62 '''Error raised for issues in L{pygeodesy.resections}.
63 '''
64 pass
67class Survey3Tuple(_NamedTuple):
68 '''3-Tuple C{(PA, PB, PC)} with distance from survey point C{P} to each of
69 the triangle corners C{A}, C{B} and C{C} in C{meter}, conventionally.
70 '''
71 _Names_ = (_PA_, _PB_, _PC_)
72 _Units_ = ( Distance, Distance, Distance)
75class Tienstra7Tuple(_NamedTuple):
76 '''7-Tuple C{(pointP, A, B, C, a, b, c)} with survey C{pointP}, interior
77 triangle angles C{A}, C{B} and C{C} in C{degrees} and triangle sides
78 C{a}, C{b} and C{c} in C{meter}, conventionally.
79 '''
80 _Names_ = (_pointP_, _A_, _B_, _C_, _a_, _b_, _c_)
81 _Units_ = (_Pass, Degrees, Degrees, Degrees, Distance, Distance, Distance)
84class TriAngle5Tuple(_NamedTuple):
85 '''5-Tuple C{(radA, radB, radC, rIn, area)} with the interior angles at
86 triangle corners C{A}, C{B} and C{C} in C{radians}, the C{InCircle}
87 radius C{rIn} aka C{inradius} in C{meter} and the triangle C{area}
88 in C{meter} I{squared}, conventionally.
89 '''
90 _Names_ = (_radA_, _radB_, _radC_, _rIn_, _area_)
91 _Units_ = ( Radians, Radians, Radians, Distance, _Pass)
94class TriSide2Tuple(_NamedTuple):
95 '''2-Tuple C{(a, radA)} with triangle side C{a} in C{meter}, conventionally
96 and angle C{radA} at the opposite triangle corner in C{radians}.
97 '''
98 _Names_ = (_a_, _radA_)
99 _Units_ = ( Distance, Radians)
102class TriSide4Tuple(_NamedTuple):
103 '''4-Tuple C{(a, b, radC, d)} with interior angle C{radC} at triangle corner
104 C{C} in C{radians}with the length of triangle sides C{a} and C{b} and
105 with triangle height C{d} perpendicular to triangle side C{c}, in the
106 same units as triangle sides C{a} and C{b}.
107 '''
108 _Names_ = (_a_, _b_, _radC_, _d_)
109 _Units_ = ( Distance, Distance, Radians, Distance)
112def _ABC3(useZ, pointA, pointB, pointC):
113 '''(INTERNAL) Helper for L{cassini} and L{tienstra7}.
114 '''
115 return (_otherV3d(useZ=useZ, pointA=pointA),
116 _otherV3d(useZ=useZ, pointB=pointB),
117 _otherV3d(useZ=useZ, pointC=pointC))
120def _B3(useZ, point1, point2, point3):
121 '''(INTERNAL) Helper for L{pierlot} and L{pierlotx}.
122 '''
123 return (_otherV3d(useZ=useZ, point1=point1),
124 _otherV3d(useZ=useZ, point2=point2),
125 _otherV3d(useZ=useZ, point3=point3))
128def cassini(pointA, pointB, pointC, alpha, beta, useZ=False, **Clas_and_kwds):
129 '''3-Point resection using U{Cassini<https://NL.WikiPedia.org/wiki/Achterwaartse_insnijding>}'s method.
131 @arg pointA: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
132 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
133 @arg pointB: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
134 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
135 @arg pointC: Center point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
136 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
137 @arg alpha: Angle subtended by triangle side B{C{pointA}} to B{C{pointC}}
138 (C{degrees}, non-negative).
139 @arg beta: Angle subtended by triangle side B{C{pointB}} to B{C{pointC}}
140 (C{degrees}, non-negative).
141 @kwarg useZ: If C{True}, use and interpolate the Z component, otherwise
142 force C{z=INT0} (C{bool}).
143 @kwarg Clas_and_kwds: Optional class C{B{Clas}=B{pointA}.classof} to
144 return the survey point with optionally other B{C{Clas}}
145 keyword arguments to instantiate the survey point.
147 @note: Typically, B{C{pointC}} is between B{C{pointA}} and B{C{pointB}}.
149 @return: The survey point, an instance of B{C{Clas}} or B{C{pointA}}'s
150 (sub-)class.
152 @raise ResectionError: Near-coincident, -colinear or -concyclic points
153 or negative or invalid B{C{alpha}} or B{C{beta}}.
155 @raise TypeError: Invalid B{C{pointA}}, B{C{pointB}} or B{C{pointM}}.
157 @see: U{Three Point Resection Problem<https://Dokumen.tips/documents/
158 three-point-resection-problem-introduction-kaestner-burkhardt-method.html>}
159 and functions L{collins5}, L{pierlot}, L{pierlotx} and L{tienstra7}.
160 '''
162 def _H(A, C, sa):
163 s, c = sincos2d(sa)
164 if isnear0(s):
165 raise ValueError(_or(_coincident_, _colinear_))
166 t = s, c, c
167 x = Fdot(t, A.x, C.y, -A.y).fover(s)
168 y = Fdot(t, A.y, -C.x, A.x).fover(s)
169 return x, y
171 A, B, C = _ABC3(useZ, pointA, pointB, pointC)
172 try:
173 sa, sb = map1(float, alpha, beta)
174 if min(sa, sb) < 0:
175 raise ValueError(_negative_)
176 if fsumf_(_360_0, -sa, -sb) < EPS0:
177 raise ValueError()
179 x1, y1 = _H(A, C, sa)
180 x2, y2 = _H(B, C, -sb)
182 x = x1 - x2
183 y = y1 - y2
184 if isnear0(x) or isnear0(y):
185 raise ValueError(_SPACE_(_concyclic_, (x, y)))
187 m = y / x
188 n = x / y
189 N = n + m
190 if isnear0(N):
191 raise ValueError(_SPACE_(_concyclic_, (m, n, N)))
193 t = n, m, _1_0, _N_1_0
194 x = Fdot(t, C.x, x1, C.y, y1).fover(N)
195 y = Fdot(t, y1, C.y, C.x, x1).fover(N)
196 z = _zidw(x, y, useZ, A, B, C)
197 return _Clas(cassini, pointA, Clas_and_kwds, x, y, z)
199 except (TypeError, ValueError) as x:
200 raise ResectionError(pointA=pointA, pointB=pointB, pointC=pointC,
201 alpha=alpha, beta=beta, cause=x)
204def _Clas(which, point, Clas_and_kwds, *args):
205 '''(INTERNAL) Return a C{B{Clas}=point.classof} survey point.
206 '''
207 Clas, kwds = _xkwds_pop2(Clas_and_kwds, Clas=point.classof)
208 return Clas(*args, **_xkwds(kwds, name=typename(which)))
211def collins5(pointA, pointB, pointC, alpha, beta, useZ=False, **Clas_and_kwds):
212 '''3-Point resection using U{Collins<https://Dokumen.tips/documents/
213 three-point-resection-problem-introduction-kaestner-burkhardt-method.html>}' method.
215 @arg pointA: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
216 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
217 @arg pointB: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
218 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
219 @arg pointC: Center point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
220 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
221 @arg alpha: Angle subtended by triangle side C{b} from B{C{pointA}} to
222 B{C{pointC}} (C{degrees}, non-negative).
223 @arg beta: Angle subtended by triangle side C{a} from B{C{pointB}} to
224 B{C{pointC}} (C{degrees}, non-negative).
225 @kwarg useZ: If C{True}, use and interpolate the Z component, otherwise
226 force C{z=INT0} (C{bool}).
227 @kwarg Clas_and_kwds: Optional class C{B{Clas}=B{pointA}.classof} to
228 return the survey point with optionally other B{C{Clas}}
229 keyword arguments to instantiate the survey point.
231 @note: Typically, B{C{pointC}} is between B{C{pointA}} and B{C{pointB}}.
233 @return: L{Collins5Tuple}C{(pointP, pointH, a, b, c)} with survey C{pointP},
234 auxiliary C{pointH}, each an instance of B{C{Clas}} or B{C{pointA}}'s
235 (sub-)class and triangle sides C{a}, C{b} and C{c} in C{meter},
236 conventionally.
238 @raise ResectionError: Near-coincident, -colinear or -concyclic points
239 or negative or invalid B{C{alpha}} or B{C{beta}}.
241 @raise TypeError: Invalid B{C{pointA}}, B{C{pointB}} or B{C{pointM}}.
243 @see: U{Collins' methode<https://NL.WikiPedia.org/wiki/Achterwaartse_insnijding>}
244 and functions L{cassini}, L{pierlot}, L{pierlotx} and L{tienstra7}.
245 '''
247 def _azi_len2(A, B, pi2=PI2):
248 v = B.minus(A)
249 r = atan2(v.x, v.y)
250 if r < 0 and pi2:
251 r += pi2
252 return r, v.length
254 def _xyz(d, r, A, B, C, useZ):
255 s, c = sincos2(r)
256 x = d * s + A.x # fma(d, s, A.x)
257 y = d * c + A.y # fma(d, c, A.y)
258 z = _zidw(x, y, useZ, A, B, C)
259 return x, y, z
261 A, B, C = _ABC3(useZ, pointA, pointB, pointC)
262 try:
263 ra, rb = radians(alpha), radians(beta)
264 if min(ra, rb) < 0:
265 raise ValueError(_negative_)
267 sra, srH = sin(ra), sin(ra + rb - PI) # rH = PI - ((PI - ra) + (PI - rb))
268 if isnear0(sra) or isnear0(srH):
269 raise ValueError(_or(_coincident_, _colinear_, _concyclic_))
271# za, a = _azi_len2(C, B)
272 zb, b = _azi_len2(C, A)
273 zc, c = _azi_len2(A, B, 0)
275# d = c * sin(PI - rb) / srH # B.minus(H).length
276 d = c * sin(PI - ra) / srH # A.minus(H).length
277 r = zc + PI - rb # zh = zc + (PI - rb)
278 H = _xyz(d, r, A, B, C, useZ)
280 zh, _ = _azi_len2(C, Vector3d(*H))
282# d = a * sin(za - zh) / sin(rb) # B.minus(P).length
283 d = b * sin(zb - zh) / sra # A.minus(P).length
284 r = zh - ra # zb - PI + (PI - ra - (zb - zh))
285 P = _xyz(d, r, A, B, C, useZ)
287 P = _Clas(collins5, pointA, Clas_and_kwds, *P)
288 H = _Clas(collins5, pointA, Clas_and_kwds, *H)
289 a = B.minus(C).length
290 return Collins5Tuple(P, H, a, b, c, name=typename(collins5))
292 except (TypeError, ValueError) as x:
293 raise ResectionError(pointA=pointA, pointB=pointB, pointC=pointC,
294 alpha=alpha, beta=beta, cause=x)
297def pierlot(point1, point2, point3, alpha12, alpha23, useZ=False, eps=EPS,
298 **Clas_and_kwds):
299 '''3-Point resection using U{Pierlot<http://www.Telecom.ULg.ac.BE/publi/publications/
300 pierlot/Pierlot2014ANewThree>}'s method C{ToTal} with I{approximate} limits for
301 the (pseudo-)singularities.
303 @arg point1: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
304 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
305 @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
306 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
307 @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
308 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
309 @arg alpha12: Angle subtended from B{C{point1}} to B{C{point2}} or
310 B{C{alpha2 - alpha1}} (C{degrees}).
311 @arg alpha23: Angle subtended from B{C{point2}} to B{C{point3}} or
312 B{C{alpha3 - alpha2}}(C{degrees}).
313 @kwarg useZ: If C{True}, interpolate the survey point's Z component,
314 otherwise use C{z=INT0} (C{bool}).
315 @kwarg eps: Tolerance for C{cot}angent (pseudo-)singularities (C{float}).
316 @kwarg Clas_and_kwds: Optional class C{B{Clas}=B{point1}.classof} to
317 return the survey point with optionally other B{C{Clas}}
318 keyword arguments to instantiate the survey point.
320 @note: Typically, B{C{point1}}, B{C{point2}} and B{C{point3}} are ordered
321 by angle, modulo 360, counter-clockwise.
323 @return: The survey (or robot) point, an instance of B{C{Clas}} or B{C{point1}}'s
324 (sub-)class.
326 @raise ResectionError: Near-coincident, -colinear or -concyclic points
327 or invalid B{C{alpha12}} or B{C{alpha23}} or
328 non-positive B{C{eps}}.
330 @raise TypeError: Invalid B{C{point1}}, B{C{point2}} or B{C{point3}}.
332 @see: I{Pierlot}'s C function U{triangulationPierlot<http://www.Telecom.ULg.ac.BE/
333 triangulation/doc/total_8c_source.html>}, U{V. Pierlot, M. Van Droogenbroeck,
334 "A New Three Object Triangulation Algorithm for Mobile Robot Positioning"
335 <https://ORBi.ULiege.BE/bitstream/2268/157469/1/Pierlot2014ANewThree.pdf>},
336 U{Vincent Pierlot, Marc Van Droogenbroeck, "18 Triangulation Algorithms for 2D
337 Positioning (also known as the Resection Problem)"<http://www.Telecom.ULg.ac.BE/
338 triangulation>} and functions L{pierlotx}, L{cassini}, L{collins5} and L{tienstra7}.
339 '''
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 t = Fmt.PARENSPACED(eps=eps)
345 raise ValueError(_SPACE_(t, _not_, _positive_))
347 B1, B2, B3 = _B3(useZ, point1, point2, point3)
348 try:
349 xyz = _pierlot3(B1, B2, B3, alpha12, alpha23, useZ, _cot)
350 return _Clas(pierlot, point1, Clas_and_kwds, *xyz)
352 except (TypeError, ValueError) as x:
353 raise ResectionError(point1=point1, point2=point2, point3=point3,
354 alpha12=alpha12, alpha23=alpha23, eps=eps, cause=x)
357def _pierlot3(B1, B2, B3, a12, a23, useZ, _cot):
358 '''(INTERNAL) Shared L{pierlot} and L{pierlotx}.
359 '''
360 x1_, y1_, _ = B1.minus(B2).xyz3
361 x3_, y3_, _ = B3.minus(B2).xyz3
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
372 K = _Fsumf_(x3_ * x1_, cot31 * (y3_ * x1_),
373 y3_ * y1_, -cot31 * (x3_ * y1_))
374 if K:
375 cot12 = _cot(s12, c12)
376 cot23 = _cot(s23, c23)
378 # x12 = x1_ + cot12 * y1_
379 # y12 = y1_ - cot12 * x1_
381 # x23 = x3_ - cot23 * y3_
382 # y23 = y3_ + cot23 * x3_
384 # x31 = x3_ + x1_ + cot31 * (y3_ - y1_)
385 # y31 = y3_ + y1_ - cot31 * (x3_ - x1_)
387 # x12 - x23 = x1_ + cot12 * y1_ - x3_ + cot23 * y3_
388 X12_23 = _Fsumf_(x1_, cot12 * y1_, -x3_, cot23 * y3_)
389 # y12 - y23 = y1_ - cot12 * x1_ - y3_ - cot23 * x3_
390 Y12_23 = _Fsumf_(y1_, -cot12 * x1_, -y3_, -cot23 * x3_)
392 # x31 - x23 = x3_ + x1_ + cot31 * (y3_ - y1_) - x3_ + cot23 * y3_
393 # = x1_ + cot31 * y3_ - cot31 * y1_ + cot23 * y3_
394 X31_23 = _Fsumf_(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 = _Fsumf_(y1_, cot31 * x1_, -cot31 * x3_, -cot23 * x3_)
399 # d = (x12 - x23) * (y23 - y31) + (x31 - x23) * (y12 - y23)
400 # = (x31 - x23) * (y12 - y23) - (x12 - x23) * (y31 - y23)
401 # x = (d * B2.x + K * Y12_23).fover(d)
402 # y = (d * B2.y - K * X12_23).fover(d)
403 x, y = _pierlotxy2(B2, -K, Y12_23, X12_23, (X31_23 * Y12_23 -
404 X12_23 * Y31_23))
405 else:
406 x, y, _ = B2.xyz3
407 return x, y, _zidw(x, y, useZ, B1, B2, B3)
410def pierlotx(point1, point2, point3, alpha1, alpha2, alpha3, useZ=False,
411 **Clas_and_kwds):
412 '''3-Point resection using U{Pierlot<http://www.Telecom.ULg.ac.BE/publi/
413 publications/pierlot/Pierlot2014ANewThree>}'s method C{ToTal} with
414 I{exact} limits for the (pseudo-)singularities.
416 @arg point1: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
417 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
418 @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
419 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
420 @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
421 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
422 @arg alpha1: Angle at B{C{point1}} (C{degrees}, counter-clockwise).
423 @arg alpha2: Angle at B{C{point2}} (C{degrees}, counter-clockwise).
424 @arg alpha3: Angle at B{C{point3}} (C{degrees}, counter-clockwise).
425 @kwarg useZ: If C{True}, interpolate the survey point's Z component,
426 otherwise use C{z=INT0} (C{bool}).
427 @kwarg Clas_and_kwds: Optional class C{B{Clas}=B{point1}.classof} to
428 return the survey point with optionally other B{C{Clas}}
429 keyword arguments to instantiate the survey point.
431 @return: The survey (or robot) point, an instance of B{C{Clas}} or
432 B{C{point1}}'s (sub-)class.
434 @raise ResectionError: Near-coincident, -colinear or -concyclic points or
435 invalid B{C{alpha1}}, B{C{alpha2}} or B{C{alpha3}}.
437 @raise TypeError: Invalid B{C{point1}}, B{C{point2}} or B{C{point3}}.
439 @see: I{Pierlot}'s C function U{triangulationPierlot2<http://www.Telecom.ULg.ac.BE/
440 triangulation/doc/total_8c_source.html>} and function L{pierlot}, L{cassini},
441 L{collins5} and L{tienstra7}.
442 '''
444 def _a_z_Bs(Bs, *alphas):
445 ds = map2(_umod_360, alphas) # 0 <= alphas < 360
446 ds, Bs = zip(*sorted(_zip(ds, Bs))) # unzip
447 for p, d, B in _zip(ds, _rotate(ds), Bs):
448 d -= p # a12 = a2 - a1, ...
449 z = isnear0(fabs(d) % _180_0)
450 yield d, z, B
452 def _cot(s, c): # I{exact} cotangent
453 try:
454 return (c / s) # if c else _copysign_0_0(s)
455 except ZeroDivisionError:
456 raise ValueError(_or(_coincident_, _colinear_))
458 Bs = _B3(useZ, point1, point2, point3)
459 try:
460 Cs = [0] # pseudo-global, passing the exception Case
461 xyz = _pierlotx3(_a_z_Bs(Bs, alpha1, alpha2, alpha3),
462 useZ, _cot, Cs.append)
463 return _Clas(pierlotx, point1, Clas_and_kwds, *xyz)
465 except (TypeError, ValueError) as x:
466 raise ResectionError(point1=point1, point2=point2, point3=point3, C=Cs.pop(),
467 alpha1=alpha1, alpha2=alpha2, alpha3=alpha3, cause=x)
470def _pierlotx3(a_z_Bs, useZ, _cot, Cs):
471 '''(INTERNAL) Core of L{pierlotx}.
472 '''
473 (a12, z12, B1), \
474 (a23, z23, B2), \
475 (a31, z31, B3) = a_z_Bs
476 if z12 and not z23:
477 Cs(1)
478 elif z23 and not z31:
479 Cs(2)
480 a23, B1, B2, B3 = a31, B2, B3, B1
481 elif z31 and not z12:
482 Cs(3)
483 a23, B2, B3 = a12, B3, B2
484 else:
485 Cs(4)
486 return _pierlot3(B1, B2, B3, a12, a23, useZ, _cot)
488 x1_, y1_, _ = B1.minus(B3).xyz3
489 x2_, y2_, _ = B2.minus(B3).xyz3
491 K = _Fsumf_(y1_ * x2_, -x1_ * y2_)
492 if K:
493 cot23 = _cot(*sincos2d(a23))
495 # x23 = x2_ + cot23 * y2_
496 # y23 = y2_ - cot23 * x2_
498 # x31 = x1_ + cot23 * y1_
499 # y31 = y1_ - cot23 * x1_
501 # x31 - x23 = x1_ + cot23 * y1_ - x2_ - cot23 * y2_
502 X31_23 = _Fsumf_(x1_, cot23 * y1_, -x2_, -cot23 * y2_)
503 # y31 - y23 = y1_ - cot23 * x1_ - y2_ + cot23 * x2_
504 Y31_23 = _Fsumf_(y1_, -cot23 * x1_, -y2_, cot23 * x2_)
506 # d = (x31 - x23) * (x2_ - x1_) + (y31 - y23) * (y2_ - y1_)
507 # x = (D * B3.x - K * Y31_23).fover(d)
508 # y = (D * B3.y + K * X31_23).fover(d)
509 x, y = _pierlotxy2(B3, K, Y31_23, X31_23, (X31_23 * _Fsumf_(x2_, -x1_) +
510 Y31_23 * _Fsumf_(y2_, -y1_)))
511 else:
512 x, y, _ = B3.xyz3
513 return x, y, _zidw(x, y, useZ, B1, B2, B3)
516def _pierlotxy2(B, K, X, Y, D):
517 '''(INTERNAL) Helper for C{_pierlot3} and C{_pierlotx3}.
518 '''
519 d = float(D)
520 if isnear0(d):
521 raise ValueError(_or(_coincident_, _colinear_, _concyclic_))
522 x = (D * B.x - K * X).fover(d)
523 y = (D * B.y + K * Y).fover(d)
524 return x, y
527def _rotate(xs, n=1):
528 '''Rotate list or tuple C{xs} by C{n} items, right if C{n > 0} else left.
529 '''
530 return xs[n:] + xs[:n]
533def snellius3(a, b, degC, alpha, beta):
534 '''Snellius' surveying using U{Snellius Pothenot<https://WikiPedia.org/wiki/Snellius–Pothenot_problem>}.
536 @arg a: Length of the triangle side between corners C{B} and C{C} and opposite of
537 triangle corner C{A} (C{scalar}, non-negative C{meter}, conventionally).
538 @arg b: Length of the triangle side between corners C{C} and C{A} and opposite of
539 triangle corner C{B} (C{scalar}, non-negative C{meter}, conventionally).
540 @arg degC: Angle at triangle corner C{C}, opposite triangle side C{c} (non-negative C{degrees}).
541 @arg alpha: Angle subtended by triangle side B{C{b}} (non-negative C{degrees}).
542 @arg beta: Angle subtended by triangle side B{C{a}} (non-negative C{degrees}).
544 @return: L{Survey3Tuple}C{(PA, PB, PC)} with distance from survey point C{P} to
545 each of the triangle corners C{A}, C{B} and C{C}, same units as triangle
546 sides B{C{a}}, B{C{b}} and B{C{c}}.
548 @raise TriangleError: Invalid B{C{a}}, B{C{b}} or B{C{degC}} or negative B{C{alpha}}
549 or B{C{beta}}.
551 @see: Function L{wildberger3}.
552 '''
553 try:
554 a, b, degC, alpha, beta = t = map1(float, a, b, degC, alpha, beta)
555 if min(t) < 0:
556 raise ValueError(_negative_)
557 ra, rb, rC = map1(radians, alpha, beta, degC)
559 r = fsum1f_(ra, rb, rC) * _0_5
560 k = PI - r
561 if min(k, r) < 0:
562 raise ValueError(_or(_coincident_, _colinear_))
564 sa, sb = map1(sin, ra, rb)
565 p = atan2(sa * a, sb * b)
566 sp, cp, sr, cr = sincos2_(PI_4 - p, r)
567 p = atan2(sp * sr, cp * cr)
568 pa = k + p
569 pb = k - p
571 if fabs(sb) > fabs(sa):
572 pc = fabs(a * sin(pb) / sb)
573 elif sa:
574 pc = fabs(b * sin(pa) / sa)
575 else:
576 raise ValueError(_or(_colinear_, _coincident_))
578 pa = _triSide(b, pc, fsumf_(PI, -ra, -pa))
579 pb = _triSide(a, pc, fsumf_(PI, -rb, -pb))
580 return Survey3Tuple(pa, pb, pc, name=typename(snellius3))
582 except (TypeError, ValueError) as x:
583 raise TriangleError(a=a, b=b, degC=degC, alpha=alpha, beta=beta, cause=x)
586def tienstra7(pointA, pointB, pointC, alpha, beta=None, gamma=None,
587 useZ=False, **Clas_and_kwds):
588 '''3-Point resection using U{Tienstra<https://WikiPedia.org/wiki/Tienstra_formula>}'s formula.
590 @arg pointA: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or
591 C{Vector2Tuple} if C{B{useZ}=False}).
592 @arg pointB: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or
593 C{Vector2Tuple} if C{B{useZ}=False}).
594 @arg pointC: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or
595 C{Vector2Tuple} if C{B{useZ}=False}).
596 @arg alpha: Angle subtended by triangle side C{a} from B{C{pointB}} to B{C{pointC}}
597 (C{degrees}, non-negative).
598 @kwarg beta: Angle subtended by triangle side C{b} from B{C{pointA}} to B{C{pointC}}
599 (C{degrees}, non-negative) or C{None} if C{B{gamma} is not None}.
600 @kwarg gamma: Angle subtended by triangle side C{c} from B{C{pointA}} to B{C{pointB}}
601 (C{degrees}, non-negative) or C{None} if C{B{beta} is not None}.
602 @kwarg useZ: If C{True}, use and interpolate the Z component, otherwise force C{z=INT0}
603 (C{bool}).
604 @kwarg Clas_and_kwds: Optional class C{B{Clas}=B{pointA}.classof} to return the survey
605 point with optionally other B{C{Clas}} keyword arguments to instantiate
606 the survey point.
608 @note: Points B{C{pointA}}, B{C{pointB}} and B{C{pointC}} are ordered clockwise.
610 @return: L{Tienstra7Tuple}C{(pointP, A, B, C, a, b, c)} with survey C{pointP}, an
611 instance of B{C{Clas}} or B{C{pointA}}'s (sub-)class, with triangle angles C{A}
612 at B{C{pointA}}, C{B} at B{C{pointB}} and C{C} at B{C{pointC}} in C{degrees}
613 and with triangle sides C{a}, C{b} and C{c} in C{meter}, conventionally.
615 @raise ResectionError: Near-coincident, -colinear or -concyclic points or sum of
616 B{C{alpha}}, B{C{beta}} and B{C{gamma}} not C{360} or negative
617 B{C{alpha}}, B{C{beta}} or B{C{gamma}}.
619 @raise TypeError: Invalid B{C{pointA}}, B{C{pointB}} or B{C{pointC}}.
621 @see: U{3-Point Resection Solver<http://MesaMike.org/geocache/GC1B0Q9/tienstra/>},
622 U{V. Pierlot, M. Van Droogenbroeck, "A New Three Object Triangulation..."
623 <http://www.Telecom.ULg.ac.BE/publi/publications/pierlot/Pierlot2014ANewThree/>},
624 U{18 Triangulation Algorithms...<http://www.Telecom.ULg.ac.BE/triangulation/>} and
625 functions L{cassini}, L{collins5}, L{pierlot} and L{pierlotx}.
626 '''
628 def _deg_ks(r, s, ks, N):
629 if isnear0(fsumf_(PI, r, -s)): # r + (PI2 - s) == PI
630 raise ValueError(Fmt.PARENSPACED(concyclic=N))
631 # k = 1 / (cot(r) - cot(s))
632 # = 1 / (cos(r) / sin(r) - cos(s) / sin(s))
633 # = 1 / (cos(r) * sin(s) - cos(s) * sin(r)) / (sin(r) * sin(s))
634 # = sin(r) * sin(s) / (cos(r) * sin(s) - cos(s) * sin(r))
635 sr, cr, ss, cs = sincos2_(r, s)
636 c = fsum1f_(cr * ss, -cs * sr)
637 if isnear0(c):
638 raise ValueError(Fmt.PARENSPACED(cotan=N))
639 ks.append(sr * ss / c)
640 return Degrees(degrees(r), name=N) # C degrees
642 A, B, C = _ABC3(useZ, pointA, pointB, pointC)
643 try:
644 sa, sb, sc = map1(radians, alpha, (beta or 0), (gamma or 0))
645 if beta is None:
646 if gamma is None:
647 raise ValueError(_and(Fmt.EQUAL(beta=beta), Fmt.EQUAL(gamma=gamma)))
648 sb = fsumf_(PI2, -sa, -sc)
649 elif gamma is None:
650 sc = fsumf_(PI2, -sa, -sb)
651 else: # subtended angles must add to 360 degrees
652 r = fsum1f_(sa, sb, sc)
653 if fabs(r - PI2) > EPS:
654 raise ValueError(Fmt.EQUAL(sum=degrees(r)))
655 if min(sa, sb, sc) < 0:
656 raise ValueError(_negative_)
658 # triangle sides
659 a = B.minus(C).length
660 b = A.minus(C).length
661 c = A.minus(B).length
663 ks = [] # 3 Ks and triangle angles
664 dA = _deg_ks(_triAngle(b, c, a), sa, ks, _A_)
665 dB = _deg_ks(_triAngle(a, c, b), sb, ks, _B_)
666 dC = _deg_ks(_triAngle(a, b, c), sc, ks, _C_)
668 k = fsum1(ks)
669 if isnear0(k):
670 raise ValueError(Fmt.EQUAL(K=k))
671 x = Fdot(ks, A.x, B.x, C.x).fover(k)
672 y = Fdot(ks, A.y, B.y, C.y).fover(k)
673 z = _zidw(x, y, useZ, A, B, C)
675 P = _Clas(tienstra7, pointA, Clas_and_kwds, x, y, z)
676 return Tienstra7Tuple(P, dA, dB, dC, a, b, c, name=typename(tienstra7))
678 except (TypeError, ValueError) as x:
679 raise ResectionError(pointA=pointA, pointB=pointB, pointC=pointC,
680 alpha=alpha, beta=beta, gamma=gamma, cause=x)
683def triAngle(a, b, c):
684 '''Compute one angle of a triangle.
686 @arg a: Adjacent triangle side length (C{scalar}, non-negative
687 C{meter}, conventionally).
688 @arg b: Adjacent triangle side length (C{scalar}, non-negative
689 C{meter}, conventionally).
690 @arg c: Opposite triangle side length (C{scalar}, non-negative
691 C{meter}, conventionally).
693 @return: Angle in C{radians} at triangle corner C{C}, opposite
694 triangle side B{C{c}}.
696 @raise TriangleError: Invalid or negative B{C{a}}, B{C{b}} or B{C{c}}.
698 @see: Functions L{triAngle5} and L{triSide}.
699 '''
700 try:
701 return _triAngle(a, b, c)
702 except (TypeError, ValueError) as x:
703 raise TriangleError(a=a, b=b, c=c, cause=x)
706def _triAngle(a, b, c):
707 # (INTERNAL) To allow callers to embellish errors
708 a, b, c = map1(float, a, b, c)
709 if a < b:
710 a, b = b, a
711 if b < 0 or c < 0:
712 raise ValueError(_negative_)
713 if a < EPS0:
714 raise ValueError(_coincident_)
715 b_a = b / a
716 if b_a < EPS0:
717 raise ValueError(_coincident_)
718 t = fsumf_(_1_0, b_a**2, -(c / a)**2) / (b_a * _2_0)
719 return acos1(t)
722def triAngle5(a, b, c):
723 '''Compute the angles of a triangle.
725 @arg a: Length of the triangle side opposite of triangle corner C{A}
726 (C{scalar}, non-negative C{meter}, conventionally).
727 @arg b: Length of the triangle side opposite of triangle corner C{B}
728 (C{scalar}, non-negative C{meter}, conventionally).
729 @arg c: Length of the triangle side opposite of triangle corner C{C}
730 (C{scalar}, non-negative C{meter}, conventionally).
732 @return: L{TriAngle5Tuple}C{(radA, radB, radC, rIn, area)} with angles
733 C{radA}, C{radB} and C{radC} at triangle corners C{A}, C{B}
734 and C{C}, all in C{radians}, the C{InCircle} radius C{rIn}
735 aka C{inradius}, same units as triangle sides B{C{a}},
736 B{C{b}} and B{C{c}} and the triangle C{area} in those same
737 units I{squared}.
739 @raise TriangleError: Invalid or negative B{C{a}}, B{C{b}} or B{C{c}}.
741 @see: Functions L{triAngle} and L{triArea}.
742 '''
743 try:
744 x, y, z = map1(float, a, b, c)
745 ab = x < y
746 if ab:
747 x, y = y, x
748 bc = y < z
749 if bc:
750 y, z = z, y
752 if z > EPS0: # z = min(a, b, c)
753 s = fsum1f_(z, y, x) * _0_5
754 sa, sb, r = (s - x), (s - y), (s - z)
755 r *= _over(sa * sb, s)
756 if r < EPS02:
757 raise ValueError(_coincident_)
758 r = sqrt(r)
759 rA = atan2(r, sa) * _2_0
760 rB = atan2(r, sb) * _2_0
761 rC = fsumf_(PI, -rA, -rB)
762 if min(rA, rB, rC) < 0:
763 raise ValueError(_colinear_)
764 s *= r # Heron's area
765 elif z < 0:
766 raise ValueError(_negative_)
767 else: # 0 <= c <= EPS0
768 rA = rB = PI_2
769 rC = r = s = _0_0
771 if bc:
772 rB, rC = rC, rB
773 if ab:
774 rA, rB = rB, rA
775 return TriAngle5Tuple(rA, rB, rC, r, s, name=typename(triAngle5))
777 except (TypeError, ValueError) as x:
778 raise TriangleError(a=a, b=b, c=c, cause=x)
781def triArea(a, b, c):
782 '''Compute the area of a triangle using U{Heron's<https://
783 WikiPedia.org/wiki/Heron%27s_formula>} C{stable} formula.
785 @arg a: Length of the triangle side opposite of triangle corner C{A}
786 (C{scalar}, non-negative C{meter}, conventionally).
787 @arg b: Length of the triangle side opposite of triangle corner C{B}
788 (C{scalar}, non-negative C{meter}, conventionally).
789 @arg c: Length of the triangle side opposite of triangle corner C{C}
790 (C{scalar}, non-negative C{meter}, conventionally).
792 @return: The triangle area (C{float}, conventionally C{meter} or
793 same units as B{C{a}}, B{C{b}} and B{C{c}} I{squared}).
795 @raise TriangleError: Invalid or negative B{C{a}}, B{C{b}} or B{C{c}}.
796 '''
797 try:
798 r, y, x = sorted(map1(float, a, b, c))
799 if r > 0: # r = min(a, b, c)
800 ab = x - y
801 bc = y - r
802 y += r
803 r = (x + y) * (r - ab) * (r + ab) * (x + bc)
804 if r:
805 r = sqrt(r / _16_0)
806 elif r < 0:
807 raise ValueError(_negative_)
808 return r
810 except (TypeError, ValueError) as x:
811 raise TriangleError(a=a, b=b, c=c, cause=x)
814def triSide(a, b, radC):
815 '''Compute one side of a triangle.
817 @arg a: Adjacent triangle side length (C{scalar},
818 non-negative C{meter}, conventionally).
819 @arg b: Adjacent triangle side length (C{scalar},
820 non-negative C{meter}, conventionally).
821 @arg radC: Angle included by sides B{C{a}} and B{C{b}},
822 opposite triangle side C{c} (C{radians}).
824 @return: Length of triangle side C{c}, opposite triangle
825 corner C{C} and angle B{C{radC}}, same units as
826 B{C{a}} and B{C{b}}.
828 @raise TriangleError: Invalid B{C{a}}, B{C{b}} or B{C{radC}}.
830 @see: Functions L{sqrt_a}, L{triAngle}, L{triSide2} and L{triSide4}.
831 '''
832 try:
833 return _triSide(a, b, radC)
834 except (TypeError, ValueError) as x:
835 raise TriangleError(a=a, b=b, radC=radC, cause=x)
838def _triSide(a, b, radC):
839 # (INTERNAL) To allow callers to embellish errors
840 a, b, r = t = map1(float, a, b, radC)
841 if min(t) < 0:
842 raise ValueError(_negative_)
844 if a < b:
845 a, b = b, a
846 if a > EPS0:
847 ba = b / a
848 c2 = fsumf_(_1_0, ba**2, _N_2_0 * ba * cos(r))
849 if c2 > EPS02:
850 return a * sqrt(c2)
851 elif c2 < 0:
852 raise ValueError(_invalid_)
853 return hypot(a, b)
856def triSide2(b, c, radB):
857 '''Compute a side and its opposite angle of a triangle.
859 @arg b: Adjacent triangle side length (C{scalar},
860 non-negative C{meter}, conventionally).
861 @arg c: Adjacent triangle side length (C{scalar},
862 non-negative C{meter}, conventionally).
863 @arg radB: Angle included by sides B{C{a}} and B{C{c}},
864 opposite triangle side C{b} (C{radians}).
866 @return: L{TriSide2Tuple}C{(a, radA)} with triangle angle
867 C{radA} in C{radians} and length of the opposite
868 triangle side C{a}, same units as B{C{b}} and B{C{c}}.
870 @raise TriangleError: Invalid B{C{b}} or B{C{c}} or either
871 B{C{b}} or B{C{radB}} near zero.
873 @see: Functions L{sqrt_a}, L{triSide} and L{triSide4}.
874 '''
875 try:
876 return _triSide2(b, c, radB)
877 except (TypeError, ValueError) as x:
878 raise TriangleError(b=b, c=c, radB=radB, cause=x)
881def _triSide2(b, c, radB):
882 # (INTERNAL) To allow callers to embellish errors
883 b, c, rB = map1(float, b, c, radB)
884 if min(b, c, rB) < 0:
885 raise ValueError(_negative_)
886 sB, cB = sincos2(rB)
887 if isnear0(sB):
888 if not isnear0(b):
889 raise ValueError(_invalid_)
890 a, rA = ((b + c), PI) if cB < 0 else (fabs(b - c), _0_0)
891 elif isnear0(b):
892 raise ValueError(_invalid_)
893 else:
894 rA = fsumf_(PI, -rB, -asin1(c * sB / b))
895 a = sin(rA) * b / sB
896 return TriSide2Tuple(a, rA, name=typename(triSide2))
899def triSide4(radA, radB, c):
900 '''Compute two sides and the height of a triangle.
902 @arg radA: Angle at triangle corner C{A}, opposite triangle side C{a}
903 (non-negative C{radians}).
904 @arg radB: Angle at triangle corner C{B}, opposite triangle side C{b}
905 (non-negative C{radians}).
906 @arg c: Length of triangle side between triangle corners C{A} and C{B},
907 (C{scalar}, non-negative C{meter}, conventionally).
909 @return: L{TriSide4Tuple}C{(a, b, radC, d)} with triangle sides C{a} and
910 C{b} and triangle height C{d} perpendicular to triangle side
911 B{C{c}}, all in the same units as B{C{c}} and interior angle
912 C{radC} in C{radians} at triangle corner C{C}, opposite
913 triangle side B{C{c}}.
915 @raise TriangleError: Invalid or negative B{C{radA}}, B{C{radB}} or B{C{c}}.
917 @see: U{Triangulation, Surveying<https://WikiPedia.org/wiki/Triangulation_(surveying)>}
918 and functions L{sqrt_a}, L{triSide} and L{triSide2}.
919 '''
920 try:
921 rA, rB, c = map1(float, radA, radB, c)
922 rC = fsumf_(PI, -rA, -rB)
923 if min(rC, rA, rB, c) < 0:
924 raise ValueError(_negative_)
925 sa, ca, sb, cb = sincos2_(rA, rB)
926 sc = fsum1f_(sa * cb, sb * ca)
927 if sc < EPS0 or min(sa, sb) < 0:
928 raise ValueError(_invalid_)
929 sc = c / sc
930 return TriSide4Tuple((sa * sc), (sb * sc), rC, (sa * sb * sc),
931 name=typename(triSide4))
933 except (TypeError, ValueError) as x:
934 raise TriangleError(radA=radA, radB=radB, c=c, cause=x)
937def wildberger3(a, b, c, alpha, beta, R3=min):
938 '''Snellius' surveying using U{Rational Trigonometry
939 <https://WikiPedia.org/wiki/Snellius–Pothenot_problem>}.
941 @arg a: Length of the triangle side between corners C{B} and C{C} and opposite of
942 triangle corner C{A} (C{scalar}, non-negative C{meter}, conventionally).
943 @arg b: Length of the triangle side between corners C{C} and C{A} and opposite of
944 triangle corner C{B} (C{scalar}, non-negative C{meter}, conventionally).
945 @arg c: Length of the triangle side between corners C{A} and C{B} and opposite of
946 triangle corner C{C} (C{scalar}, non-negative C{meter}, conventionally).
947 @arg alpha: Angle subtended by triangle side B{C{b}} (C{degrees}, non-negative).
948 @arg beta: Angle subtended by triangle side B{C{a}} (C{degrees}, non-negative).
949 @kwarg R3: Callable to determine C{R3} from C{(R3 - C)**2 = D}, typically standard
950 Python function C{min} or C{max}, invoked with 2 arguments.
952 @return: L{Survey3Tuple}C{(PA, PB, PC)} with distance from survey point C{P} to
953 each of the triangle corners C{A}, C{B} and C{C}, same units as B{C{a}},
954 B{C{b}} and B{C{c}}.
956 @raise TriangleError: Invalid B{C{a}}, B{C{b}} or B{C{c}} or negative B{C{alpha}} or
957 B{C{beta}} or B{C{R3}} not C{callable}.
959 @see: U{Wildberger, Norman J.<https://Math.Sc.Chula.ac.TH/cjm/content/
960 survey-article-greek-geometry-rational-trigonometry-and-snellius-–-pothenot-surveying>},
961 U{Devine Proportions, page 252<http://www.MS.LT/derlius/WildbergerDivineProportions.pdf>}
962 and function L{snellius3}.
963 '''
964 def _s(x):
965 return sin(x)**2
967 def _vpa(r3, q2, q3, s2, s3):
968 r1 = s2 * q3 / s3
969 r = r1 * r3 * _4_0
970 n = (r - _Fsumf_(r1, r3, -q2)**2).fover(s3)
971 if n < 0 or r < EPS0:
972 raise ValueError(_coincident_)
973 return sqrt((n / r) * q3) if n else _0_0
975 try:
976 a, b, c, da, db = q = map1(float, a, b, c, alpha, beta)
977 if min(q) < 0:
978 raise ValueError(_negative_)
980 q1, q2, q3 = q = a**2, b**2, c**2
981 if min(q) < EPS02:
982 raise ValueError(_coincident_)
984 ra, rb = map1(radians, da, db)
985 s1, s2, s3 = s = map1(_s, rb, ra, ra + rb) # rb, ra!
986 if min(s) < EPS02:
987 raise ValueError(_or(_coincident_, _colinear_))
989 q4 = hypot2_(*q) * _2_0 # a**4 + ...
990 Qs = _Fsumf_(*q) # == hypot2_(a, b, c)
991 d0 = (Qs**2 - q4).fmul(s1 * s2).fover(s3)
992 if d0 < 0:
993 raise ValueError(_negative_)
994 s += _Fsumf_(*s), # == fsum1(s),
995 C0 = Fdot(s, q1, q2, q3, -Qs * _0_5)
996 r3 = C0.fover(-s3) # C0 /= -s3
997 if d0 > EPS02: # > c0
998 _xcallable(R3=R3)
999 d0 = sqrt(d0)
1000 r3 = R3(float(C0 + d0), float(C0 - d0)) # XXX min or max
1002 pa = _vpa(r3, q2, q3, s2, s3)
1003 pb = _vpa(r3, q1, q3, s1, s3)
1004 pc = favg(_triSide2(b, pa, ra).a,
1005 _triSide2(a, pb, rb).a)
1006 return Survey3Tuple(pa, pb, pc, name=typename(wildberger3))
1008 except (TypeError, ValueError) as x:
1009 raise TriangleError(a=a, b=b, c=c, alpha=alpha, beta=beta, R3=R3, cause=x)
1012def _zidw(x, y, useZ, *ABC):
1013 if useZ: # interpolate z or coplanar with A, B and C?
1014 t = tuple(_.z for _ in ABC)
1015 v = Vector3d(x, y, fmean(t))
1016 z = fidw(t, (v.minus(T).length for T in ABC))
1017 else:
1018 z = INT0
1019 return z
1021# **) MIT License
1022#
1023# Copyright (C) 2016-2025 -- mrJean1 at Gmail -- All Rights Reserved.
1024#
1025# Permission is hereby granted, free of charge, to any person obtaining a
1026# copy of this software and associated documentation files (the "Software"),
1027# to deal in the Software without restriction, including without limitation
1028# the rights to use, copy, modify, merge, publish, distribute, sublicense,
1029# and/or sell copies of the Software, and to permit persons to whom the
1030# Software is furnished to do so, subject to the following conditions:
1031#
1032# The above copyright notice and this permission notice shall be included
1033# in all copies or substantial portions of the Software.
1034#
1035# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
1036# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
1037# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
1038# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
1039# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
1040# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
1041# OTHER DEALINGS IN THE SOFTWARE.