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, NEG0, PI, PI2, PI_2, PI_4, \
17 _0_0, _0_5, _1_0, _N_1_0, _2_0, _N_2_0, _4_0, _16_0, \
18 _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 Fsum, fsumf_, fsum1, fsum1f_
23from pygeodesy.interns import _a_, _A_, _area_, _b_, _B_, _c_, _C_, _coincident_, \
24 _colinear_, _d_, _eps_, _invalid_, _negative_, _not_, \
25 _rIn_, _SPACE_
26# from pygeodesy.lazily import _ALL_LAZY # from .basics
27from pygeodesy.named import _NamedTuple, _Pass, Fmt
28# from pygeodesy.streprs import Fmt # from .named
29from pygeodesy.units import Degrees, Distance, Radians
30from pygeodesy.utily import acos1, asin1, sincos2, sincos2_, sincos2d, sincos2d_
31from pygeodesy.vector3d import _otherV3d, Vector3d
33from math import cos, atan2, degrees, fabs, radians, sin, sqrt
35__all__ = _ALL_LAZY.resections
36__version__ = '24.03.26'
38_concyclic_ = 'concyclic'
39_PA_ = 'PA'
40_PB_ = 'PB'
41_PC_ = 'PC'
42_pointH_ = 'pointH'
43_pointP_ = 'pointP'
44_positive_ = 'positive'
45_radA_ = 'radA'
46_radB_ = 'radB'
47_radC_ = 'radC'
50class Collins5Tuple(_NamedTuple):
51 '''5-Tuple C{(pointP, pointH, a, b, c)} with survey C{pointP}, auxiliary
52 C{pointH}, each an instance of B{C{pointA}}'s (sub-)class and triangle
53 sides C{a}, C{b} and C{c} in C{meter}, conventionally.
54 '''
55 _Names_ = (_pointP_, _pointH_, _a_, _b_, _c_)
56 _Units_ = (_Pass, _Pass, Distance, Distance, Distance)
59def _F1(*xs): # class
60 '''(INTERNAL) An L{Fsum}, 1-primed.
61 '''
62 F = Fsum(_1_0, *xs)
63 F += _N_1_0
64 return F
67class ResectionError(_ValueError):
68 '''Error raised for issues in L{pygeodesy.resections}.
69 '''
70 pass
73class Survey3Tuple(_NamedTuple):
74 '''3-Tuple C{(PA, PB, PC)} with distance from survey point C{P} to each of
75 the triangle corners C{A}, C{B} and C{C} in C{meter}, conventionally.
76 '''
77 _Names_ = (_PA_, _PB_, _PC_)
78 _Units_ = ( Distance, Distance, Distance)
81class Tienstra7Tuple(_NamedTuple):
82 '''7-Tuple C{(pointP, A, B, C, a, b, c)} with survey C{pointP}, interior
83 triangle angles C{A}, C{B} and C{C} in C{degrees} and triangle sides
84 C{a}, C{b} and C{c} in C{meter}, conventionally.
85 '''
86 _Names_ = (_pointP_, _A_, _B_, _C_, _a_, _b_, _c_)
87 _Units_ = (_Pass, Degrees, Degrees, Degrees, Distance, Distance, Distance)
90class TriAngle5Tuple(_NamedTuple):
91 '''5-Tuple C{(radA, radB, radC, rIn, area)} with the interior angles at
92 triangle corners C{A}, C{B} and C{C} in C{radians}, the C{InCircle}
93 radius C{rIn} aka C{inradius} in C{meter} and the triangle C{area}
94 in C{meter} I{squared}, conventionally.
95 '''
96 _Names_ = (_radA_, _radB_, _radC_, _rIn_, _area_)
97 _Units_ = ( Radians, Radians, Radians, Distance, _Pass)
100class TriSide2Tuple(_NamedTuple):
101 '''2-Tuple C{(a, radA)} with triangle side C{a} in C{meter}, conventionally
102 and angle C{radA} at the opposite triangle corner in C{radians}.
103 '''
104 _Names_ = (_a_, _radA_)
105 _Units_ = ( Distance, Radians)
108class TriSide4Tuple(_NamedTuple):
109 '''4-Tuple C{(a, b, radC, d)} with interior angle C{radC} at triangle corner
110 C{C} in C{radians}with the length of triangle sides C{a} and C{b} and
111 with triangle height C{d} perpendicular to triangle side C{c}, in the
112 same units as triangle sides C{a} and C{b}.
113 '''
114 _Names_ = (_a_, _b_, _radC_, _d_)
115 _Units_ = ( Distance, Distance, Radians, Distance)
118def _ABC3(useZ, pointA, pointB, pointC):
119 '''(INTERNAL) Helper for L{cassini} and L{tienstra7}.
120 '''
121 return (_otherV3d(useZ=useZ, pointA=pointA),
122 _otherV3d(useZ=useZ, pointB=pointB),
123 _otherV3d(useZ=useZ, pointC=pointC))
126def _B3(useZ, point1, point2, point3):
127 '''(INTERNAL) Helper for L{pierlot} and L{pierlotx}.
128 '''
129 return (_otherV3d(useZ=useZ, point1=point1),
130 _otherV3d(useZ=useZ, point2=point2),
131 _otherV3d(useZ=useZ, point3=point3))
134def cassini(pointA, pointB, pointC, alpha, beta, useZ=False, **Clas_and_kwds):
135 '''3-Point resection using U{Cassini<https://NL.WikiPedia.org/wiki/Achterwaartse_insnijding>}'s method.
137 @arg pointA: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
138 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
139 @arg pointB: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
140 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
141 @arg pointC: Center point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
142 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
143 @arg alpha: Angle subtended by triangle side B{C{pointA}} to B{C{pointC}}
144 (C{degrees}, non-negative).
145 @arg beta: Angle subtended by triangle side B{C{pointB}} to B{C{pointC}}
146 (C{degrees}, non-negative).
147 @kwarg useZ: If C{True}, use and interpolate the Z component, otherwise
148 force C{z=INT0} (C{bool}).
149 @kwarg Clas_and_kwds: Optional class C{B{Clas}=B{pointA}.classof} to
150 return the survey point with optionally other B{C{Clas}}
151 keyword arguments to instantiate the survey point.
153 @note: Typically, B{C{pointC}} is between B{C{pointA}} and B{C{pointB}}.
155 @return: The survey point, an instance of B{C{Clas}} or B{C{pointA}}'s
156 (sub-)class.
158 @raise ResectionError: Near-coincident, -colinear or -concyclic points
159 or negative or invalid B{C{alpha}} or B{C{beta}}.
161 @raise TypeError: Invalid B{C{pointA}}, B{C{pointB}} or B{C{pointM}}.
163 @see: U{Three Point Resection Problem<https://Dokumen.tips/documents/
164 three-point-resection-problem-introduction-kaestner-burkhardt-method.html>}
165 and functions L{collins5}, L{pierlot}, L{pierlotx} and L{tienstra7}.
166 '''
168 def _H(A, C, sa):
169 s, c = sincos2d(sa)
170 if isnear0(s):
171 raise ValueError(_or(_coincident_, _colinear_))
172 t = s, c, c
173 x = Fdot(t, A.x, C.y, -A.y).fover(s)
174 y = Fdot(t, A.y, -C.x, A.x).fover(s)
175 return x, y
177 A, B, C = _ABC3(useZ, pointA, pointB, pointC)
178 try:
179 sa, sb = map1(float, alpha, beta)
180 if min(sa, sb) < 0:
181 raise ValueError(_negative_)
182 if fsumf_(_360_0, -sa, -sb) < EPS0:
183 raise ValueError()
185 x1, y1 = _H(A, C, sa)
186 x2, y2 = _H(B, C, -sb)
188 x = x1 - x2
189 y = y1 - y2
190 if isnear0(x) or isnear0(y):
191 raise ValueError(_SPACE_(_concyclic_, (x, y)))
193 m = y / x
194 n = x / y
195 N = n + m
196 if isnear0(N):
197 raise ValueError(_SPACE_(_concyclic_, (m, n, N)))
199 t = n, m, _1_0, _N_1_0
200 x = Fdot(t, C.x, x1, C.y, y1).fover(N)
201 y = Fdot(t, y1, C.y, C.x, x1).fover(N)
202 z = _zidw(x, y, useZ, A, B, C)
203 return _Clas(cassini, pointA, Clas_and_kwds, x, y, z)
205 except (TypeError, ValueError) as x:
206 raise ResectionError(pointA=pointA, pointB=pointB, pointC=pointC,
207 alpha=alpha, beta=beta, cause=x)
210def _Clas(where, point, Clas_and_kwds, *args):
211 '''(INTERNAL) Return a C{B{Clas}=point.classof} survey point.
212 '''
213 Clas, kwds = _xkwds_pop2(Clas_and_kwds, Clas=point.classof)
214 return Clas(*args, **_xkwds(kwds, name=where.__name__))
217def collins5(pointA, pointB, pointC, alpha, beta, useZ=False, **Clas_and_kwds):
218 '''3-Point resection using U{Collins<https://Dokumen.tips/documents/
219 three-point-resection-problem-introduction-kaestner-burkhardt-method.html>}' method.
221 @arg pointA: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
222 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
223 @arg pointB: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
224 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
225 @arg pointC: Center point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
226 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
227 @arg alpha: Angle subtended by triangle side C{b} from B{C{pointA}} to
228 B{C{pointC}} (C{degrees}, non-negative).
229 @arg beta: Angle subtended by triangle side C{a} from B{C{pointB}} to
230 B{C{pointC}} (C{degrees}, non-negative).
231 @kwarg useZ: If C{True}, use and interpolate the Z component, otherwise
232 force C{z=INT0} (C{bool}).
233 @kwarg Clas_and_kwds: Optional class C{B{Clas}=B{pointA}.classof} to
234 return the survey point with optionally other B{C{Clas}}
235 keyword arguments to instantiate the survey point.
237 @note: Typically, B{C{pointC}} is between B{C{pointA}} and B{C{pointB}}.
239 @return: L{Collins5Tuple}C{(pointP, pointH, a, b, c)} with survey C{pointP},
240 auxiliary C{pointH}, each an instance of B{C{Clas}} or B{C{pointA}}'s
241 (sub-)class and triangle sides C{a}, C{b} and C{c} in C{meter},
242 conventionally.
244 @raise ResectionError: Near-coincident, -colinear or -concyclic points
245 or negative or invalid B{C{alpha}} or B{C{beta}}.
247 @raise TypeError: Invalid B{C{pointA}}, B{C{pointB}} or B{C{pointM}}.
249 @see: U{Collins' methode<https://NL.WikiPedia.org/wiki/Achterwaartse_insnijding>}
250 and functions L{cassini}, L{pierlot}, L{pierlotx} and L{tienstra7}.
251 '''
253 def _azi_len2(A, B, pi2=PI2):
254 v = B.minus(A)
255 r = atan2(v.x, v.y)
256 if r < 0 and pi2:
257 r += pi2
258 return r, v.length
260 def _xyz(d, r, A, B, C, useZ):
261 s, c = sincos2(r)
262 x = A.x + d * s
263 y = A.y + d * c
264 z = _zidw(x, y, useZ, A, B, C)
265 return x, y, z
267 A, B, C = _ABC3(useZ, pointA, pointB, pointC)
268 try:
269 ra, rb = radians(alpha), radians(beta)
270 if min(ra, rb) < 0:
271 raise ValueError(_negative_)
273 sra, srH = sin(ra), sin(ra + rb - PI) # rH = PI - ((PI - ra) + (PI - rb))
274 if isnear0(sra) or isnear0(srH):
275 raise ValueError(_or(_coincident_, _colinear_, _concyclic_))
277# za, a = _azi_len2(C, B)
278 zb, b = _azi_len2(C, A)
279 zc, c = _azi_len2(A, B, 0)
281# d = c * sin(PI - rb) / srH # B.minus(H).length
282 d = c * sin(PI - ra) / srH # A.minus(H).length
283 r = zc + PI - rb # zh = zc + (PI - rb)
284 H = _xyz(d, r, A, B, C, useZ)
286 zh, _ = _azi_len2(C, Vector3d(*H))
288# d = a * sin(za - zh) / sin(rb) # B.minus(P).length
289 d = b * sin(zb - zh) / sra # A.minus(P).length
290 r = zh - ra # zb - PI + (PI - ra - (zb - zh))
291 P = _xyz(d, r, A, B, C, useZ)
292 P = _Clas(collins5, pointA, Clas_and_kwds, *P)
294 H = _Clas(collins5, pointA, Clas_and_kwds, *H)
295 a = B.minus(C).length
297 return Collins5Tuple(P, H, a, b, c, name=collins5.__name__)
299 except (TypeError, ValueError) as x:
300 raise ResectionError(pointA=pointA, pointB=pointB, pointC=pointC,
301 alpha=alpha, beta=beta, cause=x)
304def pierlot(point1, point2, point3, alpha12, alpha23, useZ=False, eps=EPS,
305 **Clas_and_kwds):
306 '''3-Point resection using U{Pierlot<http://www.Telecom.ULg.ac.BE/publi/publications/
307 pierlot/Pierlot2014ANewThree>}'s method C{ToTal} with I{approximate} limits for
308 the (pseudo-)singularities.
310 @arg point1: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
311 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
312 @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
313 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
314 @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
315 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
316 @arg alpha12: Angle subtended from B{C{point1}} to B{C{point2}} or
317 B{C{alpha2 - alpha1}} (C{degrees}).
318 @arg alpha23: Angle subtended from B{C{point2}} to B{C{point3}} or
319 B{C{alpha3 - alpha2}}(C{degrees}).
320 @kwarg useZ: If C{True}, interpolate the survey point's Z component,
321 otherwise use C{z=INT0} (C{bool}).
322 @kwarg eps: Tolerance for C{cot} (pseudo-)singularities (C{float}).
323 @kwarg Clas_and_kwds: Optional class C{B{Clas}=B{point1}.classof} to
324 return the survey point with optionally other B{C{Clas}}
325 keyword arguments to instantiate the survey point.
327 @note: Typically, B{C{point1}}, B{C{point2}} and B{C{point3}} are ordered
328 by angle, modulo 360, counter-clockwise.
330 @return: The survey (or robot) point, an instance of B{C{Clas}} or B{C{point1}}'s
331 (sub-)class.
333 @raise ResectionError: Near-coincident, -colinear or -concyclic points
334 or invalid B{C{alpha12}} or B{C{alpha23}} or
335 non-positive B{C{eps}}.
337 @raise TypeError: Invalid B{C{point1}}, B{C{point2}} or B{C{point3}}.
339 @see: I{Pierlot}'s C function U{triangulationPierlot<http://www.Telecom.ULg.ac.BE/
340 triangulation/doc/total_8c_source.html>}, U{V. Pierlot, M. Van Droogenbroeck,
341 "A New Three Object Triangulation Algorithm for Mobile Robot Positioning"
342 <https://ORBi.ULiege.BE/bitstream/2268/157469/1/Pierlot2014ANewThree.pdf>},
343 U{Vincent Pierlot, Marc Van Droogenbroeck, "18 Triangulation Algorithms for 2D
344 Positioning (also known as the Resection Problem)"<http://www.Telecom.ULg.ac.BE/
345 triangulation>} and functions L{pierlotx}, L{cassini}, L{collins5} and L{tienstra7}.
346 '''
348 def _cot(s, c): # -eps < I{approximate} cotangent < eps
349 if eps > 0:
350 return c / (min(s, -eps) if s < 0 else max(s, eps))
351 raise ValueError(_SPACE_(_eps_, _not_, _positive_))
353 B1, B2, B3 = _B3(useZ, point1, point2, point3)
354 try:
355 xyz = _pierlot3(B1, B2, B3, alpha12, alpha23, useZ, _cot)
356 return _Clas(pierlot, point1, Clas_and_kwds, *xyz)
358 except (TypeError, ValueError) as x:
359 raise ResectionError(point1=point1, point2=point2, point3=point3,
360 alpha12=alpha12, alpha23=alpha23, eps=eps, cause=x)
363def _pierlot3(B1, B2, B3, a12, a23, useZ, cot):
364 '''(INTERNAL) Shared L{pierlot} and L{pierlotx}.
365 '''
366 x1_, y1_, _ = B1.minus(B2).xyz
367 x3_, y3_, _ = B3.minus(B2).xyz
369 s12, c12, s23, c23 = sincos2d_(a12, a23)
370 # cot31 = (1 - cot12 * cot23) / (cot12 + cot32)
371 # = (1 - c12 / s12 * c23 / s23) / (c12 / s12 + c23 / s23)
372 # = (1 - (c12 * c23) / (s12 * s23)) / (c12 * s23 + s12 * c23) / (s12 * s23)
373 # = (s12 * s23 - c12 * c23) / (c12 * s23 + s12 * c23)
374 # = c31 / s31
375 cot31 = cot(fsum1f_(c12 * s23, s12 * c23), # s31
376 fsum1f_(s12 * s23, -c12 * c23)) # c31
378 K = _F1(x3_ * x1_, cot31 * (y3_ * x1_),
379 y3_ * y1_, -cot31 * (x3_ * y1_))
380 if K:
381 cot12 = cot(s12, c12)
382 cot23 = cot(s23, c23)
384 # x12 = x1_ + cot12 * y1_
385 # y12 = y1_ - cot12 * x1_
387 # x23 = x3_ - cot23 * y3_
388 # y23 = y3_ + cot23 * x3_
390 # x31 = x3_ + x1_ + cot31 * (y3_ - y1_)
391 # y31 = y3_ + y1_ - cot31 * (x3_ - x1_)
393 # x12 - x23 = x1_ + cot12 * y1_ - x3_ + cot23 * y3_
394 X12_23 = _F1(x1_, cot12 * y1_, -x3_, cot23 * y3_)
395 # y12 - y23 = y1_ - cot12 * x1_ - y3_ - cot23 * x3_
396 Y12_23 = _F1(y1_, -cot12 * x1_, -y3_, -cot23 * x3_)
398 # x31 - x23 = x3_ + x1_ + cot31 * (y3_ - y1_) - x3_ + cot23 * y3_
399 # = x1_ + cot31 * y3_ - cot31 * y1_ + cot23 * y3_
400 X31_23 = _F1(x1_, -cot31 * y1_, cot31 * y3_, cot23 * y3_)
401 # y31 - y23 = y3_ + y1_ - cot31 * (x3_ - x1_) - y3_ - cot23 * x3_
402 # = y1_ - cot31 * x3_ + cot31 * x1_ - cot23 * x3_
403 Y31_23 = _F1(y1_, cot31 * x1_, -cot31 * x3_, -cot23 * x3_)
405 # d = (x12 - x23) * (y23 - y31) + (x31 - x23) * (y12 - y23)
406 # = (x31 - x23) * (y12 - y23) - (x12 - x23) * (y31 - y23)
407 # x = (d * B2.x + K * Y12_23).fover(d)
408 # y = (d * B2.y - K * X12_23).fover(d)
409 x, y = _pierlotxy2(B2, -K, Y12_23, X12_23, (X31_23 * Y12_23 -
410 X12_23 * Y31_23))
411 else:
412 x, y, _ = B2.xyz
413 return x, y, _zidw(x, y, useZ, B1, B2, B3)
416def pierlotx(point1, point2, point3, alpha1, alpha2, alpha3, useZ=False,
417 **Clas_and_kwds):
418 '''3-Point resection using U{Pierlot<http://www.Telecom.ULg.ac.BE/publi/
419 publications/pierlot/Pierlot2014ANewThree>}'s method C{ToTal} with
420 I{exact} limits for the (pseudo-)singularities.
422 @arg point1: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
423 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
424 @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
425 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
426 @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
427 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
428 @arg alpha1: Angle at B{C{point1}} (C{degrees}, counter-clockwise).
429 @arg alpha2: Angle at B{C{point2}} (C{degrees}, counter-clockwise).
430 @arg alpha3: Angle at B{C{point3}} (C{degrees}, counter-clockwise).
431 @kwarg useZ: If C{True}, interpolate the survey point's Z component,
432 otherwise use C{z=INT0} (C{bool}).
433 @kwarg Clas_and_kwds: Optional class C{B{Clas}=B{point1}.classof} to
434 return the survey point with optionally other B{C{Clas}}
435 keyword arguments to instantiate the survey point.
437 @return: The survey (or robot) point, an instance of B{C{Clas}} or B{C{point1}}'s
438 (sub-)class.
440 @raise ResectionError: Near-coincident, -colinear or -concyclic points or
441 invalid B{C{alpha1}}, B{C{alpha2}} or B{C{alpha3}}.
443 @raise TypeError: Invalid B{C{point1}}, B{C{point2}} or B{C{point3}}.
445 @see: I{Pierlot}'s C function U{triangulationPierlot2<http://www.Telecom.ULg.ac.BE/
446 triangulation/doc/total_8c_source.html>} and function L{pierlot}, L{cassini},
447 L{collins5} and L{tienstra7}.
448 '''
450 def _a_z_Bs(Bs, *alphas):
451 ds = map2(_umod_360, alphas) # 0 <= alphas < 360
452 ds, Bs = zip(*sorted(_zip(ds, Bs))) # unzip
453 for p, d, B in _zip(ds, _rotate(ds), Bs):
454 d -= p # a12 = a2 - a1, ...
455 z = isnear0(fabs(d) % _180_0)
456 yield d, z, B
458 def _cot(s, c): # I{exact} cotangent
459 try:
460 return (c / s) if c else (NEG0 if s < 0 else _0_0)
461 except ZeroDivisionError:
462 raise ValueError(_or(_coincident_, _colinear_))
464 Bs = _B3(useZ, point1, point2, point3)
465 try:
466 Cs = [0] # pseudo-global, passing the exception Case
467 xyz = _pierlotx3(_a_z_Bs(Bs, alpha1, alpha2, alpha3),
468 useZ, _cot, Cs.append)
469 return _Clas(pierlotx, point1, Clas_and_kwds, *xyz)
471 except (TypeError, ValueError) as x:
472 raise ResectionError(point1=point1, point2=point2, point3=point3, C=Cs.pop(),
473 alpha1=alpha1, alpha2=alpha2, alpha3=alpha3, cause=x)
476def _pierlotx3(a_z_Bs, useZ, cot, Cs):
477 '''(INTERNAL) Core of L{pierlotx}.
478 '''
479 (a12, z12, B1), \
480 (a23, z23, B2), \
481 (a31, z31, B3) = a_z_Bs
482 if z12 and not z23:
483 Cs(1)
484 elif z23 and not z31:
485 Cs(2)
486 a23, B1, B2, B3 = a31, B2, B3, B1
487 elif z31 and not z12:
488 Cs(3)
489 a23, B2, B3 = a12, B3, B2
490 else:
491 Cs(4)
492 return _pierlot3(B1, B2, B3, a12, a23, useZ, cot)
494 x1_, y1_, _ = B1.minus(B3).xyz
495 x2_, y2_, _ = B2.minus(B3).xyz
497 K = _F1(y1_ * x2_, -x1_ * y2_)
498 if K:
499 cot23 = cot(*sincos2d(a23))
501 # x23 = x2_ + cot23 * y2_
502 # y23 = y2_ - cot23 * x2_
504 # x31 = x1_ + cot23 * y1_
505 # y31 = y1_ - cot23 * x1_
507 # x31 - x23 = x1_ + cot23 * y1_ - x2_ - cot23 * y2_
508 X31_23 = _F1(x1_, cot23 * y1_, -x2_, -cot23 * y2_)
509 # y31 - y23 = y1_ - cot23 * x1_ - y2_ + cot23 * x2_
510 Y31_23 = _F1(y1_, -cot23 * x1_, -y2_, cot23 * x2_)
512 # d = (x31 - x23) * (x2_ - x1_) + (y31 - y23) * (y2_ - y1_)
513 # x = (D * B3.x - K * Y31_23).fover(d)
514 # y = (D * B3.y + K * X31_23).fover(d)
515 x, y = _pierlotxy2(B3, K, Y31_23, X31_23, (X31_23 * _F1(x2_, -x1_) +
516 Y31_23 * _F1(y2_, -y1_)))
517 else:
518 x, y, _ = B3.xyz
519 return x, y, _zidw(x, y, useZ, B1, B2, B3)
522def _pierlotxy2(B, K, X, Y, D):
523 '''(INTERNAL) Helper for C{_pierlot3} and C{_pierlotx3}.
524 '''
525 d = float(D)
526 if isnear0(d):
527 raise ValueError(_or(_coincident_, _colinear_, _concyclic_))
528 x = (D * B.x - K * X).fover(d)
529 y = (D * B.y + K * Y).fover(d)
530 return x, y
533def _rotate(xs, n=1):
534 '''Rotate list or tuple C{xs} by C{n} items, right if C{n > 0} else left.
535 '''
536 return xs[n:] + xs[:n]
539def snellius3(a, b, degC, alpha, beta):
540 '''Snellius' surveying using U{Snellius Pothenot<https://WikiPedia.org/wiki/Snellius–Pothenot_problem>}.
542 @arg a: Length of the triangle side between corners C{B} and C{C} and opposite of
543 triangle corner C{A} (C{scalar}, non-negative C{meter}, conventionally).
544 @arg b: Length of the triangle side between corners C{C} and C{A} and opposite of
545 triangle corner C{B} (C{scalar}, non-negative C{meter}, conventionally).
546 @arg degC: Angle at triangle corner C{C}, opposite triangle side C{c} (non-negative C{degrees}).
547 @arg alpha: Angle subtended by triangle side B{C{b}} (non-negative C{degrees}).
548 @arg beta: Angle subtended by triangle side B{C{a}} (non-negative C{degrees}).
550 @return: L{Survey3Tuple}C{(PA, PB, PC)} with distance from survey point C{P} to
551 each of the triangle corners C{A}, C{B} and C{C}, same units as triangle
552 sides B{C{a}}, B{C{b}} and B{C{c}}.
554 @raise TriangleError: Invalid B{C{a}}, B{C{b}} or B{C{degC}} or negative B{C{alpha}}
555 or B{C{beta}}.
557 @see: Function L{wildberger3}.
558 '''
559 try:
560 a, b, degC, alpha, beta = t = map1(float, a, b, degC, alpha, beta)
561 if min(t) < 0:
562 raise ValueError(_negative_)
563 ra, rb, rC = map1(radians, alpha, beta, degC)
565 r = fsum1f_(ra, rb, rC) * _0_5
566 k = PI - r
567 if min(k, r) < 0:
568 raise ValueError(_or(_coincident_, _colinear_))
570 sa, sb = map1(sin, ra, rb)
571 p = atan2(sa * a, sb * b)
572 sp, cp, sr, cr = sincos2_(PI_4 - p, r)
573 p = atan2(sp * sr, cp * cr)
574 pa = k + p
575 pb = k - p
577 if fabs(sb) > fabs(sa):
578 pc = fabs(a * sin(pb) / sb)
579 elif sa:
580 pc = fabs(b * sin(pa) / sa)
581 else:
582 raise ValueError(_or(_colinear_, _coincident_))
584 pa = _triSide(b, pc, fsumf_(PI, -ra, -pa))
585 pb = _triSide(a, pc, fsumf_(PI, -rb, -pb))
586 return Survey3Tuple(pa, pb, pc, name=snellius3.__name__)
588 except (TypeError, ValueError) as x:
589 raise TriangleError(a=a, b=b, degC=degC, alpha=alpha, beta=beta, cause=x)
592def tienstra7(pointA, pointB, pointC, alpha, beta=None, gamma=None,
593 useZ=False, **Clas_and_kwds):
594 '''3-Point resection using U{Tienstra<https://WikiPedia.org/wiki/Tienstra_formula>}'s formula.
596 @arg pointA: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or
597 C{Vector2Tuple} if C{B{useZ}=False}).
598 @arg pointB: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or
599 C{Vector2Tuple} if C{B{useZ}=False}).
600 @arg pointC: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or
601 C{Vector2Tuple} if C{B{useZ}=False}).
602 @arg alpha: Angle subtended by triangle side C{a} from B{C{pointB}} to B{C{pointC}}
603 (C{degrees}, non-negative).
604 @kwarg beta: Angle subtended by triangle side C{b} from B{C{pointA}} to B{C{pointC}}
605 (C{degrees}, non-negative) or C{None} if C{B{gamma} is not None}.
606 @kwarg gamma: Angle subtended by triangle side C{c} from B{C{pointA}} to B{C{pointB}}
607 (C{degrees}, non-negative) or C{None} if C{B{beta} is not None}.
608 @kwarg useZ: If C{True}, use and interpolate the Z component, otherwise force C{z=INT0}
609 (C{bool}).
610 @kwarg Clas_and_kwds: Optional class C{B{Clas}=B{pointA}.classof} to return the survey
611 point with optionally other B{C{Clas}} keyword arguments to instantiate
612 the survey point.
614 @note: Points B{C{pointA}}, B{C{pointB}} and B{C{pointC}} are ordered clockwise.
616 @return: L{Tienstra7Tuple}C{(pointP, A, B, C, a, b, c)} with survey C{pointP}, an
617 instance of B{C{Clas}} or B{C{pointA}}'s (sub-)class, with triangle angles C{A}
618 at B{C{pointA}}, C{B} at B{C{pointB}} and C{C} at B{C{pointC}} in C{degrees}
619 and with triangle sides C{a}, C{b} and C{c} in C{meter}, conventionally.
621 @raise ResectionError: Near-coincident, -colinear or -concyclic points or sum of
622 B{C{alpha}}, B{C{beta}} and B{C{gamma}} not C{360} or negative
623 B{C{alpha}}, B{C{beta}} or B{C{gamma}}.
625 @raise TypeError: Invalid B{C{pointA}}, B{C{pointB}} or B{C{pointC}}.
627 @see: U{3-Point Resection Solver<http://MesaMike.org/geocache/GC1B0Q9/tienstra/>},
628 U{V. Pierlot, M. Van Droogenbroeck, "A New Three Object Triangulation..."
629 <http://www.Telecom.ULg.ac.BE/publi/publications/pierlot/Pierlot2014ANewThree/>},
630 U{18 Triangulation Algorithms...<http://www.Telecom.ULg.ac.BE/triangulation/>} and
631 functions L{cassini}, L{collins5}, L{pierlot} and L{pierlotx}.
632 '''
634 def _deg_ks(r, s, ks, N):
635 if isnear0(fsumf_(PI, r, -s)): # r + (PI2 - s) == PI
636 raise ValueError(Fmt.PARENSPACED(concyclic=N))
637 # k = 1 / (cot(r) - cot(s))
638 # = 1 / (cos(r) / sin(r) - cos(s) / sin(s))
639 # = 1 / (cos(r) * sin(s) - cos(s) * sin(r)) / (sin(r) * sin(s))
640 # = sin(r) * sin(s) / (cos(r) * sin(s) - cos(s) * sin(r))
641 sr, cr, ss, cs = sincos2_(r, s)
642 c = fsum1f_(cr * ss, -cs * sr)
643 if isnear0(c):
644 raise ValueError(Fmt.PARENSPACED(cotan=N))
645 ks.append(sr * ss / c)
646 return Degrees(degrees(r), name=N) # C degrees
648 A, B, C = _ABC3(useZ, pointA, pointB, pointC)
649 try:
650 sa, sb, sc = map1(radians, alpha, (beta or 0), (gamma or 0))
651 if beta is None:
652 if gamma is None:
653 raise ValueError(_and(Fmt.EQUAL(beta=beta), Fmt.EQUAL(gamma=gamma)))
654 sb = fsumf_(PI2, -sa, -sc)
655 elif gamma is None:
656 sc = fsumf_(PI2, -sa, -sb)
657 else: # subtended angles must add to 360 degrees
658 r = fsum1f_(sa, sb, sc)
659 if fabs(r - PI2) > EPS:
660 raise ValueError(Fmt.EQUAL(sum=degrees(r)))
661 if min(sa, sb, sc) < 0:
662 raise ValueError(_negative_)
664 # triangle sides
665 a = B.minus(C).length
666 b = A.minus(C).length
667 c = A.minus(B).length
669 ks = [] # 3 Ks and triangle angles
670 dA = _deg_ks(_triAngle(b, c, a), sa, ks, _A_)
671 dB = _deg_ks(_triAngle(a, c, b), sb, ks, _B_)
672 dC = _deg_ks(_triAngle(a, b, c), sc, ks, _C_)
674 k = fsum1(ks, floats=True)
675 if isnear0(k):
676 raise ValueError(Fmt.EQUAL(K=k))
677 x = Fdot(ks, A.x, B.x, C.x).fover(k)
678 y = Fdot(ks, A.y, B.y, C.y).fover(k)
679 z = _zidw(x, y, useZ, A, B, C)
681 P = _Clas(tienstra7, pointA, Clas_and_kwds, x, y, z)
682 return Tienstra7Tuple(P, dA, dB, dC, a, b, c, name=tienstra7.__name__)
684 except (TypeError, ValueError) as x:
685 raise ResectionError(pointA=pointA, pointB=pointB, pointC=pointC,
686 alpha=alpha, beta=beta, gamma=gamma, cause=x)
689def triAngle(a, b, c):
690 '''Compute one angle of a triangle.
692 @arg a: Adjacent triangle side length (C{scalar}, non-negative
693 C{meter}, conventionally).
694 @arg b: Adjacent triangle side length (C{scalar}, non-negative
695 C{meter}, conventionally).
696 @arg c: Opposite triangle side length (C{scalar}, non-negative
697 C{meter}, conventionally).
699 @return: Angle in C{radians} at triangle corner C{C}, opposite
700 triangle side B{C{c}}.
702 @raise TriangleError: Invalid or negative B{C{a}}, B{C{b}} or B{C{c}}.
704 @see: Functions L{triAngle5} and L{triSide}.
705 '''
706 try:
707 return _triAngle(a, b, c)
708 except (TypeError, ValueError) as x:
709 raise TriangleError(a=a, b=b, c=c, cause=x)
712def _triAngle(a, b, c):
713 # (INTERNAL) To allow callers to embellish errors
714 a, b, c = map1(float, a, b, c)
715 if a < b:
716 a, b = b, a
717 if b < 0 or c < 0:
718 raise ValueError(_negative_)
719 if a < EPS0:
720 raise ValueError(_coincident_)
721 b_a = b / a
722 if b_a < EPS0:
723 raise ValueError(_coincident_)
724 t = fsumf_(_1_0, b_a**2, -(c / a)**2) / (b_a * _2_0)
725 return acos1(t)
728def triAngle5(a, b, c):
729 '''Compute the angles of a triangle.
731 @arg a: Length of the triangle side opposite of triangle corner C{A}
732 (C{scalar}, non-negative C{meter}, conventionally).
733 @arg b: Length of the triangle side opposite of triangle corner C{B}
734 (C{scalar}, non-negative C{meter}, conventionally).
735 @arg c: Length of the triangle side opposite of triangle corner C{C}
736 (C{scalar}, non-negative C{meter}, conventionally).
738 @return: L{TriAngle5Tuple}C{(radA, radB, radC, rIn, area)} with angles
739 C{radA}, C{radB} and C{radC} at triangle corners C{A}, C{B}
740 and C{C}, all in C{radians}, the C{InCircle} radius C{rIn}
741 aka C{inradius}, same units as triangle sides B{C{a}},
742 B{C{b}} and B{C{c}} and the triangle C{area} in those same
743 units I{squared}.
745 @raise TriangleError: Invalid or negative B{C{a}}, B{C{b}} or B{C{c}}.
747 @see: Functions L{triAngle} and L{triArea}.
748 '''
749 try:
750 x, y, z = map1(float, a, b, c)
751 ab = x < y
752 if ab:
753 x, y = y, x
754 bc = y < z
755 if bc:
756 y, z = z, y
758 if z > EPS0: # z = min(a, b, c)
759 s = fsum1f_(z, y, x) * _0_5
760 sa, sb, r = (s - x), (s - y), (s - z)
761 r *= _over(sa * sb, s)
762 if r < EPS02:
763 raise ValueError(_coincident_)
764 r = sqrt(r)
765 rA = atan2(r, sa) * _2_0
766 rB = atan2(r, sb) * _2_0
767 rC = fsumf_(PI, -rA, -rB)
768 if min(rA, rB, rC) < 0:
769 raise ValueError(_colinear_)
770 s *= r # Heron's area
771 elif z < 0:
772 raise ValueError(_negative_)
773 else: # 0 <= c <= EPS0
774 rA = rB = PI_2
775 rC = r = s = _0_0
777 if bc:
778 rB, rC = rC, rB
779 if ab:
780 rA, rB = rB, rA
781 return TriAngle5Tuple(rA, rB, rC, r, s, name=triAngle5.__name__)
783 except (TypeError, ValueError) as x:
784 raise TriangleError(a=a, b=b, c=c, cause=x)
787def triArea(a, b, c):
788 '''Compute the area of a triangle using U{Heron's<https://
789 WikiPedia.org/wiki/Heron%27s_formula>} C{stable} formula.
791 @arg a: Length of the triangle side opposite of triangle corner C{A}
792 (C{scalar}, non-negative C{meter}, conventionally).
793 @arg b: Length of the triangle side opposite of triangle corner C{B}
794 (C{scalar}, non-negative C{meter}, conventionally).
795 @arg c: Length of the triangle side opposite of triangle corner C{C}
796 (C{scalar}, non-negative C{meter}, conventionally).
798 @return: The triangle area (C{float}, conventionally C{meter} or
799 same units as B{C{a}}, B{C{b}} and B{C{c}} I{squared}).
801 @raise TriangleError: Invalid or negative B{C{a}}, B{C{b}} or B{C{c}}.
802 '''
803 try:
804 r, y, x = sorted(map1(float, a, b, c))
805 if r > 0: # r = min(a, b, c)
806 ab = x - y
807 bc = y - r
808 y += r
809 r = (x + y) * (r - ab) * (r + ab) * (x + bc)
810 if r:
811 r = sqrt(r / _16_0)
812 elif r < 0:
813 raise ValueError(_negative_)
814 return r
816 except (TypeError, ValueError) as x:
817 raise TriangleError(a=a, b=b, c=c, cause=x)
820def triSide(a, b, radC):
821 '''Compute one side of a triangle.
823 @arg a: Adjacent triangle side length (C{scalar},
824 non-negative C{meter}, conventionally).
825 @arg b: Adjacent triangle side length (C{scalar},
826 non-negative C{meter}, conventionally).
827 @arg radC: Angle included by sides B{C{a}} and B{C{b}},
828 opposite triangle side C{c} (C{radians}).
830 @return: Length of triangle side C{c}, opposite triangle
831 corner C{C} and angle B{C{radC}}, same units as
832 B{C{a}} and B{C{b}}.
834 @raise TriangleError: Invalid B{C{a}}, B{C{b}} or B{C{radC}}.
836 @see: Functions L{sqrt_a}, L{triAngle}, L{triSide2} and L{triSide4}.
837 '''
838 try:
839 return _triSide(a, b, radC)
840 except (TypeError, ValueError) as x:
841 raise TriangleError(a=a, b=b, radC=radC, cause=x)
844def _triSide(a, b, radC):
845 # (INTERNAL) To allow callers to embellish errors
846 a, b, r = t = map1(float, a, b, radC)
847 if min(t) < 0:
848 raise ValueError(_negative_)
850 if a < b:
851 a, b = b, a
852 if a > EPS0:
853 ba = b / a
854 c2 = fsumf_(_1_0, ba**2, _N_2_0 * ba * cos(r))
855 if c2 > EPS02:
856 return a * sqrt(c2)
857 elif c2 < 0:
858 raise ValueError(_invalid_)
859 return hypot(a, b)
862def triSide2(b, c, radB):
863 '''Compute a side and its opposite angle of a triangle.
865 @arg b: Adjacent triangle side length (C{scalar},
866 non-negative C{meter}, conventionally).
867 @arg c: Adjacent triangle side length (C{scalar},
868 non-negative C{meter}, conventionally).
869 @arg radB: Angle included by sides B{C{a}} and B{C{c}},
870 opposite triangle side C{b} (C{radians}).
872 @return: L{TriSide2Tuple}C{(a, radA)} with triangle angle
873 C{radA} in C{radians} and length of the opposite
874 triangle side C{a}, same units as B{C{b}} and B{C{c}}.
876 @raise TriangleError: Invalid B{C{b}} or B{C{c}} or either
877 B{C{b}} or B{C{radB}} near zero.
879 @see: Functions L{sqrt_a}, L{triSide} and L{triSide4}.
880 '''
881 try:
882 return _triSide2(b, c, radB)
883 except (TypeError, ValueError) as x:
884 raise TriangleError(b=b, c=c, radB=radB, cause=x)
887def _triSide2(b, c, radB):
888 # (INTERNAL) To allow callers to embellish errors
889 b, c, rB = map1(float, b, c, radB)
890 if min(b, c, rB) < 0:
891 raise ValueError(_negative_)
892 sB, cB = sincos2(rB)
893 if isnear0(sB):
894 if not isnear0(b):
895 raise ValueError(_invalid_)
896 a, rA = ((b + c), PI) if cB < 0 else (fabs(b - c), _0_0)
897 elif isnear0(b):
898 raise ValueError(_invalid_)
899 else:
900 rA = fsumf_(PI, -rB, -asin1(c * sB / b))
901 a = sin(rA) * b / sB
902 return TriSide2Tuple(a, rA, name=triSide2.__name__)
905def triSide4(radA, radB, c):
906 '''Compute two sides and the height of a triangle.
908 @arg radA: Angle at triangle corner C{A}, opposite triangle side C{a}
909 (non-negative C{radians}).
910 @arg radB: Angle at triangle corner C{B}, opposite triangle side C{b}
911 (non-negative C{radians}).
912 @arg c: Length of triangle side between triangle corners C{A} and C{B},
913 (C{scalar}, non-negative C{meter}, conventionally).
915 @return: L{TriSide4Tuple}C{(a, b, radC, d)} with triangle sides C{a} and
916 C{b} and triangle height C{d} perpendicular to triangle side
917 B{C{c}}, all in the same units as B{C{c}} and interior angle
918 C{radC} in C{radians} at triangle corner C{C}, opposite
919 triangle side B{C{c}}.
921 @raise TriangleError: Invalid or negative B{C{radA}}, B{C{radB}} or B{C{c}}.
923 @see: U{Triangulation, Surveying<https://WikiPedia.org/wiki/Triangulation_(surveying)>}
924 and functions L{sqrt_a}, L{triSide} and L{triSide2}.
925 '''
926 try:
927 rA, rB, c = map1(float, radA, radB, c)
928 rC = fsumf_(PI, -rA, -rB)
929 if min(rC, rA, rB, c) < 0:
930 raise ValueError(_negative_)
931 sa, ca, sb, cb = sincos2_(rA, rB)
932 sc = fsum1f_(sa * cb, sb * ca)
933 if sc < EPS0 or min(sa, sb) < 0:
934 raise ValueError(_invalid_)
935 sc = c / sc
936 return TriSide4Tuple((sa * sc), (sb * sc), rC, (sa * sb * sc),
937 name=triSide4.__name__)
939 except (TypeError, ValueError) as x:
940 raise TriangleError(radA=radA, radB=radB, c=c, cause=x)
943def wildberger3(a, b, c, alpha, beta, R3=min):
944 '''Snellius' surveying using U{Rational Trigonometry
945 <https://WikiPedia.org/wiki/Snellius–Pothenot_problem>}.
947 @arg a: Length of the triangle side between corners C{B} and C{C} and opposite of
948 triangle corner C{A} (C{scalar}, non-negative C{meter}, conventionally).
949 @arg b: Length of the triangle side between corners C{C} and C{A} and opposite of
950 triangle corner C{B} (C{scalar}, non-negative C{meter}, conventionally).
951 @arg c: Length of the triangle side between corners C{A} and C{B} and opposite of
952 triangle corner C{C} (C{scalar}, non-negative C{meter}, conventionally).
953 @arg alpha: Angle subtended by triangle side B{C{b}} (C{degrees}, non-negative).
954 @arg beta: Angle subtended by triangle side B{C{a}} (C{degrees}, non-negative).
955 @kwarg R3: Callable to determine C{R3} from C{(R3 - C)**2 = D}, typically standard
956 Python function C{min} or C{max}, invoked with 2 arguments.
958 @return: L{Survey3Tuple}C{(PA, PB, PC)} with distance from survey point C{P} to
959 each of the triangle corners C{A}, C{B} and C{C}, same units as B{C{a}},
960 B{C{b}} and B{C{c}}.
962 @raise TriangleError: Invalid B{C{a}}, B{C{b}} or B{C{c}} or negative B{C{alpha}} or
963 B{C{beta}} or B{C{R3}} not C{callable}.
965 @see: U{Wildberger, Norman J.<https://Math.Sc.Chula.ac.TH/cjm/content/
966 survey-article-greek-geometry-rational-trigonometry-and-snellius-–-pothenot-surveying>},
967 U{Devine Proportions, page 252<http://www.MS.LT/derlius/WildbergerDivineProportions.pdf>}
968 and function L{snellius3}.
969 '''
970 def _s(x):
971 return sin(x)**2
973 def _vpa(r3, q2, q3, s2, s3):
974 r1 = s2 * q3 / s3
975 r = r1 * r3 * _4_0
976 n = (r - _F1(r1, r3, -q2)**2).fover(s3)
977 if n < 0 or r < EPS0:
978 raise ValueError(_coincident_)
979 return sqrt((n / r) * q3) if n else _0_0
981 try:
982 a, b, c, da, db = q = map1(float, a, b, c, alpha, beta)
983 if min(q) < 0:
984 raise ValueError(_negative_)
986 q1, q2, q3 = q = a**2, b**2, c**2
987 if min(q) < EPS02:
988 raise ValueError(_coincident_)
990 ra, rb = map1(radians, da, db)
991 s1, s2, s3 = s = map1(_s, rb, ra, ra + rb) # rb, ra!
992 if min(s) < EPS02:
993 raise ValueError(_or(_coincident_, _colinear_))
995 q4 = hypot2_(*q) * _2_0 # a**4 + ...
996 Qs = _F1(*q) # == hypot2_(a, b, c)
997 d0 = (Qs**2 - q4).fmul(s1 * s2).fover(s3)
998 if d0 < 0:
999 raise ValueError(_negative_)
1000 s += _F1(*s), # == fsum1(s),
1001 C0 = Fdot(s, q1, q2, q3, -Qs * _0_5)
1002 r3 = C0.fover(-s3) # C0 /= -s3
1003 if d0 > EPS02: # > c0
1004 _xcallable(R3=R3)
1005 d0 = sqrt(d0)
1006 r3 = R3(float(C0 + d0), float(C0 - d0)) # XXX min or max
1008 pa = _vpa(r3, q2, q3, s2, s3)
1009 pb = _vpa(r3, q1, q3, s1, s3)
1010 pc = favg(_triSide2(b, pa, ra).a,
1011 _triSide2(a, pb, rb).a)
1012 return Survey3Tuple(pa, pb, pc, name=wildberger3.__name__)
1014 except (TypeError, ValueError) as x:
1015 raise TriangleError(a=a, b=b, c=c, alpha=alpha, beta=beta, R3=R3, cause=x)
1018def _zidw(x, y, useZ, *ABC):
1019 if useZ: # interpolate z or coplanar with A, B and C?
1020 t = tuple(_.z for _ in ABC)
1021 v = Vector3d(x, y, fmean(t))
1022 z = fidw(t, (v.minus(T).length for T in ABC))
1023 else:
1024 z = INT0
1025 return z
1027# **) MIT License
1028#
1029# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved.
1030#
1031# Permission is hereby granted, free of charge, to any person obtaining a
1032# copy of this software and associated documentation files (the "Software"),
1033# to deal in the Software without restriction, including without limitation
1034# the rights to use, copy, modify, merge, publish, distribute, sublicense,
1035# and/or sell copies of the Software, and to permit persons to whom the
1036# Software is furnished to do so, subject to the following conditions:
1037#
1038# The above copyright notice and this permission notice shall be included
1039# in all copies or substantial portions of the Software.
1040#
1041# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
1042# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
1043# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
1044# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
1045# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
1046# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
1047# OTHER DEALINGS IN THE SOFTWARE.