Coverage for pygeodesy/resections.py: 97%
360 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{triAngle4}, 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, _ALL_LAZY
16from pygeodesy.constants import EPS, EPS0, EPS02, INT0, NEG0, PI, PI2, PI_2, PI_4, \
17 isnear0, _umod_360, _0_0, _0_5, _1_0, _N_1_0, _2_0, \
18 _N_2_0, _4_0, _180_0, _360_0
19from pygeodesy.errors import _and, _or, TriangleError, _ValueError, _xkwds
20from pygeodesy.fmath import favg, Fdot, fidw, fmean, hypot, hypot2_
21from pygeodesy.fsums import Fsum, fsumf_, fsum1, fsum1f_
22from pygeodesy.interns import _a_, _A_, _b_, _B_, _c_, _C_, _coincident_, _colinear_, \
23 _d_, _eps_, _invalid_, _negative_, _not_, _rIn_, _SPACE_
24# from pygeodesy.lazily import _ALL_LAZY # from .basics
25from pygeodesy.named import Fmt, _NamedTuple, _Pass
26# from pygeodesy.streprs import Fmt # from .named
27from pygeodesy.units import Degrees, Distance, Radians
28from pygeodesy.utily import acos1, asin1, sincos2, sincos2_, sincos2d, sincos2d_
29from pygeodesy.vector3d import _otherV3d, Vector3d
31from math import cos, atan2, degrees, fabs, radians, sin, sqrt
33__all__ = _ALL_LAZY.resections
34__version__ = '23.06.07'
36_concyclic_ = 'concyclic'
37_PA_ = 'PA'
38_PB_ = 'PB'
39_PC_ = 'PC'
40_pointH_ = 'pointH'
41_pointP_ = 'pointP'
42_positive_ = 'positive'
43_R3__ = 'R3 '
44_radA_ = 'radA'
45_radB_ = 'radB'
46_radC_ = 'radC'
49class Collins5Tuple(_NamedTuple):
50 '''5-Tuple C{(pointP, pointH, a, b, c)} with survey C{pointP}, auxiliary
51 C{pointH}, each an instance of B{C{pointA}}'s (sub-)class and triangle
52 sides C{a}, C{b} and C{c} in C{meter}, conventionally.
53 '''
54 _Names_ = (_pointP_, _pointH_, _a_, _b_, _c_)
55 _Units_ = (_Pass, _Pass, Distance, Distance, Distance)
58class ResectionError(_ValueError):
59 '''Error raised for issues in L{pygeodesy.resections}.
60 '''
61 pass
64class Survey3Tuple(_NamedTuple):
65 '''3-Tuple C{(PA, PB, PC)} with distance from survey point C{P} to each of
66 the triangle corners C{A}, C{B} and C{C} in C{meter}, conventionally.
67 '''
68 _Names_ = (_PA_, _PB_, _PC_)
69 _Units_ = ( Distance, Distance, Distance)
72class Tienstra7Tuple(_NamedTuple):
73 '''7-Tuple C{(pointP, A, B, C, a, b, c)} with survey C{pointP}, interior
74 triangle angles C{A}, C{B} and C{C} in C{degrees} and triangle sides
75 C{a}, C{b} and C{c} in C{meter}, conventionally.
76 '''
77 _Names_ = (_pointP_, _A_, _B_, _C_, _a_, _b_, _c_)
78 _Units_ = (_Pass, Degrees, Degrees, Degrees, Distance, Distance, Distance)
81class TriAngle4Tuple(_NamedTuple):
82 '''4-Tuple C{(radA, radB, radC, rIn)} with the interior angles at triangle
83 corners C{A}, C{B} and C{C} in C{radians} and the C{InCircle} radius
84 C{rIn} aka C{inradius} in C{meter}, conventionally.
85 '''
86 _Names_ = (_radA_, _radB_, _radC_, _rIn_)
87 _Units_ = ( Radians, Radians, Radians, Distance)
90class TriSide2Tuple(_NamedTuple):
91 '''2-Tuple C{(a, radA)} with triangle side C{a} in C{meter}, conventionally
92 and angle C{radA} at the opposite triangle corner in C{radians}.
93 '''
94 _Names_ = (_a_, _radA_)
95 _Units_ = ( Distance, Radians)
98class TriSide4Tuple(_NamedTuple):
99 '''4-Tuple C{(a, b, radC, d)} with interior angle C{radC} at triangle corner
100 C{C} in C{radians}with the length of triangle sides C{a} and C{b} and
101 with triangle height C{d} perpendicular to triangle side C{c}, in the
102 same units as triangle sides C{a} and C{b}.
103 '''
104 _Names_ = (_a_, _b_, _radC_, _d_)
105 _Units_ = ( Distance, Distance, Radians, Distance)
108def _ABC3(useZ, pointA, pointB, pointC):
109 '''(INTERNAL) Helper for L{cassini} and L{tienstra7}.
110 '''
111 return (_otherV3d(useZ=useZ, pointA=pointA),
112 _otherV3d(useZ=useZ, pointB=pointB),
113 _otherV3d(useZ=useZ, pointC=pointC))
116def _B3(useZ, point1, point2, point3):
117 '''(INTERNAL) Helper for L{pierlot} and L{pierlotx}.
118 '''
119 return (_otherV3d(useZ=useZ, point1=point1),
120 _otherV3d(useZ=useZ, point2=point2),
121 _otherV3d(useZ=useZ, point3=point3))
124def cassini(pointA, pointB, pointC, alpha, beta, useZ=False, Clas=None, **Clas_kwds):
125 '''3-Point resection using U{Cassini<https://NL.WikiPedia.org/wiki/Achterwaartse_insnijding>}'s method.
127 @arg pointA: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
128 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
129 @arg pointB: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
130 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
131 @arg pointC: Center point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
132 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
133 @arg alpha: Angle subtended by triangle side B{C{pointA}} to B{C{pointC}}
134 (C{degrees}, non-negative).
135 @arg beta: Angle subtended by triangle side B{C{pointB}} to B{C{pointC}}
136 (C{degrees}, non-negative).
137 @kwarg useZ: If C{True}, use and interpolate the Z component, otherwise
138 force C{z=INT0} (C{bool}).
139 @kwarg Clas: Optional class to return the survey point or C{None} for
140 B{C{pointA}}'s (sub-)class.
141 @kwarg Clas_kwds: Optional, additional keyword argument for the survey
142 point instance.
144 @note: Typically, B{C{pointC}} is between B{C{pointA}} and B{C{pointB}}.
146 @return: The survey point, an instance of B{C{Clas}} or if C{B{Clas} is
147 None} an instance of B{C{pointA}}'s (sub-)class.
149 @raise ResectionError: Near-coincident, -colinear or -concyclic points
150 or negative or invalid B{C{alpha}} or B{C{beta}}.
152 @raise TypeError: Invalid B{C{pointA}}, B{C{pointB}} or B{C{pointM}}.
154 @see: U{Three Point Resection Problem<https://Dokumen.tips/documents/
155 three-point-resection-problem-introduction-kaestner-burkhardt-method.html>}
156 and functions L{pygeodesy.collins5}, L{pygeodesy.pierlot} and
157 L{pygeodesy.tienstra7}.
158 '''
160 def _H(A, C, sa):
161 s, c = sincos2d(sa)
162 if isnear0(s):
163 raise ValueError(_or(_coincident_, _colinear_))
164 t = s, c, c
165 x = Fdot(t, A.x, C.y, -A.y).fover(s)
166 y = Fdot(t, A.y, -C.x, A.x).fover(s)
167 return x, y
169 A, B, C = _ABC3(useZ, pointA, pointB, pointC)
170 try:
171 sa, sb = map1(float, alpha, beta)
172 if min(sa, sb) < 0:
173 raise ValueError(_negative_)
174 if fsumf_(_360_0, -sa, -sb) < EPS0:
175 raise ValueError()
177 x1, y1 = _H(A, C, sa)
178 x2, y2 = _H(B, C, -sb)
180 x = x1 - x2
181 y = y1 - y2
182 if isnear0(x) or isnear0(y):
183 raise ValueError(_SPACE_(_concyclic_, (x, y)))
185 m = y / x
186 n = x / y
187 N = n + m
188 if isnear0(N):
189 raise ValueError(_SPACE_(_concyclic_, (m, n, N)))
191 t = n, m, _1_0, _N_1_0
192 x = Fdot(t, C.x, x1, C.y, y1).fover(N)
193 y = Fdot(t, y1, C.y, C.x, x1).fover(N)
194 z = _zidw(x, y, useZ, A, B, C)
196 clas = Clas or pointA.classof
197 return clas(x, y, z, **_xkwds(Clas_kwds, name=cassini.__name__))
199 except (TypeError, ValueError) as x:
200 raise ResectionError(pointA=pointA, pointB=pointB, pointC=pointC,
201 alpha=alpha, beta=beta, cause=x)
204def collins5(pointA, pointB, pointC, alpha, beta, useZ=False, Clas=None, **Clas_kwds):
205 '''3-Point resection using U{Collins<https://Dokumen.tips/documents/
206 three-point-resection-problem-introduction-kaestner-burkhardt-method.html>}' method.
208 @arg pointA: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
209 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
210 @arg pointB: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
211 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
212 @arg pointC: Center point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
213 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
214 @arg alpha: Angle subtended by triangle side C{b} from B{C{pointA}} to
215 B{C{pointC}} (C{degrees}, non-negative).
216 @arg beta: Angle subtended by triangle side C{a} from B{C{pointB}} to
217 B{C{pointC}} (C{degrees}, non-negative).
218 @kwarg useZ: If C{True}, use and interpolate the Z component, otherwise
219 force C{z=INT0} (C{bool}).
220 @kwarg Clas: Optional class to return the survey and auxiliary point
221 or C{None} for B{C{pointA}}'s (sub-)class.
222 @kwarg Clas_kwds: Optional, additional keyword argument for the survey
223 and auxiliary point instance.
225 @note: Typically, B{C{pointC}} is between B{C{pointA}} and B{C{pointB}}.
227 @return: L{Collins5Tuple}C{(pointP, pointH, a, b, c)} with survey C{pointP},
228 auxiliary C{pointH}, each an instance of B{C{Clas}} or if C{B{Clas}
229 is None} an instance of B{C{pointA}}'s (sub-)class and triangle
230 sides C{a}, C{b} and C{c} in C{meter}, conventionally.
232 @raise ResectionError: Near-coincident, -colinear or -concyclic points
233 or negative or invalid B{C{alpha}} or B{C{beta}}.
235 @raise TypeError: Invalid B{C{pointA}}, B{C{pointB}} or B{C{pointM}}.
237 @see: U{Collins' methode<https://NL.WikiPedia.org/wiki/Achterwaartse_insnijding>}
238 and functions L{pygeodesy.cassini}, L{pygeodesy.pierlot} and
239 L{pygeodesy.tienstra7}.
240 '''
242 def _azi_len2(A, B, pi2):
243 v = B.minus(A)
244 r = atan2(v.x, v.y)
245 if pi2 and r < 0:
246 r += pi2
247 return r, v.length
249 def _cV3(d, r, A, B, C, useZ, V3, **kwds):
250 s, c = sincos2(r)
251 x = A.x + d * s
252 y = A.y + d * c
253 z = _zidw(x, y, useZ, A, B, C)
254 return V3(x, y, z, **kwds)
256 A, B, C = _ABC3(useZ, pointA, pointB, pointC)
257 try:
258 ra, rb = radians(alpha), radians(beta)
259 if min(ra, rb) < 0:
260 raise ValueError(_negative_)
262 sra, srH = sin(ra), sin(ra + rb - PI) # rH = PI - ((PI - ra) + (PI - rb))
263 if isnear0(sra) or isnear0(srH):
264 raise ValueError(_or(_coincident_, _colinear_, _concyclic_))
266 clas = Clas or pointA.classof
267 kwds = _xkwds(Clas_kwds, name=collins5.__name__)
269# za, a = _azi_len2(C, B, PI2)
270 zb, b = _azi_len2(C, A, PI2)
271 zc, c = _azi_len2(A, B, 0)
273# d = c * sin(PI - rb) / srH # B.minus(H).length
274 d = c * sin(PI - ra) / srH # A.minus(H).length
275 r = zc + PI - rb # zh = zc + (PI - rb)
276 H = _cV3(d, r, A, B, C, useZ, Vector3d)
278 zh, _ = _azi_len2(C, H, PI2)
280# d = a * sin(za - zh) / sin(rb) # B.minus(P).length
281 d = b * sin(zb - zh) / sra # A.minus(P).length
282 r = zh - ra # zb - PI + (PI - ra - (zb - zh))
283 P = _cV3(d, r, A, B, C, useZ, clas, **kwds)
285 H = clas(H.x, H.y, H.z, **kwds)
286 a = B.minus(C).length
288 return Collins5Tuple(P, H, a, b, c, name=collins5.__name__)
290 except (TypeError, ValueError) as x:
291 raise ResectionError(pointA=pointA, pointB=pointB, pointC=pointC,
292 alpha=alpha, beta=beta, cause=x)
295def pierlot(point1, point2, point3, alpha12, alpha23, useZ=False, eps=EPS,
296 Clas=None, **Clas_kwds):
297 '''3-Point resection using U{Pierlot<http://www.Telecom.ULg.ac.BE/publi/publications/
298 pierlot/Pierlot2014ANewThree>}'s method C{ToTal} with I{approximate} limits for
299 the (pseudo-)singularities.
301 @arg point1: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
302 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
303 @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
304 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
305 @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
306 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
307 @arg alpha12: Angle subtended from B{C{point1}} to B{C{point2}} or
308 B{C{alpha2 - alpha1}} (C{degrees}).
309 @arg alpha23: Angle subtended from B{C{point2}} to B{C{point3}} or
310 B{C{alpha3 - alpha2}}(C{degrees}).
311 @kwarg useZ: If C{True}, interpolate the survey point's Z component,
312 otherwise use C{z=INT0} (C{bool}).
313 @kwarg eps: Tolerance for C{cot} (pseudo-)singularities (C{float}).
314 @kwarg Clas: Optional class to return the survey point, if C{None} use
315 B{C{point1}}'s (sub-)class.
316 @kwarg Clas_kwds: Optional, additional keyword arguments for the survey
317 point instance.
319 @note: Typically, B{C{point1}}, B{C{point2}} and B{C{point3}} are ordered
320 by angle, modulo 360, counter-clockwise.
322 @return: The survey (or robot) point, an instance of B{C{Clas}} or if
323 C{B{Clas} is None} an instance of B{C{point1}}'s (sub-)class.
325 @raise ResectionError: Near-coincident, -colinear or -concyclic points
326 or invalid B{C{alpha12}} or B{C{alpha23}} or
327 non-positive B{C{eps}}.
329 @raise TypeError: Invalid B{C{point1}}, B{C{point2}} or B{C{point3}}.
331 @see: I{Pierlot's} C function U{triangulationPierlot<http://www.Telecom.ULg.ac.BE/
332 triangulation/doc/total_8c_source.html>}, U{V. Pierlot, M. Van Droogenbroeck,
333 "A New Three Object Triangulation Algorithm for Mobile Robot Positioning"
334 <https://ORBi.ULiege.BE/bitstream/2268/157469/1/Pierlot2014ANewThree.pdf>},
335 U{Vincent Pierlot, Marc Van Droogenbroeck, "18 Triangulation Algorithms for 2D
336 Positioning (also known as the Resection Problem)"<http://www.Telecom.ULg.ac.BE/
337 triangulation>} and functions L{pygeodesy.pierlotx}, L{pygeodesy.cassini},
338 L{pygeodesy.collins5} and L{pygeodesy.tienstra7}.
339 '''
341 def _cot(s, c): # -eps < I{approximate} cotangent < eps
342 if eps > 0:
343 return c / (min(s, -eps) if s < 0 else max(s, eps))
344 raise ValueError(_SPACE_(_eps_, _not_, _positive_))
346 B1, B2, B3 = _B3(useZ, point1, point2, point3)
347 try:
348 x, y, z = _pierlot3(B1, B2, B3, alpha12, alpha23, useZ, _cot)
349 clas = Clas or point1.classof
350 return clas(x, y, z, **_xkwds(Clas_kwds, name=pierlot.__name__))
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).xyz
361 x3_, y3_, _ = B3.minus(B2).xyz
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 = Fsum(x3_ * x1_, cot31 * (y3_ * x1_),
373 y3_ * y1_, -cot31 * (x3_ * y1_))
374 if K:
375 cot12 = cot(s12, c12)
376 cot23 = cot(s23, c23)
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 = Fsum(x1_, cot12 * y1_, -x3_, cot23 * y3_)
389 # y12 - y23 = y1_ - cot12 * x1_ - y3_ - cot23 * x3_
390 Y12_23 = Fsum(y1_, -cot12 * x1_, -y3_, -cot23 * x3_)
392 # x31 - x23 = x3_ + x1_ + cot31 * (y3_ - y1_) - x3_ + cot23 * y3_
393 # = x1_ + cot31 * y3_ - cot31 * y1_ + cot23 * y3_
394 X31_23 = Fsum(x1_, -cot31 * y1_, cot31 * y3_, cot23 * y3_)
395 # y31 - y23 = y3_ + y1_ - cot31 * (x3_ - x1_) - y3_ - cot23 * x3_
396 # = y1_ - cot31 * x3_ + cot31 * x1_ - cot23 * x3_
397 Y31_23 = Fsum(y1_, cot31 * x1_, -cot31 * x3_, -cot23 * x3_)
399 # d = (x12 - x23) * (y23 - y31) + (x31 - x23) * (y12 - y23)
400 # = (x31 - x23) * (y12 - y23) - (x12 - x23) * (y12 - y23)
401 d = float(X31_23 * Y12_23 - X12_23 * Y31_23)
402 if isnear0(d):
403 raise ValueError(_or(_coincident_, _colinear_, _concyclic_))
405 x = (B2.x * d + K * Y12_23).fover(d)
406 y = (B2.y * d - K * X12_23).fover(d)
407 else:
408 x, y, _ = B2.xyz
409 return x, y, _zidw(x, y, useZ, B1, B2, B3)
412def _pierlotx3(a_z_Bs, useZ, cot, C):
413 '''(INTERNAL) Core of L{pierlotx}.
414 '''
415 (a12, z12, B1), \
416 (a23, z23, B2), \
417 (a31, z31, B3) = a_z_Bs
418 if z12 and not z23:
419 C(1)
420 elif z23 and not z31:
421 C(2)
422 a23, B1, B2, B3 = a31, B2, B3, B1
423 elif z31 and not z12:
424 C(3)
425 a23, B2, B3 = a12, B3, B2
426 else:
427 C(4)
428 return _pierlot3(B1, B2, B3, a12, a23, useZ, cot)
430 x1_, y1_, _ = B1.minus(B3).xyz
431 x2_, y2_, _ = B2.minus(B3).xyz
433 K = Fsum(_1_0, y1_ * x2_, -x1_ * y2_, _N_1_0) # 1-primed
434 if K:
435 cot23 = cot(*sincos2d(a23))
437 # x23 = x2_ + cot23 * y2_
438 # y23 = y2_ - cot23 * x2_
440 # x31 = x1_ + cot23 * y1_
441 # y31 = y1_ - cot23 * x1_
443 # x31 - x23 = x1_ + cot23 * y1_ - x2_ - cot23 * y2_
444 X31_23 = Fsum(x1_, cot23 * y1_, -x2_, -cot23 * y2_)
445 # y31 - y23 = y1_ - cot23 * x1_ - y2_ + cot23 * x2_
446 Y31_23 = Fsum(y1_, -cot23 * x1_, -y2_, cot23 * x2_)
448 # d = (x31 - x23) * (x2_ - x1_) - (y31 - y23) * (y1_ - y2_)
449 d = float(X31_23 * x2_ - X31_23 * x1_ +
450 Y31_23 * y2_ - Y31_23 * y1_)
451 if isnear0(d):
452 raise ValueError(_or(_coincident_, _colinear_, _concyclic_))
454 x = (B3.x * d - K * Y31_23).fover(d)
455 y = (B3.y * d + K * X31_23).fover(d)
456 else:
457 x, y, _ = B3.xyz
458 return x, y, _zidw(x, y, useZ, B1, B2, B3)
461def pierlotx(point1, point2, point3, alpha1, alpha2, alpha3, useZ=False,
462 Clas=None, **Clas_kwds):
463 '''3-Point resection using U{Pierlot<http://www.Telecom.ULg.ac.BE/publi/
464 publications/pierlot/Pierlot2014ANewThree>}'s method C{ToTal} with
465 I{exact} limits for the (pseudo-)singularities.
467 @arg point1: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
468 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
469 @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
470 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
471 @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
472 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
473 @arg alpha1: Angle at B{C{point1}} (C{degrees}, counter-clockwise).
474 @arg alpha2: Angle at B{C{point2}} (C{degrees}, counter-clockwise).
475 @arg alpha3: Angle at B{C{point3}} (C{degrees}, counter-clockwise).
476 @kwarg useZ: If C{True}, interpolate the survey point's Z component,
477 otherwise use C{z=INT0} (C{bool}).
478 @kwarg Clas: Optional class to return the survey point, if C{None} use
479 B{C{point1}}'s (sub-)class.
480 @kwarg Clas_kwds: Optional, additional keyword arguments for the survey
481 point instance.
483 @return: The survey (or robot) point, an instance of B{C{Clas}} or if
484 C{B{Clas} is None} an instance of B{C{point1}}'s (sub-)class.
486 @raise ResectionError: Near-coincident, -colinear or -concyclic points or
487 invalid B{C{alpha1}}, B{C{alpha2}} or B{C{alpha3}}.
489 @raise TypeError: Invalid B{C{point1}}, B{C{point2}} or B{C{point3}}.
491 @see: I{Pierlot's} C function U{triangulationPierlot2<http://www.Telecom.ULg.ac.BE/
492 triangulation/doc/total_8c_source.html>} and function L{pygeodesy.pierlot}.
493 '''
495 def _a_z_Bs(Bs, *alphas):
496 a3 = map(_umod_360, alphas) # 0 <= alphas < 360
497 (a1, a2, a3), (B1, B2, B3) = zip(*sorted(zip(a3, Bs)))
498 for a, d, B in ((a1, a2, B1), (a2, a3, B2), (a3, a1, B3)):
499 d -= a # a12 = a2 - a1, ...
500 z = isnear0(fabs(d) % _180_0)
501 yield d, z, B
503 def _cot(s, c): # I{exact} cotangent
504 try:
505 return (c / s) if c else (NEG0 if s < 0 else _0_0)
506 except ZeroDivisionError:
507 raise ValueError(_or(_coincident_, _colinear_))
509 Bs = _B3(useZ, point1, point2, point3)
510 try:
511 C = [0] # pseudo-global, passing the exception Case
512 x, y, z = _pierlotx3(_a_z_Bs(Bs, alpha1, alpha2, alpha3),
513 useZ, _cot, C.append)
514 clas = Clas or point1.classof
515 return clas(x, y, z, **_xkwds(Clas_kwds, name=pierlotx.__name__))
517 except (TypeError, ValueError) as x:
518 raise ResectionError(point1=point1, point2=point2, point3=point3, C=C.pop(),
519 alpha1=alpha1, alpha2=alpha2, alpha3=alpha3, cause=x)
522def snellius3(a, b, degC, alpha, beta):
523 '''Snellius' surveying using U{Snellius Pothenot<https://WikiPedia.org/wiki/Snellius–Pothenot_problem>}.
525 @arg a: Length of the triangle side between corners C{B} and C{C} and opposite of
526 triangle corner C{A} (C{scalar}, non-negative C{meter}, conventionally).
527 @arg b: Length of the triangle side between corners C{C} and C{A} and opposite of
528 triangle corner C{B} (C{scalar}, non-negative C{meter}, conventionally).
529 @arg degC: Angle at triangle corner C{C}, opposite triangle side C{c} (non-negative C{degrees}).
530 @arg alpha: Angle subtended by triangle side B{C{b}} (non-negative C{degrees}).
531 @arg beta: Angle subtended by triangle side B{C{a}} (non-negative C{degrees}).
533 @return: L{Survey3Tuple}C{(PA, PB, PC)} with distance from survey point C{P} to
534 each of the triangle corners C{A}, C{B} and C{C}, same units as triangle
535 sides B{C{a}}, B{C{b}} and B{C{c}}.
537 @raise TriangleError: Invalid B{C{a}}, B{C{b}} or B{C{degC}} or negative B{C{alpha}}
538 or B{C{beta}}.
540 @see: Function L{pygeodesy.wildberger3}.
541 '''
542 try:
543 a, b, degC, alpha, beta = t = map1(float, a, b, degC, alpha, beta)
544 if min(t) < 0:
545 raise ValueError(_negative_)
546 ra, rb, rC = map1(radians, alpha, beta, degC)
548 r = fsumf_(ra, rb, rC) * _0_5
549 k = PI - r
550 if min(k, r) < 0:
551 raise ValueError(_or(_coincident_, _colinear_))
553 sa, sb = sin(ra), sin(rb)
554 p = atan2(a * sa, b * sb)
555 sp, cp, sr, cr = sincos2_(PI_4 - p, r)
556 w = atan2(sp * sr, cp * cr)
557 x = k + w
558 y = k - w
560 s = fabs(sa)
561 if fabs(sb) > s:
562 pc = fabs(a * sin(y) / sb)
563 elif s:
564 pc = fabs(b * sin(x) / sa)
565 else:
566 raise ValueError(_or(_colinear_, _coincident_))
568 pa = _triSide(b, pc, fsumf_(PI, -ra, -x))
569 pb = _triSide(a, pc, fsumf_(PI, -rb, -y))
570 return Survey3Tuple(pa, pb, pc, name=snellius3.__name__)
572 except (TypeError, ValueError) as x:
573 raise TriangleError(a=a, b=b, degC=degC, alpha=alpha, beta=beta, cause=x)
576def tienstra7(pointA, pointB, pointC, alpha, beta=None, gamma=None,
577 useZ=False, Clas=None, **Clas_kwds):
578 '''3-Point resection using U{Tienstra<https://WikiPedia.org/wiki/Tienstra_formula>}'s formula.
580 @arg pointA: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or
581 C{Vector2Tuple} if C{B{useZ}=False}).
582 @arg pointB: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or
583 C{Vector2Tuple} if C{B{useZ}=False}).
584 @arg pointC: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or
585 C{Vector2Tuple} if C{B{useZ}=False}).
586 @arg alpha: Angle subtended by triangle side C{a} from B{C{pointB}} to B{C{pointC}}
587 (C{degrees}, non-negative).
588 @kwarg beta: Angle subtended by triangle side C{b} from B{C{pointA}} to B{C{pointC}}
589 (C{degrees}, non-negative) or C{None} if C{B{gamma} is not None}.
590 @kwarg gamma: Angle subtended by triangle side C{c} from B{C{pointA}} to B{C{pointB}}
591 (C{degrees}, non-negative) or C{None} if C{B{beta} is not None}.
592 @kwarg useZ: If C{True}, use and interpolate the Z component, otherwise force C{z=INT0}
593 (C{bool}).
594 @kwarg Clas: Optional class to return the survey point or C{None} for B{C{pointA}}'s
595 (sub-)class.
596 @kwarg Clas_kwds: Optional, additional keyword arguments for the survey point instance.
598 @note: Points B{C{pointA}}, B{C{pointB}} and B{C{pointC}} are ordered clockwise.
600 @return: L{Tienstra7Tuple}C{(pointP, A, B, C, a, b, c)} with survey C{pointP}, an
601 instance of B{C{Clas}} or if C{B{Clas} is None} an instance of B{C{pointA}}'s
602 (sub-)class, with triangle angles C{A} at B{C{pointA}}, C{B} at B{C{pointB}}
603 and C{C} at B{C{pointC}} in C{degrees} and with triangle sides C{a}, C{b} and
604 C{c} in C{meter}, conventionally.
606 @raise ResectionError: Near-coincident, -colinear or -concyclic points or sum of
607 B{C{alpha}}, B{C{beta}} and B{C{gamma}} not C{360} or negative
608 B{C{alpha}}, B{C{beta}} or B{C{gamma}}.
610 @raise TypeError: Invalid B{C{pointA}}, B{C{pointB}} or B{C{pointC}}.
612 @see: U{3-Point Resection Solver<http://MesaMike.org/geocache/GC1B0Q9/tienstra/>},
613 U{V. Pierlot, M. Van Droogenbroeck, "A New Three Object Triangulation..."
614 <http://www.Telecom.ULg.ac.BE/publi/publications/pierlot/Pierlot2014ANewThree/>},
615 U{18 Triangulation Algorithms...<http://www.Telecom.ULg.ac.BE/triangulation/>} and
616 functions L{pygeodesy.cassini}, L{pygeodesy.collins5} and L{pygeodesy.pierlot}.
617 '''
619 def _deg_ks(r, s, ks, N):
620 if isnear0(fsumf_(PI, r, -s)): # r + (PI2 - s) == PI
621 raise ValueError(Fmt.PARENSPACED(concyclic=N))
622 # k = 1 / (cot(r) - cot(s))
623 # = 1 / (cos(r) / sin(r) - cos(s) / sin(s))
624 # = 1 / (cos(r) * sin(s) - cos(s) * sin(r)) / (sin(r) * sin(s))
625 # = sin(r) * sin(s) / (cos(r) * sin(s) - cos(s) * sin(r))
626 sr, cr, ss, cs = sincos2_(r, s)
627 c = fsum1f_(cr * ss, -cs * sr)
628 if isnear0(c):
629 raise ValueError(Fmt.PARENSPACED(cotan=N))
630 ks.append(sr * ss / c)
631 return Degrees(degrees(r), name=N) # C degrees
633 A, B, C = _ABC3(useZ, pointA, pointB, pointC)
634 try:
635 sa, sb, sc = map1(radians, alpha, (beta or 0), (gamma or 0))
636 if beta is None:
637 if gamma is None:
638 raise ValueError(_and(Fmt.EQUAL(beta=beta), Fmt.EQUAL(gamma=gamma)))
639 sb = fsumf_(PI2, -sa, -sc)
640 elif gamma is None:
641 sc = fsumf_(PI2, -sa, -sb)
642 else: # subtended angles must add to 360 degrees
643 r = fsum1f_(sa, sb, sc)
644 if fabs(r - PI2) > EPS:
645 raise ValueError(Fmt.EQUAL(sum=degrees(r)))
646 if min(sa, sb, sc) < 0:
647 raise ValueError(_negative_)
649 # triangle sides
650 a = B.minus(C).length
651 b = A.minus(C).length
652 c = A.minus(B).length
654 ks = [] # 3 Ks and triangle angles
655 dA = _deg_ks(_triAngle(b, c, a), sa, ks, _A_)
656 dB = _deg_ks(_triAngle(a, c, b), sb, ks, _B_)
657 dC = _deg_ks(_triAngle(a, b, c), sc, ks, _C_)
659 k = fsum1(ks, floats=True)
660 if isnear0(k):
661 raise ValueError(Fmt.EQUAL(K=k))
662 x = Fdot(ks, A.x, B.x, C.x).fover(k)
663 y = Fdot(ks, A.y, B.y, C.y).fover(k)
664 z = _zidw(x, y, useZ, A, B, C)
666 n = tienstra7.__name__
667 clas = Clas or pointA.classof
668 P = clas(x, y, z, **_xkwds(Clas_kwds, name=n))
669 return Tienstra7Tuple(P, dA, dB, dC, a, b, c, name=n)
671 except (TypeError, ValueError) as x:
672 raise ResectionError(pointA=pointA, pointB=pointB, pointC=pointC,
673 alpha=alpha, beta=beta, gamma=gamma, cause=x)
676def triAngle(a, b, c):
677 '''Compute one angle of a triangle.
679 @arg a: Adjacent triangle side length (C{scalar}, non-negative
680 C{meter}, conventionally).
681 @arg b: Adjacent triangle side length (C{scalar}, non-negative
682 C{meter}, conventionally).
683 @arg c: Opposite triangle side length (C{scalar}, non-negative
684 C{meter}, conventionally).
686 @return: Angle in C{radians} at triangle corner C{C}, opposite
687 triangle side B{C{c}}.
689 @raise TriangleError: Invalid or negative B{C{a}}, B{C{b}} or B{C{c}}.
691 @see: Functions L{pygeodesy.triAngle4} and L{pygeodesy.triSide}.
692 '''
693 try:
694 return _triAngle(a, b, c)
695 except (TypeError, ValueError) as x:
696 raise TriangleError(a=a, b=b, c=c, cause=x)
699def _triAngle(a, b, c):
700 # (INTERNAL) To allow callers to embellish errors
701 a, b, c = map1(float, a, b, c)
702 if a < b:
703 a, b = b, a
704 if b < 0 or c < 0:
705 raise ValueError(_negative_)
706 if a < EPS0:
707 raise ValueError(_coincident_)
708 b_a = b / a
709 if b_a < EPS0:
710 raise ValueError(_coincident_)
711 t = fsumf_(_1_0, b_a**2, -(c / a)**2) / (b_a * _2_0)
712 return acos1(t)
715def triAngle4(a, b, c):
716 '''Compute the angles of a triangle.
718 @arg a: Length of the triangle side opposite of triangle corner C{A}
719 (C{scalar}, non-negative C{meter}, conventionally).
720 @arg b: Length of the triangle side opposite of triangle corner C{B}
721 (C{scalar}, non-negative C{meter}, conventionally).
722 @arg c: Length of the triangle side opposite of triangle corner C{C}
723 (C{scalar}, non-negative C{meter}, conventionally).
725 @return: L{TriAngle4Tuple}C{(radA, radB, radC, rIn)} with angles C{radA},
726 C{radB} and C{radC} at triangle corners C{A}, C{B} and C{C}, all
727 in C{radians} and the C{InCircle} radius C{rIn} aka C{inradius},
728 same units as triangle sides B{C{a}}, B{C{b}} and B{C{c}}.
730 @raise TriangleError: Invalid or negative B{C{a}}, B{C{b}} or B{C{c}}.
732 @see: Function L{pygeodesy.triAngle}.
733 '''
734 try:
735 a, b, c = map1(float, a, b, c)
736 ab = a < b
737 if ab:
738 a, b = b, a
739 bc = b < c
740 if bc:
741 b, c = c, b
743 if c > EPS0: # c = min(a, b, c)
744 s = fsumf_(a, b, c) * _0_5
745 if s < EPS0:
746 raise ValueError(_negative_)
747 sa, sb, sc = (s - a), (s - b), (s - c)
748 r = sa * sb * sc / s
749 if r < EPS02:
750 raise ValueError(_coincident_)
751 r = sqrt(r)
752 rA = atan2(r, sa) * _2_0
753 rB = atan2(r, sb) * _2_0
754 rC = fsumf_(PI, -rA, -rB)
755 if min(rA, rB, rC) < 0:
756 raise ValueError(_colinear_)
757 elif c < 0:
758 raise ValueError(_negative_)
759 else: # 0 <= c <= EPS0
760 rA = rB = PI_2
761 rC = r = _0_0
763 if bc:
764 rB, rC = rC, rB
765 if ab:
766 rA, rB = rB, rA
767 return TriAngle4Tuple(rA, rB, rC, r, name=triAngle4.__name__)
769 except (TypeError, ValueError) as x:
770 raise TriangleError(a=a, b=b, c=c, cause=x)
773def triSide(a, b, radC):
774 '''Compute one side of a triangle.
776 @arg a: Adjacent triangle side length (C{scalar},
777 non-negative C{meter}, conventionally).
778 @arg b: Adjacent triangle side length (C{scalar},
779 non-negative C{meter}, conventionally).
780 @arg radC: Angle included by sides B{C{a}} and B{C{b}},
781 opposite triangle side C{c} (C{radians}).
783 @return: Length of triangle side C{c}, opposite triangle
784 corner C{C} and angle B{C{radC}}, same units as
785 B{C{a}} and B{C{b}}.
787 @raise TriangleError: Invalid B{C{a}}, B{C{b}} or B{C{radC}}.
789 @see: Functions L{pygeodesy.sqrt_a}, L{pygeodesy.triAngle},
790 L{pygeodesy.triSide2} and L{pygeodesy.triSide4}.
791 '''
792 try:
793 return _triSide(a, b, radC)
794 except (TypeError, ValueError) as x:
795 raise TriangleError(a=a, b=b, radC=radC, cause=x)
798def _triSide(a, b, radC):
799 # (INTERNAL) To allow callers to embellish errors
800 a, b, r = t = map1(float, a, b, radC)
801 if min(t) < 0:
802 raise ValueError(_negative_)
804 if a < b:
805 a, b = b, a
806 if a > EPS0:
807 ba = b / a
808 c2 = fsumf_(_1_0, ba**2, _N_2_0 * ba * cos(r))
809 if c2 > EPS02:
810 return a * sqrt(c2)
811 elif c2 < 0:
812 raise ValueError(_invalid_)
813 return hypot(a, b)
816def triSide2(b, c, radB):
817 '''Compute one side and the opposite angle of a triangle.
819 @arg b: Adjacent triangle side length (C{scalar},
820 non-negative C{meter}, conventionally).
821 @arg c: Adjacent triangle side length (C{scalar},
822 non-negative C{meter}, conventionally).
823 @arg radB: Angle included by sides B{C{a}} and B{C{c}},
824 opposite triangle side C{b} (C{radians}).
826 @return: L{TriSide2Tuple}C{(a, radA)} with triangle angle
827 C{radA} in C{radians} and length of the opposite
828 triangle side C{a}, same units as B{C{b}} and B{C{c}}.
830 @raise TriangleError: Invalid B{C{b}} or B{C{c}} or either
831 B{C{b}} or B{C{radB}} near zero.
833 @see: Functions L{pygeodesy.sqrt_a}, L{pygeodesy.triSide}
834 and L{pygeodesy.triSide4}.
835 '''
836 try:
837 return _triSide2(b, c, radB)
838 except (TypeError, ValueError) as x:
839 raise TriangleError(b=b, c=c, radB=radB, cause=x)
842def _triSide2(b, c, radB):
843 # (INTERNAL) To allow callers to embellish errors
844 b, c, rB = map1(float, b, c, radB)
845 if min(b, c, rB) < 0:
846 raise ValueError(_negative_)
847 sB, cB = sincos2(rB)
848 if isnear0(sB):
849 if not isnear0(b):
850 raise ValueError(_invalid_)
851 if cB < 0:
852 a, rA = (b + c), PI
853 else:
854 a, rA = fabs(b - c), _0_0
855 elif isnear0(b):
856 raise ValueError(_invalid_)
857 else:
858 rA = fsumf_(PI, -rB, -asin1(c * sB / b))
859 a = sin(rA) * b / sB
860 return TriSide2Tuple(a, rA, name=triSide2.__name__)
863def triSide4(radA, radB, c):
864 '''Compute two sides and the height of a triangle.
866 @arg radA: Angle at triangle corner C{A}, opposite triangle side C{a}
867 (non-negative C{radians}).
868 @arg radB: Angle at triangle corner C{B}, opposite triangle side C{b}
869 (non-negative C{radians}).
870 @arg c: Length of triangle side between triangle corners C{A} and C{B},
871 (C{scalar}, non-negative C{meter}, conventionally).
873 @return: L{TriSide4Tuple}C{(a, b, radC, d)} with triangle sides C{a} and
874 C{b} and triangle height C{d} perpendicular to triangle side
875 B{C{c}}, all in the same units as B{C{c}} and interior angle
876 C{radC} in C{radians} at triangle corner C{C}, opposite
877 triangle side B{C{c}}.
879 @raise TriangleError: Invalid or negative B{C{radA}}, B{C{radB}} or B{C{c}}.
881 @see: U{Triangulation, Surveying<https://WikiPedia.org/wiki/Triangulation_(surveying)>}
882 and functions L{pygeodesy.sqrt_a}, L{pygeodesy.triSide} and L{pygeodesy.triSide2}.
883 '''
884 try:
885 rA, rB, c = map1(float, radA, radB, c)
886 rC = fsumf_(PI, -rA, -rB)
887 if min(rC, rA, rB, c) < 0:
888 raise ValueError(_negative_)
889 sa, ca, sb, cb = sincos2_(rA, rB)
890 sc = fsum1f_(sa * cb, sb * ca)
891 if sc < EPS0 or min(sa, sb) < 0:
892 raise ValueError(_invalid_)
893 sc = c / sc
894 return TriSide4Tuple((sa * sc), (sb * sc), rC, (sa * sb * sc),
895 name=triSide4.__name__)
897 except (TypeError, ValueError) as x:
898 raise TriangleError(radA=radA, radB=radB, c=c, cause=x)
901def wildberger3(a, b, c, alpha, beta, R3=min):
902 '''Snellius' surveying using U{Rational Trigonometry
903 <https://WikiPedia.org/wiki/Snellius–Pothenot_problem>}.
905 @arg a: Length of the triangle side between corners C{B} and C{C} and opposite of
906 triangle corner C{A} (C{scalar}, non-negative C{meter}, conventionally).
907 @arg b: Length of the triangle side between corners C{C} and C{A} and opposite of
908 triangle corner C{B} (C{scalar}, non-negative C{meter}, conventionally).
909 @arg c: Length of the triangle side between corners C{A} and C{B} and opposite of
910 triangle corner C{C} (C{scalar}, non-negative C{meter}, conventionally).
911 @arg alpha: Angle subtended by triangle side B{C{b}} (C{degrees}, non-negative).
912 @arg beta: Angle subtended by triangle side B{C{a}} (C{degrees}, non-negative).
913 @kwarg R3: Callable to determine C{R3} from C{(R3 - C)**2 = D}, typically standard
914 Python function C{min} or C{max}, invoked with 2 arguments.
916 @return: L{Survey3Tuple}C{(PA, PB, PC)} with distance from survey point C{P} to
917 each of the triangle corners C{A}, C{B} and C{C}, same units as B{C{a}},
918 B{C{b}} and B{C{c}}.
920 @raise TriangleError: Invalid B{C{a}}, B{C{b}} or B{C{c}} or negative B{C{alpha}} or
921 B{C{beta}} or B{C{R3}} not C{callable}.
923 @see: U{Wildberger, Norman J.<https://Math.Sc.Chula.ac.TH/cjm/content/
924 survey-article-greek-geometry-rational-trigonometry-and-snellius-–-pothenot-surveying>},
925 U{Devine Proportions, page 252<http://www.MS.LT/derlius/WildbergerDivineProportions.pdf>}
926 and function L{pygeodesy.snellius3}.
927 '''
928 def _s(x):
929 return sin(x)**2
931 def _vpa(r1, r3, q2, q3, s3):
932 r = r1 * r3 * _4_0
933 n = (r - Fsum(r1, r3, -q2).fpow(2)).fover(s3)
934 if n < 0 or isnear0(r):
935 raise ValueError(_coincident_)
936 return sqrt((n / r) * q3) if n else _0_0
938 try:
939 a, b, c, da, db = t = map1(float, a, b, c, alpha, beta)
940 if min(t) < 0:
941 raise ValueError(_negative_)
943 ra, rb = radians(da), radians(db)
944 s1, s2, s3 = s = map1(_s, rb, ra, ra + rb) # rb, ra!
945 if min(s) < EPS02:
946 raise ValueError(_or(_coincident_, _colinear_))
948 q1, q2, q3 = q = a**2, b**2, c**2
949 if min(q) < EPS02:
950 raise ValueError(_coincident_)
952 r1 = s2 * q3 / s3 # s2!
953 r2 = s1 * q3 / s3 # s1!
954 Qs = Fsum(*q) # == hypot2_(a, b, c)
955 Ss = Fsum(*s) # == fsum1(s, floats=True)
956 s += Qs * _0_5, # tuple!
957 C0 = Fdot(s, q1, q2, q3, -Ss)
958 r3 = C0.fover(-s3)
959 d0 = Qs.fpow(2).fsub_(hypot2_(*q) * _2_0).fmul(s1 * s2).fover(s3)
960 if d0 > EPS02: # > c0
961 d0 = sqrt(d0)
962 if not callable(R3):
963 raise ValueError(_R3__ + _not_(callable.__name__))
964 r3 = R3(float(C0 + d0), float(C0 - d0)) # XXX min or max
965 elif d0 < 0:
966 raise ValueError(_negative_)
968 pa = _vpa(r1, r3, q2, q3, s3)
969 pb = _vpa(r2, r3, q1, q3, s3)
970 pc = favg(_triSide2(b, pa, ra).a,
971 _triSide2(a, pb, rb).a)
972 return Survey3Tuple(pa, pb, pc, name=wildberger3.__name__)
974 except (TypeError, ValueError) as x:
975 raise TriangleError(a=a, b=b, c=c, alpha=alpha, beta=beta, R3=R3, cause=x)
978def _zidw(x, y, useZ, *ABC):
979 if useZ: # interpolate z or coplanar with A, B and C?
980 t = tuple(_.z for _ in ABC)
981 v = Vector3d(x, y, fmean(t))
982 z = fidw(t, (v.minus(T).length for T in ABC))
983 else:
984 z = INT0
985 return z
987# **) MIT License
988#
989# Copyright (C) 2016-2023 -- mrJean1 at Gmail -- All Rights Reserved.
990#
991# Permission is hereby granted, free of charge, to any person obtaining a
992# copy of this software and associated documentation files (the "Software"),
993# to deal in the Software without restriction, including without limitation
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