Coverage for pygeodesy/resections.py: 99%
312 statements
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« prev ^ index » next coverage.py v7.2.2, created at 2023-04-05 15:46 -0400
2# -*- coding: utf-8 -*-
4u'''3-Point resection functions L{cassini}, L{collins5}, L{pierlot} and L{tienstra7},
5survey functions L{snellius3} and L{wildberger3} and triangle functions L{triAngle},
6L{triAngle4}, L{triSide}, L{triSide2} and L{triSide4}.
8@note: Function L{pierlot} is transcoded with permission from U{triangulationPierlot
9 <http://www.Telecom.ULg.ac.Be/triangulation/doc/total_8c.html>} and U{Pierlot
10 <http://www.Telecom.ULg.ac.Be/publi/publications/pierlot/Pierlot2014ANewThree>}.
11'''
13from pygeodesy.basics import map1, _ALL_LAZY
14from pygeodesy.constants import EPS, EPS0, EPS02, INT0, PI, PI2, PI_2, PI_4, isnear0, \
15 _0_0, _0_5, _1_0, _N_1_0, _2_0, _N_2_0, _4_0, _360_0
16from pygeodesy.errors import _and, _or, TriangleError, _ValueError, _xkwds
17from pygeodesy.fmath import favg, Fdot, fidw, fmean, hypot, hypot2_
18from pygeodesy.fsums import Fsum, fsum_, fsum1, fsum1_
19from pygeodesy.interns import _a_, _A_, _b_, _B_, _c_, _C_, _coincident_, _colinear_, \
20 _d_, _invalid_, _negative_, _not_, _rIn_, _SPACE_
21# from pygeodesy.lazily import _ALL_LAZY # from .basics
22from pygeodesy.named import Fmt, _NamedTuple, _Pass
23# from pygeodesy.streprs import Fmt # from .named
24from pygeodesy.units import Degrees, Distance, Radians
25from pygeodesy.utily import acos1, asin1, sincos2, sincos2_, sincos2d, sincos2d_
26from pygeodesy.vector3d import _otherV3d, Vector3d
28from math import cos, atan2, degrees, fabs, radians, sin, sqrt
30__all__ = _ALL_LAZY.resections
31__version__ = '23.04.02'
33_concyclic_ = 'concyclic'
34_PA_ = 'PA'
35_PB_ = 'PB'
36_PC_ = 'PC'
37_pointH_ = 'pointH'
38_pointP_ = 'pointP'
39_R3__ = 'R3 '
40_radA_ = 'radA'
41_radB_ = 'radB'
42_radC_ = 'radC'
45class Collins5Tuple(_NamedTuple):
46 '''5-Tuple C{(pointP, pointH, a, b, c)} with survey C{pointP}, auxiliary
47 C{pointH}, each an instance of B{C{pointA}}'s (sub-)class and triangle
48 sides C{a}, C{b} and C{c} in C{meter}, conventionally.
49 '''
50 _Names_ = (_pointP_, _pointH_, _a_, _b_, _c_)
51 _Units_ = (_Pass, _Pass, Distance, Distance, Distance)
54class ResectionError(_ValueError):
55 '''Error raised for issues in L{pygeodesy.resections}.
56 '''
57 pass
60class Survey3Tuple(_NamedTuple):
61 '''3-Tuple C{(PA, PB, PC)} with distance from survey point C{P} to each of
62 the triangle corners C{A}, C{B} and C{C} in C{meter}, conventionally.
63 '''
64 _Names_ = (_PA_, _PB_, _PC_)
65 _Units_ = ( Distance, Distance, Distance)
68class Tienstra7Tuple(_NamedTuple):
69 '''7-Tuple C{(pointP, A, B, C, a, b, c)} with survey C{pointP}, interior
70 triangle angles C{A}, C{B} and C{C} in C{degrees} and triangle sides
71 C{a}, C{b} and C{c} in C{meter}, conventionally.
72 '''
73 _Names_ = (_pointP_, _A_, _B_, _C_, _a_, _b_, _c_)
74 _Units_ = (_Pass, Degrees, Degrees, Degrees, Distance, Distance, Distance)
77class TriAngle4Tuple(_NamedTuple):
78 '''4-Tuple C{(radA, radB, radC, rIn)} with the interior angles at triangle
79 corners C{A}, C{B} and C{C} in C{radians} and the C{InCircle} radius
80 C{rIn} aka C{inradius} in C{meter}, conventionally.
81 '''
82 _Names_ = (_radA_, _radB_, _radC_, _rIn_)
83 _Units_ = ( Radians, Radians, Radians, Distance)
86class TriSide2Tuple(_NamedTuple):
87 '''2-Tuple C{(a, radA)} with triangle side C{a} in C{meter}, conventionally
88 and angle C{radA} at the opposite triangle corner in C{radians}.
89 '''
90 _Names_ = (_a_, _radA_)
91 _Units_ = ( Distance, Radians)
94class TriSide4Tuple(_NamedTuple):
95 '''4-Tuple C{(a, b, radC, d)} with interior angle C{radC} at triangle corner
96 C{C} in C{radians}with the length of triangle sides C{a} and C{b} and
97 with triangle height C{d} perpendicular to triangle side C{c}, in the
98 same units as triangle sides C{a} and C{b}.
99 '''
100 _Names_ = (_a_, _b_, _radC_, _d_)
101 _Units_ = ( Distance, Distance, Radians, Distance)
104def cassini(pointA, pointB, pointC, alpha, beta, useZ=False, Clas=None, **Clas_kwds):
105 '''3-Point resection using U{Cassini<https://NL.WikiPedia.org/wiki/Achterwaartse_insnijding>}'s method.
107 @arg pointA: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
108 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
109 @arg pointB: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
110 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
111 @arg pointC: Center point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
112 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
113 @arg alpha: Angle subtended by triangle side B{C{pointA}} to B{C{pointC}}
114 (C{degrees}, non-negative).
115 @arg beta: Angle subtended by triangle side B{C{pointB}} to B{C{pointC}}
116 (C{degrees}, non-negative).
117 @kwarg useZ: If C{True}, use and interpolate the Z component, otherwise
118 force C{z=INT0} (C{bool}).
119 @kwarg Clas: Optional class to return the survey point or C{None} for
120 B{C{pointA}}'s (sub-)class.
121 @kwarg Clas_kwds: Optional additional keyword argument for the survey
122 point instance.
124 @note: Typically, B{C{pointC}} is between B{C{pointA}} and B{C{pointB}}.
126 @return: The survey point, an instance of B{C{Clas}} or if C{B{Clas} is
127 None} of B{C{pointA}}'s (sub-)class.
129 @raise ResectionError: Near-coincident, -colinear or -concyclic points
130 or negative or invalid B{C{alpha}} or B{C{beta}}.
132 @raise TypeError: Invalid B{C{pointA}}, B{C{pointB}} or B{C{pointM}}.
134 @see: U{Three Point Resection Problem<https://Dokumen.tips/documents/
135 three-point-resection-problem-introduction-kaestner-burkhardt-method.html>}
136 and functions L{pygeodesy.collins5}, L{pygeodesy.pierlot} and
137 L{pygeodesy.tienstra7}.
138 '''
140 def _H(A, C, sa):
141 s, c = sincos2d(sa)
142 if isnear0(s):
143 raise ValueError(_or(_coincident_, _colinear_))
144 t = s, c, c
145 x = Fdot(t, A.x, C.y, -A.y).fover(s)
146 y = Fdot(t, A.y, -C.x, A.x).fover(s)
147 return Vector3d(x, y, 0)
149 A = _otherV3d(useZ=useZ, pointA=pointA)
150 B = _otherV3d(useZ=useZ, pointB=pointB)
151 C = _otherV3d(useZ=useZ, pointC=pointC)
153 try:
154 sa, sb = map1(float, alpha, beta)
155 if min(sa, sb) < 0:
156 raise ValueError(_negative_)
157 if fsum_(_360_0, -sa, -sb) < EPS0:
158 raise ValueError()
160 H1 = _H(A, C, sa)
161 H2 = _H(B, C, -sb)
163 x = H1.x - H2.x
164 y = H1.y - H2.y
165 # x, y, _ = H1.minus(H2).xyz
166 if isnear0(x) or isnear0(y):
167 raise ValueError(_SPACE_(_concyclic_, (x, y)))
169 m = y / x
170 n = x / y
171 N = n + m
172 if isnear0(N):
173 raise ValueError(_SPACE_(_concyclic_, (m, n, N)))
175 t = n, m, _1_0, _N_1_0
176 x = Fdot(t, C.x, H1.x, C.y, H1.y).fover(N)
177 y = Fdot(t, H1.y, C.y, C.x, H1.x).fover(N)
178 z = _zidw(A, B, C, x, y) if useZ else INT0
180 clas = Clas or pointA.classof
181 return clas(x, y, z, **_xkwds(Clas_kwds, name=cassini.__name__))
183 except (TypeError, ValueError) as x:
184 raise ResectionError(pointA=pointA, pointB=pointB, pointC=pointC,
185 alpha=alpha, beta=beta, cause=x)
188def collins5(pointA, pointB, pointC, alpha, beta, useZ=False, Clas=None, **Clas_kwds):
189 '''3-Point resection using U{Collins<https://Dokumen.tips/documents/
190 three-point-resection-problem-introduction-kaestner-burkhardt-method.html>}' method.
192 @arg pointA: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
193 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
194 @arg pointB: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
195 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
196 @arg pointC: Center point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
197 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
198 @arg alpha: Angle subtended by triangle side C{b} from B{C{pointA}} to
199 B{C{pointC}} (C{degrees}, non-negative).
200 @arg beta: Angle subtended by triangle side C{a} from B{C{pointB}} to
201 B{C{pointC}} (C{degrees}, non-negative).
202 @kwarg useZ: If C{True}, use and interpolate the Z component, otherwise
203 force C{z=INT0} (C{bool}).
204 @kwarg Clas: Optional class to return the survey and auxiliary point
205 or C{None} for B{C{pointA}}'s (sub-)class.
206 @kwarg Clas_kwds: Optional additional keyword argument for the survey
207 and auxiliary point instance.
209 @note: Typically, B{C{pointC}} is between B{C{pointA}} and B{C{pointB}}.
211 @return: L{Collins5Tuple}C{(pointP, pointH, a, b, c)} with survey C{pointP},
212 auxiliary C{pointH}, each an instance of B{C{Clas}} or if C{B{Clas}
213 is None} of B{C{pointA}}'s (sub-)class and triangle sides C{a},
214 C{b} and C{c} in C{meter}, conventionally.
216 @raise ResectionError: Near-coincident, -colinear or -concyclic points
217 or negative or invalid B{C{alpha}} or B{C{beta}}.
219 @raise TypeError: Invalid B{C{pointA}}, B{C{pointB}} or B{C{pointM}}.
221 @see: U{Collins' methode<https://NL.WikiPedia.org/wiki/Achterwaartse_insnijding>}
222 and functions L{pygeodesy.cassini}, L{pygeodesy.pierlot} and
223 L{pygeodesy.tienstra7}.
224 '''
226 def _azi_len2(A, B, pi2):
227 v = B.minus(A)
228 r = atan2(v.x, v.y)
229 if pi2 and r < 0:
230 r += pi2
231 return r, v.length
233 def _cV3(d, r, A, B, C, useZ, V3, **kwds):
234 s, c = sincos2(r)
235 x = A.x + d * s
236 y = A.y + d * c
237 z = _zidw(A, B, C, x, y) if useZ else INT0
238 return V3(x, y, z, **kwds)
240 A = _otherV3d(useZ=useZ, pointA=pointA)
241 B = _otherV3d(useZ=useZ, pointB=pointB)
242 C = _otherV3d(useZ=useZ, pointC=pointC)
244 try:
245 ra, rb = radians(alpha), radians(beta)
246 if min(ra, rb) < 0:
247 raise ValueError(_negative_)
249 sra, srH = sin(ra), sin(ra + rb - PI) # rH = PI - ((PI - ra) + (PI - rb))
250 if isnear0(sra) or isnear0(srH):
251 raise ValueError(_or(_coincident_, _colinear_, _concyclic_))
253 clas = Clas or pointA.classof
254 kwds = _xkwds(Clas_kwds, name=collins5.__name__)
256# za, a = _azi_len2(C, B, PI2)
257 zb, b = _azi_len2(C, A, PI2)
258 zc, c = _azi_len2(A, B, 0)
260# d = c * sin(PI - rb) / srH # B.minus(H).length
261 d = c * sin(PI - ra) / srH # A.minus(H).length
262 r = zc + PI - rb # zh = zc + (PI - rb)
263 H = _cV3(d, r, A, B, C, useZ, Vector3d)
265 zh, _ = _azi_len2(C, H, PI2)
267# d = a * sin(za - zh) / sin(rb) # B.minus(P).length
268 d = b * sin(zb - zh) / sra # A.minus(P).length
269 r = zh - ra # zb - PI + (PI - ra - (zb - zh))
270 P = _cV3(d, r, A, B, C, useZ, clas, **kwds)
272 H = clas(H.x, H.y, H.z, **kwds)
273 a = B.minus(C).length
275 return Collins5Tuple(P, H, a, b, c, name=collins5.__name__)
277 except (TypeError, ValueError) as x:
278 raise ResectionError(pointA=pointA, pointB=pointB, pointC=pointC,
279 alpha=alpha, beta=beta, cause=x)
282def pierlot(point1, point2, point3, alpha12, alpha23, useZ=False, Clas=None, **Clas_kwds):
283 '''3-Point resection using U{Pierlot<http://www.Telecom.ULg.ac.Be/publi/publications/
284 pierlot/Pierlot2014ANewThree>}'s method C{ToTal}.
286 @arg point1: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
287 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
288 @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
289 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
290 @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
291 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
292 @arg alpha12: Angle subtended from B{C{point1}} to B{C{point2}} (C{degrees}).
293 @arg alpha23: Angle subtended from B{C{point2}} to B{C{point3}} (C{degrees}).
294 @kwarg useZ: If C{True}, interpolate the Z component, otherwise use C{z=INT0}
295 (C{bool}).
296 @kwarg Clas: Optional class to return the survey point or C{None} for
297 B{C{point1}}'s (sub-)class.
298 @kwarg Clas_kwds: Optional additional keyword arguments for the survey
299 point instance.
301 @note: Points B{C{point1}}, B{C{point2}} and B{C{point3}} are ordered
302 counter-clockwise, typically.
304 @return: The survey (or robot) point, an instance of B{C{Clas}} or if
305 C{B{Clas} is None} of B{C{point1}}'s (sub-)class.
307 @raise ResectionError: Near-coincident, -colinear or -concyclic points
308 or invalid B{C{alpha12}} or B{C{alpha23}}.
310 @raise TypeError: Invalid B{C{point1}}, B{C{point2}} or B{C{point3}}.
312 @see: U{V. Pierlot, M. Van Droogenbroeck, "A New Three Object Triangulation
313 Algorithm for Mobile Robot Positioning"<https://ORBi.ULiege.Be/
314 bitstream/2268/157469/1/Pierlot2014ANewThree.pdf>}, U{Vincent Pierlot,
315 Marc Van Droogenbroeck, "18 Triangulation Algorithms for 2D Positioning
316 (also known as the Resection Problem)"<http://www.Telecom.ULg.ac.Be/
317 triangulation>} and functions L{pygeodesy.cassini}, L{pygeodesy.collins5}
318 and L{pygeodesy.tienstra7}.
319 '''
320 B1 = _otherV3d(useZ=useZ, point1=point1)
321 B2 = _otherV3d(useZ=useZ, point2=point2)
322 B3 = _otherV3d(useZ=useZ, point3=point3)
324 try: # (INTERNAL) Raises error for (pseudo-)singularities
325 s12, c12, s23, c23 = sincos2d_(alpha12, alpha23)
326 if isnear0(s12) or isnear0(s23):
327 raise ValueError(_or(_coincident_, _colinear_))
328 cot12 = c12 / s12
329 cot23 = c23 / s23
330# cot31 = (1 - cot12 * cot23) / (cot12 + cot32)
331 d = fsum1_(c12 * s23, s12 * c23)
332 if isnear0(d):
333 raise ValueError(_or(_colinear_, _coincident_))
334 cot31 = Fsum(_1_0, s12 * s23, -c12 * c23, _N_1_0).fover(d)
336 x1_, y1_, _ = B1.minus(B2).xyz
337 x3_, y3_, _ = B3.minus(B2).xyz
339# x23 = x3_ - cot23 * y3_
340# y23 = y3_ + cot23 * x3_
342 X12_23 = Fsum(x1_, cot12 * y1_, -x3_, cot23 * y3_)
343 Y12_23 = Fsum(y1_, -cot12 * x1_, -y3_, -cot23 * x3_)
345 X31_23 = Fsum(x1_, -cot31 * y1_, cot31 * y3_, cot23 * y3_)
346 Y31_23 = Fsum(y1_, cot31 * x1_, -cot31 * x3_, -cot23 * x3_)
348 d = float(X31_23 * Y12_23 - X12_23 * Y31_23)
349 if isnear0(d):
350 raise ValueError(_or(_coincident_, _colinear_, _concyclic_))
351 K = Fsum(x3_ * x1_, cot31 * (y3_ * x1_),
352 y3_ * y1_, -cot31 * (x3_ * y1_))
354 x = (B2.x * d + K * Y12_23).fover(d)
355 y = (B2.y * d - K * X12_23).fover(d)
356 z = _zidw(B1, B2, B3, x, y) if useZ else INT0
358 clas = Clas or point1.classof
359 return clas(x, y, z, **_xkwds(Clas_kwds, name=pierlot.__name__))
361 except (TypeError, ValueError) as x:
362 raise ResectionError(point1=point1, point2=point2, point3=point3,
363 alpha12=alpha12, alpha23=alpha23, cause=x)
366def snellius3(a, b, degC, alpha, beta):
367 '''Snellius' surveying using U{Snellius Pothenot<https://WikiPedia.org/wiki/Snellius–Pothenot_problem>}.
369 @arg a: Length of the triangle side between corners C{B} and C{C} and opposite of
370 triangle corner C{A} (C{scalar}, non-negative C{meter}, conventionally).
371 @arg b: Length of the triangle side between corners C{C} and C{A} and opposite of
372 triangle corner C{B} (C{scalar}, non-negative C{meter}, conventionally).
373 @arg degC: Angle at triangle corner C{C}, opposite triangle side C{c} (non-negative C{degrees}).
374 @arg alpha: Angle subtended by triangle side B{C{b}} (non-negative C{degrees}).
375 @arg beta: Angle subtended by triangle side B{C{a}} (non-negative C{degrees}).
377 @return: L{Survey3Tuple}C{(PA, PB, PC)} with distance from survey point C{P} to
378 each of the triangle corners C{A}, C{B} and C{C}, same units as triangle
379 sides B{C{a}}, B{C{b}} and B{C{c}}.
381 @raise TriangleError: Invalid B{C{a}}, B{C{b}} or B{C{degC}} or negative B{C{alpha}}
382 or B{C{beta}}.
384 @see: Function L{pygeodesy.wildberger3}.
385 '''
386 try:
387 a, b, degC, alpha, beta = map1(float, a, b, degC, alpha, beta)
388 ra, rb, rC = map1(radians, alpha, beta, degC)
389 if min(ra, rb, rC, a, b) < 0:
390 raise ValueError(_negative_)
392 r = fsum_(ra, rb, rC) * _0_5
393 k = PI - r
394 if min(k, r) < 0:
395 raise ValueError(_or(_coincident_, _colinear_))
397 sa, sb = sin(ra), sin(rb)
398 p = atan2(a * sa, b * sb)
399 sp, cp, sr, cr = sincos2_(PI_4 - p, r)
400 w = atan2(sp * sr, cp * cr)
401 x = k + w
402 y = k - w
404 s = fabs(sa)
405 if fabs(sb) > s:
406 pc = fabs(a * sin(y) / sb)
407 elif s:
408 pc = fabs(b * sin(x) / sa)
409 else:
410 raise ValueError(_or(_colinear_, _coincident_))
412 pa = _triSide(b, pc, fsum_(PI, -ra, -x))
413 pb = _triSide(a, pc, fsum_(PI, -rb, -y))
414 return Survey3Tuple(pa, pb, pc, name=snellius3.__name__)
416 except (TypeError, ValueError) as x:
417 raise TriangleError(a=a, b=b, degC=degC, alpha=alpha, beta=beta, cause=x)
420def tienstra7(pointA, pointB, pointC, alpha, beta=None, gamma=None,
421 useZ=False, Clas=None, **Clas_kwds):
422 '''3-Point resection using U{Tienstra<https://WikiPedia.org/wiki/Tienstra_formula>}'s formula.
424 @arg pointA: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or
425 C{Vector2Tuple} if C{B{useZ}=False}).
426 @arg pointB: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or
427 C{Vector2Tuple} if C{B{useZ}=False}).
428 @arg pointC: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or
429 C{Vector2Tuple} if C{B{useZ}=False}).
430 @arg alpha: Angle subtended by triangle side C{a} from B{C{pointB}} to B{C{pointC}}
431 (C{degrees}, non-negative).
432 @kwarg beta: Angle subtended by triangle side C{b} from B{C{pointA}} to B{C{pointC}}
433 (C{degrees}, non-negative) or C{None} if C{B{gamma} is not None}.
434 @kwarg gamma: Angle subtended by triangle side C{c} from B{C{pointA}} to B{C{pointB}}
435 (C{degrees}, non-negative) or C{None} if C{B{beta} is not None}.
436 @kwarg useZ: If C{True}, use and interpolate the Z component, otherwise force C{z=INT0}
437 (C{bool}).
438 @kwarg Clas: Optional class to return the survey point or C{None} for B{C{pointA}}'s
439 (sub-)class.
440 @kwarg Clas_kwds: Optional additional keyword arguments for the survey point instance.
442 @note: Points B{C{pointA}}, B{C{pointB}} and B{C{pointC}} are ordered clockwise.
444 @return: L{Tienstra7Tuple}C{(pointP, A, B, C, a, b, c)} with survey C{pointP}, an
445 instance of B{C{Clas}} or if C{B{Clas} is None} of B{C{pointA}}'s (sub-)class,
446 with triangle angles C{A} at B{C{pointA}}, C{B} at B{C{pointB}} and C{C} at
447 B{C{pointC}} in C{degrees} and with triangle sides C{a}, C{b} and C{c} in
448 C{meter}, conventionally.
450 @raise ResectionError: Near-coincident, -colinear or -concyclic points or sum of
451 B{C{alpha}}, B{C{beta}} and B{C{gamma}} not C{360} or
452 negative B{C{alpha}}, B{C{beta}} or B{C{gamma}}.
454 @raise TypeError: Invalid B{C{pointA}}, B{C{pointB}} or B{C{pointC}}.
456 @see: U{3-Point Resection Solver<http://MesaMike.org/geocache/GC1B0Q9/tienstra/>},
457 U{V. Pierlot, M. Van Droogenbroeck, "A New Three Object Triangulation..."
458 <http://www.Telecom.ULg.ac.Be/publi/publications/pierlot/Pierlot2014ANewThree/>},
459 U{18 Triangulation Algorithms...<http://www.Telecom.ULg.ac.Be/triangulation/>} and
460 functions L{pygeodesy.cassini}, L{pygeodesy.collins5} and L{pygeodesy.pierlot}.
461 '''
463 def _deg_ks(r, s, ks, N):
464 if isnear0(fsum_(PI, r, -s)): # r + (PI2 - s) == PI
465 raise ValueError(Fmt.PARENSPACED(concyclic=N))
466 # k = 1 / (cot(r) - cot(s))
467 # = 1 / (cos(r) / sin(r) - cos(s) / sin(s))
468 # = 1 / (cos(r) * sin(s) - cos(s) * sin(r)) / (sin(r) * sin(s))
469 # = sin(r) * sin(s) / (cos(r) * sin(s) - cos(s) * sin(r))
470 sr, cr, ss, cs = sincos2_(r, s)
471 c = cr * ss - cs * sr
472 if isnear0(c):
473 raise ValueError(Fmt.PARENSPACED(cotan=N))
474 ks.append(sr * ss / c)
475 return Degrees(degrees(r), name=N) # C degrees
477 A = _otherV3d(useZ=useZ, pointA=pointA)
478 B = _otherV3d(useZ=useZ, pointB=pointB)
479 C = _otherV3d(useZ=useZ, pointC=pointC)
481 try:
482 sa, sb, sc = map1(radians, alpha, (beta or 0), (gamma or 0))
483 if beta is None:
484 if gamma is None:
485 raise ValueError(_and(Fmt.EQUAL(beta=beta), Fmt.EQUAL(gamma=gamma)))
486 sb = fsum_(PI2, -sa, -sc)
487 elif gamma is None:
488 sc = fsum_(PI2, -sa, -sb)
489 else: # subtended angles must add to 360 degrees
490 r = fsum1_(sa, sb, sc)
491 if fabs(r - PI2) > EPS:
492 raise ValueError(Fmt.EQUAL(sum=degrees(r)))
493 if min(sa, sb, sc) < 0:
494 raise ValueError(_negative_)
496 # triangle sides
497 a = B.minus(C).length
498 b = A.minus(C).length
499 c = A.minus(B).length
501 ks = [] # 3 Ks and triangle angles
502 dA = _deg_ks(_triAngle(b, c, a), sa, ks, _A_)
503 dB = _deg_ks(_triAngle(a, c, b), sb, ks, _B_)
504 dC = _deg_ks(_triAngle(a, b, c), sc, ks, _C_)
506 k = fsum1(ks, floats=True)
507 if isnear0(k):
508 raise ValueError(Fmt.EQUAL(K=k))
509 x = Fdot(ks, A.x, B.x, C.x).fover(k)
510 y = Fdot(ks, A.y, B.y, C.y).fover(k)
511 z = _zidw(A, B, C, x, y) if useZ else INT0
513 n = tienstra7.__name__
514 clas = Clas or pointA.classof
515 P = clas(x, y, z, **_xkwds(Clas_kwds, name=n))
516 return Tienstra7Tuple(P, dA, dB, dC, a, b, c, name=n)
518 except (TypeError, ValueError) as x:
519 raise ResectionError(pointA=pointA, pointB=pointB, pointC=pointC,
520 alpha=alpha, beta=beta, gamma=gamma, cause=x)
523def triAngle(a, b, c):
524 '''Compute one angle of a triangle.
526 @arg a: Adjacent triangle side length (C{scalar}, non-negative
527 C{meter}, conventionally).
528 @arg b: Adjacent triangle side length (C{scalar}, non-negative
529 C{meter}, conventionally).
530 @arg c: Opposite triangle side length (C{scalar}, non-negative
531 C{meter}, conventionally).
533 @return: Angle in C{radians} at triangle corner C{C}, opposite
534 triangle side B{C{c}}.
536 @raise TriangleError: Invalid or negative B{C{a}}, B{C{b}} or B{C{c}}.
538 @see: Functions L{pygeodesy.triAngle4} and L{pygeodesy.triSide}.
539 '''
540 try:
541 return _triAngle(a, b, c)
542 except (TypeError, ValueError) as x:
543 raise TriangleError(a=a, b=b, c=c, cause=x)
546def _triAngle(a, b, c):
547 # (INTERNAL) To allow callers to embellish errors
548 a, b, c = map1(float, a, b, c)
549 if a < b:
550 a, b = b, a
551 if b < 0 or c < 0:
552 raise ValueError(_negative_)
553 if a < EPS0:
554 raise ValueError(_coincident_)
555 b_a = b / a
556 if b_a < EPS0:
557 raise ValueError(_coincident_)
558 return acos1(fsum_(_1_0, b_a**2, -(c / a)**2) / (b_a * _2_0))
561def triAngle4(a, b, c):
562 '''Compute the angles of a triangle.
564 @arg a: Length of the triangle side opposite of triangle corner C{A}
565 (C{scalar}, non-negative C{meter}, conventionally).
566 @arg b: Length of the triangle side opposite of triangle corner C{B}
567 (C{scalar}, non-negative C{meter}, conventionally).
568 @arg c: Length of the triangle side opposite of triangle corner C{C}
569 (C{scalar}, non-negative C{meter}, conventionally).
571 @return: L{TriAngle4Tuple}C{(radA, radB, radC, rIn)} with angles C{radA},
572 C{radB} and C{radC} at triangle corners C{A}, C{B} and C{C}, all
573 in C{radians} and the C{InCircle} radius C{rIn} aka C{inradius},
574 same units as triangle sides B{C{a}}, B{C{b}} and B{C{c}}.
576 @raise TriangleError: Invalid or negative B{C{a}}, B{C{b}} or B{C{c}}.
578 @see: Function L{pygeodesy.triAngle}.
579 '''
580 try:
581 a, b, c = map1(float, a, b, c)
582 ab = a < b
583 if ab:
584 a, b = b, a
585 bc = b < c
586 if bc:
587 b, c = c, b
589 if c > EPS0: # c = min(a, b, c)
590 s = float(Fsum(a, b, c) * _0_5)
591 if s < EPS0:
592 raise ValueError(_negative_)
593 sa, sb, sc = (s - a), (s - b), (s - c)
594 r = sa * sb * sc / s
595 if r < EPS02:
596 raise ValueError(_coincident_)
597 r = sqrt(r)
598 rA = atan2(r, sa) * _2_0
599 rB = atan2(r, sb) * _2_0
600 rC = fsum_(PI, -rA, -rB)
601 if min(rA, rB, rC) < 0:
602 raise ValueError(_colinear_)
603 elif c < 0:
604 raise ValueError(_negative_)
605 else: # 0 <= c <= EPS0
606 rA = rB = PI_2
607 rC = r = _0_0
609 if bc:
610 rB, rC = rC, rB
611 if ab:
612 rA, rB = rB, rA
613 return TriAngle4Tuple(rA, rB, rC, r, name=triAngle4.__name__)
615 except (TypeError, ValueError) as x:
616 raise TriangleError(a=a, b=b, c=c, cause=x)
619def triSide(a, b, radC):
620 '''Compute one side of a triangle.
622 @arg a: Adjacent triangle side length (C{scalar},
623 non-negative C{meter}, conventionally).
624 @arg b: Adjacent triangle side length (C{scalar},
625 non-negative C{meter}, conventionally).
626 @arg radC: Angle included by sides B{C{a}} and B{C{b}},
627 opposite triangle side C{c} (C{radians}).
629 @return: Length of triangle side C{c}, opposite triangle
630 corner C{C} and angle B{C{radC}}, same units as
631 B{C{a}} and B{C{b}}.
633 @raise TriangleError: Invalid B{C{a}}, B{C{b}} or B{C{radC}}.
635 @see: Functions L{pygeodesy.sqrt_a}, L{pygeodesy.triAngle},
636 L{pygeodesy.triSide2} and L{pygeodesy.triSide4}.
637 '''
638 try:
639 return _triSide(a, b, radC)
640 except (TypeError, ValueError) as x:
641 raise TriangleError(a=a, b=b, radC=radC, cause=x)
644def _triSide(a, b, radC):
645 # (INTERNAL) To allow callers to embellish errors
646 a, b, r = map1(float, a, b, radC)
647 if min(a, b, r) < 0:
648 raise ValueError(_negative_)
650 if a < b:
651 a, b = b, a
652 if a > EPS0:
653 ba = b / a
654 c2 = fsum_(_1_0, ba**2, _N_2_0 * ba * cos(r))
655 if c2 > EPS02:
656 return a * sqrt(c2)
657 elif c2 < 0:
658 raise ValueError(_invalid_)
659 return hypot(a, b)
662def triSide2(b, c, radB):
663 '''Compute one side and the opposite angle of a triangle.
665 @arg b: Adjacent triangle side length (C{scalar},
666 non-negative C{meter}, conventionally).
667 @arg c: Adjacent triangle side length (C{scalar},
668 non-negative C{meter}, conventionally).
669 @arg radB: Angle included by sides B{C{a}} and B{C{c}},
670 opposite triangle side C{b} (C{radians}).
672 @return: L{TriSide2Tuple}C{(a, radA)} with triangle angle
673 C{radA} in C{radians} and length of the opposite
674 triangle side C{a}, same units as B{C{b}} and B{C{c}}.
676 @raise TriangleError: Invalid B{C{b}} or B{C{c}} or either
677 B{C{b}} or B{C{radB}} near zero.
679 @see: Functions L{pygeodesy.sqrt_a}, L{pygeodesy.triSide}
680 and L{pygeodesy.triSide4}.
681 '''
682 try:
683 return _triSide2(b, c, radB)
684 except (TypeError, ValueError) as x:
685 raise TriangleError(b=b, c=c, radB=radB, cause=x)
688def _triSide2(b, c, radB):
689 # (INTERNAL) To allow callers to embellish errors
690 b, c, rB = map1(float, b, c, radB)
691 if min(b, c, rB) < 0:
692 raise ValueError(_negative_)
693 sB, cB = sincos2(rB)
694 if isnear0(sB):
695 if not isnear0(b):
696 raise ValueError(_invalid_)
697 if cB < 0:
698 a, rA = (b + c), PI
699 else:
700 a, rA = fabs(b - c), _0_0
701 elif isnear0(b):
702 raise ValueError(_invalid_)
703 else:
704 rA = fsum_(PI, -rB, -asin1(c * sB / b))
705 a = sin(rA) * b / sB
706 return TriSide2Tuple(a, rA, name=triSide2.__name__)
709def triSide4(radA, radB, c):
710 '''Compute two sides and the height of a triangle.
712 @arg radA: Angle at triangle corner C{A}, opposite triangle side C{a}
713 (non-negative C{radians}).
714 @arg radB: Angle at triangle corner C{B}, opposite triangle side C{b}
715 (non-negative C{radians}).
716 @arg c: Length of triangle side between triangle corners C{A} and C{B},
717 (C{scalar}, non-negative C{meter}, conventionally).
719 @return: L{TriSide4Tuple}C{(a, b, radC, d)} with triangle sides C{a} and
720 C{b} and triangle height C{d} perpendicular to triangle side
721 B{C{c}}, all in the same units as B{C{c}} and interior angle
722 C{radC} in C{radians} at triangle corner C{C}, opposite
723 triangle side B{C{c}}.
725 @raise TriangleError: Invalid or negative B{C{radA}}, B{C{radB}} or B{C{c}}.
727 @see: U{Triangulation, Surveying<https://WikiPedia.org/wiki/Triangulation_(surveying)>}
728 and functions L{pygeodesy.sqrt_a}, L{pygeodesy.triSide} and L{pygeodesy.triSide2}.
729 '''
730 try:
731 rA, rB, c = map1(float, radA, radB, c)
732 rC = fsum_(PI, -rA, -rB)
733 if min(rC, rA, rB, c) < 0:
734 raise ValueError(_negative_)
735 sa, ca, sb, cb = sincos2_(rA, rB)
736 sc = fsum1_(sa * cb, sb * ca)
737 if sc < EPS0 or min(sa, sb) < 0:
738 raise ValueError(_invalid_)
739 sc = c / sc
740 return TriSide4Tuple((sa * sc), (sb * sc), rC, (sa * sb * sc),
741 name=triSide4.__name__)
743 except (TypeError, ValueError) as x:
744 raise TriangleError(radA=radA, radB=radB, c=c, cause=x)
747def wildberger3(a, b, c, alpha, beta, R3=min):
748 '''Snellius' surveying using U{Rational Trigonometry<https://WikiPedia.org/wiki/Snellius–Pothenot_problem>}.
750 @arg a: Length of the triangle side between corners C{B} and C{C} and opposite of
751 triangle corner C{A} (C{scalar}, non-negative C{meter}, conventionally).
752 @arg b: Length of the triangle side between corners C{C} and C{A} and opposite of
753 triangle corner C{B} (C{scalar}, non-negative C{meter}, conventionally).
754 @arg c: Length of the triangle side between corners C{A} and C{B} and opposite of
755 triangle corner C{C} (C{scalar}, non-negative C{meter}, conventionally).
756 @arg alpha: Angle subtended by triangle side B{C{b}} (C{degrees}, non-negative).
757 @arg beta: Angle subtended by triangle side B{C{a}} (C{degrees}, non-negative).
758 @kwarg R3: Callable to determine C{R3} from C{(R3 - C)**2 = D}, typically standard
759 function C{min} or C{max}, invoked with 2 arguments.
761 @return: L{Survey3Tuple}C{(PA, PB, PC)} with distance from survey point C{P} to
762 each of the triangle corners C{A}, C{B} and C{C}, same units as B{C{a}},
763 B{C{b}} and B{C{c}}.
765 @raise TriangleError: Invalid B{C{a}}, B{C{b}} or B{C{c}} or negative B{C{alpha}} or
766 B{C{beta}} or B{C{R3}} not C{callable}.
768 @see: U{Wildberger, Norman J.<https://math.sc.Chula.ac.TH/cjm/content/
769 survey-article-greek-geometry-rational-trigonometry-and-snellius-–-pothenot-surveying>},
770 U{Devine Proportions, page 252<http://www.ms.LT/derlius/WildbergerDivineProportions.pdf>}
771 and function L{pygeodesy.snellius3}.
772 '''
773 def _s(x):
774 return sin(x)**2
776 def _vpa(r1, r3, q2, q3, s3):
777 r = r1 * r3 * _4_0
778 n = (r - Fsum(r1, r3, -q2).fpow(2)).fover(s3)
779 if n < 0 or isnear0(r):
780 raise ValueError(_coincident_)
781 return sqrt((n / r) * q3) if n else _0_0
783 try:
784 a, b, c, da, db = map1(float, a, b, c, alpha, beta)
785 if min(a, b, c, da, db) < 0:
786 raise ValueError(_negative_)
788 ra, rb = radians(da), radians(db)
789 s1, s2, s3 = s = map1(_s, rb, ra, ra + rb) # rb, ra!
790 if min(s) < EPS02:
791 raise ValueError(_or(_coincident_, _colinear_))
793 q1, q2, q3 = q = a**2, b**2, c**2
794 if min(q) < EPS02:
795 raise ValueError(_coincident_)
797 r1 = s2 * q3 / s3 # s2!
798 r2 = s1 * q3 / s3 # s1!
799 Qs = Fsum(*q) # == hypot2_(a, b, c)
800 Ss = Fsum(*s) # == fsum1(s, floats=True)
801 s += Qs * _0_5, # tuple!
802 C0 = Fdot(s, q1, q2, q3, -Ss)
803 r3 = C0.fover(-s3)
804 d0 = Qs.fpow(2).fsub_(hypot2_(*q) * _2_0).fmul(s1 * s2).fover(s3)
805 if d0 > EPS02: # > c0
806 d0 = sqrt(d0)
807 if not callable(R3):
808 raise ValueError(_R3__ + _not_(callable.__name__))
809 r3 = R3(float(C0 + d0), float(C0 - d0)) # XXX min or max
810 elif d0 < 0:
811 raise ValueError(_negative_)
813 pa = _vpa(r1, r3, q2, q3, s3)
814 pb = _vpa(r2, r3, q1, q3, s3)
815 pc = favg(_triSide2(b, pa, ra).a,
816 _triSide2(a, pb, rb).a)
817 return Survey3Tuple(pa, pb, pc, name=wildberger3.__name__)
819 except (TypeError, ValueError) as x:
820 raise TriangleError(a=a, b=b, c=c, alpha=alpha, beta=beta, R3=R3, cause=x)
823def _zidw(A, B, C, x, y):
824 # interpolate z or coplanar with A, B and C?
825 t = A.z, B.z, C.z
826 m = Vector3d(x, y, fmean(t)).minus
827 return fidw(t, (m(T).length for T in (A, B, C)))
829# **) MIT License
830#
831# Copyright (C) 2016-2023 -- mrJean1 at Gmail -- All Rights Reserved.
832#
833# Permission is hereby granted, free of charge, to any person obtaining a
834# copy of this software and associated documentation files (the "Software"),
835# to deal in the Software without restriction, including without limitation
836# the rights to use, copy, modify, merge, publish, distribute, sublicense,
837# and/or sell copies of the Software, and to permit persons to whom the
838# Software is furnished to do so, subject to the following conditions:
839#
840# The above copyright notice and this permission notice shall be included
841# in all copies or substantial portions of the Software.
842#
843# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
844# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
845# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
846# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
847# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
848# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
849# OTHER DEALINGS IN THE SOFTWARE.