Coverage for pygeodesy/vector2d.py: 97%
319 statements
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2# -*- coding: utf-8 -*-
4u'''2- or 3-D vectorial functions L{circin6}, L{circum3}, L{circum4_},
5L{iscolinearWith}, L{meeus2}, L{nearestOn}, L{radii11} and L{soddy4}.
6'''
8from pygeodesy.basics import len2, map2, _xnumpy
9from pygeodesy.constants import EPS, EPS0, EPS02, EPS4, INF, INT0, \
10 _EPS4e8, isnear0, _0_0, _0_25, _0_5, _N_0_5, \
11 _1_0, _1_0_1T, _N_1_0, _2_0, _N_2_0, _4_0
12from pygeodesy.errors import _and, _AssertionError, IntersectionError, NumPyError, \
13 PointsError, TriangleError, _xError, _xkwds
14from pygeodesy.fmath import fabs, fdot, hypot, hypot2_, sqrt
15from pygeodesy.fsums import _Fsumf_, fsumf_, fsum1f_
16from pygeodesy.interns import NN, _a_, _and_, _b_, _c_, _center_, _coincident_, \
17 _colinear_, _COMMASPACE_, _concentric_, _few_, \
18 _intersection_, _invalid_, _near_, _no_, _of_, \
19 _radius_, _rIn_, _s_, _SPACE_, _too_, _with_
20# from pygeodesy.lazily import _ALL_LAZY # from .named
21from pygeodesy.named import _ALL_LAZY, _NamedTuple, _Pass, Property_RO
22from pygeodesy.namedTuples import LatLon3Tuple, Vector2Tuple
23# from pygeodesy.props import Property_RO # from .named
24from pygeodesy.streprs import Fmt, unstr
25from pygeodesy.units import Float, Int, Meter, Radius, Radius_
26from pygeodesy.vector3d import iscolinearWith, nearestOn, _nearestOn2, _nVc, _otherV3d, \
27 trilaterate2d2, trilaterate3d2, Vector3d # PYCHOK unused
29from contextlib import contextmanager
30# from math import fabs, sqrt # from .fmath
32__all__ = _ALL_LAZY.vector2d
33__version__ = '24.04.18'
35_cA_ = 'cA'
36_cB_ = 'cB'
37_cC_ = 'cC'
38_deltas_ = 'deltas'
39_outer_ = 'outer'
40_raise_ = 'raise' # PYCHOK used!
41_rank_ = 'rank'
42_residuals_ = 'residuals'
43_Type_ = 'Type'
46class Circin6Tuple(_NamedTuple):
47 '''6-Tuple C{(radius, center, deltas, cA, cB, cC)} with the C{radius}, the
48 trilaterated C{center} and contact points of the I{inscribed} aka I{In-
49 circle} of a triangle. The C{center} is I{un}ambiguous if C{deltas} is
50 C{None}, otherwise C{center} is the mean and C{deltas} the differences of
51 the L{pygeodesy.trilaterate3d2} results. Contact points C{cA}, C{cB} and
52 C{cC} are the points of tangency, aka the corners of the U{Contact Triangle
53 <https://MathWorld.Wolfram.com/ContactTriangle.html>}.
54 '''
55 _Names_ = (_radius_, _center_, _deltas_, _cA_, _cB_, _cC_)
56 _Units_ = ( Radius, _Pass, _Pass, _Pass, _Pass, _Pass)
59class Circum3Tuple(_NamedTuple): # in .latlonBase
60 '''3-Tuple C{(radius, center, deltas)} with the C{circumradius} and trilaterated
61 C{circumcenter} of the C{circumcircle} through 3 points (aka {Meeus}' Type II
62 circle) or the C{radius} and C{center} of the smallest I{Meeus}' Type I circle.
63 The C{center} is I{un}ambiguous if C{deltas} is C{None}, otherwise C{center}
64 is the mean and C{deltas} the differences of the L{pygeodesy.trilaterate3d2}
65 results.
66 '''
67 _Names_ = (_radius_, _center_, _deltas_)
68 _Units_ = ( Radius, _Pass, _Pass)
71class Circum4Tuple(_NamedTuple):
72 '''4-Tuple C{(radius, center, rank, residuals)} with C{radius} and C{center}
73 of a sphere I{least-squares} fitted through given points and the C{rank}
74 and C{residuals} -if any- from U{numpy.linalg.lstsq
75 <https://NumPy.org/doc/stable/reference/generated/numpy.linalg.lstsq.html>}.
76 '''
77 _Names_ = (_radius_, _center_, _rank_, _residuals_)
78 _Units_ = ( Radius, _Pass, Int, _Pass)
81class Meeus2Tuple(_NamedTuple):
82 '''2-Tuple C{(radius, Type)} with C{radius} and I{Meeus}' C{Type} of the smallest
83 circle I{containing} 3 points. C{Type} is C{None} for a I{Meeus}' Type II
84 C{circumcircle} passing through all 3 points. Otherwise C{Type} is the center
85 of a I{Meeus}' Type I circle with 2 points on (a diameter of) and 1 point
86 inside the circle.
87 '''
88 _Names_ = (_radius_, _Type_)
89 _Units_ = ( Radius, _Pass)
92class Radii11Tuple(_NamedTuple):
93 '''11-Tuple C{(rA, rB, rC, cR, rIn, riS, roS, a, b, c, s)} with the C{Tangent}
94 circle radii C{rA}, C{rB} and C{rC}, the C{circumradius} C{cR}, the C{Incircle}
95 radius C{rIn} aka C{inradius}, the inner and outer I{Soddy} circle radii C{riS}
96 and C{roS}, the sides C{a}, C{b} and C{c} and semi-perimeter C{s} of a triangle,
97 all in C{meter} conventionally.
99 @note: C{Circumradius} C{cR} and outer I{Soddy} radius C{roS} may be C{INF}.
100 '''
101 _Names_ = ('rA', 'rB', 'rC', 'cR', _rIn_, 'riS', 'roS', _a_, _b_, _c_, _s_)
102 _Units_ = ( Meter,) * len(_Names_)
105class Soddy4Tuple(_NamedTuple):
106 '''4-Tuple C{(radius, center, deltas, outer)} with C{radius} and (trilaterated)
107 C{center} of the I{inner} I{Soddy} circle and the radius of the C{outer}
108 I{Soddy} circle. The C{center} is I{un}ambiguous if C{deltas} is C{None},
109 otherwise C{center} is the mean and C{deltas} the differences of the
110 L{pygeodesy.trilaterate3d2} results.
112 @note: The outer I{Soddy} radius C{outer} may be C{INF}.
113 '''
114 _Names_ = (_radius_, _center_, _deltas_, _outer_)
115 _Units_ = ( Radius, _Pass, _Pass, Radius)
118def circin6(point1, point2, point3, eps=EPS4, useZ=True):
119 '''Return the radius and center of the I{inscribed} aka I{Incircle} of
120 a (2- or 3-D) triangle.
122 @arg point1: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
123 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
124 @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
125 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
126 @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
127 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
128 @kwarg eps: Tolerance for function L{pygeodesy.trilaterate3d2} if
129 C{B{useZ} is True} else L{pygeodesy.trilaterate2d2}.
130 @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}).
132 @return: L{Circin6Tuple}C{(radius, center, deltas, cA, cB, cC)}. The
133 C{center} and contact points C{cA}, C{cB} and C{cC}, each an
134 instance of B{C{point1}}'s (sub-)class, are co-planar with
135 the three given points.
137 @raise ImportError: Package C{numpy} not found, not installed or older
138 than version 1.10 and C{B{useZ} is True}.
140 @raise IntersectionError: Near-coincident or -colinear points or
141 a trilateration or C{numpy} issue.
143 @raise TypeError: Invalid B{C{point1}}, B{C{point2}} or B{C{point3}}.
145 @see: Functions L{radii11} and L{circum3}, U{Contact Triangle
146 <https://MathWorld.Wolfram.com/ContactTriangle.html>} and
147 U{Incircle<https://MathWorld.Wolfram.com/Incircle.html>}.
148 '''
149 try:
150 return _circin6(point1, point2, point3, eps=eps, useZ=useZ)
151 except (AssertionError, TypeError, ValueError) as x:
152 raise _xError(x, point1=point1, point2=point2, point3=point3)
155def _circin6(point1, point2, point3, eps=EPS4, useZ=True, dLL3=False, **Vector_kwds):
156 # (INTERNAL) Radius, center, deltas, 3 contact points
158 def _fraction(r, a):
159 return (r / a) if a > EPS0 else _0_5
161 def _contact2(a, p2, r2, p3, r3, V, V_kwds):
162 c = p2.intermediateTo(p3, _fraction(r2, a)) if r2 > r3 else \
163 p3.intermediateTo(p2, _fraction(r3, a))
164 C = V(c.x, c.y, c.z, **V_kwds)
165 return c, C
167 t, p1, p2, p3 = _radii11ABC(point1, point2, point3, useZ=useZ)
168 V, r1, r2, r3 = point1.classof, t.rA, t.rB, t.rC
170 c1, cA = _contact2(t.a, p2, r2, p3, r3, V, _xkwds(Vector_kwds, name=_cA_))
171 c2, cB = _contact2(t.b, p3, r3, p1, r1, V, _xkwds(Vector_kwds, name=_cB_))
172 c3, cC = _contact2(t.c, p1, r1, p2, r2, V, _xkwds(Vector_kwds, name=_cC_))
174 r = t.rIn
175 c, d = _tricenter3d2(c1, r, c2, r, c3, r, eps=eps, useZ=useZ, dLL3=dLL3,
176 **_xkwds(Vector_kwds, Vector=V, name=circin6.__name__))
177 return Circin6Tuple(r, c, d, cA, cB, cC)
180def circum3(point1, point2, point3, circum=True, eps=EPS4, useZ=True):
181 '''Return the radius and center of the smallest circle I{through} or
182 I{containing} three (2- or 3-D) points.
184 @arg point1: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or
185 C{Vector4Tuple}).
186 @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or
187 C{Vector4Tuple}).
188 @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or
189 C{Vector4Tuple}).
190 @kwarg circum: If C{True} return the C{circumradius} and C{circumcenter}
191 always, ignoring the I{Meeus}' Type I case (C{bool}).
192 @kwarg eps: Tolerance for function L{pygeodesy.trilaterate3d2} if C{B{useZ}
193 is True} else L{pygeodesy.trilaterate2d2}.
194 @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}).
196 @return: A L{Circum3Tuple}C{(radius, center, deltas)}. The C{center}, an
197 instance of B{C{point1}}'s (sub-)class, is co-planar with the three
198 given points.
200 @raise ImportError: Package C{numpy} not found, not installed or older
201 than version 1.10 and C{B{useZ} is True}.
203 @raise IntersectionError: Near-coincident or -colinear points or
204 a trilateration or C{numpy} issue.
206 @raise TypeError: Invalid B{C{point1}}, B{C{point2}} or B{C{point3}}.
208 @see: Functions L{pygeodesy.circum4_} and L{pygeodesy.meeus2} and Meeus, J.
209 U{I{Astronomical Algorithms}<http://www.Agopax.IT/Libri_astronomia/pdf/
210 Astronomical%20Algorithms.pdf>}, 2nd Ed. 1998, page 127ff, U{circumradius
211 <https://MathWorld.Wolfram.com/Circumradius.html>} and U{circumcircle
212 <https://MathWorld.Wolfram.com/Circumcircle.html>}.
213 '''
214 try:
215 p1 = _otherV3d(useZ=useZ, point1=point1)
216 return _circum3(p1, point2, point3, circum=circum, eps=eps, useZ=useZ,
217 clas=point1.classof)
218 except (AssertionError, TypeError, ValueError) as x:
219 raise _xError(x, point1=point1, point2=point2, point3=point3, circum=circum)
222def _circum3(p1, point2, point3, circum=True, eps=EPS4, useZ=True, dLL3=False,
223 clas=Vector3d, **clas_kwds): # in .latlonBase
224 # (INTERNAL) Radius, center, deltas
225 r, d, p2, p3 = _meeus4(p1, point2, point3, circum=circum, useZ=useZ,
226 clas=clas, **clas_kwds)
227 if d is None: # Meeus' Type II or circum=True
228 kwds = _xkwds(clas_kwds, eps=eps, Vector=clas, name=circum3.__name__)
229 c, d = _tricenter3d2(p1, r, p2, r, p3, r, useZ=useZ, dLL3=dLL3, **kwds)
230 else: # Meeus' Type I
231 c, d = d, None
232 return Circum3Tuple(r, c, d)
235def circum4_(*points, **useZ_Vector_and_kwds):
236 '''Best-fit a sphere through three or more (3-D) points.
238 @arg points: The points (each a C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
239 or C{Vector4Tuple}).
240 @kwarg useZ_Vector_and_kwds: Keyword arguments C{B{useZ}=True} (C{bool})
241 to use the Z components, otherwise force all C{z=INT0}, class
242 C{B{Vector}=None} to return the center point with optionally,
243 additional nB{C{Vector}} keyword arguments, otherwise the
244 first B{C{points}}' (sub-)class is used.
246 @return: L{Circum4Tuple}C{(radius, center, rank, residuals)} with C{center} an
247 instance of C{B{points}[0]}' (sub-)class or B{C{Vector}} if specified.
249 @raise ImportError: Package C{numpy} not found, not installed or older than
250 version 1.10.
252 @raise NumPyError: Some C{numpy} issue.
254 @raise PointsError: Too few B{C{points}}.
256 @raise TypeError: One of the B{C{points}} is invalid.
258 @see: Functions L{pygeodesy.circum3} and L{pygeodesy.meeus2}, Jekel, Charles F. U{I{Least
259 Squares Sphere Fit}<https://Jekel.me/2015/Least-Squares-Sphere-Fit/>} Sep 13, 2015,
260 U{Appendix A<https://hdl.handle.net/10019.1/98627>}, U{numpy.linalg.lstsq<https://
261 NumPy.org/doc/stable/reference/generated/numpy.linalg.lstsq.html>} and U{Eberly 6
262 <https://www.sci.Utah.EDU/~balling/FEtools/doc_files/LeastSquaresFitting.pdf>}.
263 '''
264 def _useZ_kwds(useZ=True, **kwds):
265 return useZ, kwds
267 n, ps = len2(points)
268 if n < 3:
269 raise PointsError(points=n, txt=_too_(_few_))
270 useZ, kwds = _useZ_kwds(**useZ_Vector_and_kwds)
272 A, b = [], []
273 for i, p in enumerate(ps):
274 v = _otherV3d(useZ=useZ, i=i, points=p)
275 A.append(v.times(_2_0).xyz + _1_0_1T)
276 b.append(v.length2)
278 with _numpy(circum4_, n=n) as _np:
279 A = _np.array(A).reshape((n, 4))
280 b = _np.array(b).reshape((n, 1))
281 C, R, rk, _ = _np.least_squares4(A, b, rcond=None) # to silence warning
282 C = map2(float, C)
283 R = map2(float, R) # empty if rk < 4 or n <= 4
285 n = circum4_.__name__
286 c = Vector3d(*C[:3], name=n)
287 r = Radius(sqrt(fsumf_(C[3], *c.x2y2z2)), name=n)
289 c = _nVc(c, **_xkwds(kwds, clas=ps[0].classof, name=n))
290 return Circum4Tuple(r, c, rk, R)
293def _iscolinearWith(p, point1, point2, eps=EPS, useZ=True):
294 # (INTERNAL) Check colinear, see L{iscolinearWith} above,
295 # separated to allow callers to embellish any exceptions
296 p1 = _otherV3d(useZ=useZ, point1=point1)
297 p2 = _otherV3d(useZ=useZ, point2=point2)
298 n, _ = _nearestOn2(p, p1, p2, within=False, eps=eps)
299 return n is p1 or n.minus(p).length2 < eps
302def meeus2(point1, point2, point3, circum=False, useZ=True):
303 '''Return the radius and I{Meeus}' Type of the smallest circle I{through}
304 or I{containing} three (2- or 3-D) points.
306 @arg point1: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
307 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
308 @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
309 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
310 @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
311 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
312 @kwarg circum: If C{True} return the C{circumradius} and C{circumcenter}
313 always, overriding I{Meeus}' Type II case (C{bool}).
314 @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}).
316 @return: L{Meeus2Tuple}C{(radius, Type)}, with C{Type} the C{circumcenter}
317 iff C{B{circum}=True}.
319 @raise IntersectionError: Near-coincident or -colinear points, iff C{B{circum}=True}.
321 @raise TypeError: Invalid B{C{point1}}, B{C{point2}} or B{C{point3}}.
323 @see: Functions L{pygeodesy.circum3} and L{pygeodesy.circum4_} and Meeus, J.
324 U{I{Astronomical Algorithms}<http://www.Agopax.IT/Libri_astronomia/pdf/
325 Astronomical%20Algorithms.pdf>}, 2nd Ed. 1998, page 127ff, U{circumradius
326 <https://MathWorld.Wolfram.com/Circumradius.html>} and U{circumcircle
327 <https://MathWorld.Wolfram.com/Circumcircle.html>}.
328 '''
329 try:
330 A = _otherV3d(useZ=useZ, point1=point1)
331 return _meeus2(A, point2, point3, circum, useZ=useZ, clas=point1.classof)
332 except (TypeError, ValueError) as x:
333 raise _xError(x, point1=point1, point2=point2, point3=point3, circum=circum)
336def _meeus2(A, point2, point3, circum, useZ=True, **clas_and_kwds): # in .vector3d
337 # (INTERNAL) Radius and center or Meeus' Type
338 f = _circum3 if circum else _meeus4
339 t = f(A, point2, point3, circum=circum, useZ=useZ, **clas_and_kwds)[:2]
340 return Meeus2Tuple(t)
343def _meeus4(A, point2, point3, circum=False, useZ=True, clas=None, **clas_kwds):
344 # (INTERNAL) Radius and Meeus' Type
345 B = p2 = _otherV3d(useZ=useZ, point2=point2)
346 C = p3 = _otherV3d(useZ=useZ, point3=point3)
348 a = B.minus(C).length2
349 b = C.minus(A).length2
350 c = A.minus(B).length2
351 if a < b:
352 a, b, A, B = b, a, B, A
353 if a < c:
354 a, c, A, C = c, a, C, A
356 if a > EPS02 and (circum or a < (b + c)): # circumradius
357 b = sqrt(b / a)
358 c = sqrt(c / a)
359 R = _Fsumf_(_1_0, b, c) * _Fsumf_(_1_0, b, -c) * \
360 _Fsumf_(_1_0, -b, c) * _Fsumf_(_N_1_0, b, c)
361 r = R.fover(a)
362 if r < EPS02:
363 raise IntersectionError(_coincident_ if b < EPS0 or c < EPS0 else (
364 _colinear_ if _iscolinearWith(A, B, C) else _invalid_))
365 r = b * c / sqrt(r)
366 t = None # Meeus' Type II
367 else: # obtuse or right angle at A
368 r = sqrt(a * _0_25) if a > EPS02 else INT0
369 t = B.plus(C).times(_0_5) # Meeus' Type I
370 if clas is not None:
371 t = clas(t.x, t.y, t.z, **_xkwds(clas_kwds, name=meeus2.__name__))
372 return r, t, p2, p3
375class _numpy(object): # see also .formy._idllmn6, .geodesicw._wargs, .latlonBase._toCartesian3
376 '''(INTERNAL) Partial C{NumPy} wrapper.
377 '''
378 @contextmanager # <https://www.Python.org/dev/peps/pep-0343/> Examples
379 def __call__(self, where, *args, **kwds):
380 '''(INTERNAL) Yield self with any errors raised as L{NumPyError}
381 embellished with all B{C{args}} and B{C{kwds}}.
382 '''
383 np = self.np
384 try: # <https://NumPy.org/doc/stable/reference/generated/numpy.seterr.html>
385 e = np.seterr(all=_raise_) # throw FloatingPointError for numpy errors
386 yield self
387 except Exception as x: # mostly FloatingPointError?
388 t = unstr(where, *args, **kwds)
389 raise NumPyError(t, cause=x) # _xError2?
390 finally: # restore numpy error handling
391 np.seterr(**e)
393 @Property_RO
394 def array(self):
395 return self.np.array
397 @Property_RO
398 def least_squares4(self):
399 '''Linear least-squares function.
400 '''
401 return self.np.linalg.lstsq
403 @Property_RO
404 def np(self):
405 '''Import numpy 1.10+ once.
406 '''
407 return _xnumpy(self.__class__, 1, 10)
409 def null_space2(self, A, rcond=None):
410 '''Return the C{null_space} and C{rank} of matrix B{C{A}}.
412 @see: U{Source<https://docs.SciPy.org/doc/scipy/reference/generated/scipy.linalg.null_space.html>}
413 U{SciPY Cookbook<https://SciPy-Cookbook.ReadTheDocs.io/items/RankNullspace.html>}, U{here
414 <https://NumPy.org/doc/stable/reference/generated/numpy.linalg.svd.html>}, U{here
415 <https://StackOverflow.com/questions/19820921>}, U{here
416 <https://StackOverflow.com/questions/2992947>} and U{here
417 <https://StackOverflow.com/questions/5889142>}.
418 '''
419 def _Error(txt=self.null_space2.__name__, **kwds):
420 return _AssertionError(txt=txt, **kwds)
422 np = self.np
423 A = np.array(A)
424 m = max(A.shape)
425 if m != 4: # for this usage
426 raise _Error(shape=m)
427 # if needed, square A, pad with zeros
428 A = np.resize(A, m * m).reshape(m, m)
429# try: # no np.linalg.null_space <https://docs.SciPy.org/doc/>
430# Z = scipy.linalg.null_space(A) # XXX no scipy.linalg?
431# return Z, ...
432# except AttributeError:
433# pass
434 U, S, V = np.linalg.svd(A)
435 s = max(EPS, rcond) if rcond else (EPS * max(U.shape[0], V.shape[1]))
436 t = max(EPS, float(np.max(S) * s)) # abs_tol, rel_tol * largest singular
437 r = int(np.sum(S > t)) # rank
438 if r == 3: # get null_space
439 Z = np.transpose(V[r:])
440 s = map2(int, Z.shape)
441 if s != (m, 1): # bad null_space shape
442 raise _Error(shape=s, m=m)
443 D = A.dot(Z) # near-zeros-vector
444 n = float(np.linalg.norm(D, INF)) # INF = max(fabs(D)), 2 = hypot_(*D)
445 if n > t: # largest exceed tol
446 raise _Error(dot=tuple(D.ravel()), norm=n, tol=t)
447 else: # coincident, colinear, concentric centers, ambiguous, etc.
448 Z = None
449 # del A, S, U, V # release numpy
450 return Z, r
452 @Property_RO
453 def pseudo_inverse(self):
454 '''Moore-Penrose pseudo-inverse function.
455 '''
456 return self.np.linalg.pinv
458 def real_roots(self, *coeffs):
459 '''Compute the real, non-complex roots of a polynomial.
460 '''
461 np = self.np
462 rs = np.polynomial.polynomial.polyroots(coeffs)
463 return tuple(float(r) for r in rs if not np.iscomplex(r))
465_numpy = _numpy() # PYCHOK singleton
468def radii11(point1, point2, point3, useZ=True):
469 '''Return the radii of the C{In-}, I{Soddy} and C{Tangent} circles of a
470 (2- or 3-D) triangle.
472 @arg point1: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
473 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
474 @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
475 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
476 @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
477 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
478 @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}).
480 @return: L{Radii11Tuple}C{(rA, rB, rC, cR, rIn, riS, roS, a, b, c, s)}.
482 @raise TriangleError: Near-coincident or -colinear points.
484 @raise TypeError: Invalid B{C{point1}}, B{C{point2}} or B{C{point3}}.
486 @see: U{Circumradius<https://MathWorld.Wolfram.com/Circumradius.html>},
487 U{Incircle<https://MathWorld.Wolfram.com/Incircle.html>}, U{Soddy
488 Circles<https://MathWorld.Wolfram.com/SoddyCircles.html>} and
489 U{Tangent Circles<https://MathWorld.Wolfram.com/TangentCircles.html>}.
490 '''
491 try:
492 return _radii11ABC(point1, point2, point3, useZ=useZ)[0]
493 except (TypeError, ValueError) as x:
494 raise _xError(x, point1=point1, point2=point2, point3=point3)
497def _radii11ABC(point1, point2, point3, useZ=True):
498 # (INTERNAL) Tangent, Circum, Incircle, Soddy radii, sides and semi-perimeter
499 A = _otherV3d(useZ=useZ, point1=point1, NN_OK=False)
500 B = _otherV3d(useZ=useZ, point2=point2, NN_OK=False)
501 C = _otherV3d(useZ=useZ, point3=point3, NN_OK=False)
503 a = B.minus(C).length
504 b = C.minus(A).length
505 c = A.minus(B).length
507 S = _Fsumf_(a, b, c) * _0_5
508 s = float(S) # semi-perimeter
509 if s > EPS0:
510 rs = float(S - a), float(S - b), float(S - c)
511 r3, r2, r1 = sorted(rs) # r3 <= r2 <= r1
512 if r3 > EPS0: # and r2 > EPS0 and r1 > EPS0
513 r3_r1 = r3 / r1
514 r3_r2 = r3 / r2
515 # t = r1 * r2 * r3 * (r1 + r2 + r3)
516 # = r1**2 * r2 * r3 * (1 + r2 / r1 + r3 / r1)
517 # = (r1 * r2)**2 * (r3 / r2) * (1 + r2 / r1 + r3 / r1)
518 t = r3_r2 * fsum1f_(_1_0, r2 / r1, r3_r1) # * (r1 * r2)**2
519 if t > EPS02:
520 t = sqrt(t) * _2_0 # * r1 * r2
521 # d = r1 * r2 + r2 * r3 + r3 * r1
522 # = r1 * (r2 + r2 * r3 / r1 + r3)
523 # = r1 * r2 * (1 + r3 / r1 + r3 / r2)
524 d = fsum1f_(_1_0, r3_r1, r3_r2) # * r1 * r2
525 # si/o = r1 * r2 * r3 / (r1 * r2 * (d +/- t))
526 # = r3 / (d +/- t)
527 si = r3 / (d + t)
528 so = (r3 / (d - t)) if d > t else INF
529 # ci = sqrt(r1 * r2 * r3 / s)
530 # = r1 * sqrt(r2 * r3 / r1 / s)
531 ci = r1 * sqrt(r2 * r3_r1 / s)
532 # co = a * b * c / (4 * ci * s)
533 t = ci * s * _4_0
534 co = (a * b * c / t) if t > EPS0 else INF
535 r1, r2, r3 = rs # original order
536 t = Radii11Tuple(r1, r2, r3, co, ci, si, so, a, b, c, s)
537 return t, A, B, C
539 raise TriangleError(_near_(_coincident_) if min(a, b, c) < EPS0 else (
540 _colinear_ if _iscolinearWith(A, B, C) else _invalid_))
543def soddy4(point1, point2, point3, eps=EPS4, useZ=True):
544 '''Return the radius and center of the C{inner} I{Soddy} circle of a
545 (2- or 3-D) triangle.
547 @arg point1: First point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
548 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
549 @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
550 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
551 @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
552 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
553 @kwarg eps: Tolerance for function L{pygeodesy.trilaterate3d2} if
554 C{B{useZ} is True} otherwise L{pygeodesy.trilaterate2d2}.
555 @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}).
557 @return: L{Soddy4Tuple}C{(radius, center, deltas, outer)}. The C{center},
558 an instance of B{C{point1}}'s (sub-)class, is co-planar with the
559 three given points. The C{outer} I{Soddy} radius may be C{INF}.
561 @raise ImportError: Package C{numpy} not found, not installed or older
562 than version 1.10 and C{B{useZ} is True}.
564 @raise IntersectionError: Near-coincident or -colinear points or
565 a trilateration or C{numpy} issue.
567 @raise TypeError: Invalid B{C{point1}}, B{C{point2}} or B{C{point3}}.
569 @see: Functions L{radii11} and L{circum3} and U{Soddy Circles
570 <https://MathWorld.Wolfram.com/SoddyCircles.html>}.
571 '''
572 t, p1, p2, p3 = _radii11ABC(point1, point2, point3, useZ=useZ)
574 r = t.riS
575 c, d = _tricenter3d2(p1, t.rA + r,
576 p2, t.rB + r,
577 p3, t.rC + r, eps=eps, useZ=useZ,
578 Vector=point1.classof, name=soddy4.__name__)
579 return Soddy4Tuple(r, c, d, t.roS)
582def _tricenter3d2(p1, r1, p2, r2, p3, r3, eps=EPS4, useZ=True, dLL3=False, **kwds):
583 # (INTERNAL) Trilaterate and disambiguate the 3-D center
584 d, kwds = None, _xkwds(kwds, eps=eps, coin=True)
585 if useZ and p1.z != p2.z != p3.z: # ignore z if all match
586 a, b = _trilaterate3d2(p1, r1, p2, r2, p3, r3, **kwds)
587 if a is b: # no unambiguity
588 c = a # == b
589 else:
590 c = a.plus(b).times(_0_5) # mean
591 if not a.isconjugateTo(b, minum=0, eps=eps):
592 if dLL3: # deltas as (lat, lon, height)
593 a = a.toLatLon()
594 b = b.toLatLon()
595 d = LatLon3Tuple(b.lat - a.lat,
596 b.lon - a.lon,
597 b.height - a.height, name=_deltas_)
598 else:
599 d = b.minus(a) # vectorial deltas
600 else:
601 if useZ: # pass z to Vector if given
602 kwds = _xkwds(kwds, z=p1.z)
603 c = _trilaterate2d2(p1.x, p1.y, r1,
604 p2.x, p2.y, r2,
605 p3.x, p3.y, r3, **kwds)
606 return c, d
609def _trilaterate2d2(x1, y1, radius1, x2, y2, radius2, x3, y3, radius3,
610 coin=False, eps=None,
611 Vector=None, **Vector_kwds):
612 # (INTERNAL) Trilaterate three circles, see L{pygeodesy.trilaterate2d2}
614 def _abct4(x1, y1, r1, x2, y2, r2):
615 a = x2 - x1
616 b = y2 - y1
617 t = _tri3near2far(r1, r2, hypot(a, b), coin)
618 c = _0_0 if t else (hypot2_(r1, x2, y2) - hypot2_(r2, x1, y1))
619 return a, b, c, t
621 def _astr(**kwds): # kwds as (name=value, ...) strings
622 return Fmt.PAREN(_COMMASPACE_(*(Fmt.EQUALg(*t) for t in kwds.items())))
624 r1 = Radius_(radius1=radius1)
625 r2 = Radius_(radius2=radius2)
626 r3 = Radius_(radius3=radius3)
628 a, b, c, t = _abct4(x1, y1, r1, x2, y2, r2)
629 if t:
630 raise IntersectionError(_and(_astr(x1=x1, y1=y1, radius1=r1),
631 _astr(x2=x2, y2=y2, radius2=r2)), txt=t)
633 d, e, f, t = _abct4(x2, y2, r2, x3, y3, r3)
634 if t:
635 raise IntersectionError(_and(_astr(x2=x2, y2=y2, radius2=r2),
636 _astr(x3=x3, y3=y3, radius3=r3)), txt=t)
638 _, _, _, t = _abct4(x3, y3, r3, x1, y1, r1)
639 if t:
640 raise IntersectionError(_and(_astr(x3=x3, y3=y3, radius3=r3),
641 _astr(x1=x1, y1=y1, radius1=r1)), txt=t)
643 q = (a * e - b * d) * _2_0
644 if isnear0(q):
645 t = _no_(_intersection_)
646 raise IntersectionError(_and(_astr(x1=x1, y1=y1, radius1=r1),
647 _astr(x2=x2, y2=y2, radius2=r2),
648 _astr(x3=x3, y3=y3, radius3=r3)), txt=t)
649 t = Vector2Tuple((c * e - b * f) / q,
650 (a * f - c * d) / q, name=trilaterate2d2.__name__)
652 if eps and eps > 0: # check distances to center vs radius
653 for x, y, r in ((x1, y1, r1), (x2, y2, r2), (x3, y3, r3)):
654 d = hypot(x - t.x, y - t.y)
655 e = fabs(d - r)
656 if e > eps:
657 t = _and(Float(delta=e).toRepr(), r.toRepr(),
658 Float(distance=d).toRepr(), t.toRepr())
659 raise IntersectionError(t, txt=Fmt.exceeds_eps(eps))
661 if Vector is not None:
662 t = Vector(t.x, t.y, **_xkwds(Vector_kwds, name=t.name))
663 return t
666def _trilaterate3d2(c1, r1, c2, r2, c3, r3, eps=EPS, coin=False,
667 **clas_Vector_and_kwds):
668 # (INTERNAL) Intersect three spheres or circles, see function
669 # L{pygeodesy.trilaterate3d2}, separated to allow callers to
670 # embellish exceptions, like C{FloatingPointError}s from C{numpy}
672 def _F3d2(F):
673 # map numpy 4-vector to floats tuple and Vector3d
674 T = map2(float, F)
675 return T, Vector3d(*T[1:])
677 def _N3(t01, x, z):
678 # compute x, y and z and return as B{C{clas}} or B{C{Vector}}
679 v = x.plus(z.times(t01))
680 n = trilaterate3d2.__name__
681 return _nVc(v, **_xkwds(clas_Vector_and_kwds, name=n))
683 c2 = _otherV3d(center2=c2, NN_OK=False)
684 c3 = _otherV3d(center3=c3, NN_OK=False)
685 rs = (r1, Radius_(radius2=r2, low=EPS),
686 Radius_(radius3=r3, low=EPS))
688 # get matrix A[3 x 4], its pseudo-inverse and null_space Z
689 A = [(_1_0_1T + c.times(_N_2_0).xyz) for c in (c1, c2, c3)]
690 with _numpy(trilaterate3d2, A=A, eps=eps) as _np:
691 Z, _ = _np.null_space2(A, eps)
692 if Z is not None:
693 Z, z = _F3d2(Z) # [4 x 1]
694 z2 = z.length2
695 A = _np.pseudo_inverse(A) # [4 x 3]
696 bs = [c.length2 for c in (c1, c2, c3)]
697 # perturbe radii and vector b slightly by eps and eps * 4
698 for p in _tri5perturbs(eps, min(rs)):
699 b = [((r + p)**2 - b) for r, b in zip(rs, bs)] # [3 x 1]
700 X, x = _F3d2(A.dot(b))
701 # quadratic polynomial, coefficients ordered (^0, ^1, ^2)
702 t = _np.real_roots(fdot(X, _N_1_0, *x.xyz),
703 fdot(Z, _N_0_5, *x.xyz) * _2_0, z2)
704 if t:
705 v = _N3(t[0], x, z)
706 if len(t) < 2: # one intersection
707 t = v, v
708 elif fabs(t[0] - t[1]) < eps: # abutting
709 t = v, v
710 else: # "lowest" intersection first (to avoid test failures)
711 u = _N3(t[1], x, z)
712 t = (u, v) if u.x < v.x else (v, u)
713 return t
715 # coincident, concentric, colinear, too distant, no intersection:
716 # create the explanation and and throw an IntersectionError
718 def _no_intersection(coin):
719 t = _no_(_intersection_)
720 if coin:
721 def _reprs(*crs):
722 return _and(*map(repr, crs))
724 r = repr(r1) if r1 == r2 == r3 else _reprs(r1, r2, r3)
725 t = _SPACE_(t, _of_, _reprs(c1, c2, c3), _with_, _radius_, r)
726 elif Z is None:
727 t = _COMMASPACE_(t, _no_(_numpy.null_space2.__name__))
728 return t
730 t = _tri4near2far(c1, r1, c2, r2, coin) or \
731 _tri4near2far(c1, r1, c3, r3, coin) or \
732 _tri4near2far(c2, r2, c3, r3, coin) or (
733 _colinear_ if _iscolinearWith(c1, c2, c3, eps=eps) else
734 _no_intersection(coin))
735 raise IntersectionError(t, txt=None)
738def _tri3near2far(r1, r2, h, coin):
739 # check for near-coincident/-concentric or too distant spheres/circles
740 return _too_(Fmt.distant(h)) if h > (r1 + r2) else (_near_(
741 _coincident_ if coin else _concentric_) if h < fabs(r1 - r2) else NN)
744def _tri4near2far(c1, r1, c2, r2, coin):
745 # check for near-coincident/-concentric or too distant spheres/circles
746 t = _tri3near2far(r1, r2, c1.minus(c2).length, coin)
747 return _SPACE_(c1.name, _and_, c2.name, t) if t else NN
750def _tri5perturbs(eps, r):
751 # perturb the radii to handle this corner case
752 # <https://GitHub.com/mrJean1/PyGeodesy/issues/49>
753 yield _0_0
754 if eps and eps > 0:
755 p = max(eps, EPS)
756 yield p
757 m = min(p, r)
758 yield -m
759 q = max(eps * _4_0, _EPS4e8)
760 if q > p:
761 yield q
762 q = min(q, r)
763 if q > m:
764 yield -q
766# **) MIT License
767#
768# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved.
769#
770# Permission is hereby granted, free of charge, to any person obtaining a
771# copy of this software and associated documentation files (the "Software"),
772# to deal in the Software without restriction, including without limitation
773# the rights to use, copy, modify, merge, publish, distribute, sublicense,
774# and/or sell copies of the Software, and to permit persons to whom the
775# Software is furnished to do so, subject to the following conditions:
776#
777# The above copyright notice and this permission notice shall be included
778# in all copies or substantial portions of the Software.
779#
780# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
781# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
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783# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
784# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
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