Coverage for pygeodesy/vector3d.py: 97%
235 statements
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2# -*- coding: utf-8 -*-
4u'''Extended 3-D vector class L{Vector3d} and functions.
6Function L{intersection3d3}, L{intersections2}, L{parse3d}, L{sumOf} and
7L{trilaterate3d2}.
8'''
10from pygeodesy.constants import EPS, EPS0, EPS1, EPS4, INT0, isnear0, \
11 _0_0, _1_0
12from pygeodesy.errors import IntersectionError, _ValueError, VectorError, \
13 _xattr, _xError, _xkwds, _xkwds_get, _xkwds_item2
14from pygeodesy.fmath import euclid, fabs, fdot, hypot, sqrt
15# from pygeodesy.fsums import fsum1_ # from _MODS
16# from pygeodesy.formy import _radical2 # _MODS
17from pygeodesy.interns import _COMMA_, _concentric_, _intersection_, \
18 _near_, _negative_, _no_, _too_
19from pygeodesy.iters import PointsIter, Fmt
20from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY, _ALL_MODS as _MODS
21from pygeodesy.named import _name__, _name2__, _xnamed, _xotherError
22from pygeodesy.namedTuples import Intersection3Tuple, NearestOn2Tuple, \
23 NearestOn6Tuple, _v2Cls, Vector3Tuple # Vector4Tuple
24# from pygeodesy.nvectorBase import _nsumOf # _MODS
25# from pygeodesy.streprs import Fmt # from .iters
26from pygeodesy.units import _fi_j2, _isDegrees, Radius, Radius_
27from pygeodesy.utily import atan2b, sincos2d
28# import pygeodesy.vector2d as _vector2d # _MODS.into
29from pygeodesy.vector3dBase import Vector3dBase
31# from math import fabs, sqrt # from .fmath
33__all__ = _ALL_LAZY.vector3d
34__version__ = '24.11.22'
36_vector2d = _MODS.into(vector2d=__name__)
39class Vector3d(Vector3dBase):
40 '''Extended 3-D vector.
42 In a geodesy context, these may be used to represent:
43 - n-vector, the normal to a point on the earth's surface
44 - Earth-Centered, Earth-Fixed (ECEF) cartesian (== spherical n-vector)
45 - great circle normal to the vector
46 - motion vector on the earth's surface
47 - etc.
48 '''
50 def bearing(self, useZ=True):
51 '''Get this vector's "bearing", the angle off the +Z axis, clockwise.
53 @kwarg useZ: If C{True}, use the Z component, otherwise ignore the
54 Z component and consider the +Y as the +Z axis.
56 @return: Bearing (compass C{degrees}).
57 '''
58 x, y = self.x, self.y
59 if useZ:
60 x, y = hypot(x, y), self.z
61 return atan2b(x, y)
63 def circin6(self, point2, point3, eps=EPS4):
64 '''Return the radius and center of the I{inscribed} aka I{In- circle}
65 of a (3-D) triangle formed by this and two other points.
67 @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
68 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
69 @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
70 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
71 @kwarg eps: Tolerance for function L{pygeodesy.trilaterate3d2} if
72 C{B{useZ} is True} otherwise L{pygeodesy.trilaterate2d2}.
74 @return: L{Circin6Tuple}C{(radius, center, deltas, cA, cB, cC)}. The
75 C{center} and contact points C{cA}, C{cB} and C{cC}, each an
76 instance of this (sub-)class, are co-planar with this and the
77 two given points.
79 @raise ImportError: Package C{numpy} not found, not installed or older
80 than version 1.10.
82 @raise IntersectionError: Near-coincident or -colinear points or
83 a trilateration or C{numpy} issue.
85 @raise TypeError: Invalid B{C{point2}} or B{C{point3}}.
87 @see: Function L{pygeodesy.circin6}, U{Incircle
88 <https://MathWorld.Wolfram.com/Incircle.html>} and U{Contact
89 Triangle<https://MathWorld.Wolfram.com/ContactTriangle.html>}.
90 '''
91 try:
92 return _vector2d._circin6(self, point2, point3, eps=eps, useZ=True)
93 except (AssertionError, TypeError, ValueError) as x:
94 raise _xError(x, point=self, point2=point2, point3=point3)
96 def circum3(self, point2, point3, circum=True, eps=EPS4):
97 '''Return the radius and center of the smallest circle I{through} or
98 I{containing} this and two other (3-D) points.
100 @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}
101 or C{Vector4Tuple}).
102 @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}
103 or C{Vector4Tuple}).
104 @kwarg circum: If C{True}, return the C{circumradius} and C{circumcenter},
105 always, ignoring the I{Meeus}' Type I case (C{bool}).
106 @kwarg eps: Tolerance passed to function L{pygeodesy.trilaterate3d2}.
108 @return: A L{Circum3Tuple}C{(radius, center, deltas)}. The C{center}, an
109 instance of this (sub-)class, is co-planar with this and the two
110 given points.
112 @raise ImportError: Package C{numpy} not found, not installed or older than
113 version 1.10.
115 @raise IntersectionError: Near-concentric, -coincident or -colinear points
116 or a trilateration or C{numpy} issue.
118 @raise TypeError: Invalid B{C{point2}} or B{C{point3}}.
120 @see: Function L{pygeodesy.circum3} and methods L{circum4_} and L{meeus2}.
121 '''
122 try:
123 return _vector2d._circum3(self, point2, point3, circum=circum,
124 eps=eps, useZ=True, clas=self.classof)
125 except (AssertionError, TypeError, ValueError) as x:
126 raise _xError(x, point=self, point2=point2, point3=point3, circum=circum)
128 def circum4_(self, *points):
129 '''Best-fit a sphere through this and two or more other (3-D) points.
131 @arg points: Other points (each a C{Cartesian}, L{Vector3d}, C{Vector3Tuple}
132 or C{Vector4Tuple}).
134 @return: L{Circum4Tuple}C{(radius, center, rank, residuals)} with C{center}
135 an instance if this (sub-)class.
137 @raise ImportError: Package C{numpy} not found, not installed or
138 older than version 1.10.
140 @raise NumPyError: Some C{numpy} issue.
142 @raise PointsError: Too few B{C{points}}.
144 @raise TypeError: One of the B{C{points}} invalid.
146 @see: Function L{pygeodesy.circum4_} and methods L{circum3} and L{meeus2}.
147 '''
148 return _vector2d.circum4_(self, *points, useZ=True, Vector=self.classof)
150 def iscolinearWith(self, point1, point2, eps=EPS):
151 '''Check whether this and two other (3-D) points are colinear.
153 @arg point1: One point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}
154 or C{Vector4Tuple}).
155 @arg point2: An other point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}
156 or C{Vector4Tuple}).
157 @kwarg eps: Tolerance (C{scalar}), same units as C{x},
158 C{y}, and C{z}.
160 @return: C{True} if this point is colinear with B{C{point1}} and
161 B{C{point2}}, C{False} otherwise.
163 @raise TypeError: Invalid B{C{point1}} or B{C{point2}}.
165 @see: Method L{nearestOn}.
166 '''
167 v = self if self.name else _otherV3d(NN_OK=False, this=self)
168 return _vector2d._iscolinearWith(v, point1, point2, eps=eps)
170 def meeus2(self, point2, point3, circum=False):
171 '''Return the radius and I{Meeus}' Type of the smallest circle I{through}
172 or I{containing} this and two other (3-D) points.
174 @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}
175 or C{Vector4Tuple}).
176 @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}
177 or C{Vector4Tuple}).
178 @kwarg circum: If C{True}, return the C{circumradius} and C{circumcenter}
179 always, overriding I{Meeus}' Type II case (C{bool}).
181 @return: L{Meeus2Tuple}C{(radius, Type)}, with C{Type} the C{circumcenter}
182 iff C{B{circum}=True}.
184 @raise IntersectionError: Coincident or colinear points, iff C{B{circum}=True}.
186 @raise TypeError: Invalid B{C{point2}} or B{C{point3}}.
188 @see: Function L{pygeodesy.meeus2} and methods L{circum3} and L{circum4_}.
189 '''
190 try:
191 return _vector2d._meeus2(self, point2, point3, circum, clas=self.classof)
192 except (TypeError, ValueError) as x:
193 raise _xError(x, point=self, point2=point2, point3=point3, circum=circum)
195 def nearestOn(self, point1, point2, within=True):
196 '''Locate the point between two points closest to this point.
198 @arg point1: Start point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or
199 C{Vector4Tuple}).
200 @arg point2: End point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or
201 C{Vector4Tuple}).
202 @kwarg within: If C{True}, return the closest point between the given
203 points, otherwise the closest point on the extended
204 line through both points (C{bool}).
206 @return: Closest point, either B{C{point1}} or B{C{point2}} or an instance
207 of this (sub-)class.
209 @raise TypeError: Invalid B{C{point1}} or B{C{point2}}.
211 @see: Method L{sphericalTrigonometry.LatLon.nearestOn3} and U{3-D Point-Line
212 Distance<https://MathWorld.Wolfram.com/Point-LineDistance3-Dimensional.html>}.
213 '''
214 return _nearestOn2(self, point1, point2, within=within).closest
216 def nearestOn6(self, points, closed=False, useZ=True): # eps=EPS
217 '''Locate the point on a path or polygon closest to this point.
219 The closest point is either on and within the extent of a polygon
220 edge or the nearest of that edge's end points.
222 @arg points: The path or polygon points (C{Cartesian}, L{Vector3d},
223 C{Vector3Tuple} or C{Vector4Tuple}[]).
224 @kwarg closed: Optionally, close the path or polygon (C{bool}).
225 @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}).
227 @return: A L{NearestOn6Tuple}C{(closest, distance, fi, j, start, end)}
228 with the C{closest}, the C{start} and the C{end} point each
229 an instance of this point's (sub-)class.
231 @raise PointsError: Insufficient number of B{C{points}}
233 @raise TypeError: Non-cartesian B{C{points}}.
235 @note: Distances measured with method L{Vector3d.equirectangular}.
237 @see: Function L{nearestOn6}.
238 '''
239 return nearestOn6(self, points, closed=closed, useZ=useZ) # Vector=self.classof
241 def parse(self, str3d, sep=_COMMA_, **name):
242 '''Parse an C{"x, y, z"} string to a L{Vector3d} instance.
244 @arg str3d: X, y and z string (C{str}), see function L{parse3d}.
245 @kwarg sep: Optional separator (C{str}).
246 @kwarg name: Optional instance C{B{name}=NN} (C{str}), overriding this name.
248 @return: The instance (L{Vector3d}).
250 @raise VectorError: Invalid B{C{str3d}}.
251 '''
252 return parse3d(str3d, sep=sep, Vector=self.classof, name=self._name__(name))
254 def radii11(self, point2, point3):
255 '''Return the radii of the C{Circum-}, C{In-}, I{Soddy} and C{Tangent}
256 circles of a (3-D) triangle.
258 @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
259 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
260 @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
261 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
263 @return: L{Radii11Tuple}C{(rA, rB, rC, cR, rIn, riS, roS, a, b, c, s)}.
265 @raise TriangleError: Near-coincident or -colinear points.
267 @raise TypeError: Invalid B{C{point2}} or B{C{point3}}.
269 @see: Function L{pygeodesy.radii11}, U{Incircle
270 <https://MathWorld.Wolfram.com/Incircle.html>}, U{Soddy Circles
271 <https://MathWorld.Wolfram.com/SoddyCircles.html>} and U{Tangent
272 Circles<https://MathWorld.Wolfram.com/TangentCircles.html>}.
273 '''
274 try:
275 return _vector2d._radii11ABC4(self, point2, point3, useZ=True)[0]
276 except (TypeError, ValueError) as x:
277 raise _xError(x, point=self, point2=point2, point3=point3)
279 def soddy4(self, point2, point3, eps=EPS4):
280 '''Return the radius and center of the C{inner} I{Soddy} circle of a
281 (3-D) triangle.
283 @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
284 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
285 @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
286 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
287 @kwarg eps: Tolerance for function L{pygeodesy.trilaterate3d2} if
288 C{B{useZ} is True} otherwise L{pygeodesy.trilaterate2d2}.
290 @return: L{Soddy4Tuple}C{(radius, center, deltas, outer)}. The C{center},
291 an instance of B{C{point1}}'s (sub-)class, is co-planar with the
292 three given points.
294 @raise ImportError: Package C{numpy} not found, not installed or older
295 than version 1.10.
297 @raise IntersectionError: Near-coincident or -colinear points or
298 a trilateration or C{numpy} issue.
300 @raise TypeError: Invalid B{C{point2}} or B{C{point3}}.
302 @see: Function L{pygeodesy.soddy4}.
303 '''
304 return _vector2d.soddy4(self, point2, point3, eps=eps, useZ=True)
306 def toCartesian(self, Cartesian, **Cartesian_kwds):
307 '''Return this C{Vector3d} as a C{Cartesian}.
309 @arg Cartesian: The C{Cartesian} class to use.
310 @kwarg Cartesian_kwds: Optional, additional C{Cartesian}
311 keyword arguments.
313 @return: The C{B{Cartesian}} instance.
314 '''
315 return _v2Cls(self, Cartesian, Cartesian_kwds)
317 def trilaterate2d2(self, radius, center2, radius2, center3, radius3, eps=EPS4, z=INT0):
318 '''Trilaterate this and two other circles, each given as a (2-D) center
319 and a radius.
321 @arg radius: Radius of this circle (same C{units} as this C{x} and C{y}.
322 @arg center2: Center of the 2nd circle (C{Cartesian}, L{Vector3d},
323 C{Vector2Tuple}, C{Vector3Tuple} or C{Vector4Tuple}).
324 @arg radius2: Radius of this circle (same C{units} as this C{x} and C{y}.
325 @arg center3: Center of the 3rd circle (C{Cartesian}, L{Vector3d},
326 C{Vector2Tuple}, C{Vector3Tuple} or C{Vector4Tuple}).
327 @arg radius3: Radius of the 3rd circle (same C{units} as this C{x} and C{y}.
328 @kwarg eps: Tolerance to check the trilaterated point I{delta} on all
329 3 circles (C{scalar}) or C{None} for no checking.
330 @kwarg z: Optional Z component of the trilaterated point (C{scalar}).
332 @return: Trilaterated point, an instance of this (sub-)class with C{z=B{z}}.
334 @raise IntersectionError: No intersection, near-concentric or -colinear
335 centers, trilateration failed some other way
336 or the trilaterated point is off one circle
337 by more than B{C{eps}}.
339 @raise TypeError: Invalid B{C{center2}} or B{C{center3}}.
341 @raise UnitError: Invalid B{C{radius1}}, B{C{radius2}} or B{C{radius3}}.
343 @see: Function L{pygeodesy.trilaterate2d2}.
344 '''
346 def _xyr3(r, **name_v):
347 v = _otherV3d(useZ=False, **name_v)
348 return v.x, v.y, r
350 try:
351 return _vector2d._trilaterate2d2(*(_xyr3(radius, center=self) +
352 _xyr3(radius2, center2=center2) +
353 _xyr3(radius3, center3=center3)),
354 eps=eps, Vector=self.classof, z=z)
355 except (AssertionError, TypeError, ValueError) as x:
356 raise _xError(x, center=self, radius=radius,
357 center2=center2, radius2=radius2,
358 center3=center3, radius3=radius3)
360 def trilaterate3d2(self, radius, center2, radius2, center3, radius3, eps=EPS4):
361 '''Trilaterate this and two other spheres, each given as a (3-D) center
362 and a radius.
364 @arg radius: Radius of this sphere (same C{units} as this C{x}, C{y}
365 and C{z}).
366 @arg center2: Center of the 2nd sphere (C{Cartesian}, L{Vector3d},
367 C{Vector3Tuple} or C{Vector4Tuple}).
368 @arg radius2: Radius of this sphere (same C{units} as this C{x}, C{y}
369 and C{z}).
370 @arg center3: Center of the 3rd sphere (C{Cartesian}, , L{Vector3d},
371 C{Vector3Tuple} or C{Vector4Tuple}).
372 @arg radius3: Radius of the 3rd sphere (same C{units} as this C{x}, C{y}
373 and C{z}).
374 @kwarg eps: Pertubation tolerance (C{scalar}), same units as C{x}, C{y}
375 and C{z} or C{None} for no pertubations.
377 @return: 2-Tuple with two trilaterated points, each an instance of this
378 (sub-)class. Both points are the same instance if all three
379 spheres intersect or abut in a single point.
381 @raise ImportError: Package C{numpy} not found, not installed or
382 older than version 1.10.
384 @raise IntersectionError: Near-concentric, -colinear, too distant or
385 non-intersecting spheres or C{numpy} issue.
387 @raise NumPyError: Some C{numpy} issue.
389 @raise TypeError: Invalid B{C{center2}} or B{C{center3}}.
391 @raise UnitError: Invalid B{C{radius}}, B{C{radius2}} or B{C{radius3}}.
393 @note: Package U{numpy<https://PyPI.org/project/numpy>} is required,
394 version 1.10 or later.
396 @see: Norrdine, A. U{I{An Algebraic Solution to the Multilateration
397 Problem}<https://www.ResearchGate.net/publication/275027725>}
398 and U{I{implementation}<https://www.ResearchGate.net/publication/288825016>}.
399 '''
400 try:
401 c1 = _otherV3d(center=self, NN_OK=False)
402 return _vector2d._trilaterate3d2(c1, Radius_(radius, low=eps),
403 center2, radius2,
404 center3, radius3,
405 eps=eps, clas=self.classof)
406 except (AssertionError, TypeError, ValueError) as x:
407 raise _xError(x, center=self, radius=radius,
408 center2=center2, radius2=radius2,
409 center3=center3, radius3=radius3)
412def _intersect3d3(start1, end1, start2, end2, eps=EPS, useZ=False): # MCCABE 16 in .formy.intersection2, .rhumbBase
413 # (INTERNAL) Intersect two lines, see L{intersection3d3} below,
414 # separated to allow callers to embellish any exceptions
416 def _corners2(s1, b1, s2, useZ):
417 # Get the C{s1'} and C{e1'} corners of a right-angle
418 # triangle with the hypotenuse thru C{s1} at bearing
419 # C{b1} and the right angle at C{s2}
420 dx, dy, d = s2.minus(s1).xyz3
421 if useZ and not isnear0(d): # not supported
422 raise IntersectionError(useZ=d, bearing=b1)
423 s, c = sincos2d(b1)
424 if s and c:
425 dx *= c / s
426 dy *= s / c
427 e1 = Vector3d(s2.x, s1.y + dx, s1.z)
428 s1 = Vector3d(s1.x + dy, s2.y, s1.z)
429 else: # orthogonal
430 d = euclid(dx, dy) # hypot?
431 e1 = Vector3d(s1.x + s * d, s1.y + c * d, s1.z)
432 return s1, e1
434 def _outside(t, d2, o): # -o before start#, +o after end#
435 return -o if t < 0 else (o if t > d2 else 0) # XXX d2 + eps?
437 s1 = t = _otherV3d(useZ=useZ, start1=start1)
438 s2 = _otherV3d(useZ=useZ, start2=start2)
439 b1 = _isDegrees(end1)
440 if b1: # bearing, make an e1
441 s1, e1 = _corners2(s1, end1, s2, useZ)
442 else:
443 e1 = _otherV3d(useZ=useZ, end1=end1)
444 b2 = _isDegrees(end2)
445 if b2: # bearing, make an e2
446 s2, e2 = _corners2(s2, end2, t, useZ)
447 else:
448 e2 = _otherV3d(useZ=useZ, end2=end2)
450 a = e1.minus(s1)
451 b = e2.minus(s2)
452 c = s2.minus(s1)
454 ab = a.cross(b)
455 d = fabs(c.dot(ab))
456 e = max(EPS0, eps or _0_0)
457 if d > EPS0 and ab.length > e: # PYCHOK no cover
458 d = d / ab.length # /= chokes PyChecker
459 if d > e: # argonic, skew lines distance
460 raise IntersectionError(skew_d=d, txt=_no_(_intersection_))
462 # co-planar, non-skew lines
463 ab2 = ab.length2
464 if ab2 < e: # colinear, parallel or null line(s)
465 x = a.length2 > b.length2
466 if x: # make C{a} the shortest
467 a, b = b, a
468 s1, s2 = s2, s1
469 e1, e2 = e2, e1
470 b1, b2 = b2, b1
471 if b.length2 < e: # PYCHOK no cover
472 if c.length < e:
473 return s1, 0, 0
474 elif e2.minus(e1).length < e:
475 return e1, 0, 0
476 elif a.length2 < e: # null (s1, e1), non-null (s2, e2)
477 # like _nearestOn2(s1, s2, e2, within=False, eps=e)
478 t = s1.minus(s2).dot(b)
479 v = s2.plus(b.times(t / b.length2))
480 if s1.minus(v).length < e:
481 o = 0 if b2 else _outside(t, b.length2, 1 if x else 2)
482 return (v, o, 0) if x else (v, 0, o)
483 raise IntersectionError(length2=ab2, txt=_no_(_intersection_))
485 cb = c.cross(b)
486 t = cb.dot(ab)
487 o1 = 0 if b1 else _outside(t, ab2, 1)
488 v = s1.plus(a.times(t / ab2))
489 t = v.minus(s2).dot(b)
490 o2 = 0 if b2 else _outside(t, b.length2, 2)
491 return v, o1, o2
494def intersection3d3(start1, end1, start2, end2, eps=EPS, useZ=True,
495 **Vector_and_kwds):
496 '''Compute the intersection point of two (2- or 3-D) lines, each defined
497 by two points or by a point and a bearing.
499 @arg start1: Start point of the first line (C{Cartesian}, L{Vector3d},
500 C{Vector3Tuple} or C{Vector4Tuple}).
501 @arg end1: End point of the first line (C{Cartesian}, L{Vector3d},
502 C{Vector3Tuple} or C{Vector4Tuple}) or the bearing at
503 B{C{start1}} (compass C{degrees}).
504 @arg start2: Start point of the second line (C{Cartesian}, L{Vector3d},
505 C{Vector3Tuple} or C{Vector4Tuple}).
506 @arg end2: End point of the second line (C{Cartesian}, L{Vector3d},
507 C{Vector3Tuple} or C{Vector4Tuple}) or the bearing at
508 B{C{start2}} (Ccompass C{degrees}).
509 @kwarg eps: Tolerance for skew line distance and length (C{EPS}).
510 @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}).
511 @kwarg Vector_and_kwds: Optional class C{B{Vector}=None} to return the intersection
512 points and optional, additional B{C{Vector}} keyword arguments,
513 otherwise B{C{start1}}'s (sub-)class.
515 @return: An L{Intersection3Tuple}C{(point, outside1, outside2)} with C{point}
516 an instance of B{C{Vector}} or B{C{start1}}'s (sub-)class.
518 @note: The C{outside} values is C{0} for lines specified by point and bearing.
520 @raise IntersectionError: Invalid, skew, non-co-planar or otherwise non-intersecting lines.
522 @see: U{Line-line intersection<https://MathWorld.Wolfram.com/Line-LineIntersection.html>}
523 and U{line-line distance<https://MathWorld.Wolfram.com/Line-LineDistance.html>},
524 U{skew lines<https://MathWorld.Wolfram.com/SkewLines.html>} and U{point-line
525 distance<https://MathWorld.Wolfram.com/Point-LineDistance3-Dimensional.html>}.
526 '''
527 try:
528 v, o1, o2 = _intersect3d3(start1, end1, start2, end2, eps=eps, useZ=useZ)
529 except (TypeError, ValueError) as x:
530 raise _xError(x, start1=start1, end1=end1, start2=start2, end2=end2)
531 v = _nVc(v, **_xkwds(Vector_and_kwds, clas=start1.classof,
532 name=intersection3d3.__name__))
533 return Intersection3Tuple(v, o1, o2)
536def intersections2(center1, radius1, center2, radius2, sphere=True, **Vector_and_kwds):
537 '''Compute the intersection of two spheres or circles, each defined by a (3-D)
538 center point and a radius.
540 @arg center1: Center of the first sphere or circle (C{Cartesian}, L{Vector3d},
541 C{Vector3Tuple} or C{Vector4Tuple}).
542 @arg radius1: Radius of the first sphere or circle (same units as the
543 B{C{center1}} coordinates).
544 @arg center2: Center of the second sphere or circle (C{Cartesian}, L{Vector3d},
545 C{Vector3Tuple} or C{Vector4Tuple}).
546 @arg radius2: Radius of the second sphere or circle (same units as the
547 B{C{center1}} and B{C{center2}} coordinates).
548 @kwarg sphere: If C{True}, compute the center and radius of the intersection of
549 two spheres. If C{False}, ignore the C{z}-component and compute
550 the intersection of two circles (C{bool}).
551 @kwarg Vector_and_kwds: Optional class C{B{Vector}=None} to return the intersection
552 points and optionally, additional B{C{Vector}} keyword arguments,
553 otherwise B{C{center1}}'s (sub-)class.
555 @return: If C{B{sphere} is True}, a 2-tuple of the C{center} and C{radius} of the
556 intersection of the I{spheres}. For abutting circles, C{radius} is C{0.0}
557 and C{center} is the I{radical center}.
559 If C{B{sphere} is False}, a 2-tuple with the two intersection points of the
560 I{circles}. For abutting circles, both points are the same instance, aka
561 the I{radical center}.
563 @raise IntersectionError: Concentric, invalid or non-intersecting spheres or circles.
565 @raise TypeError: Invalid B{C{center1}} or B{C{center2}}.
567 @raise UnitError: Invalid B{C{radius1}} or B{C{radius2}}.
569 @see: U{Sphere-Sphere<https://MathWorld.Wolfram.com/Sphere-SphereIntersection.html>} and
570 U{Circle-Circle<https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>}
571 Intersection.
572 '''
573 try:
574 return _intersects2(center1, Radius_(radius1=radius1),
575 center2, Radius_(radius2=radius2), sphere=sphere,
576 clas=center1.classof, **Vector_and_kwds)
577 except (TypeError, ValueError) as x:
578 raise _xError(x, center1=center1, radius1=radius1, center2=center2, radius2=radius2)
581def _intersects2(center1, r1, center2, r2, sphere=True, too_d=None, # in CartesianEllipsoidalBase.intersections2,
582 **clas_Vector_and_kwds): # .ellipsoidalBaseDI._intersections2, .formy.intersections2
583 # (INTERNAL) Intersect two spheres or circles, see L{intersections2}
584 # above, separated to allow callers to embellish any exceptions
586 def _nV3(x, y, z):
587 v = Vector3d(x, y, z)
588 n = intersections2.__name__
589 return _nVc(v, **_xkwds(clas_Vector_and_kwds, name=n))
591 def _xV3(c1, u, x, y):
592 xy1 = x, y, _1_0 # transform to original space
593 return _nV3(fdot(xy1, u.x, -u.y, c1.x),
594 fdot(xy1, u.y, u.x, c1.y), _0_0)
596 c1 = _otherV3d(useZ=sphere, center1=center1)
597 c2 = _otherV3d(useZ=sphere, center2=center2)
599 if r1 < r2: # r1, r2 == R, r
600 c1, c2 = c2, c1
601 r1, r2 = r2, r1
603 m = c2.minus(c1)
604 d = m.length
605 if d < max(r2 - r1, EPS):
606 raise IntersectionError(_near_(_concentric_)) # XXX ConcentricError?
608 o = _MODS.fsums.fsum1_(-d, r1, r2) # overlap == -(d - (r1 + r2))
609 # compute intersections with c1 at (0, 0) and c2 at (d, 0), like
610 # <https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>
611 if o > EPS: # overlapping, r1, r2 == R, r
612 x = _MODS.formy._radical2(d, r1, r2).xline
613 y = _1_0 - (x / r1)**2
614 if y > EPS:
615 y = r1 * sqrt(y) # y == a / 2
616 elif y < 0: # PYCHOK no cover
617 raise IntersectionError(_negative_)
618 else: # abutting
619 y = _0_0
620 elif o < 0: # PYCHOK no cover
621 if too_d is not None:
622 d = too_d
623 raise IntersectionError(_too_(Fmt.distant(d)))
624 else: # abutting
625 x, y = r1, _0_0
627 u = m.unit()
628 if sphere: # sphere center and radius
629 c = c1 if x < EPS else (
630 c2 if x > EPS1 else c1.plus(u.times(x)))
631 t = _nV3(c.x, c.y, c.z), Radius(y)
633 elif y > 0: # intersecting circles
634 t = _xV3(c1, u, x, y), _xV3(c1, u, x, -y)
635 else: # abutting circles
636 t = _xV3(c1, u, x, 0)
637 t = t, t
638 return t
641def iscolinearWith(point, point1, point2, eps=EPS, useZ=True):
642 '''Check whether a point is colinear with two other (2- or 3-D) points.
644 @arg point: The point (L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}).
645 @arg point1: First point (L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}).
646 @arg point2: Second point (L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}).
647 @kwarg eps: Tolerance (C{scalar}), same units as C{x}, C{y} and C{z}.
648 @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}).
650 @return: C{True} if B{C{point}} is colinear B{C{point1}} and B{C{point2}}, C{False}
651 otherwise.
653 @raise TypeError: Invalid B{C{point}}, B{C{point1}} or B{C{point2}}.
655 @see: Function L{nearestOn}.
656 '''
657 p = _otherV3d(useZ=useZ, point=point)
658 return _vector2d._iscolinearWith(p, point1, point2, eps=eps, useZ=useZ)
661def nearestOn(point, point1, point2, within=True, useZ=True, Vector=None, **Vector_kwds):
662 '''Locate the point between two points closest to a reference (2- or 3-D).
664 @arg point: Reference point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}
665 or C{Vector4Tuple}).
666 @arg point1: Start point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or
667 C{Vector4Tuple}).
668 @arg point2: End point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or
669 C{Vector4Tuple}).
670 @kwarg within: If C{True}, return the closest point between both given
671 points, otherwise the closest point on the extended line
672 through both points (C{bool}).
673 @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}).
674 @kwarg Vector: Class to return closest point (C{Cartesian}, L{Vector3d} or
675 C{Vector3Tuple}) or C{None}.
676 @kwarg Vector_kwds: Optional, additional B{C{Vector}} keyword arguments,
677 ignored if C{B{Vector} is None}.
679 @return: Closest point, either B{C{point1}} or B{C{point2}} or an instance
680 of the B{C{point}}'s (sub-)class or B{C{Vector}} if not C{None}.
682 @raise TypeError: Invalid B{C{point}}, B{C{point1}} or B{C{point2}}.
684 @see: U{3-D Point-Line Distance<https://MathWorld.Wolfram.com/Point-LineDistance3-Dimensional.html>},
685 C{Cartesian} and C{LatLon} methods C{nearestOn}, method L{sphericalTrigonometry.LatLon.nearestOn3}
686 and function L{sphericalTrigonometry.nearestOn3}.
687 '''
688 p0 = _otherV3d(useZ=useZ, point =point)
689 p1 = _otherV3d(useZ=useZ, point1=point1)
690 p2 = _otherV3d(useZ=useZ, point2=point2)
692 p, _ = _nearestOn2(p0, p1, p2, within=within)
693 if Vector is not None:
694 p = Vector(p.x, p.y, **_xkwds(Vector_kwds, z=p.z, name__=nearestOn))
695 elif p is p1:
696 p = point1
697 elif p is p2:
698 p = point2
699 else: # ignore Vector_kwds
700 p = point.classof(p.x, p.y, _xkwds_get(Vector_kwds, z=p.z), name__=nearestOn)
701 return p
704def _nearestOn2(p0, p1, p2, within=True, eps=EPS):
705 # (INTERNAL) Closest point and fraction, see L{nearestOn} above,
706 # separated to allow callers to embellish any exceptions
707 p21 = p2.minus(p1)
708 d2 = p21.length2
709 if d2 < eps: # coincident
710 p = p1 # ~= p2
711 t = 0
712 else: # see comments in .points.nearestOn5
713 t = p0.minus(p1).dot(p21) / d2
714 if within and t < eps:
715 p = p1
716 t = 0
717 elif within and t > (_1_0 - eps):
718 p = p2
719 t = 1
720 else:
721 p = p1.plus(p21.times(t))
722 return NearestOn2Tuple(p, t)
725def nearestOn6(point, points, closed=False, useZ=True, **Vector_and_kwds): # eps=EPS
726 '''Locate the point on a path or polygon closest to a reference point.
728 The closest point on each polygon edge is either the nearest of that
729 edge's end points or a point in between.
731 @arg point: Reference point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or
732 C{Vector4Tuple}).
733 @arg points: The path or polygon points (C{Cartesian}, L{Vector3d},
734 C{Vector3Tuple} or C{Vector4Tuple}[]).
735 @kwarg closed: Optionally, close the path or polygon (C{bool}).
736 @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}).
737 @kwarg Vector_and_kwds: Optional class C{B{Vector}=None} to return the closest
738 point and optionally, additional B{C{Vector}} keyword arguments,
739 otherwise B{C{point}}'s (sub-)class.
741 @return: A L{NearestOn6Tuple}C{(closest, distance, fi, j, start, end)} with the
742 C{closest}, the C{start} and the C{end} point each an instance of the
743 B{C{Vector}} keyword argument or if {B{Vector}=None} or not specified,
744 an instance of the reference B{C{point}}'s (sub-)class.
746 @raise PointsError: Insufficient number of B{C{points}}
748 @raise TypeError: Non-cartesian B{C{point}} and B{C{points}}.
750 @note: Distances measured with method L{Vector3d.equirectangular}. For
751 geodetic distances use function L{nearestOn5} or one of the
752 C{LatLon.nearestOn6} methods.
753 '''
754 r = _otherV3d(useZ=useZ, point=point)
755 D2 = r.equirectangular # distance squared
757 Ps = PointsIter(points, loop=1, name=nearestOn6.__name__)
758 p1 = c = s = e = _otherV3d(useZ=useZ, i=0, points=Ps[0])
759 c2 = D2(c) # == r.minus(c).length2
761 f = i = 0 # p1..p2 == points[i]..[j]
762 for j, p2 in Ps.enumerate(closed=closed):
763 p2 = _otherV3d(useZ=useZ, i=j, points=p2)
764 p, t = _nearestOn2(r, p1, p2) # within=True, eps=EPS
765 d2 = D2(p) # == r.minus(p).length2
766 if d2 < c2:
767 c2, c, s, e, f = d2, p, p1, p2, (i + t)
768 p1, i = p2, j
770 f, j = _fi_j2(f, len(Ps)) # like .ellipsoidalBaseDI._nearestOn2_
772 kwds = _xkwds(Vector_and_kwds, clas=point.classof, name=Ps.name)
773 v = _nVc(c, **kwds)
774 s = _nVc(s, **kwds) if s is not c else v
775 e = _nVc(e, **kwds) if e is not c else v
776 return NearestOn6Tuple(v, sqrt(c2), f, j, s, e)
779def _nVc(v, clas=None, Vector=None, **name_Vector_kwds): # in .vector2d
780 # return a named C{Vector} or C{clas} instance
781 name, kwds = _name2__(**name_Vector_kwds)
782 if Vector is not None:
783 v = Vector(v.x, v.y, v.z, **kwds)
784 elif clas is not None:
785 v = clas(v.x, v.y, v.z) # ignore Vector_kwds
786 return _xnamed(v, name) if name else v
789def _otherV3d(useZ=True, NN_OK=True, i=None, **name_vector):
790 # check named vector instance, return Vector3d
791 n, v = _xkwds_item2(name_vector)
792 n = Fmt.INDEX(n, i)
793 if useZ and isinstance(v, Vector3dBase):
794 return v if NN_OK or v.name else v.copy(name=n)
795 try:
796 return Vector3d(v.x, v.y, (v.z if useZ else INT0), name=n)
797 except AttributeError: # no .x, .y or .z attr
798 pass
799 raise _xotherError(Vector3d(0, 0, 0), v, name=n, up=2)
802def parse3d(str3d, sep=_COMMA_, Vector=Vector3d, **Vector_kwds):
803 '''Parse an C{"x, y, z"} string.
805 @arg str3d: X, y and z values (C{str}).
806 @kwarg sep: Optional separator (C{str}).
807 @kwarg Vector: Optional class (L{Vector3d}).
808 @kwarg Vector_kwds: Optional B{C{Vector}} keyword arguments,
809 ignored if C{B{Vector} is None}.
811 @return: A B{C{Vector}} instance or if C{B{Vector} is None},
812 a named L{Vector3Tuple}C{(x, y, z)}.
814 @raise VectorError: Invalid B{C{str3d}}.
815 '''
816 try:
817 v = [float(v.strip()) for v in str3d.split(sep)]
818 n = len(v)
819 if n != 3:
820 raise _ValueError(len=n)
821 except (TypeError, ValueError) as x:
822 raise VectorError(str3d=str3d, cause=x)
823 return _xnamed((Vector3Tuple(v) if Vector is None else # *v
824 Vector(*v, **Vector_kwds)), name__=parse3d) # .__name__
827def sumOf(vectors, Vector=Vector3d, **Vector_kwds):
828 '''Compute the I{vectorial} sum of two oe more vectors.
830 @arg vectors: Vectors to be added (L{Vector3d}[]).
831 @kwarg Vector: Optional class for the vectorial sum (L{Vector3d}).
832 @kwarg Vector_kwds: Optional B{C{Vector}} keyword arguments, ignored
833 if C{B{Vector} is None}.
835 @return: Vectorial sum as B{C{Vector}} or if B{C{Vector} is None},
836 a named L{Vector3Tuple}C{(x, y, z)}.
838 @raise VectorError: No B{C{vectors}}.
839 '''
840 try:
841 t = _MODS.nvectorBase._nsumOf(vectors, 0, None, {}) # no H
842 except (TypeError, ValueError) as x:
843 raise VectorError(vectors=vectors, Vector=Vector, cause=x)
844 x, y, z = t[:3]
845 return Vector3Tuple(x, y, z, name__=sumOf) if Vector is None else \
846 Vector(x, y, z, **_xkwds(Vector_kwds, name__=sumOf)) # .__name__
849def trilaterate3d2(center1, radius1, center2, radius2, center3, radius3,
850 eps=EPS, **Vector_and_kwds):
851 '''Trilaterate three spheres, each given as a (3-D) center and a radius.
853 @arg center1: Center of the 1st sphere (C{Cartesian}, L{Vector3d},
854 C{Vector3Tuple} or C{Vector4Tuple}).
855 @arg radius1: Radius of the 1st sphere (same C{units} as C{x}, C{y}
856 and C{z}).
857 @arg center2: Center of the 2nd sphere (C{Cartesian}, L{Vector3d},
858 C{Vector3Tuple} or C{Vector4Tuple}).
859 @arg radius2: Radius of this sphere (same C{units} as C{x}, C{y}
860 and C{z}).
861 @arg center3: Center of the 3rd sphere (C{Cartesian}, L{Vector3d},
862 C{Vector3Tuple} or C{Vector4Tuple}).
863 @arg radius3: Radius of the 3rd sphere (same C{units} as C{x}, C{y}
864 and C{z}).
865 @kwarg eps: Pertubation tolerance (C{scalar}), same units as C{x},
866 C{y} and C{z} or C{None} for no pertubations.
867 @kwarg Vector_and_kwds: Optional class C{B{Vector}=None} to return the
868 trilateration and optionally, additional B{C{Vector}}
869 keyword arguments, otherwise B{C{center1}}'s (sub-)class.
871 @return: 2-Tuple with two trilaterated points, each a B{C{Vector}}
872 instance. Both points are the same instance if all three
873 spheres abut/intersect in a single point.
875 @raise ImportError: Package C{numpy} not found, not installed or older
876 than version 1.10.
878 @raise IntersectionError: Near-concentric, -colinear, too distant or
879 non-intersecting spheres.
881 @raise NumPyError: Some C{numpy} issue.
883 @raise TypeError: Invalid B{C{center1}}, B{C{center2}} or B{C{center3}}.
885 @raise UnitError: Invalid B{C{radius1}}, B{C{radius2}} or B{C{radius3}}.
887 @see: Norrdine, A. U{I{An Algebraic Solution to the Multilateration
888 Problem}<https://www.ResearchGate.net/publication/275027725>},
889 the U{I{implementation}<https://www.ResearchGate.net/publication/
890 288825016>} and function L{pygeodesy.trilaterate2d2}.
891 '''
892 try:
893 return _vector2d._trilaterate3d2(_otherV3d(center1=center1, NN_OK=False),
894 Radius_(radius1=radius1, low=eps),
895 center2, radius2, center3, radius3, eps=eps,
896 clas=center1.classof, **Vector_and_kwds)
897 except (AssertionError, TypeError, ValueError) as x:
898 raise _xError(x, center1=center1, radius1=radius1,
899 center2=center2, radius2=radius2,
900 center3=center3, radius3=radius3)
903def _xyzhdlln4(xyz, height, datum, ll=None, **name): # in .cartesianBase, .nvectorBase
904 '''(INTERNAL) Get a C{(h, d, ll, name)} 4-tuple.
905 '''
906 _x = _xattr
907 h = height or _x(xyz, height=None) or _x(xyz, h=None) or _x(ll, height=None)
908 d = datum or _x(xyz, datum=None) or _x(ll, datum=None)
909 return h, d, ll, _name__(name, _or_nameof=ll)
912__all__ += _ALL_DOCS(intersections2, sumOf, Vector3dBase)
914# **) MIT License
915#
916# Copyright (C) 2016-2025 -- mrJean1 at Gmail -- All Rights Reserved.
917#
918# Permission is hereby granted, free of charge, to any person obtaining a
919# copy of this software and associated documentation files (the "Software"),
920# to deal in the Software without restriction, including without limitation
921# the rights to use, copy, modify, merge, publish, distribute, sublicense,
922# and/or sell copies of the Software, and to permit persons to whom the
923# Software is furnished to do so, subject to the following conditions:
924#
925# The above copyright notice and this permission notice shall be included
926# in all copies or substantial portions of the Software.
927#
928# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
929# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
930# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
931# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
932# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
933# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
934# OTHER DEALINGS IN THE SOFTWARE.