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, typename
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
17# from pygeodesy.internals import typename # from .constants
18from pygeodesy.interns import _COMMA_, _concentric_, _intersection_, \
19 _near_, _negative_, _no_, _too_
20from pygeodesy.iters import PointsIter, Fmt
21from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY, _ALL_MODS as _MODS
22from pygeodesy.named import _name__, _name2__, _xnamed, _xotherError
23from pygeodesy.namedTuples import Intersection3Tuple, NearestOn2Tuple, \
24 NearestOn6Tuple, _v2Cls, Vector3Tuple # Vector4Tuple
25# from pygeodesy.nvectorBase import _nsumOf # _MODS
26# from pygeodesy.streprs import Fmt # from .iters
27from pygeodesy.units import _fi_j2, _isDegrees, Radius, Radius_
28from pygeodesy.utily import atan2b, sincos2d
29# from pygeodesy import vector2d as _vector2d # _MODS.into
30from pygeodesy.vector3dBase import Vector3dBase
32# from math import fabs, sqrt # from .fmath
34__all__ = _ALL_LAZY.vector3d
35__version__ = '25.04.21'
37_vector2d = _MODS.into(vector2d=__name__)
40class Vector3d(Vector3dBase):
41 '''Extended 3-D vector.
43 In a geodesy context, these may be used to represent:
44 - n-vector, the normal to a point on the earth's surface
45 - Earth-Centered, Earth-Fixed (ECEF) cartesian (== spherical n-vector)
46 - great circle normal to the vector
47 - motion vector on the earth's surface
48 - etc.
49 '''
51 def bearing(self, useZ=True):
52 '''Get this vector's "bearing", the angle off the +Z axis, clockwise.
54 @kwarg useZ: If C{True}, use the Z component, otherwise ignore the
55 Z component and consider the +Y as the +Z axis.
57 @return: Bearing (compass C{degrees}).
58 '''
59 x, y = self.x, self.y
60 if useZ:
61 x, y = hypot(x, y), self.z
62 return atan2b(x, y)
64 def circin6(self, point2, point3, eps=EPS4):
65 '''Return the radius and center of the I{inscribed} aka I{In- circle}
66 of a (3-D) triangle formed by this and two other points.
68 @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
69 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
70 @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
71 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
72 @kwarg eps: Tolerance for function L{pygeodesy.trilaterate3d2} if
73 C{B{useZ} is True} otherwise L{pygeodesy.trilaterate2d2}.
75 @return: L{Circin6Tuple}C{(radius, center, deltas, cA, cB, cC)}. The
76 C{center} and contact points C{cA}, C{cB} and C{cC}, each an
77 instance of this (sub-)class, are co-planar with this and the
78 two given points.
80 @raise ImportError: Package C{numpy} not found, not installed or older
81 than version 1.10.
83 @raise IntersectionError: Near-coincident or -colinear points or
84 a trilateration or C{numpy} issue.
86 @raise TypeError: Invalid B{C{point2}} or B{C{point3}}.
88 @see: Function L{pygeodesy.circin6}, U{Incircle
89 <https://MathWorld.Wolfram.com/Incircle.html>} and U{Contact
90 Triangle<https://MathWorld.Wolfram.com/ContactTriangle.html>}.
91 '''
92 try:
93 return _vector2d._circin6(self, point2, point3, eps=eps, useZ=True)
94 except (AssertionError, TypeError, ValueError) as x:
95 raise _xError(x, point=self, point2=point2, point3=point3)
97 def circum3(self, point2, point3, circum=True, eps=EPS4):
98 '''Return the radius and center of the smallest circle I{through} or
99 I{containing} this and two other (3-D) points.
101 @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}
102 or C{Vector4Tuple}).
103 @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}
104 or C{Vector4Tuple}).
105 @kwarg circum: If C{True}, return the C{circumradius} and C{circumcenter},
106 always, ignoring the I{Meeus}' Type I case (C{bool}).
107 @kwarg eps: Tolerance passed to function L{pygeodesy.trilaterate3d2}.
109 @return: A L{Circum3Tuple}C{(radius, center, deltas)}. The C{center}, an
110 instance of this (sub-)class, is co-planar with this and the two
111 given points.
113 @raise ImportError: Package C{numpy} not found, not installed or older than
114 version 1.10.
116 @raise IntersectionError: Near-concentric, -coincident or -colinear points
117 or a trilateration or C{numpy} issue.
119 @raise TypeError: Invalid B{C{point2}} or B{C{point3}}.
121 @see: Function L{pygeodesy.circum3} and methods L{circum4_} and L{meeus2}.
122 '''
123 try:
124 return _vector2d._circum3(self, point2, point3, circum=circum,
125 eps=eps, useZ=True, clas=self.classof)
126 except (AssertionError, TypeError, ValueError) as x:
127 raise _xError(x, point=self, point2=point2, point3=point3, circum=circum)
129 def circum4_(self, *points):
130 '''Best-fit a sphere through this and two or more other (3-D) points.
132 @arg points: Other points (each a C{Cartesian}, L{Vector3d}, C{Vector3Tuple}
133 or C{Vector4Tuple}).
135 @return: L{Circum4Tuple}C{(radius, center, rank, residuals)} with C{center}
136 an instance if this (sub-)class.
138 @raise ImportError: Package C{numpy} not found, not installed or
139 older than version 1.10.
141 @raise NumPyError: Some C{numpy} issue.
143 @raise PointsError: Too few B{C{points}}.
145 @raise TypeError: One of the B{C{points}} invalid.
147 @see: Function L{pygeodesy.circum4_} and methods L{circum3} and L{meeus2}.
148 '''
149 return _vector2d.circum4_(self, *points, useZ=True, Vector=self.classof)
151 def iscolinearWith(self, point1, point2, eps=EPS):
152 '''Check whether this and two other (3-D) points are colinear.
154 @arg point1: One point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}
155 or C{Vector4Tuple}).
156 @arg point2: An other point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}
157 or C{Vector4Tuple}).
158 @kwarg eps: Tolerance (C{scalar}), same units as C{x},
159 C{y}, and C{z}.
161 @return: C{True} if this point is colinear with B{C{point1}} and
162 B{C{point2}}, C{False} otherwise.
164 @raise TypeError: Invalid B{C{point1}} or B{C{point2}}.
166 @see: Method L{nearestOn}.
167 '''
168 v = self if self.name else _otherV3d(NN_OK=False, this=self)
169 return _vector2d._iscolinearWith(v, point1, point2, eps=eps)
171 def meeus2(self, point2, point3, circum=False):
172 '''Return the radius and I{Meeus}' Type of the smallest circle I{through}
173 or I{containing} this and two other (3-D) points.
175 @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}
176 or C{Vector4Tuple}).
177 @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}
178 or C{Vector4Tuple}).
179 @kwarg circum: If C{True}, return the C{circumradius} and C{circumcenter}
180 always, overriding I{Meeus}' Type II case (C{bool}).
182 @return: L{Meeus2Tuple}C{(radius, Type)}, with C{Type} the C{circumcenter}
183 iff C{B{circum}=True}.
185 @raise IntersectionError: Coincident or colinear points, iff C{B{circum}=True}.
187 @raise TypeError: Invalid B{C{point2}} or B{C{point3}}.
189 @see: Function L{pygeodesy.meeus2} and methods L{circum3} and L{circum4_}.
190 '''
191 try:
192 return _vector2d._meeus2(self, point2, point3, circum, clas=self.classof)
193 except (TypeError, ValueError) as x:
194 raise _xError(x, point=self, point2=point2, point3=point3, circum=circum)
196 def nearestOn(self, point1, point2, within=True):
197 '''Locate the point between two points closest to this point.
199 @arg point1: Start point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or
200 C{Vector4Tuple}).
201 @arg point2: End point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or
202 C{Vector4Tuple}).
203 @kwarg within: If C{True}, return the closest point between the given
204 points, otherwise the closest point on the extended
205 line through both points (C{bool}).
207 @return: Closest point, either B{C{point1}} or B{C{point2}} or an instance
208 of this (sub-)class.
210 @raise TypeError: Invalid B{C{point1}} or B{C{point2}}.
212 @see: Method L{sphericalTrigonometry.LatLon.nearestOn3} and U{3-D Point-Line
213 Distance<https://MathWorld.Wolfram.com/Point-LineDistance3-Dimensional.html>}.
214 '''
215 return _nearestOn2(self, point1, point2, within=within).closest
217 def nearestOn6(self, points, closed=False, useZ=True): # eps=EPS
218 '''Locate the point on a path or polygon closest to this point.
220 The closest point is either on and within the extent of a polygon
221 edge or the nearest of that edge's end points.
223 @arg points: The path or polygon points (C{Cartesian}, L{Vector3d},
224 C{Vector3Tuple} or C{Vector4Tuple}[]).
225 @kwarg closed: Optionally, close the path or polygon (C{bool}).
226 @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}).
228 @return: A L{NearestOn6Tuple}C{(closest, distance, fi, j, start, end)}
229 with the C{closest}, the C{start} and the C{end} point each
230 an instance of this point's (sub-)class.
232 @raise PointsError: Insufficient number of B{C{points}}
234 @raise TypeError: Non-cartesian B{C{points}}.
236 @note: Distances measured with method L{Vector3d.equirectangular}.
238 @see: Function L{nearestOn6}.
239 '''
240 return nearestOn6(self, points, closed=closed, useZ=useZ) # Vector=self.classof
242 def parse(self, str3d, sep=_COMMA_, **name):
243 '''Parse an C{"x, y, z"} string to a L{Vector3d} instance.
245 @arg str3d: X, y and z string (C{str}), see function L{parse3d}.
246 @kwarg sep: Optional separator (C{str}).
247 @kwarg name: Optional instance C{B{name}=NN} (C{str}), overriding this name.
249 @return: The instance (L{Vector3d}).
251 @raise VectorError: Invalid B{C{str3d}}.
252 '''
253 return parse3d(str3d, sep=sep, Vector=self.classof, name=self._name__(name))
255 def radii11(self, point2, point3):
256 '''Return the radii of the C{Circum-}, C{In-}, I{Soddy} and C{Tangent}
257 circles of a (3-D) triangle.
259 @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
260 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
261 @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
262 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
264 @return: L{Radii11Tuple}C{(rA, rB, rC, cR, rIn, riS, roS, a, b, c, s)}.
266 @raise TriangleError: Near-coincident or -colinear points.
268 @raise TypeError: Invalid B{C{point2}} or B{C{point3}}.
270 @see: Function L{pygeodesy.radii11}, U{Incircle
271 <https://MathWorld.Wolfram.com/Incircle.html>}, U{Soddy Circles
272 <https://MathWorld.Wolfram.com/SoddyCircles.html>} and U{Tangent
273 Circles<https://MathWorld.Wolfram.com/TangentCircles.html>}.
274 '''
275 try:
276 return _vector2d._radii11ABC4(self, point2, point3, useZ=True)[0]
277 except (TypeError, ValueError) as x:
278 raise _xError(x, point=self, point2=point2, point3=point3)
280 def soddy4(self, point2, point3, eps=EPS4):
281 '''Return the radius and center of the C{inner} I{Soddy} circle of a
282 (3-D) triangle.
284 @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
285 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
286 @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
287 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
288 @kwarg eps: Tolerance for function L{pygeodesy.trilaterate3d2} if
289 C{B{useZ} is True} otherwise L{pygeodesy.trilaterate2d2}.
291 @return: L{Soddy4Tuple}C{(radius, center, deltas, outer)}. The C{center},
292 an instance of B{C{point1}}'s (sub-)class, is co-planar with the
293 three given points.
295 @raise ImportError: Package C{numpy} not found, not installed or older
296 than version 1.10.
298 @raise IntersectionError: Near-coincident or -colinear points or
299 a trilateration or C{numpy} issue.
301 @raise TypeError: Invalid B{C{point2}} or B{C{point3}}.
303 @see: Function L{pygeodesy.soddy4}.
304 '''
305 return _vector2d.soddy4(self, point2, point3, eps=eps, useZ=True)
307 def toCartesian(self, Cartesian, **Cartesian_kwds):
308 '''Return this C{Vector3d} as a C{Cartesian}.
310 @arg Cartesian: The C{Cartesian} class to use.
311 @kwarg Cartesian_kwds: Optional, additional C{Cartesian}
312 keyword arguments.
314 @return: The C{B{Cartesian}} instance.
315 '''
316 return _v2Cls(self, Cartesian, Cartesian_kwds)
318 def trilaterate2d2(self, radius, center2, radius2, center3, radius3, eps=EPS4, z=INT0):
319 '''Trilaterate this and two other circles, each given as a (2-D) center
320 and a radius.
322 @arg radius: Radius of this circle (same C{units} as this C{x} and C{y}.
323 @arg center2: Center of the 2nd circle (C{Cartesian}, L{Vector3d},
324 C{Vector2Tuple}, C{Vector3Tuple} or C{Vector4Tuple}).
325 @arg radius2: Radius of this circle (same C{units} as this C{x} and C{y}.
326 @arg center3: Center of the 3rd circle (C{Cartesian}, L{Vector3d},
327 C{Vector2Tuple}, C{Vector3Tuple} or C{Vector4Tuple}).
328 @arg radius3: Radius of the 3rd circle (same C{units} as this C{x} and C{y}.
329 @kwarg eps: Tolerance to check the trilaterated point I{delta} on all
330 3 circles (C{scalar}) or C{None} for no checking.
331 @kwarg z: Optional Z component of the trilaterated point (C{scalar}).
333 @return: Trilaterated point, an instance of this (sub-)class with C{z=B{z}}.
335 @raise IntersectionError: No intersection, near-concentric or -colinear
336 centers, trilateration failed some other way
337 or the trilaterated point is off one circle
338 by more than B{C{eps}}.
340 @raise TypeError: Invalid B{C{center2}} or B{C{center3}}.
342 @raise UnitError: Invalid B{C{radius1}}, B{C{radius2}} or B{C{radius3}}.
344 @see: Function L{pygeodesy.trilaterate2d2}.
345 '''
347 def _xyr3(r, **name_v):
348 v = _otherV3d(useZ=False, **name_v)
349 return v.x, v.y, r
351 try:
352 return _vector2d._trilaterate2d2(*(_xyr3(radius, center=self) +
353 _xyr3(radius2, center2=center2) +
354 _xyr3(radius3, center3=center3)),
355 eps=eps, Vector=self.classof, z=z)
356 except (AssertionError, TypeError, ValueError) as x:
357 raise _xError(x, center=self, radius=radius,
358 center2=center2, radius2=radius2,
359 center3=center3, radius3=radius3)
361 def trilaterate3d2(self, radius, center2, radius2, center3, radius3, eps=EPS4):
362 '''Trilaterate this and two other spheres, each given as a (3-D) center
363 and a radius.
365 @arg radius: Radius of this sphere (same C{units} as this C{x}, C{y}
366 and C{z}).
367 @arg center2: Center of the 2nd sphere (C{Cartesian}, L{Vector3d},
368 C{Vector3Tuple} or C{Vector4Tuple}).
369 @arg radius2: Radius of this sphere (same C{units} as this C{x}, C{y}
370 and C{z}).
371 @arg center3: Center of the 3rd sphere (C{Cartesian}, , L{Vector3d},
372 C{Vector3Tuple} or C{Vector4Tuple}).
373 @arg radius3: Radius of the 3rd sphere (same C{units} as this C{x}, C{y}
374 and C{z}).
375 @kwarg eps: Pertubation tolerance (C{scalar}), same units as C{x}, C{y}
376 and C{z} or C{None} for no pertubations.
378 @return: 2-Tuple with two trilaterated points, each an instance of this
379 (sub-)class. Both points are the same instance if all three
380 spheres intersect or abut in a single point.
382 @raise ImportError: Package C{numpy} not found, not installed or
383 older than version 1.10.
385 @raise IntersectionError: Near-concentric, -colinear, too distant or
386 non-intersecting spheres or C{numpy} issue.
388 @raise NumPyError: Some C{numpy} issue.
390 @raise TypeError: Invalid B{C{center2}} or B{C{center3}}.
392 @raise UnitError: Invalid B{C{radius}}, B{C{radius2}} or B{C{radius3}}.
394 @note: Package U{numpy<https://PyPI.org/project/numpy>} is required,
395 version 1.10 or later.
397 @see: Norrdine, A. U{I{An Algebraic Solution to the Multilateration
398 Problem}<https://www.ResearchGate.net/publication/275027725>}
399 and U{I{implementation}<https://www.ResearchGate.net/publication/288825016>}.
400 '''
401 try:
402 c1 = _otherV3d(center=self, NN_OK=False)
403 return _vector2d._trilaterate3d2(c1, Radius_(radius, low=eps),
404 center2, radius2,
405 center3, radius3,
406 eps=eps, clas=self.classof)
407 except (AssertionError, TypeError, ValueError) as x:
408 raise _xError(x, center=self, radius=radius,
409 center2=center2, radius2=radius2,
410 center3=center3, radius3=radius3)
413def _intersect3d3(start1, end1, start2, end2, eps=EPS, useZ=False): # MCCABE 16 in .formy.intersection2, .rhumbBase
414 # (INTERNAL) Intersect two lines, see L{intersection3d3} below,
415 # separated to allow callers to embellish any exceptions
417 def _corners2(s1, b1, s2, useZ):
418 # Get the C{s1'} and C{e1'} corners of a right-angle
419 # triangle with the hypotenuse thru C{s1} at bearing
420 # C{b1} and the right angle at C{s2}
421 dx, dy, d = s2.minus(s1).xyz3
422 if useZ and not isnear0(d): # not supported
423 raise IntersectionError(useZ=d, bearing=b1)
424 s, c = sincos2d(b1)
425 if s and c:
426 dx *= c / s
427 dy *= s / c
428 e1 = Vector3d(s2.x, s1.y + dx, s1.z)
429 s1 = Vector3d(s1.x + dy, s2.y, s1.z)
430 else: # orthogonal
431 d = euclid(dx, dy) # hypot?
432 e1 = Vector3d(s1.x + s * d, s1.y + c * d, s1.z)
433 return s1, e1
435 def _outside(t, d2, o): # -o before start#, +o after end#
436 return -o if t < 0 else (o if t > d2 else 0) # XXX d2 + eps?
438 s1 = t = _otherV3d(useZ=useZ, start1=start1)
439 s2 = _otherV3d(useZ=useZ, start2=start2)
440 b1 = _isDegrees(end1)
441 if b1: # bearing, make an e1
442 s1, e1 = _corners2(s1, end1, s2, useZ)
443 else:
444 e1 = _otherV3d(useZ=useZ, end1=end1)
445 b2 = _isDegrees(end2)
446 if b2: # bearing, make an e2
447 s2, e2 = _corners2(s2, end2, t, useZ)
448 else:
449 e2 = _otherV3d(useZ=useZ, end2=end2)
451 a = e1.minus(s1)
452 b = e2.minus(s2)
453 c = s2.minus(s1)
455 ab = a.cross(b)
456 d = fabs(c.dot(ab))
457 e = max(EPS0, eps or _0_0)
458 if d > EPS0 and ab.length > e: # PYCHOK no cover
459 d = d / ab.length # /= chokes PyChecker
460 if d > e: # argonic, skew lines distance
461 raise IntersectionError(skew_d=d, txt=_no_(_intersection_))
463 # co-planar, non-skew lines
464 ab2 = ab.length2
465 if ab2 < e: # colinear, parallel or null line(s)
466 x = a.length2 > b.length2
467 if x: # make C{a} the shortest
468 a, b = b, a
469 s1, s2 = s2, s1
470 e1, e2 = e2, e1
471 b1, b2 = b2, b1
472 if b.length2 < e: # PYCHOK no cover
473 if c.length < e:
474 return s1, 0, 0
475 elif e2.minus(e1).length < e:
476 return e1, 0, 0
477 elif a.length2 < e: # null (s1, e1), non-null (s2, e2)
478 # like _nearestOn2(s1, s2, e2, within=False, eps=e)
479 t = s1.minus(s2).dot(b)
480 v = s2.plus(b.times(t / b.length2))
481 if s1.minus(v).length < e:
482 o = 0 if b2 else _outside(t, b.length2, 1 if x else 2)
483 return (v, o, 0) if x else (v, 0, o)
484 raise IntersectionError(length2=ab2, txt=_no_(_intersection_))
486 cb = c.cross(b)
487 t = cb.dot(ab)
488 o1 = 0 if b1 else _outside(t, ab2, 1)
489 v = s1.plus(a.times(t / ab2))
490 t = v.minus(s2).dot(b)
491 o2 = 0 if b2 else _outside(t, b.length2, 2)
492 return v, o1, o2
495def intersection3d3(start1, end1, start2, end2, eps=EPS, useZ=True,
496 **Vector_and_kwds):
497 '''Compute the intersection point of two (2- or 3-D) lines, each defined
498 by two points or by a point and a bearing.
500 @arg start1: Start point of the first line (C{Cartesian}, L{Vector3d},
501 C{Vector3Tuple} or C{Vector4Tuple}).
502 @arg end1: End point of the first line (C{Cartesian}, L{Vector3d},
503 C{Vector3Tuple} or C{Vector4Tuple}) or the bearing at
504 B{C{start1}} (compass C{degrees}).
505 @arg start2: Start point of the second line (C{Cartesian}, L{Vector3d},
506 C{Vector3Tuple} or C{Vector4Tuple}).
507 @arg end2: End point of the second line (C{Cartesian}, L{Vector3d},
508 C{Vector3Tuple} or C{Vector4Tuple}) or the bearing at
509 B{C{start2}} (Ccompass C{degrees}).
510 @kwarg eps: Tolerance for skew line distance and length (C{EPS}).
511 @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}).
512 @kwarg Vector_and_kwds: Optional class C{B{Vector}=None} to return the intersection
513 points and optional, additional B{C{Vector}} keyword arguments,
514 otherwise B{C{start1}}'s (sub-)class.
516 @return: An L{Intersection3Tuple}C{(point, outside1, outside2)} with C{point}
517 an instance of B{C{Vector}} or B{C{start1}}'s (sub-)class.
519 @note: The C{outside} values is C{0} for lines specified by point and bearing.
521 @raise IntersectionError: Invalid, skew, non-co-planar or otherwise non-intersecting lines.
523 @see: U{Line-line intersection<https://MathWorld.Wolfram.com/Line-LineIntersection.html>}
524 and U{line-line distance<https://MathWorld.Wolfram.com/Line-LineDistance.html>},
525 U{skew lines<https://MathWorld.Wolfram.com/SkewLines.html>} and U{point-line
526 distance<https://MathWorld.Wolfram.com/Point-LineDistance3-Dimensional.html>}.
527 '''
528 try:
529 v, o1, o2 = _intersect3d3(start1, end1, start2, end2, eps=eps, useZ=useZ)
530 except (TypeError, ValueError) as x:
531 raise _xError(x, start1=start1, end1=end1, start2=start2, end2=end2)
532 v = _nVc(v, **_xkwds(Vector_and_kwds, clas=start1.classof,
533 name=typename(intersection3d3)))
534 return Intersection3Tuple(v, o1, o2)
537def intersections2(center1, radius1, center2, radius2, sphere=True, **Vector_and_kwds):
538 '''Compute the intersection of two spheres or circles, each defined by a (3-D)
539 center point and a radius.
541 @arg center1: Center of the first sphere or circle (C{Cartesian}, L{Vector3d},
542 C{Vector3Tuple} or C{Vector4Tuple}).
543 @arg radius1: Radius of the first sphere or circle (same units as the
544 B{C{center1}} coordinates).
545 @arg center2: Center of the second sphere or circle (C{Cartesian}, L{Vector3d},
546 C{Vector3Tuple} or C{Vector4Tuple}).
547 @arg radius2: Radius of the second sphere or circle (same units as the
548 B{C{center1}} and B{C{center2}} coordinates).
549 @kwarg sphere: If C{True}, compute the center and radius of the intersection of
550 two spheres. If C{False}, ignore the C{z}-component and compute
551 the intersection of two circles (C{bool}).
552 @kwarg Vector_and_kwds: Optional class C{B{Vector}=None} to return the intersection
553 points and optionally, additional B{C{Vector}} keyword arguments,
554 otherwise B{C{center1}}'s (sub-)class.
556 @return: If C{B{sphere} is True}, a 2-tuple of the C{center} and C{radius} of the
557 intersection of the I{spheres}. For abutting circles, C{radius} is C{0.0}
558 and C{center} is the I{radical center}.
560 If C{B{sphere} is False}, a 2-tuple with the two intersection points of the
561 I{circles}. For abutting circles, both points are the same instance, aka
562 the I{radical center}.
564 @raise IntersectionError: Concentric, invalid or non-intersecting spheres or circles.
566 @raise TypeError: Invalid B{C{center1}} or B{C{center2}}.
568 @raise UnitError: Invalid B{C{radius1}} or B{C{radius2}}.
570 @see: U{Sphere-Sphere<https://MathWorld.Wolfram.com/Sphere-SphereIntersection.html>} and
571 U{Circle-Circle<https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>}
572 Intersection.
573 '''
574 try:
575 return _intersects2(center1, Radius_(radius1=radius1),
576 center2, Radius_(radius2=radius2), sphere=sphere,
577 clas=center1.classof, **Vector_and_kwds)
578 except (TypeError, ValueError) as x:
579 raise _xError(x, center1=center1, radius1=radius1, center2=center2, radius2=radius2)
582def _intersects2(center1, r1, center2, r2, sphere=True, too_d=None, # in CartesianEllipsoidalBase.intersections2,
583 **clas_Vector_and_kwds): # .ellipsoidalBaseDI._intersections2, .formy.intersections2
584 # (INTERNAL) Intersect two spheres or circles, see L{intersections2}
585 # above, separated to allow callers to embellish any exceptions
587 def _nV3(x, y, z):
588 v = Vector3d(x, y, z)
589 n = typename(intersections2)
590 return _nVc(v, **_xkwds(clas_Vector_and_kwds, name=n))
592 def _xV3(c1, u, x, y):
593 xy1 = x, y, _1_0 # transform to original space
594 return _nV3(fdot(xy1, u.x, -u.y, c1.x),
595 fdot(xy1, u.y, u.x, c1.y), _0_0)
597 c1 = _otherV3d(useZ=sphere, center1=center1)
598 c2 = _otherV3d(useZ=sphere, center2=center2)
600 if r1 < r2: # r1, r2 == R, r
601 c1, c2 = c2, c1
602 r1, r2 = r2, r1
604 m = c2.minus(c1)
605 d = m.length
606 if d < max(r2 - r1, EPS):
607 raise IntersectionError(_near_(_concentric_)) # XXX ConcentricError?
609 o = _MODS.fsums.fsum1_(-d, r1, r2) # overlap == -(d - (r1 + r2))
610 # compute intersections with c1 at (0, 0) and c2 at (d, 0), like
611 # <https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>
612 if o > EPS: # overlapping, r1, r2 == R, r
613 x = _MODS.formy._radical2(d, r1, r2).xline
614 y = _1_0 - (x / r1)**2
615 if y > EPS:
616 y = r1 * sqrt(y) # y == a / 2
617 elif y < 0: # PYCHOK no cover
618 raise IntersectionError(_negative_)
619 else: # abutting
620 y = _0_0
621 elif o < 0: # PYCHOK no cover
622 if too_d is not None:
623 d = too_d
624 raise IntersectionError(_too_(Fmt.distant(d)))
625 else: # abutting
626 x, y = r1, _0_0
628 u = m.unit()
629 if sphere: # sphere center and radius
630 c = c1 if x < EPS else (
631 c2 if x > EPS1 else c1.plus(u.times(x)))
632 t = _nV3(c.x, c.y, c.z), Radius(y)
634 elif y > 0: # intersecting circles
635 t = _xV3(c1, u, x, y), _xV3(c1, u, x, -y)
636 else: # abutting circles
637 t = _xV3(c1, u, x, 0)
638 t = t, t
639 return t
642def iscolinearWith(point, point1, point2, eps=EPS, useZ=True):
643 '''Check whether a point is colinear with two other (2- or 3-D) points.
645 @arg point: The point (L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}).
646 @arg point1: First point (L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}).
647 @arg point2: Second point (L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}).
648 @kwarg eps: Tolerance (C{scalar}), same units as C{x}, C{y} and C{z}.
649 @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}).
651 @return: C{True} if B{C{point}} is colinear B{C{point1}} and B{C{point2}}, C{False}
652 otherwise.
654 @raise TypeError: Invalid B{C{point}}, B{C{point1}} or B{C{point2}}.
656 @see: Function L{nearestOn}.
657 '''
658 p = _otherV3d(useZ=useZ, point=point)
659 return _vector2d._iscolinearWith(p, point1, point2, eps=eps, useZ=useZ)
662def nearestOn(point, point1, point2, within=True, useZ=True, Vector=None, **Vector_kwds):
663 '''Locate the point between two points closest to a reference (2- or 3-D).
665 @arg point: Reference point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}
666 or C{Vector4Tuple}).
667 @arg point1: Start point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or
668 C{Vector4Tuple}).
669 @arg point2: End point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or
670 C{Vector4Tuple}).
671 @kwarg within: If C{True}, return the closest point between both given
672 points, otherwise the closest point on the extended line
673 through both points (C{bool}).
674 @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}).
675 @kwarg Vector: Class to return closest point (C{Cartesian}, L{Vector3d} or
676 C{Vector3Tuple}) or C{None}.
677 @kwarg Vector_kwds: Optional, additional B{C{Vector}} keyword arguments,
678 ignored if C{B{Vector} is None}.
680 @return: Closest point, either B{C{point1}} or B{C{point2}} or an instance
681 of the B{C{point}}'s (sub-)class or B{C{Vector}} if not C{None}.
683 @raise TypeError: Invalid B{C{point}}, B{C{point1}} or B{C{point2}}.
685 @see: U{3-D Point-Line Distance<https://MathWorld.Wolfram.com/Point-LineDistance3-Dimensional.html>},
686 C{Cartesian} and C{LatLon} methods C{nearestOn}, method L{sphericalTrigonometry.LatLon.nearestOn3}
687 and function L{sphericalTrigonometry.nearestOn3}.
688 '''
689 p0 = _otherV3d(useZ=useZ, point =point)
690 p1 = _otherV3d(useZ=useZ, point1=point1)
691 p2 = _otherV3d(useZ=useZ, point2=point2)
693 p, _ = _nearestOn2(p0, p1, p2, within=within)
694 if Vector is not None:
695 p = Vector(p.x, p.y, **_xkwds(Vector_kwds, z=p.z, name__=nearestOn))
696 elif p is p1:
697 p = point1
698 elif p is p2:
699 p = point2
700 else: # ignore Vector_kwds
701 p = point.classof(p.x, p.y, _xkwds_get(Vector_kwds, z=p.z), name__=nearestOn)
702 return p
705def _nearestOn2(p0, p1, p2, within=True, eps=EPS):
706 # (INTERNAL) Closest point and fraction, see L{nearestOn} above,
707 # separated to allow callers to embellish any exceptions
708 p21 = p2.minus(p1)
709 d2 = p21.length2
710 if d2 < eps: # coincident
711 p = p1 # ~= p2
712 t = 0
713 else: # see comments in .points.nearestOn5
714 t = p0.minus(p1).dot(p21) / d2
715 if within and t < eps:
716 p = p1
717 t = 0
718 elif within and t > (_1_0 - eps):
719 p = p2
720 t = 1
721 else:
722 p = p1.plus(p21.times(t))
723 return NearestOn2Tuple(p, t)
726def nearestOn6(point, points, closed=False, useZ=True, **Vector_and_kwds): # eps=EPS
727 '''Locate the point on a path or polygon closest to a reference point.
729 The closest point on each polygon edge is either the nearest of that
730 edge's end points or a point in between.
732 @arg point: Reference point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or
733 C{Vector4Tuple}).
734 @arg points: The path or polygon points (C{Cartesian}, L{Vector3d},
735 C{Vector3Tuple} or C{Vector4Tuple}[]).
736 @kwarg closed: Optionally, close the path or polygon (C{bool}).
737 @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}).
738 @kwarg Vector_and_kwds: Optional class C{B{Vector}=None} to return the closest
739 point and optionally, additional B{C{Vector}} keyword arguments,
740 otherwise B{C{point}}'s (sub-)class.
742 @return: A L{NearestOn6Tuple}C{(closest, distance, fi, j, start, end)} with the
743 C{closest}, the C{start} and the C{end} point each an instance of the
744 B{C{Vector}} keyword argument or if {B{Vector}=None} or not specified,
745 an instance of the reference B{C{point}}'s (sub-)class.
747 @raise PointsError: Insufficient number of B{C{points}}
749 @raise TypeError: Non-cartesian B{C{point}} and B{C{points}}.
751 @note: Distances measured with method L{Vector3d.equirectangular}. For
752 geodetic distances use function L{nearestOn5} or one of the
753 C{LatLon.nearestOn6} methods.
754 '''
755 r = _otherV3d(useZ=useZ, point=point)
756 D2 = r.equirectangular # distance squared
758 Ps = PointsIter(points, loop=1, name=typename(nearestOn6))
759 p1 = c = s = e = _otherV3d(useZ=useZ, i=0, points=Ps[0])
760 c2 = D2(c) # == r.minus(c).length2
762 f = i = 0 # p1..p2 == points[i]..[j]
763 for j, p2 in Ps.enumerate(closed=closed):
764 p2 = _otherV3d(useZ=useZ, i=j, points=p2)
765 p, t = _nearestOn2(r, p1, p2) # within=True, eps=EPS
766 d2 = D2(p) # == r.minus(p).length2
767 if d2 < c2:
768 c2, c, s, e, f = d2, p, p1, p2, (i + t)
769 p1, i = p2, j
771 f, j = _fi_j2(f, len(Ps)) # like .ellipsoidalBaseDI._nearestOn2_
773 kwds = _xkwds(Vector_and_kwds, clas=point.classof, name=Ps.name)
774 v = _nVc(c, **kwds)
775 s = _nVc(s, **kwds) if s is not c else v
776 e = _nVc(e, **kwds) if e is not c else v
777 return NearestOn6Tuple(v, sqrt(c2), f, j, s, e)
780def _nVc(v, clas=None, Vector=None, **name_Vector_kwds): # in .vector2d
781 # return a named C{Vector} or C{clas} instance
782 name, kwds = _name2__(**name_Vector_kwds)
783 if Vector is not None:
784 v = Vector(v.x, v.y, v.z, **kwds)
785 elif clas is not None:
786 v = clas(v.x, v.y, v.z) # ignore Vector_kwds
787 return _xnamed(v, name) if name else v
790def _otherV3d(useZ=True, NN_OK=True, i=None, **name_vector):
791 # check named vector instance, return Vector3d
792 n, v = _xkwds_item2(name_vector)
793 n = Fmt.INDEX(n, i)
794 if useZ and isinstance(v, Vector3dBase):
795 return v if NN_OK or v.name else v.copy(name=n)
796 try:
797 return Vector3d(v.x, v.y, (v.z if useZ else INT0), name=n)
798 except AttributeError: # no .x, .y or .z attr
799 pass
800 raise _xotherError(Vector3d(0, 0, 0), v, name=n, up=2)
803def parse3d(str3d, sep=_COMMA_, Vector=Vector3d, **Vector_kwds):
804 '''Parse an C{"x, y, z"} string.
806 @arg str3d: X, y and z values (C{str}).
807 @kwarg sep: Optional separator (C{str}).
808 @kwarg Vector: Optional class (L{Vector3d}).
809 @kwarg Vector_kwds: Optional B{C{Vector}} keyword arguments,
810 ignored if C{B{Vector} is None}.
812 @return: A B{C{Vector}} instance or if C{B{Vector} is None},
813 a named L{Vector3Tuple}C{(x, y, z)}.
815 @raise VectorError: Invalid B{C{str3d}}.
816 '''
817 try:
818 v = [float(v.strip()) for v in str3d.split(sep)]
819 n = len(v)
820 if n != 3:
821 raise _ValueError(len=n)
822 except (TypeError, ValueError) as x:
823 raise VectorError(str3d=str3d, cause=x)
824 return _xnamed((Vector3Tuple(v) if Vector is None else # *v
825 Vector(*v, **Vector_kwds)), name__=parse3d) # .__name__
828def sumOf(vectors, Vector=Vector3d, **Vector_kwds):
829 '''Compute the I{vectorial} sum of two oe more vectors.
831 @arg vectors: Vectors to be added (L{Vector3d}[]).
832 @kwarg Vector: Optional class for the vectorial sum (L{Vector3d}).
833 @kwarg Vector_kwds: Optional B{C{Vector}} keyword arguments, ignored
834 if C{B{Vector} is None}.
836 @return: Vectorial sum as B{C{Vector}} or if B{C{Vector} is None},
837 a named L{Vector3Tuple}C{(x, y, z)}.
839 @raise VectorError: No B{C{vectors}}.
840 '''
841 try:
842 t = _MODS.nvectorBase._nsumOf(vectors, 0, None, {}) # no H
843 except (TypeError, ValueError) as x:
844 raise VectorError(vectors=vectors, Vector=Vector, cause=x)
845 x, y, z = t[:3]
846 return Vector3Tuple(x, y, z, name__=sumOf) if Vector is None else \
847 Vector(x, y, z, **_xkwds(Vector_kwds, name__=sumOf)) # .__name__
850def trilaterate3d2(center1, radius1, center2, radius2, center3, radius3,
851 eps=EPS, **Vector_and_kwds):
852 '''Trilaterate three spheres, each given as a (3-D) center and a radius.
854 @arg center1: Center of the 1st sphere (C{Cartesian}, L{Vector3d},
855 C{Vector3Tuple} or C{Vector4Tuple}).
856 @arg radius1: Radius of the 1st sphere (same C{units} as C{x}, C{y}
857 and C{z}).
858 @arg center2: Center of the 2nd sphere (C{Cartesian}, L{Vector3d},
859 C{Vector3Tuple} or C{Vector4Tuple}).
860 @arg radius2: Radius of this sphere (same C{units} as C{x}, C{y}
861 and C{z}).
862 @arg center3: Center of the 3rd sphere (C{Cartesian}, L{Vector3d},
863 C{Vector3Tuple} or C{Vector4Tuple}).
864 @arg radius3: Radius of the 3rd sphere (same C{units} as C{x}, C{y}
865 and C{z}).
866 @kwarg eps: Pertubation tolerance (C{scalar}), same units as C{x},
867 C{y} and C{z} or C{None} for no pertubations.
868 @kwarg Vector_and_kwds: Optional class C{B{Vector}=None} to return the
869 trilateration and optionally, additional B{C{Vector}}
870 keyword arguments, otherwise B{C{center1}}'s (sub-)class.
872 @return: 2-Tuple with two trilaterated points, each a B{C{Vector}}
873 instance. Both points are the same instance if all three
874 spheres abut/intersect in a single point.
876 @raise ImportError: Package C{numpy} not found, not installed or older
877 than version 1.10.
879 @raise IntersectionError: Near-concentric, -colinear, too distant or
880 non-intersecting spheres.
882 @raise NumPyError: Some C{numpy} issue.
884 @raise TypeError: Invalid B{C{center1}}, B{C{center2}} or B{C{center3}}.
886 @raise UnitError: Invalid B{C{radius1}}, B{C{radius2}} or B{C{radius3}}.
888 @see: Norrdine, A. U{I{An Algebraic Solution to the Multilateration
889 Problem}<https://www.ResearchGate.net/publication/275027725>},
890 the U{I{implementation}<https://www.ResearchGate.net/publication/
891 288825016>} and function L{pygeodesy.trilaterate2d2}.
892 '''
893 try:
894 return _vector2d._trilaterate3d2(_otherV3d(center1=center1, NN_OK=False),
895 Radius_(radius1=radius1, low=eps),
896 center2, radius2, center3, radius3, eps=eps,
897 clas=center1.classof, **Vector_and_kwds)
898 except (AssertionError, TypeError, ValueError) as x:
899 raise _xError(x, center1=center1, radius1=radius1,
900 center2=center2, radius2=radius2,
901 center3=center3, radius3=radius3)
904def _xyzhdlln4(xyz, height, datum, ll=None, **name): # in .cartesianBase, .nvectorBase
905 '''(INTERNAL) Get a C{(h, d, ll, name)} 4-tuple.
906 '''
907 _x = _xattr
908 h = height or _x(xyz, height=None) or _x(xyz, h=None) or _x(ll, height=None)
909 d = datum or _x(xyz, datum=None) or _x(ll, datum=None)
910 return h, d, ll, _name__(name, _or_nameof=ll)
913__all__ += _ALL_DOCS(intersections2, sumOf, Vector3dBase)
915# **) MIT License
916#
917# Copyright (C) 2016-2025 -- mrJean1 at Gmail -- All Rights Reserved.
918#
919# Permission is hereby granted, free of charge, to any person obtaining a
920# copy of this software and associated documentation files (the "Software"),
921# to deal in the Software without restriction, including without limitation
922# the rights to use, copy, modify, merge, publish, distribute, sublicense,
923# and/or sell copies of the Software, and to permit persons to whom the
924# Software is furnished to do so, subject to the following conditions:
925#
926# The above copyright notice and this permission notice shall be included
927# in all copies or substantial portions of the Software.
928#
929# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
930# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
931# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
932# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
933# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
934# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
935# OTHER DEALINGS IN THE SOFTWARE.