Coverage for pygeodesy/vector3d.py: 98%
234 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},
7L{trilaterate2d2} and L{trilaterate3d2}.
8'''
10from pygeodesy.basics import isscalar, len2
11from pygeodesy.constants import EPS, EPS0, EPS1, EPS4, INT0, isnear0, \
12 _0_0, _1_0
13from pygeodesy.errors import IntersectionError, _ValueError, VectorError, \
14 _xError, _xkwds, _xkwds_popitem
15from pygeodesy.fmath import euclid, fabs, fdot, fsum, fsum1_, hypot, sqrt
16# from pygeodesy.fsums import fsum, fsum1_ # from .fmath
17# from pygeodesy.formy import _radical2 # in _intersects2 below
18from pygeodesy.interns import MISSING, NN, _COMMA_, _concentric_, _datum_, \
19 _h_, _height_, _intersection_, _name_, _near_, \
20 _negative_, _no_, _too_, _z_
21from pygeodesy.iters import Fmt, PointsIter
22from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY, _ALL_MODS as _MODS
23from pygeodesy.named import _xnamed, _xotherError
24from pygeodesy.namedTuples import Intersection3Tuple, NearestOn2Tuple, \
25 NearestOn6Tuple, Vector3Tuple # Vector4Tuple
26# from pygeodesy.streprs import Fmt # from .iters
27from pygeodesy.units import _fi_j2, Radius, Radius_
28from pygeodesy.utily import atan2b, sincos2d
29# from pygeodesy.vector2d import .... # in .... below
30from pygeodesy.vector3dBase import Vector3dBase
32# from math import fabs, sqrt # from .fmath
34__all__ = _ALL_LAZY.vector3d
35__version__ = '22.10.23'
38class Vector3d(Vector3dBase):
39 '''Extended 3-D vector.
41 In a geodesy context, these may be used to represent:
42 - earth-centered, earth-fixed cartesian (ECEF)
43 - n-vector representing a normal to a point on earth's surface
44 - great circle normal to vector
45 - motion vector on earth's surface
46 - etc.
47 '''
49 def bearing(self, useZ=True):
50 '''Get the "bearing" of this vector.
52 @kwarg useZ: If C{True}, use the Z component, otherwise
53 consider the Y as +Z axis.
55 @return: Bearing (compass C{degrees}), the counter-clockwise
56 angle off the +Z axis.
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 _MODS.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 _MODS.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 _MODS.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 _MODS.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 _MODS.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=NN):
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 name (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=name or self.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 _MODS.vector2d._radii11ABC(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 _MODS.vector2d.soddy4(self, point2, point3, eps=eps, useZ=True)
306 def trilaterate2d2(self, radius, center2, radius2, center3, radius3, eps=EPS, z=INT0):
307 '''Trilaterate this and two other circles, each given as a (2-D) center
308 and a radius.
310 @arg radius: Radius of this circle (same C{units} as this C{x} and C{y}.
311 @arg center2: Center of the 2nd circle (C{Cartesian}, L{Vector3d},
312 C{Vector2Tuple}, C{Vector3Tuple} or C{Vector4Tuple}).
313 @arg radius2: Radius of this circle (same C{units} as this C{x} and C{y}.
314 @arg center3: Center of the 3rd circle (C{Cartesian}, L{Vector3d},
315 C{Vector2Tuple}, C{Vector3Tuple} or C{Vector4Tuple}).
316 @arg radius3: Radius of the 3rd circle (same C{units} as this C{x} and C{y}.
317 @kwarg eps: Tolerance to check the trilaterated point I{delta} on all
318 3 circles (C{scalar}) or C{None} for no checking.
319 @kwarg z: Optional Z component of the trilaterated point (C{scalar}).
321 @return: Trilaterated point, an instance of this (sub-)class with C{z=B{z}}.
323 @raise IntersectionError: No intersection, near-concentric or -colinear
324 centers, trilateration failed some other way
325 or the trilaterated point is off one circle
326 by more than B{C{eps}}.
328 @raise TypeError: Invalid B{C{center2}} or B{C{center3}}.
330 @raise UnitError: Invalid B{C{radius1}}, B{C{radius2}} or B{C{radius3}}.
332 @see: Function L{pygeodesy.trilaterate2d2}.
333 '''
335 def _xyr3(r, **name_v):
336 v = _otherV3d(useZ=False, **name_v)
337 return v.x, v.y, r
339 try:
340 return _MODS.vector2d._trilaterate2d2(*(_xyr3(radius, center=self) +
341 _xyr3(radius2, center2=center2) +
342 _xyr3(radius3, center3=center3)),
343 eps=eps, Vector=self.classof, z=z)
344 except (AssertionError, TypeError, ValueError) as x:
345 raise _xError(x, center=self, radius=radius,
346 center2=center2, radius2=radius2,
347 center3=center3, radius3=radius3)
349 def trilaterate3d2(self, radius, center2, radius2, center3, radius3, eps=EPS):
350 '''Trilaterate this and two other spheres, each given as a (3-D) center
351 and a radius.
353 @arg radius: Radius of this sphere (same C{units} as this C{x}, C{y}
354 and C{z}).
355 @arg center2: Center of the 2nd sphere (C{Cartesian}, L{Vector3d},
356 C{Vector3Tuple} or C{Vector4Tuple}).
357 @arg radius2: Radius of this sphere (same C{units} as this C{x}, C{y}
358 and C{z}).
359 @arg center3: Center of the 3rd sphere (C{Cartesian}, , L{Vector3d},
360 C{Vector3Tuple} or C{Vector4Tuple}).
361 @arg radius3: Radius of the 3rd sphere (same C{units} as this C{x}, C{y}
362 and C{z}).
363 @kwarg eps: Pertubation tolerance (C{scalar}), same units as C{x}, C{y}
364 and C{z} or C{None} for no pertubations.
366 @return: 2-Tuple with two trilaterated points, each an instance of this
367 (sub-)class. Both points are the same instance if all three
368 spheres intersect or abut in a single point.
370 @raise ImportError: Package C{numpy} not found, not installed or
371 older than version 1.10.
373 @raise IntersectionError: Near-concentric, -colinear, too distant or
374 non-intersecting spheres or C{numpy} issue.
376 @raise NumPyError: Some C{numpy} issue.
378 @raise TypeError: Invalid B{C{center2}} or B{C{center3}}.
380 @raise UnitError: Invalid B{C{radius}}, B{C{radius2}} or B{C{radius3}}.
382 @note: Package U{numpy<https://PyPI.org/project/numpy>} is required,
383 version 1.10 or later.
385 @see: Norrdine, A. U{I{An Algebraic Solution to the Multilateration
386 Problem}<https://www.ResearchGate.net/publication/
387 275027725_An_Algebraic_Solution_to_the_Multilateration_Problem>}
388 and U{I{implementation}<https://www.ResearchGate.net/publication/
389 288825016_Trilateration_Matlab_Code>}.
390 '''
391 try:
392 c1 = _otherV3d(center=self, NN_OK=False)
393 return _MODS.vector2d._trilaterate3d2(c1, Radius_(radius, low=eps),
394 center2, radius2,
395 center3, radius3,
396 eps=eps, clas=self.classof)
397 except (AssertionError, TypeError, ValueError) as x:
398 raise _xError(x, center=self, radius=radius,
399 center2=center2, radius2=radius2,
400 center3=center3, radius3=radius3)
403def _intersect3d3(start1, end1, start2, end2, eps=EPS, useZ=False): # MCCABE 16 in .rhumbx._RhumbLine
404 # (INTERNAL) Intersect two lines, see L{intersection3d3} below,
405 # separated to allow callers to embellish any exceptions
407 def _outside(t, d2, o): # -o before start#, +o after end#
408 return -o if t < 0 else (o if t > d2 else 0) # XXX d2 + eps?
410 def _rightangle2(s1, b1, s2, useZ):
411 # Get the C{s1'} and C{e1'}, corners of a right-angle
412 # triangle with the hypotenuse thru C{s1} at bearing
413 # C{b1} and the right angle at C{s2}
414 dx, dy, d = s2.minus(s1).xyz
415 if useZ and not isnear0(d): # not supported
416 raise IntersectionError(useZ=d, bearing=b1)
417 s, c = sincos2d(b1)
418 if s and c:
419 dx *= c / s
420 dy *= s / c
421 e1 = Vector3d(s2.x, s1.y + dx, s1.z)
422 s1 = Vector3d(s1.x + dy, s2.y, s1.z)
423 else: # orthogonal
424 d = euclid(dx, dy) # hypot?
425 e1 = Vector3d(s1.x + s * d, s1.y + c * d, s1.z)
426 return s1, e1
428 s1 = x = _otherV3d(useZ=useZ, start1=start1)
429 s2 = _otherV3d(useZ=useZ, start2=start2)
430 b1 = isscalar(end1)
431 if b1: # bearing, make an e1
432 s1, e1 = _rightangle2(s1, end1, s2, useZ)
433 else:
434 e1 = _otherV3d(useZ=useZ, end1=end1)
435 b2 = isscalar(end2)
436 if b2: # bearing, make an e2
437 s2, e2 = _rightangle2(s2, end2, x, useZ)
438 else:
439 e2 = _otherV3d(useZ=useZ, end2=end2)
441 a = e1.minus(s1)
442 b = e2.minus(s2)
443 c = s2.minus(s1)
445 ab = a.cross(b)
446 d = fabs(c.dot(ab))
447 e = max(EPS0, eps or _0_0)
448 if d > EPS0 and ab.length > e: # PYCHOK no cover
449 d = d / ab.length # /= chokes PyChecker
450 if d > e: # argonic, skew lines distance
451 raise IntersectionError(skew_d=d, txt=_no_(_intersection_))
453 # co-planar, non-skew lines
454 ab2 = ab.length2
455 if ab2 < e: # colinear, parallel or null line(s)
456 x = b.length2 < a.length2
457 if x: # make C{a} the shortest
458 a, b = b, a
459 s1, s2 = s2, s1
460 e1, e2 = e2, e1
461 b1, b2 = b2, b1
462 if b.length2 < e: # PYCHOK no cover
463 if c.length < e:
464 return s1, 0, 0
465 elif e2.minus(e1).length < e:
466 return e1, 0, 0
467 elif a.length2 < e: # null (s1, e1), non-null (s2, e2)
468 # like _nearestOn2(s1, s2, e2, within=False, eps=e)
469 t = s1.minus(s2).dot(b)
470 v = s2.plus(b.times(t / b.length2))
471 if s1.minus(v).length < e:
472 o = 0 if b2 else _outside(t, b.length2, 1 if x else 2)
473 return (v, o, 0) if x else (v, 0, o)
474 raise IntersectionError(length2=ab2, txt=_no_(_intersection_))
476 cb = c.cross(b)
477 t = cb.dot(ab)
478 o1 = 0 if b1 else _outside(t, ab2, 1)
479 v = s1.plus(a.times(t / ab2))
480 o2 = 0 if b2 else _outside(v.minus(s2).dot(b), b.length2, 2)
481 return v, o1, o2
484def intersection3d3(start1, end1, start2, end2, eps=EPS, useZ=True,
485 **Vector_and_kwds):
486 '''Compute the intersection point of two lines, each defined by two
487 points or by a point and a bearing.
489 @arg start1: Start point of the first line (C{Cartesian}, L{Vector3d},
490 C{Vector3Tuple} or C{Vector4Tuple}).
491 @arg end1: End point of the first line (C{Cartesian}, L{Vector3d},
492 C{Vector3Tuple} or C{Vector4Tuple}) or the bearing at
493 B{C{start1}} (compass C{degrees}).
494 @arg start2: Start point of the second line (C{Cartesian}, L{Vector3d},
495 C{Vector3Tuple} or C{Vector4Tuple}).
496 @arg end2: End point of the second line (C{Cartesian}, L{Vector3d},
497 C{Vector3Tuple} or C{Vector4Tuple}) or the bearing at
498 B{C{start2}} (Ccompass C{degrees}).
499 @kwarg eps: Tolerance for skew line distance and length (C{EPS}).
500 @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}).
501 @kwarg Vector_and_kwds: Optional class C{B{Vector}=None} to return the
502 intersection points and optional, additional B{C{Vector}}
503 keyword arguments, otherwise B{C{start1}}'s (sub-)class.
505 @return: An L{Intersection3Tuple}C{(point, outside1, outside2)} with
506 C{point} an instance of B{C{Vector}} or B{C{start1}}'s (sub-)class.
508 @note: The C{outside} values is C{0} for lines specified by point and bearing.
510 @raise IntersectionError: Invalid, skew, non-co-planar or otherwise
511 non-intersecting lines.
513 @see: U{Line-line intersection<https://MathWorld.Wolfram.com/Line-LineIntersection.html>}
514 and U{line-line distance<https://MathWorld.Wolfram.com/Line-LineDistance.html>},
515 U{skew lines<https://MathWorld.Wolfram.com/SkewLines.html>} and U{point-line
516 distance<https://MathWorld.Wolfram.com/Point-LineDistance3-Dimensional.html>}.
517 '''
518 try:
519 v, o1, o2 = _intersect3d3(start1, end1, start2, end2, eps=eps, useZ=useZ)
520 except (TypeError, ValueError) as x:
521 raise _xError(x, start1=start1, end1=end1, start2=start2, end2=end2)
522 v = _nVc(v, **_xkwds(Vector_and_kwds, clas=start1.classof,
523 name=intersection3d3.__name__))
524 return Intersection3Tuple(v, o1, o2)
527def intersections2(center1, radius1, center2, radius2, sphere=True, **Vector_and_kwds):
528 '''Compute the intersection of two spheres or circles, each defined by a
529 (3-D) center point and a radius.
531 @arg center1: Center of the first sphere or circle (C{Cartesian}, L{Vector3d},
532 C{Vector3Tuple} or C{Vector4Tuple}).
533 @arg radius1: Radius of the first sphere or circle (same units as the
534 B{C{center1}} coordinates).
535 @arg center2: Center of the second sphere or circle (C{Cartesian}, L{Vector3d},
536 C{Vector3Tuple} or C{Vector4Tuple}).
537 @arg radius2: Radius of the second sphere or circle (same units as the
538 B{C{center1}} and B{C{center2}} coordinates).
539 @kwarg sphere: If C{True} compute the center and radius of the intersection of
540 two spheres. If C{False}, ignore the C{z}-component and compute
541 the intersection of two circles (C{bool}).
542 @kwarg Vector_and_kwds: Optional class C{B{Vector}=None} to return the
543 intersection points and optional, additional B{C{Vector}}
544 keyword arguments, otherwise B{C{center1}}'s (sub-)class.
546 @return: If B{C{sphere}} is C{True}, a 2-tuple of the C{center} and C{radius}
547 of the intersection of the I{spheres}. The C{radius} is C{0.0} for
548 abutting spheres (and the C{center} is aka I{radical center}).
550 If B{C{sphere}} is C{False}, a 2-tuple with the two intersection
551 points of the I{circles}. For abutting circles, both points are
552 the same instance, aka I{radical center}.
554 @raise IntersectionError: Concentric, invalid or non-intersecting spheres
555 or circles.
557 @raise TypeError: Invalid B{C{center1}} or B{C{center2}}.
559 @raise UnitError: Invalid B{C{radius1}} or B{C{radius2}}.
561 @see: U{Sphere-Sphere<https://MathWorld.Wolfram.com/Sphere-SphereIntersection.html>} and
562 U{Circle-Circle<https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>}
563 Intersection.
564 '''
565 try:
566 return _intersects2(center1, Radius_(radius1=radius1),
567 center2, Radius_(radius2=radius2), sphere=sphere,
568 clas=center1.classof, **Vector_and_kwds)
569 except (TypeError, ValueError) as x:
570 raise _xError(x, center1=center1, radius1=radius1, center2=center2, radius2=radius2)
573def _intersects2(center1, r1, center2, r2, sphere=True, too_d=None, # in CartesianEllipsoidalBase.intersections2,
574 **clas_Vector_and_kwds): # .ellipsoidalBaseDI._intersections2
575 # (INTERNAL) Intersect two spheres or circles, see L{intersections2}
576 # above, separated to allow callers to embellish any exceptions
578 def _nV3(x, y, z):
579 v = Vector3d(x, y, z)
580 n = intersections2.__name__
581 return _nVc(v, **_xkwds(clas_Vector_and_kwds, name=n))
583 def _xV3(c1, u, x, y):
584 xy1 = x, y, _1_0 # transform to original space
585 return _nV3(fdot(xy1, u.x, -u.y, c1.x),
586 fdot(xy1, u.y, u.x, c1.y), _0_0)
588 c1 = _otherV3d(useZ=sphere, center1=center1)
589 c2 = _otherV3d(useZ=sphere, center2=center2)
591 if r1 < r2: # r1, r2 == R, r
592 c1, c2 = c2, c1
593 r1, r2 = r2, r1
595 m = c2.minus(c1)
596 d = m.length
597 if d < max(r2 - r1, EPS):
598 raise IntersectionError(_near_(_concentric_)) # XXX ConcentricError?
600 o = fsum1_(-d, r1, r2) # overlap == -(d - (r1 + r2))
601 # compute intersections with c1 at (0, 0) and c2 at (d, 0), like
602 # <https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>
603 if o > EPS: # overlapping, r1, r2 == R, r
604 x = _MODS.formy._radical2(d, r1, r2).xline
605 y = _1_0 - (x / r1)**2
606 if y > EPS:
607 y = r1 * sqrt(y) # y == a / 2
608 elif y < 0: # PYCHOK no cover
609 raise IntersectionError(_negative_)
610 else: # abutting
611 y = _0_0
612 elif o < 0: # PYCHOK no cover
613 t = d if too_d is None else too_d
614 raise IntersectionError(_too_(Fmt.distant(t)))
615 else: # abutting
616 x, y = r1, _0_0
618 u = m.unit()
619 if sphere: # sphere center and radius
620 c = c1 if x < EPS else (
621 c2 if x > EPS1 else c1.plus(u.times(x)))
622 t = _nV3(c.x, c.y, c.z), Radius(y)
624 elif y > 0: # intersecting circles
625 t = _xV3(c1, u, x, y), _xV3(c1, u, x, -y)
626 else: # abutting circles
627 t = _xV3(c1, u, x, 0)
628 t = t, t
629 return t
632def iscolinearWith(point, point1, point2, eps=EPS, useZ=True):
633 '''Check whether a point is colinear with two other (2- or 3-D) points.
635 @arg point: The point (L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}).
636 @arg point1: First point (L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}).
637 @arg point2: Second point (L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}).
638 @kwarg eps: Tolerance (C{scalar}), same units as C{x}, C{y} and C{z}.
639 @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}).
641 @return: C{True} if B{C{point}} is colinear B{C{point1}} and B{C{point2}},
642 C{False} otherwise.
644 @raise TypeError: Invalid B{C{point}}, B{C{point1}} or B{C{point2}}.
646 @see: Function L{nearestOn}.
647 '''
648 p = _otherV3d(useZ=useZ, point=point)
649 return _MODS.vector2d._iscolinearWith(p, point1, point2, eps=eps, useZ=useZ)
652def nearestOn(point, point1, point2, within=True, useZ=True, Vector=None, **Vector_kwds):
653 '''Locate the point between two points closest to a reference (2- or 3-D).
655 @arg point: Reference point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}
656 or C{Vector4Tuple}).
657 @arg point1: Start point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or
658 C{Vector4Tuple}).
659 @arg point2: End point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or
660 C{Vector4Tuple}).
661 @kwarg within: If C{True} return the closest point between both given
662 points, otherwise the closest point on the extended line
663 through both points (C{bool}).
664 @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}).
665 @kwarg Vector: Class to return closest point (C{Cartesian}, L{Vector3d}
666 or C{Vector3Tuple}) or C{None}.
667 @kwarg Vector_kwds: Optional, additional B{C{Vector}} keyword arguments,
668 ignored if C{B{Vector} is None}.
670 @return: Closest point, either B{C{point1}} or B{C{point2}} or an instance
671 of the B{C{point}}'s (sub-)class or B{C{Vector}} if not C{None}.
673 @raise TypeError: Invalid B{C{point}}, B{C{point1}} or B{C{point2}}.
675 @see: U{3-D Point-Line Distance<https://MathWorld.Wolfram.com/Point-LineDistance3-Dimensional.html>},
676 C{Cartesian} and C{LatLon} methods C{nearestOn}, method L{sphericalTrigonometry.LatLon.nearestOn3}
677 and function L{sphericalTrigonometry.nearestOn3}.
678 '''
679 p0 = _otherV3d(useZ=useZ, point =point)
680 p1 = _otherV3d(useZ=useZ, point1=point1)
681 p2 = _otherV3d(useZ=useZ, point2=point2)
683 p, _ = _nearestOn2(p0, p1, p2, within=within)
684 if Vector is not None:
685 p = Vector(p.x, p.y, **_xkwds(Vector_kwds, z=p.z, name=nearestOn.__name__))
686 elif p is p1:
687 p = point1
688 elif p is p2:
689 p = point2
690 else: # ignore Vector_kwds
691 p = point.classof(p.x, p.y, Vector_kwds.get(_z_, p.z), name=nearestOn.__name__)
692 return p
695def _nearestOn2(p0, p1, p2, within=True, eps=EPS):
696 # (INTERNAL) Closest point and fraction, see L{nearestOn} above,
697 # separated to allow callers to embellish any exceptions
698 p21 = p2.minus(p1)
699 d2 = p21.length2
700 if d2 < eps: # coincident
701 p = p1 # ~= p2
702 t = 0
703 else: # see comments in .points.nearestOn5
704 t = p0.minus(p1).dot(p21) / d2
705 if within and t < eps:
706 p = p1
707 t = 0
708 elif within and t > (_1_0 - eps):
709 p = p2
710 t = 1
711 else:
712 p = p1.plus(p21.times(t))
713 return NearestOn2Tuple(p, t)
716def nearestOn6(point, points, closed=False, useZ=True, **Vector_and_kwds): # eps=EPS
717 '''Locate the point on a path or polygon closest to a reference point.
719 The closest point is either on and within the extent of a polygon edge or
720 the nearest of that edge's end points.
722 @arg point: Reference point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or
723 C{Vector4Tuple}).
724 @arg points: The path or polygon points (C{Cartesian}, L{Vector3d},
725 C{Vector3Tuple} or C{Vector4Tuple}[]).
726 @kwarg closed: Optionally, close the path or polygon (C{bool}).
727 @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}).
728 @kwarg Vector_and_kwds: Optional class C{B{Vector}=None} to return the closest
729 point and optional, additional B{C{Vector}} keyword
730 arguments, otherwise B{C{point}}'s (sub-)class.
732 @return: A L{NearestOn6Tuple}C{(closest, distance, fi, j, start, end)} with the
733 C{closest}, the C{start} and the C{end} point each an instance of the
734 B{C{Vector}} keyword argument of if {B{Vector}=None} or not specified,
735 an instance of the reference B{C{point}}'s (sub-)class.
737 @raise PointsError: Insufficient number of B{C{points}}
739 @raise TypeError: Non-cartesian B{C{point}} and B{C{points}}.
741 @note: Distances measured with method L{Vector3d.equirectangular}.
743 @see: Method C{LatLon.nearestOn6} or function L{nearestOn5} for geodetic points.
744 '''
745 r = _otherV3d(useZ=useZ, point=point)
746 D2 = r.equirectangular # distance squared
748 Ps = PointsIter(points, loop=1, name=nearestOn6.__name__)
749 p1 = c = s = e = _otherV3d(useZ=useZ, i=0, points=Ps[0])
750 c2 = D2(c) # == r.minus(c).length2
752 f = i = 0 # p1..p2 == points[i]..[j]
753 for j, p2 in Ps.enumerate(closed=closed):
754 p2 = _otherV3d(useZ=useZ, i=j, points=p2)
755 p, t = _nearestOn2(r, p1, p2) # within=True, eps=EPS
756 d2 = D2(p) # == r.minus(p).length2
757 if d2 < c2:
758 c2, c, s, e, f = d2, p, p1, p2, (i + t)
759 p1, i = p2, j
761 f, j = _fi_j2(f, len(Ps)) # like .ellipsoidalBaseDI._nearestOn2_
763 kwds = _xkwds(Vector_and_kwds, clas=point.classof, name=Ps.name)
764 v = _nVc(c, **kwds)
765 s = _nVc(s, **kwds) if s is not c else v
766 e = _nVc(e, **kwds) if e is not c else v
767 return NearestOn6Tuple(v, sqrt(c2), f, j, s, e)
770def _nVc(v, clas=None, name=NN, Vector=None, **Vector_kwds): # in .vector2d
771 # return a named C{Vector} or C{clas} instance
772 if Vector is not None:
773 v = Vector(v.x, v.y, v.z, **Vector_kwds)
774 elif clas is not None:
775 v = clas(v.x, v.y, v.z) # ignore Vector_kwds
776 return _xnamed(v, name) if name else v
779def _otherV3d(useZ=True, NN_OK=True, i=None, **name_v): # in .CartesianEllipsoidalBase.intersections2,
780 # check named vector instance, return Vector3d .Ellipsoid.height4, .formy.hartzell, .vector2d
781 def _name_i(name, i):
782 return name if i is None else Fmt.SQUARE(name, i)
784 name, v = _xkwds_popitem(name_v)
785 if useZ and isinstance(v, Vector3dBase):
786 return v if NN_OK or v.name else v.copy(name=_name_i(name, i))
787 try:
788 return Vector3d(v.x, v.y, (v.z if useZ else INT0), name=_name_i(name, i))
789 except AttributeError: # no .x, .y or .z attr
790 pass
791 raise _xotherError(Vector3d(0, 0, 0), v, name=_name_i(name, i), up=2)
794def parse3d(str3d, sep=_COMMA_, Vector=Vector3d, **Vector_kwds):
795 '''Parse an C{"x, y, z"} string.
797 @arg str3d: X, y and z values (C{str}).
798 @kwarg sep: Optional separator (C{str}).
799 @kwarg Vector: Optional class (L{Vector3d}).
800 @kwarg Vector_kwds: Optional B{C{Vector}} keyword arguments,
801 ignored if C{B{Vector} is None}.
803 @return: A B{C{Vector}} instance or if B{C{Vector}} is C{None},
804 a named L{Vector3Tuple}C{(x, y, z)}.
806 @raise VectorError: Invalid B{C{str3d}}.
807 '''
808 try:
809 v = [float(v.strip()) for v in str3d.split(sep)]
810 n = len(v)
811 if n != 3:
812 raise _ValueError(len=n)
813 except (TypeError, ValueError) as x:
814 raise VectorError(str3d=str3d, cause=x)
815 return _xnamed((Vector3Tuple(v) if Vector is None else # *v
816 Vector(*v, **Vector_kwds)), parse3d.__name__)
819def sumOf(vectors, Vector=Vector3d, **Vector_kwds):
820 '''Compute the vectorial sum of several vectors.
822 @arg vectors: Vectors to be added (L{Vector3d}[]).
823 @kwarg Vector: Optional class for the vectorial sum (L{Vector3d}).
824 @kwarg Vector_kwds: Optional B{C{Vector}} keyword arguments,
825 ignored if C{B{Vector} is None}.
827 @return: Vectorial sum as B{C{Vector}} or if B{C{Vector}} is
828 C{None}, a named L{Vector3Tuple}C{(x, y, z)}.
830 @raise VectorError: No B{C{vectors}}.
831 '''
832 n, vectors = len2(vectors)
833 if n < 1:
834 raise VectorError(vectors=n, txt=MISSING)
836 v = Vector3Tuple(fsum(v.x for v in vectors),
837 fsum(v.y for v in vectors),
838 fsum(v.z for v in vectors))
839 return _xnamed((v if Vector is None else
840 Vector(*v, **Vector_kwds)), sumOf.__name__)
843def trilaterate2d2(x1, y1, radius1, x2, y2, radius2, x3, y3, radius3,
844 eps=None, **Vector_and_kwds):
845 '''Trilaterate three circles, each given as a (2-D) center and a radius.
847 @arg x1: Center C{x} coordinate of the 1st circle (C{scalar}).
848 @arg y1: Center C{y} coordinate of the 1st circle (C{scalar}).
849 @arg radius1: Radius of the 1st circle (C{scalar}).
850 @arg x2: Center C{x} coordinate of the 2nd circle (C{scalar}).
851 @arg y2: Center C{y} coordinate of the 2nd circle (C{scalar}).
852 @arg radius2: Radius of the 2nd circle (C{scalar}).
853 @arg x3: Center C{x} coordinate of the 3rd circle (C{scalar}).
854 @arg y3: Center C{y} coordinate of the 3rd circle (C{scalar}).
855 @arg radius3: Radius of the 3rd circle (C{scalar}).
856 @kwarg eps: Tolerance to check the trilaterated point I{delta} on all
857 3 circles (C{scalar}) or C{None} for no checking.
858 @kwarg Vector_and_kwds: Optional class C{B{Vector}=None} to return the
859 trilateration and optional, additional B{C{Vector}}
860 keyword arguments, otherwise (L{Vector3d}).
862 @return: Trilaterated point as C{B{Vector}(x, y, **B{Vector_kwds})}
863 or L{Vector2Tuple}C{(x, y)} if C{B{Vector} is None}..
865 @raise IntersectionError: No intersection, near-concentric or -colinear
866 centers, trilateration failed some other way
867 or the trilaterated point is off one circle
868 by more than B{C{eps}}.
870 @raise UnitError: Invalid B{C{radius1}}, B{C{radius2}} or B{C{radius3}}.
872 @see: U{Issue #49<https://GitHub.com/mrJean1/PyGeodesy/issues/49>},
873 U{Find X location using 3 known (X,Y) location using trilateration
874 <https://math.StackExchange.com/questions/884807>} and function
875 L{pygeodesy.trilaterate3d2}.
876 '''
877 return _MODS.vector2d._trilaterate2d2(x1, y1, radius1,
878 x2, y2, radius2,
879 x3, y3, radius3, eps=eps, **Vector_and_kwds)
882def trilaterate3d2(center1, radius1, center2, radius2, center3, radius3,
883 eps=EPS, **Vector_and_kwds):
884 '''Trilaterate three spheres, each given as a (3-D) center and a radius.
886 @arg center1: Center of the 1st sphere (C{Cartesian}, L{Vector3d},
887 C{Vector3Tuple} or C{Vector4Tuple}).
888 @arg radius1: Radius of the 1st sphere (same C{units} as C{x}, C{y}
889 and C{z}).
890 @arg center2: Center of the 2nd sphere (C{Cartesian}, L{Vector3d},
891 C{Vector3Tuple} or C{Vector4Tuple}).
892 @arg radius2: Radius of this sphere (same C{units} as C{x}, C{y}
893 and C{z}).
894 @arg center3: Center of the 3rd sphere (C{Cartesian}, L{Vector3d},
895 C{Vector3Tuple} or C{Vector4Tuple}).
896 @arg radius3: Radius of the 3rd sphere (same C{units} as C{x}, C{y}
897 and C{z}).
898 @kwarg eps: Pertubation tolerance (C{scalar}), same units as C{x},
899 C{y} and C{z} or C{None} for no pertubations.
900 @kwarg Vector_and_kwds: Optional class C{B{Vector}=None} to return the
901 trilateration and optional, additional B{C{Vector}}
902 keyword arguments, otherwise B{C{center1}}'s
903 (sub-)class.
905 @return: 2-Tuple with two trilaterated points, each a B{C{Vector}}
906 instance. Both points are the same instance if all three
907 spheres abut/intersect in a single point.
909 @raise ImportError: Package C{numpy} not found, not installed or
910 older than version 1.10.
912 @raise IntersectionError: Near-concentric, -colinear, too distant or
913 non-intersecting spheres.
915 @raise NumPyError: Some C{numpy} issue.
917 @raise TypeError: Invalid B{C{center1}}, B{C{center2}} or B{C{center3}}.
919 @raise UnitError: Invalid B{C{radius1}}, B{C{radius2}} or B{C{radius3}}.
921 @see: Norrdine, A. U{I{An Algebraic Solution to the Multilateration
922 Problem}<https://www.ResearchGate.net/publication/
923 275027725_An_Algebraic_Solution_to_the_Multilateration_Problem>},
924 the U{I{implementation}<https://www.ResearchGate.net/publication/
925 288825016_Trilateration_Matlab_Code>} and function
926 L{pygeodesy.trilaterate2d2}.
927 '''
928 try:
929 return _MODS.vector2d._trilaterate3d2(_otherV3d(center1=center1, NN_OK=False),
930 Radius_(radius1=radius1, low=eps),
931 center2, radius2, center3, radius3, eps=eps,
932 clas=center1.classof, **Vector_and_kwds)
933 except (AssertionError, TypeError, ValueError) as x:
934 raise _xError(x, center1=center1, radius1=radius1,
935 center2=center2, radius2=radius2,
936 center3=center3, radius3=radius3)
939def _xyzhdn3(xyz, height, datum, ll): # in .cartesianBase, .nvectorBase
940 '''(INTERNAL) Get a C{(h, d, name)} 3-tuple.
941 '''
942 h = height or getattr(xyz, _height_, None) \
943 or getattr(xyz, _h_, None) \
944 or getattr(ll, _height_, None)
946 d = datum or getattr(xyz, _datum_, None) \
947 or getattr(ll, _datum_, None)
949 return h, d, getattr(xyz, _name_, NN)
952__all__ += _ALL_DOCS(intersections2, sumOf, Vector3dBase)
954# **) MIT License
955#
956# Copyright (C) 2016-2023 -- mrJean1 at Gmail -- All Rights Reserved.
957#
958# Permission is hereby granted, free of charge, to any person obtaining a
959# copy of this software and associated documentation files (the "Software"),
960# to deal in the Software without restriction, including without limitation
961# the rights to use, copy, modify, merge, publish, distribute, sublicense,
962# and/or sell copies of the Software, and to permit persons to whom the
963# Software is furnished to do so, subject to the following conditions:
964#
965# The above copyright notice and this permission notice shall be included
966# in all copies or substantial portions of the Software.
967#
968# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
969# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
970# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
971# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
972# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
973# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
974# OTHER DEALINGS IN THE SOFTWARE.