Coverage for pygeodesy/vector3d.py: 96%
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.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, fsum1_
15# from pygeodesy.fsums import fsum1_ # from .fmath
16# from pygeodesy.formy import _radical2 # in _intersects2 below
17from pygeodesy.interns import NN, _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 _xnamed, _xotherError
22from pygeodesy.namedTuples import Intersection3Tuple, NearestOn2Tuple, \
23 NearestOn6Tuple, 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# from pygeodesy.vector2d import .... # in .... below
29from pygeodesy.vector3dBase import Vector3dBase
31# from math import fabs, sqrt # from .fmath
33__all__ = _ALL_LAZY.vector3d
34__version__ = '24.02.20'
37class Vector3d(Vector3dBase):
38 '''Extended 3-D vector.
40 In a geodesy context, these may be used to represent:
41 - n-vector, the normal to a point on the earth's surface
42 - Earth-Centered, Earth-Fixed (ECEF) cartesian (== spherical n-vector)
43 - great circle normal to the vector
44 - motion vector on the earth's surface
45 - etc.
46 '''
48 def bearing(self, useZ=True):
49 '''Get this vector's "bearing", the angle off the +Z axis, clockwise.
51 @kwarg useZ: If C{True}, use the Z component, otherwise ignore the
52 Z component and consider the +Y as the +Z axis.
54 @return: Bearing (compass C{degrees}).
55 '''
56 x, y = self.x, self.y
57 if useZ:
58 x, y = hypot(x, y), self.z
59 return atan2b(x, y)
61 def circin6(self, point2, point3, eps=EPS4):
62 '''Return the radius and center of the I{inscribed} aka I{In- circle}
63 of a (3-D) triangle formed by this and two other points.
65 @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
66 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
67 @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
68 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
69 @kwarg eps: Tolerance for function L{pygeodesy.trilaterate3d2} if
70 C{B{useZ} is True} otherwise L{pygeodesy.trilaterate2d2}.
72 @return: L{Circin6Tuple}C{(radius, center, deltas, cA, cB, cC)}. The
73 C{center} and contact points C{cA}, C{cB} and C{cC}, each an
74 instance of this (sub-)class, are co-planar with this and the
75 two given points.
77 @raise ImportError: Package C{numpy} not found, not installed or older
78 than version 1.10.
80 @raise IntersectionError: Near-coincident or -colinear points or
81 a trilateration or C{numpy} issue.
83 @raise TypeError: Invalid B{C{point2}} or B{C{point3}}.
85 @see: Function L{pygeodesy.circin6}, U{Incircle
86 <https://MathWorld.Wolfram.com/Incircle.html>} and U{Contact
87 Triangle<https://MathWorld.Wolfram.com/ContactTriangle.html>}.
88 '''
89 try:
90 return _MODS.vector2d._circin6(self, point2, point3, eps=eps, useZ=True)
91 except (AssertionError, TypeError, ValueError) as x:
92 raise _xError(x, point=self, point2=point2, point3=point3)
94 def circum3(self, point2, point3, circum=True, eps=EPS4):
95 '''Return the radius and center of the smallest circle I{through} or
96 I{containing} this and two other (3-D) points.
98 @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}
99 or C{Vector4Tuple}).
100 @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}
101 or C{Vector4Tuple}).
102 @kwarg circum: If C{True} return the C{circumradius} and C{circumcenter},
103 always, ignoring the I{Meeus}' Type I case (C{bool}).
104 @kwarg eps: Tolerance passed to function L{pygeodesy.trilaterate3d2}.
106 @return: A L{Circum3Tuple}C{(radius, center, deltas)}. The C{center}, an
107 instance of this (sub-)class, is co-planar with this and the two
108 given points.
110 @raise ImportError: Package C{numpy} not found, not installed or older than
111 version 1.10.
113 @raise IntersectionError: Near-concentric, -coincident or -colinear points
114 or a trilateration or C{numpy} issue.
116 @raise TypeError: Invalid B{C{point2}} or B{C{point3}}.
118 @see: Function L{pygeodesy.circum3} and methods L{circum4_} and L{meeus2}.
119 '''
120 try:
121 return _MODS.vector2d._circum3(self, point2, point3, circum=circum,
122 eps=eps, useZ=True, clas=self.classof)
123 except (AssertionError, TypeError, ValueError) as x:
124 raise _xError(x, point=self, point2=point2, point3=point3, circum=circum)
126 def circum4_(self, *points):
127 '''Best-fit a sphere through this and two or more other (3-D) points.
129 @arg points: Other points (each a C{Cartesian}, L{Vector3d}, C{Vector3Tuple}
130 or C{Vector4Tuple}).
132 @return: L{Circum4Tuple}C{(radius, center, rank, residuals)} with C{center}
133 an instance if this (sub-)class.
135 @raise ImportError: Package C{numpy} not found, not installed or
136 older than version 1.10.
138 @raise NumPyError: Some C{numpy} issue.
140 @raise PointsError: Too few B{C{points}}.
142 @raise TypeError: One of the B{C{points}} invalid.
144 @see: Function L{pygeodesy.circum4_} and methods L{circum3} and L{meeus2}.
145 '''
146 return _MODS.vector2d.circum4_(self, *points, useZ=True, Vector=self.classof)
148 def iscolinearWith(self, point1, point2, eps=EPS):
149 '''Check whether this and two other (3-D) points are colinear.
151 @arg point1: One point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}
152 or C{Vector4Tuple}).
153 @arg point2: An other point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}
154 or C{Vector4Tuple}).
155 @kwarg eps: Tolerance (C{scalar}), same units as C{x},
156 C{y}, and C{z}.
158 @return: C{True} if this point is colinear with B{C{point1}} and
159 B{C{point2}}, C{False} otherwise.
161 @raise TypeError: Invalid B{C{point1}} or B{C{point2}}.
163 @see: Method L{nearestOn}.
164 '''
165 v = self if self.name else _otherV3d(NN_OK=False, this=self)
166 return _MODS.vector2d._iscolinearWith(v, point1, point2, eps=eps)
168 def meeus2(self, point2, point3, circum=False):
169 '''Return the radius and I{Meeus}' Type of the smallest circle I{through}
170 or I{containing} this and two other (3-D) points.
172 @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}
173 or C{Vector4Tuple}).
174 @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}
175 or C{Vector4Tuple}).
176 @kwarg circum: If C{True} return the C{circumradius} and C{circumcenter}
177 always, overriding I{Meeus}' Type II case (C{bool}).
179 @return: L{Meeus2Tuple}C{(radius, Type)}, with C{Type} the C{circumcenter}
180 iff C{B{circum}=True}.
182 @raise IntersectionError: Coincident or colinear points, iff C{B{circum}=True}.
184 @raise TypeError: Invalid B{C{point2}} or B{C{point3}}.
186 @see: Function L{pygeodesy.meeus2} and methods L{circum3} and L{circum4_}.
187 '''
188 try:
189 return _MODS.vector2d._meeus2(self, point2, point3, circum, clas=self.classof)
190 except (TypeError, ValueError) as x:
191 raise _xError(x, point=self, point2=point2, point3=point3, circum=circum)
193 def nearestOn(self, point1, point2, within=True):
194 '''Locate the point between two points closest to this point.
196 @arg point1: Start point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or
197 C{Vector4Tuple}).
198 @arg point2: End point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or
199 C{Vector4Tuple}).
200 @kwarg within: If C{True} return the closest point between the given
201 points, otherwise the closest point on the extended
202 line through both points (C{bool}).
204 @return: Closest point, either B{C{point1}} or B{C{point2}} or an instance
205 of this (sub-)class.
207 @raise TypeError: Invalid B{C{point1}} or B{C{point2}}.
209 @see: Method L{sphericalTrigonometry.LatLon.nearestOn3} and U{3-D Point-Line
210 Distance<https://MathWorld.Wolfram.com/Point-LineDistance3-Dimensional.html>}.
211 '''
212 return _nearestOn2(self, point1, point2, within=within).closest
214 def nearestOn6(self, points, closed=False, useZ=True): # eps=EPS
215 '''Locate the point on a path or polygon closest to this point.
217 The closest point is either on and within the extent of a polygon
218 edge or the nearest of that edge's end points.
220 @arg points: The path or polygon points (C{Cartesian}, L{Vector3d},
221 C{Vector3Tuple} or C{Vector4Tuple}[]).
222 @kwarg closed: Optionally, close the path or polygon (C{bool}).
223 @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}).
225 @return: A L{NearestOn6Tuple}C{(closest, distance, fi, j, start, end)}
226 with the C{closest}, the C{start} and the C{end} point each
227 an instance of this point's (sub-)class.
229 @raise PointsError: Insufficient number of B{C{points}}
231 @raise TypeError: Non-cartesian B{C{points}}.
233 @note: Distances measured with method L{Vector3d.equirectangular}.
235 @see: Function L{nearestOn6}.
236 '''
237 return nearestOn6(self, points, closed=closed, useZ=useZ) # Vector=self.classof
239 def parse(self, str3d, sep=_COMMA_, name=NN):
240 '''Parse an C{"x, y, z"} string to a L{Vector3d} instance.
242 @arg str3d: X, y and z string (C{str}), see function L{parse3d}.
243 @kwarg sep: Optional separator (C{str}).
244 @kwarg name: Optional instance name (C{str}), overriding this name.
246 @return: The instance (L{Vector3d}).
248 @raise VectorError: Invalid B{C{str3d}}.
249 '''
250 return parse3d(str3d, sep=sep, Vector=self.classof, name=name or self.name)
252 def radii11(self, point2, point3):
253 '''Return the radii of the C{Circum-}, C{In-}, I{Soddy} and C{Tangent}
254 circles of a (3-D) triangle.
256 @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
257 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
258 @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
259 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
261 @return: L{Radii11Tuple}C{(rA, rB, rC, cR, rIn, riS, roS, a, b, c, s)}.
263 @raise TriangleError: Near-coincident or -colinear points.
265 @raise TypeError: Invalid B{C{point2}} or B{C{point3}}.
267 @see: Function L{pygeodesy.radii11}, U{Incircle
268 <https://MathWorld.Wolfram.com/Incircle.html>}, U{Soddy Circles
269 <https://MathWorld.Wolfram.com/SoddyCircles.html>} and U{Tangent
270 Circles<https://MathWorld.Wolfram.com/TangentCircles.html>}.
271 '''
272 try:
273 return _MODS.vector2d._radii11ABC(self, point2, point3, useZ=True)[0]
274 except (TypeError, ValueError) as x:
275 raise _xError(x, point=self, point2=point2, point3=point3)
277 def soddy4(self, point2, point3, eps=EPS4):
278 '''Return the radius and center of the C{inner} I{Soddy} circle of a
279 (3-D) triangle.
281 @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
282 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
283 @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple},
284 C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}).
285 @kwarg eps: Tolerance for function L{pygeodesy.trilaterate3d2} if
286 C{B{useZ} is True} otherwise L{pygeodesy.trilaterate2d2}.
288 @return: L{Soddy4Tuple}C{(radius, center, deltas, outer)}. The C{center},
289 an instance of B{C{point1}}'s (sub-)class, is co-planar with the
290 three given points.
292 @raise ImportError: Package C{numpy} not found, not installed or older
293 than version 1.10.
295 @raise IntersectionError: Near-coincident or -colinear points or
296 a trilateration or C{numpy} issue.
298 @raise TypeError: Invalid B{C{point2}} or B{C{point3}}.
300 @see: Function L{pygeodesy.soddy4}.
301 '''
302 return _MODS.vector2d.soddy4(self, point2, point3, eps=eps, useZ=True)
304 def trilaterate2d2(self, radius, center2, radius2, center3, radius3, eps=EPS, z=INT0):
305 '''Trilaterate this and two other circles, each given as a (2-D) center
306 and a radius.
308 @arg radius: Radius of this circle (same C{units} as this C{x} and C{y}.
309 @arg center2: Center of the 2nd circle (C{Cartesian}, L{Vector3d},
310 C{Vector2Tuple}, C{Vector3Tuple} or C{Vector4Tuple}).
311 @arg radius2: Radius of this circle (same C{units} as this C{x} and C{y}.
312 @arg center3: Center of the 3rd circle (C{Cartesian}, L{Vector3d},
313 C{Vector2Tuple}, C{Vector3Tuple} or C{Vector4Tuple}).
314 @arg radius3: Radius of the 3rd circle (same C{units} as this C{x} and C{y}.
315 @kwarg eps: Tolerance to check the trilaterated point I{delta} on all
316 3 circles (C{scalar}) or C{None} for no checking.
317 @kwarg z: Optional Z component of the trilaterated point (C{scalar}).
319 @return: Trilaterated point, an instance of this (sub-)class with C{z=B{z}}.
321 @raise IntersectionError: No intersection, near-concentric or -colinear
322 centers, trilateration failed some other way
323 or the trilaterated point is off one circle
324 by more than B{C{eps}}.
326 @raise TypeError: Invalid B{C{center2}} or B{C{center3}}.
328 @raise UnitError: Invalid B{C{radius1}}, B{C{radius2}} or B{C{radius3}}.
330 @see: Function L{pygeodesy.trilaterate2d2}.
331 '''
333 def _xyr3(r, **name_v):
334 v = _otherV3d(useZ=False, **name_v)
335 return v.x, v.y, r
337 try:
338 return _MODS.vector2d._trilaterate2d2(*(_xyr3(radius, center=self) +
339 _xyr3(radius2, center2=center2) +
340 _xyr3(radius3, center3=center3)),
341 eps=eps, Vector=self.classof, z=z)
342 except (AssertionError, TypeError, ValueError) as x:
343 raise _xError(x, center=self, radius=radius,
344 center2=center2, radius2=radius2,
345 center3=center3, radius3=radius3)
347 def trilaterate3d2(self, radius, center2, radius2, center3, radius3, eps=EPS):
348 '''Trilaterate this and two other spheres, each given as a (3-D) center
349 and a radius.
351 @arg radius: Radius of this sphere (same C{units} as this C{x}, C{y}
352 and C{z}).
353 @arg center2: Center of the 2nd sphere (C{Cartesian}, L{Vector3d},
354 C{Vector3Tuple} or C{Vector4Tuple}).
355 @arg radius2: Radius of this sphere (same C{units} as this C{x}, C{y}
356 and C{z}).
357 @arg center3: Center of the 3rd sphere (C{Cartesian}, , L{Vector3d},
358 C{Vector3Tuple} or C{Vector4Tuple}).
359 @arg radius3: Radius of the 3rd sphere (same C{units} as this C{x}, C{y}
360 and C{z}).
361 @kwarg eps: Pertubation tolerance (C{scalar}), same units as C{x}, C{y}
362 and C{z} or C{None} for no pertubations.
364 @return: 2-Tuple with two trilaterated points, each an instance of this
365 (sub-)class. Both points are the same instance if all three
366 spheres intersect or abut in a single point.
368 @raise ImportError: Package C{numpy} not found, not installed or
369 older than version 1.10.
371 @raise IntersectionError: Near-concentric, -colinear, too distant or
372 non-intersecting spheres or C{numpy} issue.
374 @raise NumPyError: Some C{numpy} issue.
376 @raise TypeError: Invalid B{C{center2}} or B{C{center3}}.
378 @raise UnitError: Invalid B{C{radius}}, B{C{radius2}} or B{C{radius3}}.
380 @note: Package U{numpy<https://PyPI.org/project/numpy>} is required,
381 version 1.10 or later.
383 @see: Norrdine, A. U{I{An Algebraic Solution to the Multilateration
384 Problem}<https://www.ResearchGate.net/publication/275027725>}
385 and U{I{implementation}<https://www.ResearchGate.net/publication/288825016>}.
386 '''
387 try:
388 c1 = _otherV3d(center=self, NN_OK=False)
389 return _MODS.vector2d._trilaterate3d2(c1, Radius_(radius, low=eps),
390 center2, radius2,
391 center3, radius3,
392 eps=eps, clas=self.classof)
393 except (AssertionError, TypeError, ValueError) as x:
394 raise _xError(x, center=self, radius=radius,
395 center2=center2, radius2=radius2,
396 center3=center3, radius3=radius3)
399def _intersect3d3(start1, end1, start2, end2, eps=EPS, useZ=False): # MCCABE 16 in .formy.intersection2, .rhumbBase
400 # (INTERNAL) Intersect two lines, see L{intersection3d3} below,
401 # separated to allow callers to embellish any exceptions
403 def _corners2(s1, b1, s2, useZ):
404 # Get the C{s1'} and C{e1'} corners of a right-angle
405 # triangle with the hypotenuse thru C{s1} at bearing
406 # C{b1} and the right angle at C{s2}
407 dx, dy, d = s2.minus(s1).xyz
408 if useZ and not isnear0(d): # not supported
409 raise IntersectionError(useZ=d, bearing=b1)
410 s, c = sincos2d(b1)
411 if s and c:
412 dx *= c / s
413 dy *= s / c
414 e1 = Vector3d(s2.x, s1.y + dx, s1.z)
415 s1 = Vector3d(s1.x + dy, s2.y, s1.z)
416 else: # orthogonal
417 d = euclid(dx, dy) # hypot?
418 e1 = Vector3d(s1.x + s * d, s1.y + c * d, s1.z)
419 return s1, e1
421 def _outside(t, d2, o): # -o before start#, +o after end#
422 return -o if t < 0 else (o if t > d2 else 0) # XXX d2 + eps?
424 s1 = t = _otherV3d(useZ=useZ, start1=start1)
425 s2 = _otherV3d(useZ=useZ, start2=start2)
426 b1 = _isDegrees(end1)
427 if b1: # bearing, make an e1
428 s1, e1 = _corners2(s1, end1, s2, useZ)
429 else:
430 e1 = _otherV3d(useZ=useZ, end1=end1)
431 b2 = _isDegrees(end2)
432 if b2: # bearing, make an e2
433 s2, e2 = _corners2(s2, end2, t, useZ)
434 else:
435 e2 = _otherV3d(useZ=useZ, end2=end2)
437 a = e1.minus(s1)
438 b = e2.minus(s2)
439 c = s2.minus(s1)
441 ab = a.cross(b)
442 d = fabs(c.dot(ab))
443 e = max(EPS0, eps or _0_0)
444 if d > EPS0 and ab.length > e: # PYCHOK no cover
445 d = d / ab.length # /= chokes PyChecker
446 if d > e: # argonic, skew lines distance
447 raise IntersectionError(skew_d=d, txt=_no_(_intersection_))
449 # co-planar, non-skew lines
450 ab2 = ab.length2
451 if ab2 < e: # colinear, parallel or null line(s)
452 x = a.length2 > b.length2
453 if x: # make C{a} the shortest
454 a, b = b, a
455 s1, s2 = s2, s1
456 e1, e2 = e2, e1
457 b1, b2 = b2, b1
458 if b.length2 < e: # PYCHOK no cover
459 if c.length < e:
460 return s1, 0, 0
461 elif e2.minus(e1).length < e:
462 return e1, 0, 0
463 elif a.length2 < e: # null (s1, e1), non-null (s2, e2)
464 # like _nearestOn2(s1, s2, e2, within=False, eps=e)
465 t = s1.minus(s2).dot(b)
466 v = s2.plus(b.times(t / b.length2))
467 if s1.minus(v).length < e:
468 o = 0 if b2 else _outside(t, b.length2, 1 if x else 2)
469 return (v, o, 0) if x else (v, 0, o)
470 raise IntersectionError(length2=ab2, txt=_no_(_intersection_))
472 cb = c.cross(b)
473 t = cb.dot(ab)
474 o1 = 0 if b1 else _outside(t, ab2, 1)
475 v = s1.plus(a.times(t / ab2))
476 o2 = 0 if b2 else _outside(v.minus(s2).dot(b), b.length2, 2)
477 return v, o1, o2
480def intersection3d3(start1, end1, start2, end2, eps=EPS, useZ=True,
481 **Vector_and_kwds):
482 '''Compute the intersection point of two (2- or 3-D) lines, each defined
483 by two points or by a point and a bearing.
485 @arg start1: Start point of the first line (C{Cartesian}, L{Vector3d},
486 C{Vector3Tuple} or C{Vector4Tuple}).
487 @arg end1: End point of the first line (C{Cartesian}, L{Vector3d},
488 C{Vector3Tuple} or C{Vector4Tuple}) or the bearing at
489 B{C{start1}} (compass C{degrees}).
490 @arg start2: Start point of the second line (C{Cartesian}, L{Vector3d},
491 C{Vector3Tuple} or C{Vector4Tuple}).
492 @arg end2: End point of the second line (C{Cartesian}, L{Vector3d},
493 C{Vector3Tuple} or C{Vector4Tuple}) or the bearing at
494 B{C{start2}} (Ccompass C{degrees}).
495 @kwarg eps: Tolerance for skew line distance and length (C{EPS}).
496 @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}).
497 @kwarg Vector_and_kwds: Optional class C{B{Vector}=None} to return the
498 intersection points and optional, additional B{C{Vector}}
499 keyword arguments, otherwise B{C{start1}}'s (sub-)class.
501 @return: An L{Intersection3Tuple}C{(point, outside1, outside2)} with
502 C{point} an instance of B{C{Vector}} or B{C{start1}}'s (sub-)class.
504 @note: The C{outside} values is C{0} for lines specified by point and bearing.
506 @raise IntersectionError: Invalid, skew, non-co-planar or otherwise
507 non-intersecting lines.
509 @see: U{Line-line intersection<https://MathWorld.Wolfram.com/Line-LineIntersection.html>}
510 and U{line-line distance<https://MathWorld.Wolfram.com/Line-LineDistance.html>},
511 U{skew lines<https://MathWorld.Wolfram.com/SkewLines.html>} and U{point-line
512 distance<https://MathWorld.Wolfram.com/Point-LineDistance3-Dimensional.html>}.
513 '''
514 try:
515 v, o1, o2 = _intersect3d3(start1, end1, start2, end2, eps=eps, useZ=useZ)
516 except (TypeError, ValueError) as x:
517 raise _xError(x, start1=start1, end1=end1, start2=start2, end2=end2)
518 v = _nVc(v, **_xkwds(Vector_and_kwds, clas=start1.classof,
519 name=intersection3d3.__name__))
520 return Intersection3Tuple(v, o1, o2)
523def intersections2(center1, radius1, center2, radius2, sphere=True, **Vector_and_kwds):
524 '''Compute the intersection of two spheres or circles, each defined by a (3-D)
525 center point and a radius.
527 @arg center1: Center of the first sphere or circle (C{Cartesian}, L{Vector3d},
528 C{Vector3Tuple} or C{Vector4Tuple}).
529 @arg radius1: Radius of the first sphere or circle (same units as the
530 B{C{center1}} coordinates).
531 @arg center2: Center of the second sphere or circle (C{Cartesian}, L{Vector3d},
532 C{Vector3Tuple} or C{Vector4Tuple}).
533 @arg radius2: Radius of the second sphere or circle (same units as the
534 B{C{center1}} and B{C{center2}} coordinates).
535 @kwarg sphere: If C{True} compute the center and radius of the intersection of
536 two spheres. If C{False}, ignore the C{z}-component and compute
537 the intersection of two circles (C{bool}).
538 @kwarg Vector_and_kwds: Optional class C{B{Vector}=None} to return the
539 intersection points and optional, additional B{C{Vector}}
540 keyword arguments, otherwise B{C{center1}}'s (sub-)class.
542 @return: If B{C{sphere}} is C{True}, a 2-tuple of the C{center} and C{radius}
543 of the intersection of the I{spheres}. The C{radius} is C{0.0} for
544 abutting spheres (and the C{center} is aka the I{radical center}).
546 If B{C{sphere}} is C{False}, a 2-tuple with the two intersection
547 points of the I{circles}. For abutting circles, both points are
548 the same instance, aka the I{radical center}.
550 @raise IntersectionError: Concentric, invalid or non-intersecting spheres
551 or circles.
553 @raise TypeError: Invalid B{C{center1}} or B{C{center2}}.
555 @raise UnitError: Invalid B{C{radius1}} or B{C{radius2}}.
557 @see: U{Sphere-Sphere<https://MathWorld.Wolfram.com/Sphere-SphereIntersection.html>} and
558 U{Circle-Circle<https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>}
559 Intersection.
560 '''
561 try:
562 return _intersects2(center1, Radius_(radius1=radius1),
563 center2, Radius_(radius2=radius2), sphere=sphere,
564 clas=center1.classof, **Vector_and_kwds)
565 except (TypeError, ValueError) as x:
566 raise _xError(x, center1=center1, radius1=radius1, center2=center2, radius2=radius2)
569def _intersects2(center1, r1, center2, r2, sphere=True, too_d=None, # in CartesianEllipsoidalBase.intersections2,
570 **clas_Vector_and_kwds): # .ellipsoidalBaseDI._intersections2, .formy.intersections2
571 # (INTERNAL) Intersect two spheres or circles, see L{intersections2}
572 # above, separated to allow callers to embellish any exceptions
574 def _nV3(x, y, z):
575 v = Vector3d(x, y, z)
576 n = intersections2.__name__
577 return _nVc(v, **_xkwds(clas_Vector_and_kwds, name=n))
579 def _xV3(c1, u, x, y):
580 xy1 = x, y, _1_0 # transform to original space
581 return _nV3(fdot(xy1, u.x, -u.y, c1.x),
582 fdot(xy1, u.y, u.x, c1.y), _0_0)
584 c1 = _otherV3d(useZ=sphere, center1=center1)
585 c2 = _otherV3d(useZ=sphere, center2=center2)
587 if r1 < r2: # r1, r2 == R, r
588 c1, c2 = c2, c1
589 r1, r2 = r2, r1
591 m = c2.minus(c1)
592 d = m.length
593 if d < max(r2 - r1, EPS):
594 raise IntersectionError(_near_(_concentric_)) # XXX ConcentricError?
596 o = fsum1_(-d, r1, r2) # overlap == -(d - (r1 + r2))
597 # compute intersections with c1 at (0, 0) and c2 at (d, 0), like
598 # <https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>
599 if o > EPS: # overlapping, r1, r2 == R, r
600 x = _MODS.formy._radical2(d, r1, r2).xline
601 y = _1_0 - (x / r1)**2
602 if y > EPS:
603 y = r1 * sqrt(y) # y == a / 2
604 elif y < 0: # PYCHOK no cover
605 raise IntersectionError(_negative_)
606 else: # abutting
607 y = _0_0
608 elif o < 0: # PYCHOK no cover
609 if too_d is not None:
610 d = too_d
611 raise IntersectionError(_too_(Fmt.distant(d)))
612 else: # abutting
613 x, y = r1, _0_0
615 u = m.unit()
616 if sphere: # sphere center and radius
617 c = c1 if x < EPS else (
618 c2 if x > EPS1 else c1.plus(u.times(x)))
619 t = _nV3(c.x, c.y, c.z), Radius(y)
621 elif y > 0: # intersecting circles
622 t = _xV3(c1, u, x, y), _xV3(c1, u, x, -y)
623 else: # abutting circles
624 t = _xV3(c1, u, x, 0)
625 t = t, t
626 return t
629def iscolinearWith(point, point1, point2, eps=EPS, useZ=True):
630 '''Check whether a point is colinear with two other (2- or 3-D) points.
632 @arg point: The point (L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}).
633 @arg point1: First point (L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}).
634 @arg point2: Second point (L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}).
635 @kwarg eps: Tolerance (C{scalar}), same units as C{x}, C{y} and C{z}.
636 @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}).
638 @return: C{True} if B{C{point}} is colinear B{C{point1}} and B{C{point2}},
639 C{False} otherwise.
641 @raise TypeError: Invalid B{C{point}}, B{C{point1}} or B{C{point2}}.
643 @see: Function L{nearestOn}.
644 '''
645 p = _otherV3d(useZ=useZ, point=point)
646 return _MODS.vector2d._iscolinearWith(p, point1, point2, eps=eps, useZ=useZ)
649def nearestOn(point, point1, point2, within=True, useZ=True, Vector=None, **Vector_kwds):
650 '''Locate the point between two points closest to a reference (2- or 3-D).
652 @arg point: Reference point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}
653 or C{Vector4Tuple}).
654 @arg point1: Start point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or
655 C{Vector4Tuple}).
656 @arg point2: End point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or
657 C{Vector4Tuple}).
658 @kwarg within: If C{True} return the closest point between both given
659 points, otherwise the closest point on the extended line
660 through both points (C{bool}).
661 @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}).
662 @kwarg Vector: Class to return closest point (C{Cartesian}, L{Vector3d}
663 or C{Vector3Tuple}) or C{None}.
664 @kwarg Vector_kwds: Optional, additional B{C{Vector}} keyword arguments,
665 ignored if C{B{Vector} is None}.
667 @return: Closest point, either B{C{point1}} or B{C{point2}} or an instance
668 of the B{C{point}}'s (sub-)class or B{C{Vector}} if not C{None}.
670 @raise TypeError: Invalid B{C{point}}, B{C{point1}} or B{C{point2}}.
672 @see: U{3-D Point-Line Distance<https://MathWorld.Wolfram.com/Point-LineDistance3-Dimensional.html>},
673 C{Cartesian} and C{LatLon} methods C{nearestOn}, method L{sphericalTrigonometry.LatLon.nearestOn3}
674 and function L{sphericalTrigonometry.nearestOn3}.
675 '''
676 p0 = _otherV3d(useZ=useZ, point =point)
677 p1 = _otherV3d(useZ=useZ, point1=point1)
678 p2 = _otherV3d(useZ=useZ, point2=point2)
680 n = nearestOn.__name__
681 p, _ = _nearestOn2(p0, p1, p2, within=within)
682 if Vector is not None:
683 p = Vector(p.x, p.y, **_xkwds(Vector_kwds, z=p.z, name=n))
684 elif p is p1:
685 p = point1
686 elif p is p2:
687 p = point2
688 else: # ignore Vector_kwds
689 p = point.classof(p.x, p.y, _xkwds_get(Vector_kwds, z=p.z), name=n)
690 return p
693def _nearestOn2(p0, p1, p2, within=True, eps=EPS):
694 # (INTERNAL) Closest point and fraction, see L{nearestOn} above,
695 # separated to allow callers to embellish any exceptions
696 p21 = p2.minus(p1)
697 d2 = p21.length2
698 if d2 < eps: # coincident
699 p = p1 # ~= p2
700 t = 0
701 else: # see comments in .points.nearestOn5
702 t = p0.minus(p1).dot(p21) / d2
703 if within and t < eps:
704 p = p1
705 t = 0
706 elif within and t > (_1_0 - eps):
707 p = p2
708 t = 1
709 else:
710 p = p1.plus(p21.times(t))
711 return NearestOn2Tuple(p, t)
714def nearestOn6(point, points, closed=False, useZ=True, **Vector_and_kwds): # eps=EPS
715 '''Locate the point on a path or polygon closest to a reference point.
717 The closest point on each polygon edge is either the nearest of that
718 edge's end points or a point in between.
720 @arg point: Reference point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or
721 C{Vector4Tuple}).
722 @arg points: The path or polygon points (C{Cartesian}, L{Vector3d},
723 C{Vector3Tuple} or C{Vector4Tuple}[]).
724 @kwarg closed: Optionally, close the path or polygon (C{bool}).
725 @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=INT0} (C{bool}).
726 @kwarg Vector_and_kwds: Optional class C{B{Vector}=None} to return the closest
727 point and optional, additional B{C{Vector}} keyword
728 arguments, otherwise B{C{point}}'s (sub-)class.
730 @return: A L{NearestOn6Tuple}C{(closest, distance, fi, j, start, end)} with the
731 C{closest}, the C{start} and the C{end} point each an instance of the
732 B{C{Vector}} keyword argument of if {B{Vector}=None} or not specified,
733 an instance of the reference B{C{point}}'s (sub-)class.
735 @raise PointsError: Insufficient number of B{C{points}}
737 @raise TypeError: Non-cartesian B{C{point}} and B{C{points}}.
739 @note: Distances measured with method L{Vector3d.equirectangular}. For
740 geodetic distances use function L{nearestOn5} or one of the
741 C{LatLon.nearestOn6} methods.
742 '''
743 r = _otherV3d(useZ=useZ, point=point)
744 D2 = r.equirectangular # distance squared
746 Ps = PointsIter(points, loop=1, name=nearestOn6.__name__)
747 p1 = c = s = e = _otherV3d(useZ=useZ, i=0, points=Ps[0])
748 c2 = D2(c) # == r.minus(c).length2
750 f = i = 0 # p1..p2 == points[i]..[j]
751 for j, p2 in Ps.enumerate(closed=closed):
752 p2 = _otherV3d(useZ=useZ, i=j, points=p2)
753 p, t = _nearestOn2(r, p1, p2) # within=True, eps=EPS
754 d2 = D2(p) # == r.minus(p).length2
755 if d2 < c2:
756 c2, c, s, e, f = d2, p, p1, p2, (i + t)
757 p1, i = p2, j
759 f, j = _fi_j2(f, len(Ps)) # like .ellipsoidalBaseDI._nearestOn2_
761 kwds = _xkwds(Vector_and_kwds, clas=point.classof, name=Ps.name)
762 v = _nVc(c, **kwds)
763 s = _nVc(s, **kwds) if s is not c else v
764 e = _nVc(e, **kwds) if e is not c else v
765 return NearestOn6Tuple(v, sqrt(c2), f, j, s, e)
768def _nVc(v, clas=None, name=NN, Vector=None, **Vector_kwds): # in .vector2d
769 # return a named C{Vector} or C{clas} instance
770 if Vector is not None:
771 v = Vector(v.x, v.y, v.z, **Vector_kwds)
772 elif clas is not None:
773 v = clas(v.x, v.y, v.z) # ignore Vector_kwds
774 return _xnamed(v, name) if name else v
777def _otherV3d(useZ=True, NN_OK=True, i=None, **name_v):
778 # check named vector instance, return Vector3d
779 n, v = _xkwds_item2(name_v)
780 if useZ and isinstance(v, Vector3dBase):
781 return v if NN_OK or v.name else v.copy(name=Fmt.INDEX(n, i))
783 n = Fmt.INDEX(n, i)
784 try:
785 return Vector3d(v.x, v.y, (v.z if useZ else INT0), name=n)
786 except AttributeError: # no .x, .y or .z attr
787 pass
788 raise _xotherError(Vector3d(0, 0, 0), v, name=n, up=2)
791def parse3d(str3d, sep=_COMMA_, Vector=Vector3d, **Vector_kwds):
792 '''Parse an C{"x, y, z"} string.
794 @arg str3d: X, y and z values (C{str}).
795 @kwarg sep: Optional separator (C{str}).
796 @kwarg Vector: Optional class (L{Vector3d}).
797 @kwarg Vector_kwds: Optional B{C{Vector}} keyword arguments,
798 ignored if C{B{Vector} is None}.
800 @return: A B{C{Vector}} instance or if B{C{Vector}} is C{None},
801 a named L{Vector3Tuple}C{(x, y, z)}.
803 @raise VectorError: Invalid B{C{str3d}}.
804 '''
805 try:
806 v = [float(v.strip()) for v in str3d.split(sep)]
807 n = len(v)
808 if n != 3:
809 raise _ValueError(len=n)
810 except (TypeError, ValueError) as x:
811 raise VectorError(str3d=str3d, cause=x)
812 return _xnamed((Vector3Tuple(v) if Vector is None else # *v
813 Vector(*v, **Vector_kwds)), parse3d.__name__)
816def sumOf(vectors, Vector=Vector3d, **Vector_kwds):
817 '''Compute the I{vectorial} sum of two oe more vectors.
819 @arg vectors: Vectors to be added (L{Vector3d}[]).
820 @kwarg Vector: Optional class for the vectorial sum (L{Vector3d}).
821 @kwarg Vector_kwds: Optional B{C{Vector}} keyword arguments,
822 ignored if C{B{Vector} is None}.
824 @return: Vectorial sum as B{C{Vector}} or if B{C{Vector}} is
825 C{None}, a named L{Vector3Tuple}C{(x, y, z)}.
827 @raise VectorError: No B{C{vectors}}.
828 '''
829 try:
830 t = _MODS.nvectorBase._nsumOf(vectors, 0, None, {}) # no H
831 except (TypeError, ValueError) as x:
832 raise VectorError(vectors=vectors, Vector=Vector, cause=x)
833 x, y, z = t[:3]
834 n = sumOf.__name__
835 return Vector3Tuple(x, y, z, name=n) if Vector is None else \
836 Vector(x, y, z, **_xkwds(Vector_kwds, name=n))
839def trilaterate2d2(x1, y1, radius1, x2, y2, radius2, x3, y3, radius3,
840 eps=None, **Vector_and_kwds):
841 '''Trilaterate three circles, each given as a (2-D) center and a radius.
843 @arg x1: Center C{x} coordinate of the 1st circle (C{scalar}).
844 @arg y1: Center C{y} coordinate of the 1st circle (C{scalar}).
845 @arg radius1: Radius of the 1st circle (C{scalar}).
846 @arg x2: Center C{x} coordinate of the 2nd circle (C{scalar}).
847 @arg y2: Center C{y} coordinate of the 2nd circle (C{scalar}).
848 @arg radius2: Radius of the 2nd circle (C{scalar}).
849 @arg x3: Center C{x} coordinate of the 3rd circle (C{scalar}).
850 @arg y3: Center C{y} coordinate of the 3rd circle (C{scalar}).
851 @arg radius3: Radius of the 3rd circle (C{scalar}).
852 @kwarg eps: Tolerance to check the trilaterated point I{delta} on all
853 3 circles (C{scalar}) or C{None} for no checking.
854 @kwarg Vector_and_kwds: Optional class C{B{Vector}=None} to return the
855 trilateration and optional, additional B{C{Vector}}
856 keyword arguments, otherwise (L{Vector3d}).
858 @return: Trilaterated point as C{B{Vector}(x, y, **B{Vector_kwds})}
859 or L{Vector2Tuple}C{(x, y)} if C{B{Vector} is None}..
861 @raise IntersectionError: No intersection, near-concentric or -colinear
862 centers, trilateration failed some other way
863 or the trilaterated point is off one circle
864 by more than B{C{eps}}.
866 @raise UnitError: Invalid B{C{radius1}}, B{C{radius2}} or B{C{radius3}}.
868 @see: U{Issue #49<https://GitHub.com/mrJean1/PyGeodesy/issues/49>},
869 U{Find X location using 3 known (X,Y) location using trilateration
870 <https://math.StackExchange.com/questions/884807>} and function
871 L{pygeodesy.trilaterate3d2}.
872 '''
873 return _MODS.vector2d._trilaterate2d2(x1, y1, radius1,
874 x2, y2, radius2,
875 x3, y3, radius3, eps=eps, **Vector_and_kwds)
878def trilaterate3d2(center1, radius1, center2, radius2, center3, radius3,
879 eps=EPS, **Vector_and_kwds):
880 '''Trilaterate three spheres, each given as a (3-D) center and a radius.
882 @arg center1: Center of the 1st sphere (C{Cartesian}, L{Vector3d},
883 C{Vector3Tuple} or C{Vector4Tuple}).
884 @arg radius1: Radius of the 1st sphere (same C{units} as C{x}, C{y}
885 and C{z}).
886 @arg center2: Center of the 2nd sphere (C{Cartesian}, L{Vector3d},
887 C{Vector3Tuple} or C{Vector4Tuple}).
888 @arg radius2: Radius of this sphere (same C{units} as C{x}, C{y}
889 and C{z}).
890 @arg center3: Center of the 3rd sphere (C{Cartesian}, L{Vector3d},
891 C{Vector3Tuple} or C{Vector4Tuple}).
892 @arg radius3: Radius of the 3rd sphere (same C{units} as C{x}, C{y}
893 and C{z}).
894 @kwarg eps: Pertubation tolerance (C{scalar}), same units as C{x},
895 C{y} and C{z} or C{None} for no pertubations.
896 @kwarg Vector_and_kwds: Optional class C{B{Vector}=None} to return the
897 trilateration and optional, additional B{C{Vector}}
898 keyword arguments, otherwise B{C{center1}}'s
899 (sub-)class.
901 @return: 2-Tuple with two trilaterated points, each a B{C{Vector}}
902 instance. Both points are the same instance if all three
903 spheres abut/intersect in a single point.
905 @raise ImportError: Package C{numpy} not found, not installed or
906 older than version 1.10.
908 @raise IntersectionError: Near-concentric, -colinear, too distant or
909 non-intersecting spheres.
911 @raise NumPyError: Some C{numpy} issue.
913 @raise TypeError: Invalid B{C{center1}}, B{C{center2}} or B{C{center3}}.
915 @raise UnitError: Invalid B{C{radius1}}, B{C{radius2}} or B{C{radius3}}.
917 @see: Norrdine, A. U{I{An Algebraic Solution to the Multilateration
918 Problem}<https://www.ResearchGate.net/publication/275027725>},
919 the U{I{implementation}<https://www.ResearchGate.net/publication/
920 288825016>} and function L{pygeodesy.trilaterate2d2}.
921 '''
922 try:
923 return _MODS.vector2d._trilaterate3d2(_otherV3d(center1=center1, NN_OK=False),
924 Radius_(radius1=radius1, low=eps),
925 center2, radius2, center3, radius3, eps=eps,
926 clas=center1.classof, **Vector_and_kwds)
927 except (AssertionError, TypeError, ValueError) as x:
928 raise _xError(x, center1=center1, radius1=radius1,
929 center2=center2, radius2=radius2,
930 center3=center3, radius3=radius3)
933def _xyzhdn3(xyz, height, datum, ll): # in .cartesianBase, .nvectorBase
934 '''(INTERNAL) Get a C{(h, d, name)} 3-tuple.
935 '''
936 h = height or _xattr(xyz, height=None) \
937 or _xattr(xyz, h=None) \
938 or _xattr(ll, height=None)
940 d = datum or _xattr(xyz, datum=None) \
941 or _xattr(ll, datum=None)
943 return h, d, _xattr(xyz, name=NN)
946__all__ += _ALL_DOCS(intersections2, sumOf, Vector3dBase)
948# **) MIT License
949#
950# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved.
951#
952# Permission is hereby granted, free of charge, to any person obtaining a
953# copy of this software and associated documentation files (the "Software"),
954# to deal in the Software without restriction, including without limitation
955# the rights to use, copy, modify, merge, publish, distribute, sublicense,
956# and/or sell copies of the Software, and to permit persons to whom the
957# Software is furnished to do so, subject to the following conditions:
958#
959# The above copyright notice and this permission notice shall be included
960# in all copies or substantial portions of the Software.
961#
962# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
963# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
964# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
965# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
966# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
967# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
968# OTHER DEALINGS IN THE SOFTWARE.