Coverage for pygeodesy/sphericalNvector.py: 97%
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2# -*- coding: utf-8 -*-
4u'''Spherical, C{N-vector}-based geodesy.
6N-vector-based classes geodetic (lat-/longitude) L{LatLon}, geocentric
7(ECEF) L{Cartesian} and L{Nvector} and functions L{areaOf}, L{intersection},
8L{meanOf}, L{nearestOn3}, L{perimeterOf}, L{sumOf}, L{triangulate} and
9L{trilaterate}, I{all spherical}.
11Pure Python implementation of n-vector-based spherical geodetic (lat-/longitude)
12methods, transcoded from JavaScript originals by I{(C) Chris Veness 2011-2016},
13published under the same MIT Licence**. See U{Vector-based geodesy
14<https://www.Movable-Type.co.UK/scripts/latlong-vectors.html>} and
15U{Module latlon-nvector-spherical
16<https://www.Movable-Type.co.UK/scripts/geodesy/docs/module-latlon-nvector-spherical.html>}.
18Tools for working with points and paths on (a spherical model of) the
19earth’s surface using using n-vectors rather than the more common
20spherical trigonometry. N-vectors make many calculations much simpler,
21and easier to follow, compared with the trigonometric equivalents.
23Based on Kenneth Gade’s U{‘Non-singular Horizontal Position Representation’
24<https://www.NavLab.net/Publications/A_Nonsingular_Horizontal_Position_Representation.pdf>},
25The Journal of Navigation (2010), vol 63, nr 3, pp 395-417.
27Note that the formulations below take x => 0°N,0°E, y => 0°N,90°E and
28z => 90°N while Gade uses x => 90°N, y => 0°N,90°E, z => 0°N,0°E.
30Also note that on a spherical earth model, an n-vector is equivalent
31to a normalised version of an (ECEF) cartesian coordinate.
32'''
33# make sure int/int division yields float quosient, see .basics
34from __future__ import division as _; del _ # PYCHOK semicolon
36from pygeodesy.basics import isscalar, _xinstanceof
37from pygeodesy.constants import EPS, EPS0, PI, PI2, PI_2, R_M, \
38 _0_0, _0_5, _1_0
39# from pygeodesy.datums import Datums # from .sphericalBase
40from pygeodesy.errors import _ValueError, _xError, _xkwds
41from pygeodesy.fmath import fmean, fsum
42# from pygeodesy.fsums import fsum # from .fmath
43from pygeodesy.interns import _composite_, _end_, _Nv00_, _other_, _point_, \
44 _points_, _pole_
45from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS, _ALL_OTHER
46# from pygeodesy.named import notImplemented # from .nvectorBase
47# from pygeodesy.namedTuples import NearestOn3Tuple # from .points
48from pygeodesy.nvectorBase import NvectorBase, NorthPole, LatLonNvectorBase, \
49 sumOf as _sumOf, _triangulate, _trilaterate, \
50 notImplemented
51from pygeodesy.points import NearestOn3Tuple, ispolar # PYCHOK exported
52from pygeodesy.props import deprecated_function, deprecated_method
53from pygeodesy.sphericalBase import _angular, CartesianSphericalBase, \
54 Datums, _intersecant2, LatLonSphericalBase
55from pygeodesy.units import Bearing, Bearing_, Height, Radius, Scalar
56from pygeodesy.utily import atan2, degrees360, fabs, sincos2, sincos2_, sincos2d
58# from math import atan2, fabs # from utily
60__all__ = _ALL_LAZY.sphericalNvector
61__version__ = '23.03.30'
63_paths_ = 'paths'
66class Cartesian(CartesianSphericalBase):
67 '''Extended to convert geocentric, L{Cartesian} points to
68 L{Nvector} and n-vector-based, spherical L{LatLon}.
69 '''
71 def toLatLon(self, **LatLon_and_kwds): # PYCHOK LatLon=LatLon
72 '''Convert this cartesian to an C{Nvector}-based geodetic point.
74 @kwarg LatLon_and_kwds: Optional L{LatLon} and L{LatLon} keyword
75 arguments, like C{datum}. Use C{B{LatLon}=...}
76 to override this L{LatLon} class or specify
77 C{B{LatLon}=None}.
79 @return: The geodetic point (L{LatLon}) or if B{C{LatLon}} is set
80 to C{None}, an L{Ecef9Tuple}C{(x, y, z, lat, lon, height,
81 C, M, datum)} with C{C} and C{M} if available.
83 @raise TypeError: Invalid B{C{LatLon_and_kwds}} argument.
84 '''
85 kwds = _xkwds(LatLon_and_kwds, LatLon=LatLon, datum=self.datum)
86 return CartesianSphericalBase.toLatLon(self, **kwds)
88 def toNvector(self, **Nvector_and_kwds): # PYCHOK Datums.WGS84
89 '''Convert this cartesian to L{Nvector} components, I{including height}.
91 @kwarg Nvector_and_kwds: Optional L{Nvector} and L{Nvector} keyword
92 arguments, like C{datum}. Use C{B{Nvector}=...}
93 to override this L{Nvector} class or specify
94 C{B{Nvector}=None}.
96 @return: The C{n-vector} components (L{Nvector}) or if B{C{Nvector}}
97 is set to C{None}, a L{Vector4Tuple}C{(x, y, z, h)}
99 @raise TypeError: Invalid B{C{Nvector_and_kwds}} argument.
100 '''
101 # ll = CartesianBase.toLatLon(self, LatLon=LatLon,
102 # datum=datum or self.datum)
103 # kwds = _xkwds(kwds, Nvector=Nvector)
104 # return ll.toNvector(**kwds)
105 kwds = _xkwds(Nvector_and_kwds, Nvector=Nvector, datum=self.datum)
106 return CartesianSphericalBase.toNvector(self, **kwds)
109class LatLon(LatLonNvectorBase, LatLonSphericalBase):
110 '''New n-vector based point on a spherical earth model.
112 Tools for working with points and paths on (a spherical
113 model of) the earth's surface using vector-based methods.
115 @example:
117 >>> from sphericalNvector import LatLon
118 >>> p = LatLon(52.205, 0.119)
119 '''
120 _Nv = None # cached_toNvector L{Nvector})
122 def _update(self, updated, *attrs, **setters): # PYCHOK args
123 '''(INTERNAL) Zap cached attributes if updated.
124 '''
125 if updated: # reset caches
126 LatLonNvectorBase._update(self, updated, _Nv=self._Nv) # special case
127 LatLonSphericalBase._update(self, updated, *attrs, **setters)
129 def alongTrackDistanceTo(self, start, end, radius=R_M):
130 '''Compute the (signed) distance from the start to the closest
131 point on the great circle path defined by a start and an
132 end point.
134 That is, if a perpendicular is drawn from this point to the
135 great circle path, the along-track distance is the distance
136 from the start point to the point where the perpendicular
137 crosses the path.
139 @arg start: Start point of great circle path (L{LatLon}).
140 @arg end: End point of great circle path (L{LatLon}) or
141 initial bearing from start point (compass
142 C{degrees360}).
143 @kwarg radius: Mean earth radius (C{meter}).
145 @return: Distance along the great circle path (positive if
146 after the start toward the end point of the path
147 or negative if before the start point).
149 @raise TypeError: If B{C{start}} or B{C{end}} point is not L{LatLon}.
151 @raise Valuerror: Some points coincide.
153 @example:
155 >>> p = LatLon(53.2611, -0.7972)
157 >>> s = LatLon(53.3206, -1.7297)
158 >>> e = LatLon(53.1887, 0.1334)
159 >>> d = p.alongTrackDistanceTo(s, e) # 62331.58
160 '''
161 self.others(start=start)
162 gc, _, _ = self._gc3(start, end, _end_)
164 p = self.toNvector()
165 a = gc.cross(p).cross(gc) # along-track point gc × p × gc
166 return start.toNvector().angleTo(a, vSign=gc) * radius
168 @deprecated_method
169 def bearingTo(self, other, **unused): # PYCHOK no cover
170 '''DEPRECATED, use method L{initialBearingTo}.
171 '''
172 return self.initialBearingTo(other)
174 def crossTrackDistanceTo(self, start, end, radius=R_M):
175 '''Compute the (signed) distance from this point to great circle
176 defined by a start and end point.
178 @arg start: Start point of great circle path (L{LatLon}).
179 @arg end: End point of great circle path (L{LatLon}) or
180 initial bearing from start point (compass
181 C{degrees360}).
182 @kwarg radius: Mean earth radius (C{meter}).
184 @return: Distance to great circle (negative if to the
185 left or positive if to the right of the path).
187 @raise TypeError: If B{C{start}} or B{C{end}} point is not L{LatLon}.
189 @raise Valuerror: Some points coincide.
191 @example:
193 >>> p = LatLon(53.2611, -0.7972)
195 >>> s = LatLon(53.3206, -1.7297)
196 >>> d = p.crossTrackDistanceTo(s, 96) # -305.7
198 >>> e = LatLon(53.1887, 0.1334)
199 >>> d = p.crossTrackDistanceTo(s, e) # -307.5
200 '''
201 self.others(start=start)
202 gc, _, _ = self._gc3(start, end, _end_)
204 p = self.toNvector()
205 return (gc.angleTo(p) - PI_2) * radius
207 def destination(self, distance, bearing, radius=R_M, height=None):
208 '''Locate the destination from this point after having
209 travelled the given distance on the given bearing.
211 @arg distance: Distance travelled (C{meter}, same units
212 as B{C{radius}}).
213 @arg bearing: Bearing from this point (compass C{degrees360}).
214 @kwarg radius: Mean earth radius (C{meter}).
215 @kwarg height: Optional height at destination, overriding the
216 default height (C{meter}, same units as B{C{radius}}).
218 @return: Destination point (L{LatLon}).
220 @raise Valuerror: Polar coincidence or invalid B{C{distance}},
221 B{C{bearing}}, B{C{radius}} or B{C{height}}.
223 @example:
225 >>> p = LatLon(51.4778, -0.0015)
226 >>> q = p.destination(7794, 300.7)
227 >>> q.toStr() # 51.513546°N, 000.098345°W
228 '''
229 a = _angular(distance, radius)
230 sa, ca, sb, cb = sincos2_(a, Bearing_(bearing))
232 p = self.toNvector()
233 e = NorthPole.cross(p, raiser=_pole_).unit() # east vector at p
234 n = p.cross(e) # north vector at p
235 q = n.times(cb).plus(e.times(sb)) # direction vector @ p
236 n = p.times(ca).plus(q.times(sa))
237 return n.toLatLon(height=height, LatLon=self.classof) # Nvector(n.x, n.y, n.z).toLatLon(...)
239 def distanceTo(self, other, radius=R_M, wrap=False):
240 '''Compute the distance from this to an other point.
242 @arg other: The other point (L{LatLon}).
243 @kwarg radius: Mean earth radius (C{meter}) or C{None}.
244 @kwarg wrap: Wrap/unroll the angular distance (C{bool}).
246 @return: Distance between this and the B{C{other}} point
247 (C{meter}, same units as B{C{radius}} or C{radians}
248 if B{C{radius}} is C{None}).
250 @raise TypeError: Invalid B{C{other}} point.
252 @example:
254 >>> p = LatLon(52.205, 0.119)
255 >>> q = LatLon(48.857, 2.351);
256 >>> d = p.distanceTo(q) # 404.3 km
257 '''
258 self.others(other)
260 r = fabs(self.toNvector().angleTo(other.toNvector(), wrap=wrap))
261 return r if radius is None else (Radius(radius) * r)
263# @Property_RO
264# def Ecef(self):
265# '''Get the ECEF I{class} (L{EcefVeness}), I{lazily}.
266# '''
267# return _ALL_MODS.ecef.EcefKarney
269 def _gc3(self, start, end, namend, raiser=_points_):
270 '''(INTERNAL) Return great circle, start and end Nvectors.
271 '''
272 s = start.toNvector()
273 if isscalar(end): # bearing
274 gc = s.greatCircle(end)
275 e = None
276 else:
277 self.others(end, name=namend)
278 e = end.toNvector()
279 gc = s.cross(e, raiser=raiser) # XXX .unit()?
280 return gc, s, e
282 def greatCircle(self, bearing):
283 '''Compute the vector normal to great circle obtained by
284 heading on the given bearing from this point.
286 Direction of vector is such that initial bearing vector
287 b = c × n, where n is an n-vector representing this point.
289 @arg bearing: Bearing from this point (compass C{degrees360}).
291 @return: N-vector representing the great circle (L{Nvector}).
292 '''
293 a, b = self.philam
294 t = Bearing_(bearing)
296 sa, ca, sb, cb, st, ct = sincos2_(a, b, t)
297 return Nvector(sb * ct - sa * cb * st,
298 -cb * ct - sa * sb * st,
299 ca * st, name=self.name) # XXX .unit()
301 def greatCircleTo(self, other):
302 '''Compute the vector normal to great circle obtained by
303 heading from this to an other point or on a given bearing.
305 Direction of vector is such that initial bearing vector
306 b = c × n, where n is an n-vector representing this point.
308 @arg other: The other point (L{LatLon}) or the bearing from
309 this point (compass C{degrees360}).
311 @return: N-vector representing the great circle (L{Nvector}).
313 @raise TypeError: The B{C{other}} point is not L{LatLon}.
315 @raise Valuerror: Points coincide.
317 @example:
319 >>> p = LatLon(53.3206, -1.7297)
320 >>> gc = p.greatCircle(96.0)
321 >>> gc.toStr() # (-0.79408, 0.12856, 0.59406)
323 >>> q = LatLon(53.1887, 0.1334)
324 >>> g = p.greatCircleTo(q)
325 >>> g.toStr() # (-0.79408, 0.12859, 0.59406)
326 '''
327 gc, _, _ = self._gc3(self, other, _other_)
328 return gc.unit()
330 def initialBearingTo(self, other, **unused):
331 '''Compute the initial bearing (forward azimuth) from this
332 to an other point.
334 @arg other: The other point (L{LatLon}).
335 @arg unused: Optional keyword argument B{C{wrap}} ignored.
337 @return: Initial bearing (compass C{degrees360}).
339 @raise Crosserror: This point coincides with the B{C{other}}
340 point or the C{NorthPole}, provided
341 L{pygeodesy.crosserrors} is C{True}.
343 @raise TypeError: The B{C{other}} point is not L{LatLon}.
345 @example:
347 >>> p1 = LatLon(52.205, 0.119)
348 >>> p2 = LatLon(48.857, 2.351)
349 >>> b = p1.initialBearingTo(p2) # 156.2
350 '''
351 self.others(other)
352 # see <https://MathForum.org/library/drmath/view/55417.html>
353 n = self.toNvector()
354# gc1 = self.greatCircleTo(other)
355 gc1 = n.cross(other.toNvector(), raiser=_points_) # .unit()
356# gc2 = self.greatCircleTo(NorthPole)
357 gc2 = n.cross(NorthPole, raiser=_pole_) # .unit()
358 return degrees360(gc1.angleTo(gc2, vSign=n))
360 def intermediateChordTo(self, other, fraction, height=None):
361 '''Locate the point projected from the point at given fraction
362 on a straight line (chord) between this and an other point.
364 @arg other: The other point (L{LatLon}).
365 @arg fraction: Fraction between both points (float, between
366 0.0 for this and 1.0 for the other point).
367 @kwarg height: Optional height at the intermediate point,
368 overriding the fractional height (C{meter}).
370 @return: Intermediate point (L{LatLon}).
372 @raise TypeError: The B{C{other}} point is not L{LatLon}.
374 @example:
376 >>> p = LatLon(52.205, 0.119)
377 >>> q = LatLon(48.857, 2.351)
378 >>> i = p.intermediateChordTo(q, 0.25) # 51.3723°N, 000.7072°E
379 '''
380 self.others(other)
382 f = Scalar(fraction=fraction)
383 i = other.toNvector().times(f).plus(
384 self.toNvector().times(1 - f))
385# i = other.toNvector() * f + \
386# self.toNvector() * (1 - f))
388 h = self._havg(other, f=f) if height is None else Height(height)
389 return i.toLatLon(height=h, LatLon=self.classof) # Nvector(i.x, i.y, i.z).toLatLon(...)
391 def intermediateTo(self, other, fraction, height=None, **unused): # wrap=False
392 '''Locate the point at a given fraction between this and an
393 other point.
395 @arg other: The other point (L{LatLon}).
396 @arg fraction: Fraction between both points (C{float}, between
397 0.0 for this and 1.0 for the other point).
398 @kwarg height: Optional height at the intermediate point,
399 overriding the fractional height (C{meter}).
401 @return: Intermediate point (L{LatLon}).
403 @raise TypeError: The B{C{other}} point is not L{LatLon}.
405 @raise Valuerror: Points coincide or invalid B{C{height}}.
407 @see: Methods C{midpointTo} and C{rhumbMidpointTo}.
409 @example:
411 >>> p = LatLon(52.205, 0.119)
412 >>> q = LatLon(48.857, 2.351)
413 >>> i = p.intermediateTo(q, 0.25) # 51.3721°N, 000.7074°E
414 '''
415 q = self.others(other).toNvector()
416 p = self.toNvector()
417 f = Scalar(fraction=fraction)
419 x = p.cross(q, raiser=_points_)
420 d = x.unit().cross(p) # unit(p × q) × p
421 # angular distance α, tan(α) = |p × q| / p ⋅ q
422 s, c = sincos2(atan2(x.length, p.dot(q)) * f) # interpolated
423 i = p.times(c).plus(d.times(s)) # p * cosα + d * sinα
425 h = self._havg(other, f=f) if height is None else Height(height)
426 return i.toLatLon(height=h, LatLon=self.classof) # Nvector(i.x, i.y, i.z).toLatLon(...)
428 def intersection(self, end1, start2, end2, height=None):
429 '''Locate the intersection point of two paths each defined
430 by two points or a start point and bearing from North.
432 @arg end1: End point of the first path (L{LatLon}) or the
433 initial bearing at this point (compass C{degrees360}).
434 @arg start2: Start point of the second path (L{LatLon}).
435 @arg end2: End point of the second path (L{LatLon}) or the
436 initial bearing at the second point (compass
437 C{degrees}).
438 @kwarg height: Optional height at the intersection point,
439 overriding the mean height (C{meter}).
441 @return: The intersection point (L{LatLon}) or C{None}
442 if no unique intersection exists.
444 @raise TypeError: If B{C{start2}}, B{C{end1}} or B{C{end2}} point
445 is not L{LatLon}.
447 @raise ValueError: Intersection is ambiguous or infinite or
448 the paths are parallel, coincident or null.
450 @example:
452 >>> s = LatLon(51.8853, 0.2545)
453 >>> e = LatLon(49.0034, 2.5735)
454 >>> i = s.intersection(108.55, e, 32.44) # 50.9076°N, 004.5086°E
455 '''
456 return intersection(self, end1, start2, end2,
457 height=height, LatLon=self.classof)
459 def isenclosedBy(self, points):
460 '''Check whether this point is enclosed by a (convex) polygon.
462 @arg points: The polygon points (L{LatLon}[]).
464 @return: C{True} if the polygon encloses this point,
465 C{False} otherwise.
467 @raise PointsError: Insufficient number of B{C{points}}.
469 @raise TypeError: Some B{C{points}} are not L{LatLon}.
471 @see: Functions L{pygeodesy.isconvex}, L{pygeodesy.isenclosedBy}
472 and L{pygeodesy.ispolar} especially if the B{C{points}} may
473 enclose a pole or wrap around the earth longitudinally.
475 @example:
477 >>> b = LatLon(45,1), LatLon(45,2), LatLon(46,2), LatLon(46,1)
478 >>> p = LatLon(45.1, 1.1)
479 >>> inside = p.isenclosedBy(b) # True
480 '''
481 # sum subtended angles of each edge (using n0, the
482 # normal vector to this point for sign of α)
483 def _subtangles(Ps, n0):
484 vs1 = n0.minus(Ps[0].toNvector())
485 for p in Ps.iterate(closed=True):
486 vs2 = n0.minus(p.toNvector())
487 yield vs1.angleTo(vs2, vSign=n0) # PYCHOK false
488 vs1 = vs2
490 # Note, this method uses angle summation test: on a plane,
491 # angles for an enclosed point will sum to 360°, angles for
492 # an exterior point will sum to 0°. On a sphere, enclosed
493 # point angles will sum to less than 360° (due to spherical
494 # excess), exterior point angles will be small but non-zero.
495 s = fsum(_subtangles(self.PointsIter(points, loop=1),
496 self.toNvector()), floats=True) # normal vector
497 # XXX are winding number optimisations equally applicable to
498 # spherical surface?
499 return fabs(s) > PI
501 @deprecated_method
502 def isEnclosedBy(self, points): # PYCHOK no cover
503 '''DEPRECATED, use method C{isenclosedBy}.'''
504 return self.isenclosedBy(points)
506 def iswithin(self, point1, point2):
507 '''Check whether this point is between two other points.
509 If this point is not on the great circle arc defined by
510 both points, return whether it is within the area bound
511 by perpendiculars to the great circle at each point (in
512 the same hemispere).
514 @arg point1: Start point of the arc (L{LatLon}).
515 @arg point2: End point of the arc (L{LatLon}).
517 @return: C{True} if this point is within the arc,
518 C{False} otherwise.
520 @raise TypeError: If B{C{point1}} or B{C{point2}} is not L{LatLon}.
521 '''
522 n0 = self.toNvector()
523 n1 = self.others(point1=point1).toNvector()
524 n2 = self.others(point2=point2).toNvector()
526 # corner case, null arc
527 if n1.isequalTo(n2):
528 return n0.isequalTo(n1) or n0.isequalTo(n2) # PYCHOK returns
530 if n0.dot(n1) < 0 or n0.dot(n2) < 0: # different hemisphere
531 return False # PYCHOK returns
533 # get vectors representing d0=p0->p1 and d2=p2->p1 and the
534 # dot product d0⋅d2 tells us if p0 is on the p2 side of p1 or
535 # on the other side (similarly for d0=p0->p2 and d1=p1->p2
536 # and dot product d0⋅d1 and p0 on the p1 side of p2 or not)
537 return n0.minus(n1).dot(n2.minus(n1)) >= 0 and \
538 n0.minus(n2).dot(n1.minus(n2)) >= 0
540 @deprecated_method
541 def isWithin(self, point1, point2): # PYCHOK no cover
542 '''DEPRECATED, use method C{iswithin}.'''
543 return self.iswithin(point1, point2)
545 def midpointTo(self, other, height=None, fraction=_0_5):
546 '''Find the midpoint between this and an other point.
548 @arg other: The other point (L{LatLon}).
549 @kwarg height: Optional height at the midpoint, overriding
550 the mean height (C{meter}).
551 @kwarg fraction: Midpoint location from this point (C{scalar}),
552 may be negative or greater than 1.0.
554 @return: Midpoint (L{LatLon}).
556 @raise TypeError: The B{C{other}} point is not L{LatLon}.
558 @see: Methods C{intermediateTo} and C{rhumbMidpointTo}.
560 @example:
562 >>> p1 = LatLon(52.205, 0.119)
563 >>> p2 = LatLon(48.857, 2.351)
564 >>> m = p1.midpointTo(p2) # '50.5363°N, 001.2746°E'
565 '''
566 if fraction is _0_5:
567 self.others(other)
569 m = self.toNvector().plus(other.toNvector())
570 h = self._havg(other) if height is None else height
571 r = m.toLatLon(height=h, LatLon=self.classof)
572 else:
573 r = self.intermediateTo(other, fraction, height=height)
574 return r
576 def nearestOn(self, point1, point2, height=None, within=True, wrap=False):
577 '''Locate the point on the great circle arc between two
578 points closest to this point.
580 @arg point1: Start point of the arc (L{LatLon}).
581 @arg point2: End point of the arc (L{LatLon}).
582 @kwarg height: Optional height, overriding the mean height
583 for the point within the arc (C{meter}), or
584 C{None} to interpolate the height.
585 @kwarg within: If C{True} return the closest point between
586 both given points, otherwise the closest
587 point elsewhere on the arc (C{bool}).
588 @kwarg wrap: Wrap and unroll longitudes (C{bool}).
590 @return: Closest point on the arc (L{LatLon}).
592 @raise NotImplementedError: Keyword argument C{B{wrap}=True}
593 not supported.
595 @raise TypeError: Invalid B{C{point1}} or B{C{point2}}.
597 @example:
599 >>> s1 = LatLon(51.0, 1.0)
600 >>> s2 = LatLon(51.0, 2.0)
602 >>> s = LatLon(51.0, 1.9)
603 >>> p = s.nearestOn(s1, s2) # 51.0004°N, 001.9000°E
605 >>> d = p.distanceTo(s) # 42.71 m
607 >>> s = LatLon(51.0, 2.1)
608 >>> p = s.nearestOn(s1, s2) # 51.0000°N, 002.0000°E
609 '''
610 if wrap: # wrap=True throws C{NotImplementedError} always.
611 notImplemented(self, wrap=wrap)
613 if self.iswithin(point1, point2) and not point1.isequalTo(point2, EPS):
614 # closer to arc than to its endpoints,
615 # find the closest point on the arc
616 gc1 = point1.toNvector().cross(point2.toNvector())
617 gc2 = self.toNvector().cross(gc1)
618 n = gc1.cross(gc2)
620 elif within: # for backward compatibility
621 return point1 if self.distanceTo(point1) < self.distanceTo(point2) else point2
623 else: # handle beyond arc extent by .vector3d.nearestOn
624 n1 = point1.toNvector()
625 n2 = point2.toNvector()
626 n = self.toNvector().nearestOn(n1, n2, within=False)
627 if n is n1:
628 return point1
629 elif n is n2:
630 return point2
632 p = n.toLatLon(height=height or 0, LatLon=self.classof)
633 if height in (None, False): # interpolate height within extent
634 d = point1.distanceTo(point2)
635 f = (point1.distanceTo(p) / d) if d > EPS0 else _0_5
636 p.height = point1._havg(point2, f=max(_0_0, min(f, _1_0)))
637 return p
639 # @deprecated_method
640 def nearestOn2(self, points, **closed_radius_height): # PYCHOK no cover
641 '''DEPRECATED, use method L{sphericalNvector.LatLon.nearestOn3}.
643 @return: ... 2-Tuple C{(closest, distance)} of the C{closest}
644 point (L{LatLon}) on the polygon and the C{distance}
645 to that point from this point ...
646 '''
647 r = self.nearestOn3(points, **closed_radius_height)
648 return r.closest, r.distance
650 def nearestOn3(self, points, closed=False, radius=R_M, height=None):
651 '''Locate the point on a path or polygon (with great circle
652 arcs joining consecutive points) closest to this point.
654 The closest point is either on within the extent of any great
655 circle arc or the nearest of the arc's end points.
657 @arg points: The path or polygon points (L{LatLon}[]).
658 @kwarg closed: Optionally, close the polygon (C{bool}).
659 @kwarg radius: Mean earth radius (C{meter}) or C{None}.
660 @kwarg height: Optional height, overriding the mean height
661 for a point within the arc (C{meter}).
663 @return: A L{NearestOn3Tuple}C{(closest, distance, angle)} of
664 the C{closest} point (L{LatLon}), the C{distance}
665 between this and the C{closest} point in C{meter},
666 same units as B{C{radius}} or in C{radians} if
667 B{C{radius}} is C{None} and the C{angle} from this
668 to the C{closest} point in compass C{degrees360}.
670 @raise TypeError: Some B{C{points}} are not C{LatLon}.
672 @raise ValueError: No B{C{points}}.
673 '''
674 Ps = self.PointsIter(points, loop=1)
676 R = self.distanceTo
677 N = self.nearestOn
679 c = p1 = Ps[0]
680 r = R(c, radius=None) # radians
681 for p2 in Ps.iterate(closed=closed):
682 p = N(p1, p2, height=height)
683 d = R(p, radius=None) # radians
684 if d < r:
685 c, r = p, d
686 p1 = p2
687 d = r if radius is None else (Radius(radius) * r)
688 return NearestOn3Tuple(c, d, degrees360(r))
690 def toCartesian(self, **Cartesian_and_kwds): # PYCHOK Cartesian=Cartesian, datum=None
691 '''Convert this point to C{Nvector}-based cartesian (ECEF) coordinates.
693 @kwarg Cartesian_and_kwds: Optional L{Cartesian} and L{Cartesian} keyword
694 arguments, like C{datum}. Use C{B{Cartesian}=...}
695 to override this L{Cartesian} class or specify
696 C{B{Cartesian}=None}.
698 @return: The cartesian point (L{Cartesian}) or if B{C{Cartesian}} is
699 set to C{None}, an L{Ecef9Tuple}C{(x, y, z, lat, lon, height,
700 C, M, datum)} with C{C} and C{M} if available.
702 @raise TypeError: Invalid B{C{Cartesian_and_kwds}} argument.
703 '''
704 kwds = _xkwds(Cartesian_and_kwds, Cartesian=Cartesian, datum=self.datum)
705 return LatLonSphericalBase.toCartesian(self, **kwds)
707 def toNvector(self, **Nvector_and_kwds): # PYCHOK signature
708 '''Convert this point to L{Nvector} components, I{including height}.
710 @kwarg Nvector_and_kwds: Optional L{Nvector} and L{Nvector} keyword
711 arguments. Use C{B{Nvector}=...} to override
712 this L{Nvector} class or specify
713 C{B{Nvector}=None}.
715 @return: The C{n-vector} components (L{Nvector}) or if B{C{Nvector}} is
716 set to C{None}, a L{Vector4Tuple}C{(x, y, z, h)} if B{C{Nvector}}.
718 @raise TypeError: Invalid B{C{Nvector_and_kwds}} argument.
720 @example:
722 >>> p = LatLon(45, 45)
723 >>> n = p.toNvector()
724 >>> n.toStr() # [0.50, 0.50, 0.70710]
725 '''
726 kwds = _xkwds(Nvector_and_kwds, Nvector=Nvector)
727 return LatLonNvectorBase.toNvector(self, **kwds)
730class Nvector(NvectorBase):
731 '''An n-vector is a position representation using a (unit) vector
732 normal to the earth's surface. Unlike lat-/longitude points,
733 n-vectors have no singularities or discontinuities.
735 For many applications, n-vectors are more convenient to work
736 with than other position representations like lat-/longitude,
737 earth-centred earth-fixed (ECEF) vectors, UTM coordinates, etc.
739 On a spherical model earth, an n-vector is equivalent to an
740 earth-centred earth-fixed (ECEF) vector.
742 Note commonality with L{ellipsoidalNvector.Nvector}.
743 '''
744 _datum = Datums.Sphere # default datum (L{Datum})
746 def toCartesian(self, **Cartesian_and_kwds): # PYCHOK Cartesian=Cartesian
747 '''Convert this n-vector to C{Nvector}-based cartesian
748 (ECEF) coordinates.
750 @kwarg Cartesian_and_kwds: Optional L{Cartesian} and L{Cartesian} keyword
751 arguments, like C{h}. Use C{B{Cartesian}=...}
752 to override this L{Cartesian} class or specify
753 C{B{Cartesian}=None}.
755 @return: The cartesian point (L{Cartesian}) or if B{C{Cartesian}} is
756 set to C{None}, an L{Ecef9Tuple}C{(x, y, z, lat, lon, height,
757 C, M, datum)} with C{C} and C{M} if available.
759 @raise TypeError: Invalid B{C{Cartesian_and_kwds}} argument.
760 '''
761 kwds = _xkwds(Cartesian_and_kwds, h=self.h, Cartesian=Cartesian)
762 return NvectorBase.toCartesian(self, **kwds) # class or .classof
764 def toLatLon(self, **LatLon_and_kwds): # PYCHOK height=None, LatLon=LatLon
765 '''Convert this n-vector to an C{Nvector}-based geodetic point.
767 @kwarg LatLon_and_kwds: Optional L{LatLon} and L{LatLon} keyword
768 arguments, like C{height}. Use C{B{LatLon}=...}
769 to override this L{LatLon} class or specify
770 C{B{LatLon}=None}.
772 @return: The geodetic point (L{LatLon}) or if B{C{LatLon}} is set
773 to C{None}, an L{Ecef9Tuple}C{(x, y, z, lat, lon, height,
774 C, M, datum)} with C{C} and C{M} if available.
776 @raise TypeError: Invalid B{C{LatLon_and_kwds}} argument.
778 @raise ValueError: Invalid B{C{height}}.
779 '''
780 kwds = _xkwds(LatLon_and_kwds, height=self.h, LatLon=LatLon)
781 return NvectorBase.toLatLon(self, **kwds) # class or .classof
783 def greatCircle(self, bearing):
784 '''Compute the n-vector normal to great circle obtained by
785 heading on given compass bearing from this point as its
786 n-vector.
788 Direction of vector is such that initial bearing vector
789 b = c × p.
791 @arg bearing: Initial compass bearing (C{degrees}).
793 @return: N-vector representing great circle (L{Nvector}).
795 @raise Valuerror: Polar coincidence.
797 @example:
799 >>> n = LatLon(53.3206, -1.7297).toNvector()
800 >>> gc = n.greatCircle(96.0) # [-0.794, 0.129, 0.594]
801 '''
802 s, c = sincos2d(Bearing(bearing))
804 e = NorthPole.cross(self, raiser=_pole_) # easting
805 n = self.cross(e, raiser=_point_) # northing
807 e = e.times(c / e.length)
808 n = n.times(s / n.length)
809 return n.minus(e)
812_Nvll = LatLon(_0_0, _0_0, name=_Nv00_) # reference instance (L{LatLon})
815def areaOf(points, radius=R_M):
816 '''Calculate the area of a (spherical) polygon or composite
817 (with great circle arcs joining consecutive points).
819 @arg points: The polygon points or clips (C{LatLon}[],
820 L{BooleanFHP} or L{BooleanGH}).
821 @kwarg radius: Mean earth radius (C{meter}) or C{None}.
823 @return: Polygon area (C{meter} I{squared} , same units as
824 B{C{radius}}, or C{radians} if B{C{radius}} is C{None}).
826 @raise PointsError: Insufficient number of B{C{points}}.
828 @raise TypeError: Some B{C{points}} are not L{LatLon}.
830 @see: Functions L{pygeodesy.areaOf}, L{sphericalTrigonometry.areaOf}
831 and L{ellipsoidalKarney.areaOf}.
833 @example:
835 >>> b = LatLon(45, 1), LatLon(45, 2), LatLon(46, 2), LatLon(46, 1)
836 >>> areaOf(b) # 8666058750.718977
837 '''
838 def _interangles(Ps):
839 # use vector to 1st point as plane normal for sign of α
840 n0 = Ps[0].toNvector()
842 v2 = Ps[0]._N_vector # XXX v2 == no?
843 v1 = Ps[1]._N_vector
844 gc = v2.cross(v1)
845 for p in Ps.iterate(closed=True):
846 v2 = p._N_vector
847 gc1 = v1.cross(v2)
848 v1 = v2
849 yield gc.angleTo(gc1, vSign=n0)
850 gc = gc1
852 if _MODS.booleans.isBoolean(points):
853 r = points._sum2(LatLon, areaOf, radius=None)
854 else:
855 # sum interior angles: depending on whether polygon is cw or ccw,
856 # angle between edges is π−α or π+α, where α is angle between
857 # great-circle vectors; so sum α, then take n·π − |Σα| (cannot
858 # use Σ(π−|α|) as concave polygons would fail)
859 s = fsum(_interangles(_Nvll.PointsIter(points, loop=2)), floats=True)
860 # using Girard’s theorem: A = [Σθᵢ − (n−2)·π]·R²
861 # (PI2 - abs(s) == (n*PI - abs(s)) - (n-2)*PI)
862 r = fabs(PI2 - fabs(s))
863 return r if radius is None else (r * Radius(radius)**2)
866def intersecant2(center, circle, point, bearing, radius=R_M, exact=False,
867 height=None, wrap=True):
868 '''Compute the intersections of a circle and a line.
870 @arg center: Center of the circle (L{LatLon}).
871 @arg circle: Radius of the circle (C{meter}, same units as B{C{radius}})
872 or a point on the circle (L{LatLon}).
873 @arg point: A point in- or outside the circle (L{LatLon}).
874 @arg bearing: Bearing at the B{C{point}} (compass C{degrees360}) or
875 a second point on the line (L{LatLon}).
876 @kwarg radius: Mean earth radius (C{meter}, conventionally).
877 @kwarg exact: If C{True} use the I{exact} rhumb methods for azimuth,
878 destination and distance, if C{False} use the basic
879 rhumb methods (C{bool}) or if C{None} use the I{great
880 circle} methods.
881 @kwarg height: Optional height for the intersection points (C{meter},
882 conventionally) or C{None}.
883 @kwarg wrap: Wrap and unroll longitudes (C{bool}).
885 @return: 2-Tuple of the intersection points (representing a chord),
886 each an instance of this class. For a tangent line, each
887 point C{is} this very instance.
889 @raise IntersectionError: The circle and line do not intersect.
891 @raise TypeError: If B{C{center}} or B{C{point}} not L{LatLon} or
892 B{C{circle}} or B{C{bearing}} invalid.
894 @raise ValueError: Invalid B{C{circle}}, B{C{bearing}}, B{C{radius}},
895 B{C{exact}} or B{C{height}}.
896 '''
897 c = _Nvll.others(center=center)
898 p = _Nvll.others(point=point)
899 try:
900 return _intersecant2(c, circle, p, bearing, radius=radius, exact=exact,
901 height=height, wrap=wrap)
902 except (TypeError, ValueError) as x:
903 raise _xError(x, center=center, circle=circle, point=point, bearing=bearing, exact=exact)
906def intersection(start1, end1, start2, end2,
907 height=None, LatLon=LatLon, **LatLon_kwds):
908 '''Locate the intersection of two paths each defined by two
909 points or by a start point and an initial bearing.
911 @arg start1: Start point of the first path (L{LatLon}).
912 @arg end1: End point of the first path (L{LatLon}) or the
913 initial bearing at the first start point
914 (compass C{degrees360}).
915 @arg start2: Start point of the second path (L{LatLon}).
916 @arg end2: End point of the second path (L{LatLon}) or the
917 initial bearing at the second start point
918 (compass C{degrees360}).
919 @kwarg height: Optional height at the intersection point,
920 overriding the mean height (C{meter}).
921 @kwarg LatLon: Optional class to return the intersection
922 point (L{LatLon}).
923 @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword
924 arguments, ignored if C{B{LatLon} is None}.
926 @return: The intersection point (B{C{LatLon}}) or 3-tuple
927 (C{degrees90}, C{degrees180}, height) if B{C{LatLon}}
928 is C{None} or C{None} if no unique intersection
929 exists.
931 @raise TypeError: If B{C{start*}} or B{C{end*}} is not L{LatLon}.
933 @raise ValueError: Intersection is ambiguous or infinite or
934 the paths are parallel, coincident or null.
936 @example:
938 >>> p = LatLon(51.8853, 0.2545)
939 >>> q = LatLon(49.0034, 2.5735)
940 >>> i = intersection(p, 108.55, q, 32.44) # 50.9076°N, 004.5086°E
941 '''
942 _Nvll.others(start1=start1)
943 _Nvll.others(start2=start2)
945 # If gc1 and gc2 are great circles through start and end points
946 # (or defined by start point and bearing), then the candidate
947 # intersections are simply gc1 × gc2 and gc2 × gc1. Most of the
948 # work is deciding the correct intersection point to select! If
949 # bearing is given, that determines the intersection, but if both
950 # paths are defined by start/end points, take closer intersection.
951 gc1, s1, e1 = _Nvll._gc3(start1, end1, 'end1')
952 gc2, s2, e2 = _Nvll._gc3(start2, end2, 'end2')
954 hs = start1.height, start2.height
955 # there are two (antipodal) candidate intersection
956 # points ... we have to choose the one to return
957 i1 = gc1.cross(gc2, raiser=_paths_)
958 # postpone computing i2 until needed
959 # i2 = gc2.cross(gc1, raiser=_paths_)
961 # selection of intersection point depends on how
962 # paths are defined (by bearings or endpoints)
963 if e1 and e2: # endpoint+endpoint
964 d = sumOf((s1, s2, e1, e2)).dot(i1)
965 hs += end1.height, end2.height
966 elif e1 and not e2: # endpoint+bearing
967 # gc2 x v2 . i1 +ve means v2 bearing points to i1
968 d = gc2.cross(s2).dot(i1)
969 hs += end1.height,
970 elif e2 and not e1: # bearing+endpoint
971 # gc1 x v1 . i1 +ve means v1 bearing points to i1
972 d = gc1.cross(s1).dot(i1)
973 hs += end2.height,
974 else: # bearing+bearing
975 # if gc x v . i1 is +ve, initial bearing is
976 # towards i1, otherwise towards antipodal i2
977 d1 = gc1.cross(s1).dot(i1) # +ve means p1 bearing points to i1
978 d2 = gc2.cross(s2).dot(i1) # +ve means p2 bearing points to i1
979 if d1 > 0 and d2 > 0:
980 d = 1 # both point to i1
981 elif d1 < 0 and d2 < 0:
982 d = -1 # both point to i2
983 else: # d1, d2 opposite signs
984 # intersection is at further-away intersection point,
985 # take opposite intersection from mid- point of v1
986 # and v2 [is this always true?] XXX changed to always
987 # get intersection p1 bearing points to, aka being
988 # located "after" p1 along the bearing at p1, like
989 # function .sphericalTrigonometry._intersect and
990 # .ellipsoidalBaseDI._intersect3
991 d = d1 # neg(s1.plus(s2).dot(i1))
993 i = i1 if d > 0 else gc2.cross(gc1, raiser=_paths_)
995 h = fmean(hs) if height is None else height
996 kwds = _xkwds(LatLon_kwds, height=h, LatLon=LatLon)
997 return i.toLatLon(**kwds) # Nvector(i.x, i.y, i.z).toLatLon(...)
1000def meanOf(points, height=None, LatLon=LatLon, **LatLon_kwds):
1001 '''Compute the geographic mean of the supplied points.
1003 @arg points: Array of points to be averaged (L{LatLon}[]).
1004 @kwarg height: Optional height, overriding the mean height
1005 (C{meter}).
1006 @kwarg LatLon: Optional class to return the mean point
1007 (L{LatLon}).
1008 @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword
1009 arguments, ignored if C{B{LatLon} is None}.
1011 @return: Point at geographic mean and mean height (B{C{LatLon}}).
1013 @raise PointsError: Insufficient number of B{C{points}}.
1015 @raise TypeError: Some B{C{points}} are not C{LatLon}.
1016 '''
1017 Ps = _Nvll.PointsIter(points)
1018 # geographic mean
1019 m = sumOf(p._N_vector for p in Ps.iterate(closed=False))
1020 kwds = _xkwds(LatLon_kwds, height=height, LatLon=LatLon,
1021 name=meanOf.__name__)
1022 return m.toLatLon(**kwds)
1025@deprecated_function
1026def nearestOn2(point, points, **closed_radius_height): # PYCHOK no cover
1027 '''DEPRECATED, use method L{sphericalNvector.nearestOn3}.
1029 @return: ... 2-Tuple C{(closest, distance)} of the C{closest}
1030 point (L{LatLon}) on the polygon and the C{distance}
1031 between the C{closest} and the given B{C{point}} ...
1032 '''
1033 r = nearestOn3(point, points, **closed_radius_height)
1034 return r.closest, r.distance
1037def nearestOn3(point, points, closed=False, radius=R_M, height=None):
1038 '''Locate the point on a polygon (with great circle arcs
1039 joining consecutive points) closest to an other point.
1041 If the given point is within the extent of any great circle
1042 arc, the closest point is on that arc. Otherwise, the
1043 closest is the nearest of the arc's end points.
1045 @arg point: The other, reference point (L{LatLon}).
1046 @arg points: The polygon points (L{LatLon}[]).
1047 @kwarg closed: Optionally, close the polygon (C{bool}).
1048 @kwarg radius: Mean earth radius (C{meter}) or C{None}.
1049 @kwarg height: Optional height, overriding the mean height
1050 for a point within the arc (C{meter}).
1052 @return: A L{NearestOn3Tuple}C{(closest, distance, angle)} of
1053 the C{closest} point (L{LatLon}) on the polygon, the
1054 C{distance} and the C{angle} between the C{closest}
1055 and the given B{C{point}}. The C{distance} is in
1056 C{meter}, same units as B{C{radius}} or in C{radians}
1057 if B{C{radius}} is C{None}, the C{angle} is in compass
1058 C{degrees360}.
1060 @raise PointsError: Insufficient number of B{C{points}}.
1062 @raise TypeError: Some B{C{points}} or B{C{point}} not C{LatLon}.
1063 '''
1064 _xinstanceof(LatLon, point=point)
1066 return point.nearestOn3(points, closed=closed, radius=radius, height=height)
1069def perimeterOf(points, closed=False, radius=R_M):
1070 '''Compute the perimeter of a (spherical) polygon or composite
1071 (with great circle arcs joining consecutive points).
1073 @arg points: The polygon points (L{LatLon}[]).
1074 @kwarg closed: Optionally, close the polygon (C{bool}).
1075 @kwarg radius: Mean earth radius (C{meter}) or C{None}.
1077 @return: Polygon perimeter (C{meter}, same units as B{C{radius}} or
1078 C{radians} if B{C{radius}} is C{None}).
1080 @raise PointsError: Insufficient number of B{C{points}}.
1082 @raise TypeError: Some B{C{points}} are not L{LatLon}.
1084 @raise ValueError: Invalid B{C{radius}} or C{B{closed}=False} with
1085 C{B{points}} a composite.
1087 @see: Functions L{pygeodesy.perimeterOf}, L{sphericalTrigonometry.perimeterOf}
1088 and L{ellipsoidalKarney.perimeterOf}.
1089 '''
1090 def _rads(Ps, closed): # angular edge lengths in radians
1091 v1 = Ps[0]._N_vector
1092 for p in Ps.iterate(closed=closed):
1093 v2 = p._N_vector
1094 yield v1.angleTo(v2)
1095 v1 = v2
1097 if _MODS.booleans.isBoolean(points):
1098 if not closed:
1099 raise _ValueError(closed=closed, points=_composite_)
1100 r = points._sum2(LatLon, perimeterOf, closed=True, radius=None)
1101 else:
1102 r = fsum(_rads(_Nvll.PointsIter(points, loop=1), closed), floats=True)
1103 return r if radius is None else (Radius(radius) * r)
1106def sumOf(nvectors, Vector=Nvector, h=None, **Vector_kwds):
1107 '''Return the vectorial sum of two or more n-vectors.
1109 @arg nvectors: Vectors to be added (L{Nvector}[]).
1110 @kwarg Vector: Optional class for the vectorial sum (L{Nvector}).
1111 @kwarg h: Optional height, overriding the mean height (C{meter}).
1112 @kwarg Vector_kwds: Optional, additional B{C{Vector}} keyword arguments.
1114 @return: Vectorial sum (B{C{Vector}}).
1116 @raise VectorError: No B{C{nvectors}}.
1117 '''
1118 return _sumOf(nvectors, Vector=Vector, h=h, **Vector_kwds)
1121def triangulate(point1, bearing1, point2, bearing2,
1122 height=None, LatLon=LatLon, **LatLon_kwds):
1123 '''Locate a point given two known points and the initial bearings
1124 from those points.
1126 @arg point1: First reference point (L{LatLon}).
1127 @arg bearing1: Bearing at the first point (compass C{degrees360}).
1128 @arg point2: Second reference point (L{LatLon}).
1129 @arg bearing2: Bearing at the second point (compass C{degrees360}).
1130 @kwarg height: Optional height at the triangulated point, overriding
1131 the mean height (C{meter}).
1132 @kwarg LatLon: Optional class to return the triangulated point
1133 (L{LatLon}).
1134 @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword
1135 arguments, ignored if C{B{LatLon} is None}.
1137 @return: Triangulated point (B{C{LatLon}}).
1139 @raise TypeError: If B{C{point1}} or B{C{point2}} is not L{LatLon}.
1141 @raise Valuerror: Points coincide.
1143 @example:
1145 >>> p = LatLon("47°18.228'N","002°34.326'W") # Basse Castouillet
1146 >>> q = LatLon("47°18.664'N","002°31.717'W") # Basse Hergo
1147 >>> t = triangulate(p, 7, q, 295) # 47.323667°N, 002.568501°W'
1148 '''
1149 return _triangulate(_Nvll.others(point1=point1), bearing1,
1150 _Nvll.others(point2=point2), bearing2,
1151 height=height, LatLon=LatLon, **LatLon_kwds)
1154def trilaterate(point1, distance1, point2, distance2, point3, distance3, # PYCHOK args
1155 radius=R_M, height=None, useZ=False,
1156 LatLon=LatLon, **LatLon_kwds):
1157 '''Locate a point at given distances from three other points.
1159 @arg point1: First point (L{LatLon}).
1160 @arg distance1: Distance to the first point (C{meter}, same units
1161 as B{C{radius}}).
1162 @arg point2: Second point (L{LatLon}).
1163 @arg distance2: Distance to the second point (C{meter}, same units
1164 as B{C{radius}}).
1165 @arg point3: Third point (L{LatLon}).
1166 @arg distance3: Distance to the third point (C{meter}, same units
1167 as B{C{radius}}).
1168 @kwarg radius: Mean earth radius (C{meter}).
1169 @kwarg height: Optional height at the trilaterated point, overriding
1170 the IDW height (C{meter}, same units as B{C{radius}}).
1171 @kwarg useZ: Include Z component iff non-NaN, non-zero (C{bool}).
1172 @kwarg LatLon: Optional class to return the trilaterated
1173 @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword arguments,
1174 ignored if C{B{LatLon} is None}.
1176 @return: Trilaterated point (B{C{LatLon}}).
1178 @raise IntersectionError: No intersection, trilateration failed.
1180 @raise TypeError: Invalid B{C{point1}}, B{C{point2}} or B{C{point3}}.
1182 @raise ValueError: Coincident B{C{points}} or invalid B{C{distance1}},
1183 B{C{distance2}}, B{C{distance3}} or B{C{radius}}.
1185 @see: U{Trilateration<https://WikiPedia.org/wiki/Trilateration>}.
1186 '''
1187 return _trilaterate(_Nvll.others(point1=point1), distance1,
1188 _Nvll.others(point2=point2), distance2,
1189 _Nvll.others(point3=point3), distance3,
1190 radius=radius, height=height, useZ=useZ,
1191 LatLon=LatLon, **LatLon_kwds)
1194__all__ += _ALL_OTHER(Cartesian, LatLon, Nvector, # classes
1195 areaOf, # functions
1196 intersecant2, intersection, ispolar,
1197 meanOf,
1198 nearestOn2, nearestOn3,
1199 perimeterOf,
1200 sumOf,
1201 triangulate, trilaterate)
1203# **) MIT License
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1205# Copyright (C) 2016-2023 -- mrJean1 at Gmail -- All Rights Reserved.
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1212# Software is furnished to do so, subject to the following conditions:
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