Coverage for pygeodesy/sphericalTrigonometry.py: 94%
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« prev ^ index » next coverage.py v7.2.2, created at 2023-12-12 14:24 -0500
2# -*- coding: utf-8 -*-
4u'''Spherical, C{trigonometry}-based geodesy.
6Trigonometric classes geodetic (lat-/longitude) L{LatLon} and
7geocentric (ECEF) L{Cartesian} and functions L{areaOf}, L{intersection},
8L{intersections2}, L{isPoleEnclosedBy}, L{meanOf}, L{nearestOn3} and
9L{perimeterOf}, I{all spherical}.
11Pure Python implementation of geodetic (lat-/longitude) methods using
12spherical trigonometry, transcoded from JavaScript originals by
13I{(C) Chris Veness 2011-2016} published under the same MIT Licence**, see
14U{Latitude/Longitude<https://www.Movable-Type.co.UK/scripts/latlong.html>}.
15'''
16# make sure int/int division yields float quotient, see .basics
17from __future__ import division as _; del _ # PYCHOK semicolon
19from pygeodesy.basics import copysign0, map1, signOf
20from pygeodesy.constants import EPS, EPS1, EPS4, PI, PI2, PI_2, PI_4, R_M, \
21 isnear0, isnear1, isnon0, _0_0, _0_5, \
22 _1_0, _2_0, _90_0
23from pygeodesy.datums import _ellipsoidal_datum, _mean_radius
24from pygeodesy.errors import _AssertionError, CrossError, crosserrors, \
25 _ValueError, IntersectionError, _xError, \
26 _xkwds, _xkwds_get, _xkwds_pop
27from pygeodesy.fmath import favg, fdot, fmean, hypot
28from pygeodesy.fsums import Fsum, fsum, fsumf_
29from pygeodesy.formy import antipode_, bearing_, _bearingTo2, excessAbc_, \
30 excessGirard_, excessLHuilier_, opposing_, _radical2, \
31 vincentys_
32from pygeodesy.interns import _1_, _2_, _coincident_, _composite_, _colinear_, \
33 _concentric_, _convex_, _end_, _infinite_, _invalid_,\
34 _line_, _near_, _not_, _null_, _parallel_, _point_, \
35 _SPACE_, _too_
36from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS, _ALL_OTHER
37# from pygeodesy.named import notImplemented # from .points
38# from pygeodesy.nvectorBase import NvectorBase, sumOf # _MODE
39from pygeodesy.namedTuples import LatLon2Tuple, LatLon3Tuple, NearestOn3Tuple, \
40 Triangle7Tuple, Triangle8Tuple
41from pygeodesy.points import ispolar, nearestOn5 as _nearestOn5, \
42 Fmt as _Fmt, notImplemented # XXX shadowed
43from pygeodesy.props import deprecated_function, deprecated_method
44from pygeodesy.sphericalBase import _m2radians, CartesianSphericalBase, \
45 _intersecant2, LatLonSphericalBase, \
46 _rads3, _radians2m, _trilaterate5
47# from pygeodesy.streprs import Fmt as _Fmt # from .points XXX shadowed
48from pygeodesy.units import Bearing_, Height, _isDegrees, _isRadius, Lam_, \
49 Phi_, Radius_, Scalar
50from pygeodesy.utily import acos1, asin1, atan1d, atan2d, degrees90, degrees180, \
51 degrees2m, m2radians, radiansPI2, sincos2_, tan_2, \
52 unrollPI, _unrollon, _unrollon3, _Wrap, wrap180, wrapPI
53from pygeodesy.vector3d import sumOf, Vector3d
55from math import asin, atan2, cos, degrees, fabs, radians, sin
57__all__ = _ALL_LAZY.sphericalTrigonometry
58__version__ = '23.12.03'
60_PI_EPS4 = PI - EPS4
61if _PI_EPS4 >= PI:
62 raise _AssertionError(EPS4=EPS4, PI=PI, PI_EPS4=_PI_EPS4)
65class Cartesian(CartesianSphericalBase):
66 '''Extended to convert geocentric, L{Cartesian} points to
67 spherical, geodetic L{LatLon}.
68 '''
70 def toLatLon(self, **LatLon_and_kwds): # PYCHOK LatLon=LatLon
71 '''Convert this cartesian point to a C{spherical} geodetic point.
73 @kwarg LatLon_and_kwds: Optional L{LatLon} and L{LatLon} keyword
74 arguments. Use C{B{LatLon}=...} to override
75 this L{LatLon} class or specify C{B{LatLon}=None}.
77 @return: The geodetic point (L{LatLon}) or if B{C{LatLon}} is C{None},
78 an L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)}
79 with C{C} and C{M} if available.
81 @raise TypeError: Invalid B{C{LatLon_and_kwds}} argument.
82 '''
83 kwds = _xkwds(LatLon_and_kwds, LatLon=LatLon, datum=self.datum)
84 return CartesianSphericalBase.toLatLon(self, **kwds)
87class LatLon(LatLonSphericalBase):
88 '''New point on spherical model earth model.
89 '''
91 def _ab1_ab2_db5(self, other, wrap):
92 '''(INTERNAL) Helper for several methods.
93 '''
94 a1, b1 = self.philam
95 a2, b2 = self.others(other, up=2).philam
96 if wrap:
97 a2, b2 = _Wrap.philam(a2, b2)
98 db, b2 = unrollPI(b1, b2, wrap=wrap)
99 else: # unrollPI shortcut
100 db = b2 - b1
101 return a1, b1, a2, b2, db
103 def alongTrackDistanceTo(self, start, end, radius=R_M, wrap=False):
104 '''Compute the (signed) distance from the start to the closest
105 point on the great circle line defined by a start and an
106 end point.
108 That is, if a perpendicular is drawn from this point to the
109 great circle line, the along-track distance is the distance
110 from the start point to the point where the perpendicular
111 crosses the line.
113 @arg start: Start point of the great circle line (L{LatLon}).
114 @arg end: End point of the great circle line (L{LatLon}).
115 @kwarg radius: Mean earth radius (C{meter}) or C{None}.
116 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
117 the B{C{start}} and B{C{end}} point (C{bool}).
119 @return: Distance along the great circle line (C{radians}
120 if C{B{radius} is None} or C{meter}, same units
121 as B{C{radius}}), positive if I{after} the
122 B{C{start}} toward the B{C{end}} point of the
123 line, I{negative} if before or C{0} if at the
124 B{C{start}} point.
126 @raise TypeError: Invalid B{C{start}} or B{C{end}} point.
128 @raise ValueError: Invalid B{C{radius}}.
129 '''
130 r, x, b = self._a_x_b3(start, end, radius, wrap)
131 cx = cos(x)
132 return _0_0 if isnear0(cx) else \
133 _radians2m(copysign0(acos1(cos(r) / cx), cos(b)), radius)
135 def _a_x_b3(self, start, end, radius, wrap):
136 '''(INTERNAL) Helper for .along-/crossTrackDistanceTo.
137 '''
138 s = self.others(start=start)
139 e = self.others(end=end)
140 s, e, w = _unrollon3(self, s, e, wrap)
142 r = Radius_(radius)
143 r = s.distanceTo(self, r, wrap=w) / r
145 b = radians(s.initialBearingTo(self, wrap=w)
146 - s.initialBearingTo(e, wrap=w))
147 x = asin(sin(r) * sin(b))
148 return r, x, -b
150 @deprecated_method
151 def bearingTo(self, other, wrap=False, raiser=False): # PYCHOK no cover
152 '''DEPRECATED, use method L{initialBearingTo}.
153 '''
154 return self.initialBearingTo(other, wrap=wrap, raiser=raiser)
156 def crossingParallels(self, other, lat, wrap=False):
157 '''Return the pair of meridians at which a great circle defined
158 by this and an other point crosses the given latitude.
160 @arg other: The other point defining great circle (L{LatLon}).
161 @arg lat: Latitude at the crossing (C{degrees}).
162 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the
163 B{C{other}} point (C{bool}).
165 @return: 2-Tuple C{(lon1, lon2)}, both in C{degrees180} or
166 C{None} if the great circle doesn't reach B{C{lat}}.
167 '''
168 a1, b1, a2, b2, db = self._ab1_ab2_db5(other, wrap)
169 sa, ca, sa1, ca1, \
170 sa2, ca2, sdb, cdb = sincos2_(radians(lat), a1, a2, db)
171 sa1 *= ca2 * ca
173 x = sa1 * sdb
174 y = sa1 * cdb - ca1 * sa2 * ca
175 z = ca1 * sdb * ca2 * sa
177 h = hypot(x, y)
178 if h < EPS or fabs(z) > h: # PYCHOK no cover
179 return None # great circle doesn't reach latitude
181 m = atan2(-y, x) + b1 # longitude at max latitude
182 d = acos1(z / h) # delta longitude to intersections
183 return degrees180(m - d), degrees180(m + d)
185 def crossTrackDistanceTo(self, start, end, radius=R_M, wrap=False):
186 '''Compute the (signed) distance from this point to
187 the great circle defined by a start and an end point.
189 @arg start: Start point of the great circle line (L{LatLon}).
190 @arg end: End point of the great circle line (L{LatLon}).
191 @kwarg radius: Mean earth radius (C{meter}) or C{None}.
192 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
193 the B{C{start}} and B{C{end}} point (C{bool}).
195 @return: Distance to the great circle (C{radians} if
196 B{C{radius}} or C{meter}, same units as
197 B{C{radius}}), I{negative} if to the left or
198 I{positive} if to the right of the line.
200 @raise TypeError: If B{C{start}} or B{C{end}} is not L{LatLon}.
202 @raise ValueError: Invalid B{C{radius}}.
203 '''
204 _, x, _ = self._a_x_b3(start, end, radius, wrap)
205 return _radians2m(x, 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 initial bearing.
211 @arg distance: Distance travelled (C{meter}, same units as
212 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 (C{meter}, same
216 units a B{C{radius}}).
218 @return: Destination point (L{LatLon}).
220 @raise ValueError: Invalid B{C{distance}}, B{C{bearing}},
221 B{C{radius}} or B{C{height}}.
222 '''
223 a, b = self.philam
224 r, t = _m2radians(distance, radius, low=None), Bearing_(bearing)
226 a, b = _destination2(a, b, r, t)
227 h = self._heigHt(height)
228 return self.classof(degrees90(a), degrees180(b), height=h)
230 def distanceTo(self, other, radius=R_M, wrap=False):
231 '''Compute the (angular) distance from this to an other point.
233 @arg other: The other point (L{LatLon}).
234 @kwarg radius: Mean earth radius (C{meter}) or C{None}.
235 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
236 the B{C{other}} point (C{bool}).
238 @return: Distance between this and the B{C{other}} point
239 (C{meter}, same units as B{C{radius}} or
240 C{radians} if B{C{radius}} is C{None}).
242 @raise TypeError: The B{C{other}} point is not L{LatLon}.
244 @raise ValueError: Invalid B{C{radius}}.
245 '''
246 a1, _, a2, _, db = self._ab1_ab2_db5(other, wrap)
247 return _radians2m(vincentys_(a2, a1, db), radius)
249# @Property_RO
250# def Ecef(self):
251# '''Get the ECEF I{class} (L{EcefVeness}), I{lazily}.
252# '''
253# return _MODS.ecef.EcefKarney
255 def greatCircle(self, bearing, Vector=Vector3d, **Vector_kwds):
256 '''Compute the vector normal to great circle obtained by heading
257 on the given initial bearing from this point.
259 Direction of vector is such that initial bearing vector
260 b = c × n, where n is an n-vector representing this point.
262 @arg bearing: Bearing from this point (compass C{degrees360}).
263 @kwarg Vector: Vector class to return the great circle,
264 overriding the default L{Vector3d}.
265 @kwarg Vector_kwds: Optional, additional keyword argunents
266 for B{C{Vector}}.
268 @return: Vector representing great circle (C{Vector}).
270 @raise ValueError: Invalid B{C{bearing}}.
271 '''
272 a, b = self.philam
273 sa, ca, sb, cb, st, ct = sincos2_(a, b, Bearing_(bearing))
275 return Vector(sb * ct - cb * sa * st,
276 -cb * ct - sb * sa * st,
277 ca * st, **Vector_kwds) # XXX .unit()?
279 def initialBearingTo(self, other, wrap=False, raiser=False):
280 '''Compute the initial bearing (forward azimuth) from this
281 to an other point.
283 @arg other: The other point (spherical L{LatLon}).
284 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
285 the B{C{other}} point (C{bool}).
286 @kwarg raiser: Optionally, raise L{CrossError} (C{bool}),
287 use C{B{raiser}=True} for behavior like
288 C{sphericalNvector.LatLon.initialBearingTo}.
290 @return: Initial bearing (compass C{degrees360}).
292 @raise CrossError: If this and the B{C{other}} point coincide,
293 provided both B{C{raiser}} is C{True} and
294 L{pygeodesy.crosserrors} is C{True}.
296 @raise TypeError: The B{C{other}} point is not L{LatLon}.
297 '''
298 a1, b1, a2, b2, db = self._ab1_ab2_db5(other, wrap)
299 # XXX behavior like sphericalNvector.LatLon.initialBearingTo
300 if raiser and crosserrors() and max(fabs(a2 - a1), fabs(db)) < EPS:
301 raise CrossError(_point_, self, other=other, wrap=wrap, txt=_coincident_)
303 return degrees(bearing_(a1, b1, a2, b2, final=False))
305 def intermediateTo(self, other, fraction, height=None, wrap=False):
306 '''Locate the point at given fraction between (or along) this
307 and an other point.
309 @arg other: The other point (L{LatLon}).
310 @arg fraction: Fraction between both points (C{scalar},
311 0.0 at this and 1.0 at the other point).
312 @kwarg height: Optional height, overriding the intermediate
313 height (C{meter}).
314 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the
315 B{C{other}} point (C{bool}).
317 @return: Intermediate point (L{LatLon}).
319 @raise TypeError: The B{C{other}} point is not L{LatLon}.
321 @raise ValueError: Invalid B{C{fraction}} or B{C{height}}.
323 @see: Methods C{midpointTo} and C{rhumbMidpointTo}.
324 '''
325 p = self
326 f = Scalar(fraction=fraction)
327 if not isnear0(f):
328 p = p.others(other)
329 if wrap:
330 p = _Wrap.point(p)
331 if not isnear1(f): # and not near0
332 a1, b1 = self.philam
333 a2, b2 = p.philam
334 db, b2 = unrollPI(b1, b2, wrap=wrap)
335 r = vincentys_(a2, a1, db)
336 sr = sin(r)
337 if isnon0(sr):
338 sa1, ca1, sa2, ca2, \
339 sb1, cb1, sb2, cb2 = sincos2_(a1, a2, b1, b2)
341 t = f * r
342 a = sin(r - t) # / sr superflous
343 b = sin( t) # / sr superflous
345 x = a * ca1 * cb1 + b * ca2 * cb2
346 y = a * ca1 * sb1 + b * ca2 * sb2
347 z = a * sa1 + b * sa2
349 a = atan1d(z, hypot(x, y))
350 b = atan2d(y, x)
352 else: # PYCHOK no cover
353 a = degrees90( favg(a1, a2, f=f)) # coincident
354 b = degrees180(favg(b1, b2, f=f))
356 h = self._havg(other, f=f, h=height)
357 p = self.classof(a, b, height=h)
358 return p
360 def intersection(self, end1, other, end2, height=None, wrap=False):
361 '''Compute the intersection point of two lines, each defined by
362 two points or a start point and bearing from North.
364 @arg end1: End point of this line (L{LatLon}) or the initial
365 bearing at this point (compass C{degrees360}).
366 @arg other: Start point of the other line (L{LatLon}).
367 @arg end2: End point of the other line (L{LatLon}) or the
368 initial bearing at the B{C{other}} point (compass
369 C{degrees360}).
370 @kwarg height: Optional height for intersection point,
371 overriding the mean height (C{meter}).
372 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
373 B{C{start2}} and both B{C{end*}} points (C{bool}).
375 @return: The intersection point (L{LatLon}). An alternate
376 intersection point might be the L{antipode} to
377 the returned result.
379 @raise IntersectionError: Ambiguous or infinite intersection
380 or colinear, parallel or otherwise
381 non-intersecting lines.
383 @raise TypeError: If B{C{other}} is not L{LatLon} or B{C{end1}}
384 or B{C{end2}} not C{scalar} nor L{LatLon}.
386 @raise ValueError: Invalid B{C{height}} or C{null} line.
387 '''
388 try:
389 s2 = self.others(other)
390 return _intersect(self, end1, s2, end2, height=height, wrap=wrap,
391 LatLon=self.classof)
392 except (TypeError, ValueError) as x:
393 raise _xError(x, start1=self, end1=end1,
394 other=other, end2=end2, wrap=wrap)
396 def intersections2(self, rad1, other, rad2, radius=R_M, eps=_0_0,
397 height=None, wrap=True):
398 '''Compute the intersection points of two circles, each defined
399 by a center point and radius.
401 @arg rad1: Radius of the this circle (C{meter} or C{radians},
402 see B{C{radius}}).
403 @arg other: Center point of the other circle (L{LatLon}).
404 @arg rad2: Radius of the other circle (C{meter} or C{radians},
405 see B{C{radius}}).
406 @kwarg radius: Mean earth radius (C{meter} or C{None} if B{C{rad1}},
407 B{C{rad2}} and B{C{eps}} are given in C{radians}).
408 @kwarg eps: Required overlap (C{meter} or C{radians}, see
409 B{C{radius}}).
410 @kwarg height: Optional height for the intersection points (C{meter},
411 conventionally) or C{None} for the I{"radical height"}
412 at the I{radical line} between both centers.
413 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the
414 B{C{other}} point (C{bool}).
416 @return: 2-Tuple of the intersection points, each a L{LatLon}
417 instance. For abutting circles, both intersection
418 points are the same instance, aka the I{radical center}.
420 @raise IntersectionError: Concentric, antipodal, invalid or
421 non-intersecting circles.
423 @raise TypeError: If B{C{other}} is not L{LatLon}.
425 @raise ValueError: Invalid B{C{rad1}}, B{C{rad2}}, B{C{radius}},
426 B{C{eps}} or B{C{height}}.
427 '''
428 try:
429 c2 = self.others(other)
430 return _intersects2(self, rad1, c2, rad2, radius=radius, eps=eps,
431 height=height, wrap=wrap,
432 LatLon=self.classof)
433 except (TypeError, ValueError) as x:
434 raise _xError(x, center=self, rad1=rad1,
435 other=other, rad2=rad2, wrap=wrap)
437 @deprecated_method
438 def isEnclosedBy(self, points): # PYCHOK no cover
439 '''DEPRECATED, use method C{isenclosedBy}.'''
440 return self.isenclosedBy(points)
442 def isenclosedBy(self, points, wrap=False):
443 '''Check whether a (convex) polygon or composite encloses this point.
445 @arg points: The polygon points or composite (L{LatLon}[],
446 L{BooleanFHP} or L{BooleanGH}).
447 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the
448 B{C{points}} (C{bool}).
450 @return: C{True} if this point is inside the polygon or
451 composite, C{False} otherwise.
453 @raise PointsError: Insufficient number of B{C{points}}.
455 @raise TypeError: Some B{C{points}} are not L{LatLon}.
457 @raise ValueError: Invalid B{C{points}}, non-convex polygon.
459 @see: Functions L{pygeodesy.isconvex}, L{pygeodesy.isenclosedBy}
460 and L{pygeodesy.ispolar} especially if the B{C{points}} may
461 enclose a pole or wrap around the earth I{longitudinally}.
462 '''
463 if _MODS.booleans.isBoolean(points):
464 return points._encloses(self.lat, self.lon, wrap=wrap)
466 Ps = self.PointsIter(points, loop=2, dedup=True, wrap=wrap)
467 n0 = self._N_vector
469 v2 = Ps[0]._N_vector
470 p1 = Ps[1]
471 v1 = p1._N_vector
472 # check whether this point on same side of all
473 # polygon edges (to the left or right depending
474 # on the anti-/clockwise polygon direction)
475 gc1 = v2.cross(v1)
476 t0 = gc1.angleTo(n0) > PI_2
477 s0 = None
478 # get great-circle vector for each edge
479 for i, p2 in Ps.enumerate(closed=True):
480 if wrap and not Ps.looped:
481 p2 = _unrollon(p1, p2)
482 p1 = p2
483 v2 = p2._N_vector
484 gc = v1.cross(v2)
485 t = gc.angleTo(n0) > PI_2
486 if t != t0: # different sides of edge i
487 return False # outside
489 # check for convex polygon: angle between
490 # gc vectors, signed by direction of n0
491 # (otherwise the test above is not reliable)
492 s = signOf(gc1.angleTo(gc, vSign=n0))
493 if s != s0:
494 if s0 is None:
495 s0 = s
496 else:
497 t = _Fmt.SQUARE(points=i)
498 raise _ValueError(t, p2, wrap=wrap, txt=_not_(_convex_))
499 gc1, v1 = gc, v2
501 return True # inside
503 def midpointTo(self, other, height=None, fraction=_0_5, wrap=False):
504 '''Find the midpoint between this and an other point.
506 @arg other: The other point (L{LatLon}).
507 @kwarg height: Optional height for midpoint, overriding
508 the mean height (C{meter}).
509 @kwarg fraction: Midpoint location from this point (C{scalar}),
510 may be negative or greater than 1.0.
511 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the
512 B{C{other}} point (C{bool}).
514 @return: Midpoint (L{LatLon}).
516 @raise TypeError: The B{C{other}} point is not L{LatLon}.
518 @raise ValueError: Invalid B{C{height}}.
520 @see: Methods C{intermediateTo} and C{rhumbMidpointTo}.
521 '''
522 if fraction is _0_5:
523 # see <https://MathForum.org/library/drmath/view/51822.html>
524 a1, b, a2, _, db = self._ab1_ab2_db5(other, wrap)
525 sa1, ca1, sa2, ca2, sdb, cdb = sincos2_(a1, a2, db)
527 x = ca2 * cdb + ca1
528 y = ca2 * sdb
530 a = atan1d(sa1 + sa2, hypot(x, y))
531 b = degrees180(b + atan2(y, x))
533 h = self._havg(other, h=height)
534 r = self.classof(a, b, height=h)
535 else:
536 r = self.intermediateTo(other, fraction, height=height, wrap=wrap)
537 return r
539 def nearestOn(self, point1, point2, radius=R_M, **wrap_adjust_limit):
540 '''Locate the point between two points closest to this point.
542 Distances are approximated by function L{pygeodesy.equirectangular_},
543 subject to the supplied B{C{options}}.
545 @arg point1: Start point (L{LatLon}).
546 @arg point2: End point (L{LatLon}).
547 @kwarg radius: Mean earth radius (C{meter}).
548 @kwarg wrap_adjust_limit: Optional keyword arguments for functions
549 L{sphericalTrigonometry.nearestOn3} and
550 L{pygeodesy.equirectangular_},
552 @return: Closest point on the great circle line (L{LatLon}).
554 @raise LimitError: Lat- and/or longitudinal delta exceeds B{C{limit}},
555 see function L{pygeodesy.equirectangular_}.
557 @raise NotImplementedError: Keyword argument C{B{within}=False}
558 is not (yet) supported.
560 @raise TypeError: Invalid B{C{point1}} or B{C{point2}}.
562 @raise ValueError: Invalid B{C{radius}} or B{C{options}}.
564 @see: Functions L{pygeodesy.equirectangular_} and L{pygeodesy.nearestOn5}
565 and method L{sphericalTrigonometry.LatLon.nearestOn3}.
566 '''
567 # remove kwarg B{C{within}} if present
568 w = _xkwds_pop(wrap_adjust_limit, within=True)
569 if not w:
570 notImplemented(self, within=w)
572# # UNTESTED - handle C{B{within}=False} and C{B{within}=True}
573# wrap = _xkwds_get(options, wrap=False)
574# a = self.alongTrackDistanceTo(point1, point2, radius=radius, wrap=wrap)
575# if fabs(a) < EPS or (within and a < EPS):
576# return point1
577# d = point1.distanceTo(point2, radius=radius, wrap=wrap)
578# if isnear0(d):
579# return point1 # or point2
580# elif fabs(d - a) < EPS or (a + EPS) > d:
581# return point2
582# f = a / d
583# if within:
584# if f > EPS1:
585# return point2
586# elif f < EPS:
587# return point1
588# return point1.intermediateTo(point2, f, wrap=wrap)
590 # without kwarg B{C{within}}, use backward compatible .nearestOn3
591 return self.nearestOn3([point1, point2], closed=False, radius=radius,
592 **wrap_adjust_limit)[0]
594 @deprecated_method
595 def nearestOn2(self, points, closed=False, radius=R_M, **options): # PYCHOK no cover
596 '''DEPRECATED, use method L{sphericalTrigonometry.LatLon.nearestOn3}.
598 @return: ... 2-Tuple C{(closest, distance)} of the closest
599 point (L{LatLon}) on the polygon and the distance
600 to that point from this point in C{meter}, same
601 units of B{C{radius}}.
602 '''
603 r = self.nearestOn3(points, closed=closed, radius=radius, **options)
604 return r.closest, r.distance
606 def nearestOn3(self, points, closed=False, radius=R_M, **wrap_adjust_limit):
607 '''Locate the point on a polygon closest to this point.
609 Distances are approximated by function L{pygeodesy.equirectangular_},
610 subject to the supplied B{C{options}}.
612 @arg points: The polygon points (L{LatLon}[]).
613 @kwarg closed: Optionally, close the polygon (C{bool}).
614 @kwarg radius: Mean earth radius (C{meter}).
615 @kwarg wrap_adjust_limit: Optional keyword arguments for function
616 L{sphericalTrigonometry.nearestOn3} and
617 L{pygeodesy.equirectangular_},
619 @return: A L{NearestOn3Tuple}C{(closest, distance, angle)} of the
620 C{closest} point (L{LatLon}), the L{pygeodesy.equirectangular_}
621 C{distance} between this and the C{closest} point converted to
622 C{meter}, same units as B{C{radius}}. The C{angle} from this
623 to the C{closest} point is in compass C{degrees360}, like
624 function L{pygeodesy.compassAngle}.
626 @raise LimitError: Lat- and/or longitudinal delta exceeds B{C{limit}},
627 see function L{pygeodesy.equirectangular_}.
629 @raise PointsError: Insufficient number of B{C{points}}.
631 @raise TypeError: Some B{C{points}} are not C{LatLon}.
633 @raise ValueError: Invalid B{C{radius}} or B{C{options}}.
635 @see: Functions L{pygeodesy.compassAngle}, L{pygeodesy.equirectangular_}
636 and L{pygeodesy.nearestOn5}.
637 '''
638 return nearestOn3(self, points, closed=closed, radius=radius,
639 LatLon=self.classof, **wrap_adjust_limit)
641 def toCartesian(self, **Cartesian_datum_kwds): # PYCHOK Cartesian=Cartesian, datum=None
642 '''Convert this point to C{Karney}-based cartesian (ECEF)
643 coordinates.
645 @kwarg Cartesian_datum_kwds: Optional L{Cartesian}, B{C{datum}}
646 and other keyword arguments, ignored
647 if C{B{Cartesian} is None}. Use
648 C{B{Cartesian}=...} to override
649 this L{Cartesian} class or specify
650 C{B{Cartesian}=None}.
652 @return: The cartesian point (L{Cartesian}) or if B{C{Cartesian}}
653 is C{None}, an L{Ecef9Tuple}C{(x, y, z, lat, lon, height,
654 C, M, datum)} with C{C} and C{M} if available.
656 @raise TypeError: Invalid B{C{Cartesian_datum_kwds}} argument.
657 '''
658 kwds = _xkwds(Cartesian_datum_kwds, Cartesian=Cartesian, datum=self.datum)
659 return LatLonSphericalBase.toCartesian(self, **kwds)
661 def triangle7(self, otherB, otherC, radius=R_M, wrap=False):
662 '''Compute the angles, sides and area of a spherical triangle.
664 @arg otherB: Second triangle point (C{LatLon}).
665 @arg otherC: Third triangle point (C{LatLon}).
666 @kwarg radius: Mean earth radius, ellipsoid or datum
667 (C{meter}, L{Ellipsoid}, L{Ellipsoid2},
668 L{Datum} or L{a_f2Tuple}) or C{None}.
669 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the
670 B{C{otherB}} and B{C{otherC}} points (C{bool}).
672 @return: L{Triangle7Tuple}C{(A, a, B, b, C, c, area)} or if
673 B{C{radius}} is C{None}, a L{Triangle8Tuple}C{(A,
674 a, B, b, C, c, D, E)}.
676 @see: Function L{triangle7} and U{Spherical trigonometry
677 <https://WikiPedia.org/wiki/Spherical_trigonometry>}.
678 '''
679 B = self.others(otherB=otherB)
680 C = self.others(otherC=otherC)
681 B, C, _ = _unrollon3(self, B, C, wrap)
683 r = self.philam + B.philam + C.philam
684 t = triangle8_(*r, wrap=wrap)
685 return self._xnamed(_t7Tuple(t, radius))
687 def trilaterate5(self, distance1, point2, distance2, point3, distance3,
688 area=True, eps=EPS1, radius=R_M, wrap=False):
689 '''Trilaterate three points by I{area overlap} or I{perimeter
690 intersection} of three corresponding circles.
692 @arg distance1: Distance to this point (C{meter}, same units
693 as B{C{radius}}).
694 @arg point2: Second center point (C{LatLon}).
695 @arg distance2: Distance to point2 (C{meter}, same units as
696 B{C{radius}}).
697 @arg point3: Third center point (C{LatLon}).
698 @arg distance3: Distance to point3 (C{meter}, same units as
699 B{C{radius}}).
700 @kwarg area: If C{True} compute the area overlap, otherwise the
701 perimeter intersection of the circles (C{bool}).
702 @kwarg eps: The required I{minimal overlap} for C{B{area}=True}
703 or the I{intersection margin} for C{B{area}=False}
704 (C{meter}, same units as B{C{radius}}).
705 @kwarg radius: Mean earth radius (C{meter}, conventionally).
706 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
707 B{C{point2}} and B{C{point3}} (C{bool}).
709 @return: A L{Trilaterate5Tuple}C{(min, minPoint, max, maxPoint, n)}
710 with C{min} and C{max} in C{meter}, same units as B{C{eps}},
711 the corresponding trilaterated points C{minPoint} and
712 C{maxPoint} as I{spherical} C{LatLon} and C{n}, the number
713 of trilatered points found for the given B{C{eps}}.
715 If only a single trilaterated point is found, C{min I{is}
716 max}, C{minPoint I{is} maxPoint} and C{n = 1}.
718 For C{B{area}=True}, C{min} and C{max} are the smallest
719 respectively largest I{radial} overlap found.
721 For C{B{area}=False}, C{min} and C{max} represent the
722 nearest respectively farthest intersection margin.
724 If C{B{area}=True} and all 3 circles are concentric, C{n =
725 0} and C{minPoint} and C{maxPoint} are both the B{C{point#}}
726 with the smallest B{C{distance#}} C{min} and C{max} the
727 largest B{C{distance#}}.
729 @raise IntersectionError: Trilateration failed for the given B{C{eps}},
730 insufficient overlap for C{B{area}=True} or
731 no intersection or all (near-)concentric for
732 C{B{area}=False}.
734 @raise TypeError: Invalid B{C{point2}} or B{C{point3}}.
736 @raise ValueError: Coincident B{C{point2}} or B{C{point3}} or invalid
737 B{C{distance1}}, B{C{distance2}}, B{C{distance3}}
738 or B{C{radius}}.
739 '''
740 return _trilaterate5(self, distance1,
741 self.others(point2=point2), distance2,
742 self.others(point3=point3), distance3,
743 area=area, radius=radius, eps=eps, wrap=wrap)
746_T00 = LatLon(0, 0, name='T00') # reference instance (L{LatLon})
749def areaOf(points, radius=R_M, wrap=False): # was=True
750 '''Calculate the area of a (spherical) polygon or composite
751 (with the pointsjoined by great circle arcs).
753 @arg points: The polygon points or clips (L{LatLon}[], L{BooleanFHP}
754 or L{BooleanGH}).
755 @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter},
756 L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple})
757 or C{None}.
758 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the B{C{points}}
759 (C{bool}).
761 @return: Polygon area (C{meter} I{quared}, same units as B{C{radius}}
762 or C{radians} if B{C{radius}} is C{None}).
764 @raise PointsError: Insufficient number of B{C{points}}.
766 @raise TypeError: Some B{C{points}} are not L{LatLon}.
768 @raise ValueError: Invalid B{C{radius}} or semi-circular polygon edge.
770 @note: The area is based on I{Karney}'s U{'Area of a spherical
771 polygon'<https://MathOverflow.net/questions/97711/
772 the-area-of-spherical-polygons>}, 3rd Answer.
774 @see: Functions L{pygeodesy.areaOf}, L{sphericalNvector.areaOf},
775 L{pygeodesy.excessKarney}, L{ellipsoidalExact.areaOf} and
776 L{ellipsoidalKarney.areaOf}.
777 '''
778 if _MODS.booleans.isBoolean(points):
779 return points._sum2(LatLon, areaOf, radius=radius, wrap=wrap)
781 _at2, _t_2 = atan2, tan_2
782 _un, _w180 = unrollPI, wrap180
784 Ps = _T00.PointsIter(points, loop=1, wrap=wrap)
785 p1 = p2 = Ps[0]
786 a1, b1 = p1.philam
787 ta1, z1 = _t_2(a1), None
789 A = Fsum() # mean phi
790 R = Fsum() # see L{pygeodesy.excessKarney_}
791 # ispolar: Summation of course deltas around pole is 0° rather than normally ±360°
792 # <https://blog.Element84.com/determining-if-a-spherical-polygon-contains-a-pole.html>
793 # XXX duplicate of function C{points.ispolar} to avoid copying all iterated points
794 D = Fsum()
795 for i, p2 in Ps.enumerate(closed=True):
796 a2, b2 = p2.philam
797 db, b2 = _un(b1, b2, wrap=wrap and not Ps.looped)
798 A += a2
799 ta2 = _t_2(a2)
800 tdb = _t_2(db, points=i)
801 R += _at2(tdb * (ta1 + ta2),
802 _1_0 + ta1 * ta2)
803 ta1, b1 = ta2, b2
805 if not p2.isequalTo(p1, eps=EPS):
806 z, z2 = _bearingTo2(p1, p2, wrap=wrap)
807 if z1 is not None:
808 D += _w180(z - z1) # (z - z1 + 540) ...
809 D += _w180(z2 - z) # (z2 - z + 540) % 360 - 180
810 p1, z1 = p2, z2
812 R = abs(R * _2_0)
813 if abs(D) < _90_0: # ispolar(points)
814 R = abs(R - PI2)
815 if radius:
816 a = degrees(A.fover(len(A))) # mean lat
817 R *= _mean_radius(radius, a)**2
818 return float(R)
821def _destination2(a, b, r, t):
822 '''(INTERNAL) Destination lat- and longitude in C{radians}.
824 @arg a: Latitude (C{radians}).
825 @arg b: Longitude (C{radians}).
826 @arg r: Angular distance (C{radians}).
827 @arg t: Bearing (compass C{radians}).
829 @return: 2-Tuple (phi, lam) of (C{radians}, C{radiansPI}).
830 '''
831 # see <https://www.EdWilliams.org/avform.htm#LL>
832 sa, ca, sr, cr, st, ct = sincos2_(a, r, t)
833 ca *= sr
835 a = asin1(ct * ca + cr * sa)
836 d = atan2(st * ca, cr - sa * sin(a))
837 # note, in EdWilliams.org/avform.htm W is + and E is -
838 return a, (b + d) # (mod(b + d + PI, PI2) - PI)
841def _int3d2(s, end, wrap, _i_, Vector, hs):
842 # see <https://www.EdWilliams.org/intersect.htm> (5) ff
843 # and similar logic in .ellipsoidalBaseDI._intersect3
844 a1, b1 = s.philam
846 if _isDegrees(end): # bearing, get pseudo-end point
847 a2, b2 = _destination2(a1, b1, PI_4, radians(end))
848 else: # must be a point
849 s.others(end, name=_end_ + _i_)
850 hs.append(end.height)
851 a2, b2 = end.philam
852 if wrap:
853 a2, b2 = _Wrap.philam(a2, b2)
855 db, b2 = unrollPI(b1, b2, wrap=wrap)
856 if max(fabs(db), fabs(a2 - a1)) < EPS:
857 raise _ValueError(_SPACE_(_line_ + _i_, _null_))
858 # note, in EdWilliams.org/avform.htm W is + and E is -
859 sb21, cb21, sb12, cb12 = sincos2_(db * _0_5,
860 -(b1 + b2) * _0_5)
861 cb21 *= sin(a1 - a2) # sa21
862 sb21 *= sin(a1 + a2) # sa12
863 x = Vector(sb12 * cb21 - cb12 * sb21,
864 cb12 * cb21 + sb12 * sb21,
865 cos(a1) * cos(a2) * sin(db)) # ll=start
866 return x.unit(), (db, (a2 - a1)) # negated d
869def _intdot(ds, a1, b1, a, b, wrap):
870 # compute dot product ds . (-b + b1, a - a1)
871 db, _ = unrollPI(b1, b, wrap=wrap)
872 return fdot(ds, db, a - a1)
875def intersecant2(center, circle, point, other, **radius_exact_height_wrap):
876 '''Compute the intersections of a circle and a (great circle) line given as
877 two points or as a point and bearing.
879 @arg center: Center of the circle (L{LatLon}).
880 @arg circle: Radius of the circle (C{meter}, same units as B{C{radius}})
881 or a point on the circle (L{LatLon}).
882 @arg point: A point on the (great circle) line (L{LatLon}).
883 @arg other: An other point on the (great circle) line (L{LatLon}) or
884 the bearing at the B{C{point}} (compass C{degrees360}).
885 @kwarg radius_exact_height_wrap: Optional keyword arguments, see
886 method L{LatLon.intersecant2} for further details.
888 @return: 2-Tuple of the intersection points (representing a chord), each
889 an instance of the B{C{point}} class. Both points are the same
890 instance if the (great circle) line is tangent to the circle.
892 @raise IntersectionError: The circle and line do not intersect.
894 @raise TypeError: If B{C{center}} or B{C{point}} not L{LatLon} or
895 B{C{circle}} or B{C{other}} invalid.
897 @raise UnitError: Invalid B{C{circle}}, B{C{other}}, B{C{radius}},
898 B{C{exact}}, B{C{height}} or B{C{napieradius}}.
899 '''
900 c = _T00.others(center=center)
901 p = _T00.others(point=point)
902 try:
903 return _intersecant2(c, circle, p, other, **radius_exact_height_wrap)
904 except (TypeError, ValueError) as x:
905 raise _xError(x, center=center, circle=circle, point=point, other=other,
906 **radius_exact_height_wrap)
909def _intersect(start1, end1, start2, end2, height=None, wrap=False, # in.ellipsoidalBaseDI._intersect3
910 LatLon=None, **LatLon_kwds):
911 # (INTERNAL) Intersect two (spherical) lines, see L{intersection}
912 # above, separated to allow callers to embellish any exceptions
914 s1, s2 = start1, start2
915 if wrap:
916 s2 = _Wrap.point(s2)
917 hs = [s1.height, s2.height]
919 a1, b1 = s1.philam
920 a2, b2 = s2.philam
921 db, b2 = unrollPI(b1, b2, wrap=wrap)
922 r12 = vincentys_(a2, a1, db)
923 if fabs(r12) < EPS: # [nearly] coincident points
924 a, b = favg(a1, a2), favg(b1, b2)
926 # see <https://www.EdWilliams.org/avform.htm#Intersection>
927 elif _isDegrees(end1) and _isDegrees(end2): # both bearings
928 sa1, ca1, sa2, ca2, sr12, cr12 = sincos2_(a1, a2, r12)
930 x1, x2 = (sr12 * ca1), (sr12 * ca2)
931 if isnear0(x1) or isnear0(x2):
932 raise IntersectionError(_parallel_)
933 # handle domain error for equivalent longitudes,
934 # see also functions asin_safe and acos_safe at
935 # <https://www.EdWilliams.org/avform.htm#Math>
936 t12, t13 = acos1((sa2 - sa1 * cr12) / x1), radiansPI2(end1)
937 t21, t23 = acos1((sa1 - sa2 * cr12) / x2), radiansPI2(end2)
938 if sin(db) > 0:
939 t21 = PI2 - t21
940 else:
941 t12 = PI2 - t12
942 sx1, cx1, sx2, cx2 = sincos2_(wrapPI(t13 - t12), # angle 2-1-3
943 wrapPI(t21 - t23)) # angle 1-2-3)
944 if isnear0(sx1) and isnear0(sx2):
945 raise IntersectionError(_infinite_)
946 sx3 = sx1 * sx2
947# XXX if sx3 < 0:
948# XXX raise ValueError(_ambiguous_)
949 x3 = acos1(cr12 * sx3 - cx2 * cx1)
950 r13 = atan2(sr12 * sx3, cx2 + cx1 * cos(x3))
952 a, b = _destination2(a1, b1, r13, t13)
953 # like .ellipsoidalBaseDI,_intersect3, if this intersection
954 # is "before" the first point, use the antipodal intersection
955 if opposing_(t13, bearing_(a1, b1, a, b, wrap=wrap)):
956 a, b = antipode_(a, b) # PYCHOK PhiLam2Tuple
958 else: # end point(s) or bearing(s)
959 _N_vector_ = _MODS.nvectorBase._N_vector_
961 x1, d1 = _int3d2(s1, end1, wrap, _1_, _N_vector_, hs)
962 x2, d2 = _int3d2(s2, end2, wrap, _2_, _N_vector_, hs)
963 x = x1.cross(x2)
964 if x.length < EPS: # [nearly] colinear or parallel lines
965 raise IntersectionError(_colinear_)
966 a, b = x.philam
967 # choose intersection similar to sphericalNvector
968 if not (_intdot(d1, a1, b1, a, b, wrap) *
969 _intdot(d2, a2, b2, a, b, wrap)) > 0:
970 a, b = antipode_(a, b) # PYCHOK PhiLam2Tuple
972 h = fmean(hs) if height is None else Height(height)
973 return _LL3Tuple(degrees90(a), degrees180(b), h,
974 intersection, LatLon, LatLon_kwds)
977def intersection(start1, end1, start2, end2, height=None, wrap=False,
978 LatLon=LatLon, **LatLon_kwds):
979 '''Compute the intersection point of two lines, each defined
980 by two points or a start point and bearing from North.
982 @arg start1: Start point of the first line (L{LatLon}).
983 @arg end1: End point of the first line (L{LatLon}) or
984 the initial bearing at the first start point
985 (compass C{degrees360}).
986 @arg start2: Start point of the second line (L{LatLon}).
987 @arg end2: End point of the second line (L{LatLon}) or
988 the initial bearing at the second start point
989 (compass C{degrees360}).
990 @kwarg height: Optional height for the intersection point,
991 overriding the mean height (C{meter}).
992 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
993 B{C{start2}} and both B{C{end*}} points (C{bool}).
994 @kwarg LatLon: Optional class to return the intersection
995 point (L{LatLon}) or C{None}.
996 @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword
997 arguments, ignored if C{B{LatLon} is None}.
999 @return: The intersection point as a (B{C{LatLon}}) or if
1000 C{B{LatLon} is None} a L{LatLon3Tuple}C{(lat, lon,
1001 height)}. An alternate intersection point might
1002 be the L{antipode} to the returned result.
1004 @raise IntersectionError: Ambiguous or infinite intersection
1005 or colinear, parallel or otherwise
1006 non-intersecting lines.
1008 @raise TypeError: A B{C{start1}}, B{C{end1}}, B{C{start2}}
1009 or B{C{end2}} point not L{LatLon}.
1011 @raise ValueError: Invalid B{C{height}} or C{null} line.
1012 '''
1013 s1 = _T00.others(start1=start1)
1014 s2 = _T00.others(start2=start2)
1015 try:
1016 return _intersect(s1, end1, s2, end2, height=height, wrap=wrap,
1017 LatLon=LatLon, **LatLon_kwds)
1018 except (TypeError, ValueError) as x:
1019 raise _xError(x, start1=start1, end1=end1, start2=start2, end2=end2)
1022def intersections2(center1, rad1, center2, rad2, radius=R_M, eps=_0_0,
1023 height=None, wrap=False, # was=True
1024 LatLon=LatLon, **LatLon_kwds):
1025 '''Compute the intersection points of two circles each defined
1026 by a center point and a radius.
1028 @arg center1: Center of the first circle (L{LatLon}).
1029 @arg rad1: Radius of the first circle (C{meter} or C{radians},
1030 see B{C{radius}}).
1031 @arg center2: Center of the second circle (L{LatLon}).
1032 @arg rad2: Radius of the second circle (C{meter} or C{radians},
1033 see B{C{radius}}).
1034 @kwarg radius: Mean earth radius (C{meter} or C{None} if B{C{rad1}},
1035 B{C{rad2}} and B{C{eps}} are given in C{radians}).
1036 @kwarg eps: Required overlap (C{meter} or C{radians}, see
1037 B{C{radius}}).
1038 @kwarg height: Optional height for the intersection points (C{meter},
1039 conventionally) or C{None} for the I{"radical height"}
1040 at the I{radical line} between both centers.
1041 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{center2}}
1042 (C{bool}).
1043 @kwarg LatLon: Optional class to return the intersection
1044 points (L{LatLon}) or C{None}.
1045 @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword
1046 arguments, ignored if C{B{LatLon} is None}.
1048 @return: 2-Tuple of the intersection points, each a B{C{LatLon}}
1049 instance or if C{B{LatLon} is None} a L{LatLon3Tuple}C{(lat,
1050 lon, height)}. For abutting circles, both intersection
1051 points are the same instance, aka the I{radical center}.
1053 @raise IntersectionError: Concentric, antipodal, invalid or
1054 non-intersecting circles.
1056 @raise TypeError: If B{C{center1}} or B{C{center2}} not L{LatLon}.
1058 @raise ValueError: Invalid B{C{rad1}}, B{C{rad2}}, B{C{radius}},
1059 B{C{eps}} or B{C{height}}.
1061 @note: Courtesy of U{Samuel Čavoj<https://GitHub.com/mrJean1/PyGeodesy/issues/41>}.
1063 @see: This U{Answer<https://StackOverflow.com/questions/53324667/
1064 find-intersection-coordinates-of-two-circles-on-earth/53331953>}.
1065 '''
1066 c1 = _T00.others(center1=center1)
1067 c2 = _T00.others(center2=center2)
1068 try:
1069 return _intersects2(c1, rad1, c2, rad2, radius=radius, eps=eps,
1070 height=height, wrap=wrap,
1071 LatLon=LatLon, **LatLon_kwds)
1072 except (TypeError, ValueError) as x:
1073 raise _xError(x, center1=center1, rad1=rad1,
1074 center2=center2, rad2=rad2, wrap=wrap)
1077def _intersects2(c1, rad1, c2, rad2, radius=R_M, eps=_0_0, # in .ellipsoidalBaseDI._intersects2
1078 height=None, too_d=None, wrap=False, # was=True
1079 LatLon=LatLon, **LatLon_kwds):
1080 # (INTERNAL) Intersect two spherical circles, see L{intersections2}
1081 # above, separated to allow callers to embellish any exceptions
1083 def _dest3(bearing, h):
1084 a, b = _destination2(a1, b1, r1, bearing)
1085 return _LL3Tuple(degrees90(a), degrees180(b), h,
1086 intersections2, LatLon, LatLon_kwds)
1088 a1, b1 = c1.philam
1089 a2, b2 = c2.philam
1090 if wrap:
1091 a2, b2 = _Wrap.philam(a2, b2)
1093 r1, r2, f = _rads3(rad1, rad2, radius)
1094 if f: # swapped radii, swap centers
1095 a1, a2 = a2, a1 # PYCHOK swap!
1096 b1, b2 = b2, b1 # PYCHOK swap!
1098 db, b2 = unrollPI(b1, b2, wrap=wrap)
1099 d = vincentys_(a2, a1, db) # radians
1100 if d < max(r1 - r2, EPS):
1101 raise IntersectionError(_near_(_concentric_)) # XXX ConcentricError?
1103 r = eps if radius is None else (m2radians(
1104 eps, radius=radius) if eps else _0_0)
1105 if r < _0_0:
1106 raise _ValueError(eps=r)
1108 x = fsumf_(r1, r2, -d) # overlap
1109 if x > max(r, EPS):
1110 sd, cd, sr1, cr1, _, cr2 = sincos2_(d, r1, r2)
1111 x = sd * sr1
1112 if isnear0(x):
1113 raise _ValueError(_invalid_)
1114 x = acos1((cr2 - cd * cr1) / x) # 0 <= x <= PI
1116 elif x < r: # PYCHOK no cover
1117 t = (d * radius) if too_d is None else too_d
1118 raise IntersectionError(_too_(_Fmt.distant(t)))
1120 if height is None: # "radical height"
1121 f = _radical2(d, r1, r2).ratio
1122 h = Height(favg(c1.height, c2.height, f=f))
1123 else:
1124 h = Height(height)
1126 b = bearing_(a1, b1, a2, b2, final=False, wrap=wrap)
1127 if x < EPS4: # externally ...
1128 r = _dest3(b, h)
1129 elif x > _PI_EPS4: # internally ...
1130 r = _dest3(b + PI, h)
1131 else:
1132 return _dest3(b + x, h), _dest3(b - x, h)
1133 return r, r # ... abutting circles
1136@deprecated_function
1137def isPoleEnclosedBy(points, wrap=False): # PYCHOK no cover
1138 '''DEPRECATED, use function L{pygeodesy.ispolar}.
1139 '''
1140 return ispolar(points, wrap=wrap)
1143def _LL3Tuple(lat, lon, height, where, LatLon, LatLon_kwds):
1144 '''(INTERNAL) Helper for L{intersection}, L{intersections2} and L{meanOf}.
1145 '''
1146 n = where.__name__
1147 if LatLon is None:
1148 r = LatLon3Tuple(lat, lon, height, name=n)
1149 else:
1150 kwds = _xkwds(LatLon_kwds, height=height, name=n)
1151 r = LatLon(lat, lon, **kwds)
1152 return r
1155def meanOf(points, height=None, wrap=False, LatLon=LatLon, **LatLon_kwds):
1156 '''Compute the I{geographic} mean of several points.
1158 @arg points: Points to be averaged (L{LatLon}[]).
1159 @kwarg height: Optional height at mean point, overriding the mean
1160 height (C{meter}).
1161 @kwarg wrap: If C{True}, wrap or I{normalize} the B{C{points}}
1162 (C{bool}).
1163 @kwarg LatLon: Optional class to return the mean point (L{LatLon})
1164 or C{None}.
1165 @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword
1166 arguments, ignored if C{B{LatLon} is None}.
1168 @return: The geographic mean and height (B{C{LatLon}}) or a
1169 L{LatLon3Tuple}C{(lat, lon, height)} if B{C{LatLon}}
1170 is C{None}.
1172 @raise TypeError: Some B{C{points}} are not L{LatLon}.
1174 @raise ValueError: No B{C{points}} or invalid B{C{height}}.
1175 '''
1176 def _N_vs(ps, w):
1177 Ps = _T00.PointsIter(ps, wrap=w)
1178 for p in Ps.iterate(closed=False):
1179 yield p._N_vector
1181 m = _MODS.nvectorBase
1182 # geographic, vectorial mean
1183 n = m.sumOf(_N_vs(points, wrap), h=height, Vector=m.NvectorBase)
1184 lat, lon, h = n.latlonheight
1185 return _LL3Tuple(lat, lon, h, meanOf, LatLon, LatLon_kwds)
1188@deprecated_function
1189def nearestOn2(point, points, **closed_radius_LatLon_options): # PYCHOK no cover
1190 '''DEPRECATED, use function L{sphericalTrigonometry.nearestOn3}.
1192 @return: ... 2-tuple C{(closest, distance)} of the C{closest}
1193 point (L{LatLon}) on the polygon and the C{distance}
1194 between the C{closest} and the given B{C{point}}. The
1195 C{closest} is a B{C{LatLon}} or a L{LatLon2Tuple}C{(lat,
1196 lon)} if B{C{LatLon}} is C{None} ...
1197 '''
1198 ll, d, _ = nearestOn3(point, points, **closed_radius_LatLon_options) # PYCHOK 3-tuple
1199 if _xkwds_get(closed_radius_LatLon_options, LatLon=LatLon) is None:
1200 ll = LatLon2Tuple(ll.lat, ll.lon)
1201 return ll, d
1204def nearestOn3(point, points, closed=False, radius=R_M, wrap=False, adjust=True,
1205 limit=9, **LatLon_and_kwds):
1206 '''Locate the point on a path or polygon closest to a reference point.
1208 Distances are I{approximated} using function L{pygeodesy.equirectangular_},
1209 subject to the supplied B{C{options}}.
1211 @arg point: The reference point (L{LatLon}).
1212 @arg points: The path or polygon points (L{LatLon}[]).
1213 @kwarg closed: Optionally, close the polygon (C{bool}).
1214 @kwarg radius: Mean earth radius (C{meter}).
1215 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the
1216 B{C{points}} (C{bool}).
1217 @kwarg adjust: See function L{pygeodesy.equirectangular_} (C{bool}).
1218 @kwarg limit: See function L{pygeodesy.equirectangular_} (C{degrees}),
1219 default C{9 degrees} is about C{1,000 Kmeter} (for mean
1220 spherical earth radius L{R_KM}).
1221 @kwarg LatLon: Optional class to return the closest point (L{LatLon})
1222 or C{None}.
1223 @kwarg options: Optional keyword arguments for function
1224 L{pygeodesy.equirectangular_}.
1226 @return: A L{NearestOn3Tuple}C{(closest, distance, angle)} with the
1227 C{closest} point as B{C{LatLon}} or L{LatLon3Tuple}C{(lat,
1228 lon, height)} if B{C{LatLon}} is C{None}. The C{distance}
1229 is the L{pygeodesy.equirectangular_} distance between the
1230 C{closest} and the given B{C{point}} converted to C{meter},
1231 same units as B{C{radius}}. The C{angle} from the given
1232 B{C{point}} to the C{closest} is in compass C{degrees360},
1233 like function L{pygeodesy.compassAngle}. The C{height} is
1234 the (interpolated) height at the C{closest} point.
1236 @raise LimitError: Lat- and/or longitudinal delta exceeds the B{C{limit}},
1237 see function L{pygeodesy.equirectangular_}.
1239 @raise PointsError: Insufficient number of B{C{points}}.
1241 @raise TypeError: Some B{C{points}} are not C{LatLon}.
1243 @raise ValueError: Invalid B{C{radius}}.
1245 @see: Functions L{pygeodesy.equirectangular_} and L{pygeodesy.nearestOn5}.
1246 '''
1247 t = _nearestOn5(point, points, closed=closed, wrap=wrap,
1248 adjust=adjust, limit=limit)
1249 d = degrees2m(t.distance, radius=radius)
1250 h = t.height
1251 n = nearestOn3.__name__
1253 kwds = _xkwds(LatLon_and_kwds, height=h, name=n)
1254 LL = _xkwds_pop(kwds, LatLon=LatLon)
1255 r = LatLon3Tuple(t.lat, t.lon, h, name=n) if LL is None else \
1256 LL(t.lat, t.lon, **kwds)
1257 return NearestOn3Tuple(r, d, t.angle, name=n)
1260def perimeterOf(points, closed=False, radius=R_M, wrap=True):
1261 '''Compute the perimeter of a (spherical) polygon or composite
1262 (with great circle arcs joining the points).
1264 @arg points: The polygon points or clips (L{LatLon}[], L{BooleanFHP}
1265 or L{BooleanGH}).
1266 @kwarg closed: Optionally, close the polygon (C{bool}).
1267 @kwarg radius: Mean earth radius (C{meter}) or C{None}.
1268 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the
1269 B{C{points}} (C{bool}).
1271 @return: Polygon perimeter (C{meter}, same units as B{C{radius}}
1272 or C{radians} if B{C{radius}} is C{None}).
1274 @raise PointsError: Insufficient number of B{C{points}}.
1276 @raise TypeError: Some B{C{points}} are not L{LatLon}.
1278 @raise ValueError: Invalid B{C{radius}} or C{B{closed}=False} with
1279 C{B{points}} a composite.
1281 @note: Distances are based on function L{pygeodesy.vincentys_}.
1283 @see: Functions L{perimeterOf<pygeodesy.perimeterOf>},
1284 L{sphericalNvector.perimeterOf} and L{ellipsoidalKarney.perimeterOf}.
1285 '''
1286 def _rads(ps, c, w): # angular edge lengths in radians
1287 Ps = _T00.PointsIter(ps, loop=1, wrap=w)
1288 a1, b1 = Ps[0].philam
1289 for p in Ps.iterate(closed=c):
1290 a2, b2 = p.philam
1291 db, b2 = unrollPI(b1, b2, wrap=w and not (c and Ps.looped))
1292 yield vincentys_(a2, a1, db)
1293 a1, b1 = a2, b2
1295 if _MODS.booleans.isBoolean(points):
1296 if not closed:
1297 raise _ValueError(closed=closed, points=_composite_)
1298 r = points._sum2(LatLon, perimeterOf, closed=True, radius=radius, wrap=wrap)
1299 else:
1300 r = fsum(_rads(points, closed, wrap), floats=True)
1301 return _radians2m(r, radius)
1304def triangle7(latA, lonA, latB, lonB, latC, lonC, radius=R_M,
1305 excess=excessAbc_,
1306 wrap=False):
1307 '''Compute the angles, sides, and area of a (spherical) triangle.
1309 @arg latA: First corner latitude (C{degrees}).
1310 @arg lonA: First corner longitude (C{degrees}).
1311 @arg latB: Second corner latitude (C{degrees}).
1312 @arg lonB: Second corner longitude (C{degrees}).
1313 @arg latC: Third corner latitude (C{degrees}).
1314 @arg lonC: Third corner longitude (C{degrees}).
1315 @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter},
1316 L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple})
1317 or C{None}.
1318 @kwarg excess: I{Spherical excess} callable (L{excessAbc_},
1319 L{excessGirard_} or L{excessLHuilier_}).
1320 @kwarg wrap: If C{True}, wrap and L{pygeodesy.unroll180}
1321 longitudes (C{bool}).
1323 @return: A L{Triangle7Tuple}C{(A, a, B, b, C, c, area)} with
1324 spherical angles C{A}, C{B} and C{C}, angular sides
1325 C{a}, C{b} and C{c} all in C{degrees} and C{area}
1326 in I{square} C{meter} or same units as B{C{radius}}
1327 I{squared} or if C{B{radius}=0} or C{None}, a
1328 L{Triangle8Tuple}C{(A, a, B, b, C, c, D, E)} all in
1329 C{radians} with the I{spherical excess} C{E} as the
1330 C{unit area} in C{radians}.
1331 '''
1332 t = triangle8_(Phi_(latA=latA), Lam_(lonA=lonA),
1333 Phi_(latB=latB), Lam_(lonB=lonB),
1334 Phi_(latC=latC), Lam_(lonC=lonC),
1335 excess=excess, wrap=wrap)
1336 return _t7Tuple(t, radius)
1339def triangle8_(phiA, lamA, phiB, lamB, phiC, lamC, excess=excessAbc_,
1340 wrap=False):
1341 '''Compute the angles, sides, I{spherical deficit} and I{spherical
1342 excess} of a (spherical) triangle.
1344 @arg phiA: First corner latitude (C{radians}).
1345 @arg lamA: First corner longitude (C{radians}).
1346 @arg phiB: Second corner latitude (C{radians}).
1347 @arg lamB: Second corner longitude (C{radians}).
1348 @arg phiC: Third corner latitude (C{radians}).
1349 @arg lamC: Third corner longitude (C{radians}).
1350 @kwarg excess: I{Spherical excess} callable (L{excessAbc_},
1351 L{excessGirard_} or L{excessLHuilier_}).
1352 @kwarg wrap: If C{True}, L{pygeodesy.unrollPI} the
1353 longitudinal deltas (C{bool}).
1355 @return: A L{Triangle8Tuple}C{(A, a, B, b, C, c, D, E)} with
1356 spherical angles C{A}, C{B} and C{C}, angular sides
1357 C{a}, C{b} and C{c}, I{spherical deficit} C{D} and
1358 I{spherical excess} C{E}, all in C{radians}.
1359 '''
1360 def _a_r(w, phiA, lamA, phiB, lamB, phiC, lamC):
1361 d, _ = unrollPI(lamB, lamC, wrap=w)
1362 a = vincentys_(phiC, phiB, d)
1363 return a, (phiB, lamB, phiC, lamC, phiA, lamA) # rotate A, B, C
1365 def _A_r(a, sa, ca, sb, cb, sc, cc):
1366 s = sb * sc
1367 A = acos1((ca - cb * cc) / s) if isnon0(s) else a
1368 return A, (sb, cb, sc, cc, sa, ca) # rotate sincos2_'s
1370 # notation: side C{a} is oposite to corner C{A}, etc.
1371 a, r = _a_r(wrap, phiA, lamA, phiB, lamB, phiC, lamC)
1372 b, r = _a_r(wrap, *r)
1373 c, _ = _a_r(wrap, *r)
1375 A, r = _A_r(a, *sincos2_(a, b, c))
1376 B, r = _A_r(b, *r)
1377 C, _ = _A_r(c, *r)
1379 D = fsumf_(PI2, -a, -b, -c) # deficit aka defect
1380 E = excessGirard_(A, B, C) if excess in (excessGirard_, True) else (
1381 excessLHuilier_(a, b, c) if excess in (excessLHuilier_, False) else
1382 excessAbc_(*max((A, b, c), (B, c, a), (C, a, b))))
1384 return Triangle8Tuple(A, a, B, b, C, c, D, E)
1387def _t7Tuple(t, radius):
1388 '''(INTERNAL) Convert a L{Triangle8Tuple} to L{Triangle7Tuple}.
1389 '''
1390 if radius: # not in (None, _0_0)
1391 r = radius if _isRadius(radius) else \
1392 _ellipsoidal_datum(radius).ellipsoid.Rmean
1393 A, B, C = map1(degrees, t.A, t.B, t.C)
1394 t = Triangle7Tuple(A, (r * t.a),
1395 B, (r * t.b),
1396 C, (r * t.c), t.E * r**2)
1397 return t
1400__all__ += _ALL_OTHER(Cartesian, LatLon, # classes
1401 areaOf, # functions
1402 intersecant2, intersection, intersections2, ispolar,
1403 isPoleEnclosedBy, # DEPRECATED, use ispolar
1404 meanOf,
1405 nearestOn2, nearestOn3,
1406 perimeterOf,
1407 sumOf, # XXX == vector3d.sumOf
1408 triangle7, triangle8_)
1410# **) MIT License
1411#
1412# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved.
1413#
1414# Permission is hereby granted, free of charge, to any person obtaining a
1415# copy of this software and associated documentation files (the "Software"),
1416# to deal in the Software without restriction, including without limitation
1417# the rights to use, copy, modify, merge, publish, distribute, sublicense,
1418# and/or sell copies of the Software, and to permit persons to whom the
1419# Software is furnished to do so, subject to the following conditions:
1420#
1421# The above copyright notice and this permission notice shall be included
1422# in all copies or substantial portions of the Software.
1423#
1424# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
1425# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
1426# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
1427# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
1428# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
1429# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
1430# OTHER DEALINGS IN THE SOFTWARE.