Coverage for pygeodesy/sphericalTrigonometry.py: 93%
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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 _TypeError, _ValueError, IntersectionError, \
26 _xError, _xkwds, _xkwds_get, _xkwds_pop2
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.nvectorBase import NvectorBase, sumOf # _MODE
38from pygeodesy.namedTuples import LatLon2Tuple, LatLon3Tuple, NearestOn3Tuple, \
39 Triangle7Tuple, Triangle8Tuple
40from pygeodesy.points import ispolar, nearestOn5 as _nearestOn5, \
41 Fmt as _Fmt # XXX shadowed
42from pygeodesy.props import deprecated_function, deprecated_method
43from pygeodesy.sphericalBase import _m2radians, CartesianSphericalBase, \
44 _intersecant2, LatLonSphericalBase, \
45 _rads3, _radians2m, _trilaterate5
46# from pygeodesy.streprs import Fmt as _Fmt # from .points XXX shadowed
47from pygeodesy.units import Bearing_, Height, _isDegrees, _isRadius, Lam_, \
48 Phi_, Radius_, Scalar
49from pygeodesy.utily import acos1, asin1, atan1d, atan2d, degrees90, degrees180, \
50 degrees2m, m2radians, radiansPI2, sincos2_, tan_2, \
51 unrollPI, _unrollon, _unrollon3, _Wrap, wrap180, wrapPI
52from pygeodesy.vector3d import sumOf, Vector3d
54from math import asin, atan2, cos, degrees, fabs, radians, sin
56__all__ = _ALL_LAZY.sphericalTrigonometry
57__version__ = '24.04.07'
59_PI_EPS4 = PI - EPS4
60if _PI_EPS4 >= PI:
61 raise _AssertionError(EPS4=EPS4, PI=PI, PI_EPS4=_PI_EPS4)
64class Cartesian(CartesianSphericalBase):
65 '''Extended to convert geocentric, L{Cartesian} points to
66 spherical, geodetic L{LatLon}.
67 '''
69 def toLatLon(self, **LatLon_and_kwds): # PYCHOK LatLon=LatLon
70 '''Convert this cartesian point to a C{spherical} geodetic point.
72 @kwarg LatLon_and_kwds: Optional L{LatLon} and L{LatLon} keyword
73 arguments. Use C{B{LatLon}=...} to override
74 this L{LatLon} class or specify C{B{LatLon}=None}.
76 @return: The geodetic point (L{LatLon}) or if B{C{LatLon}} is C{None},
77 an L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)}
78 with C{C} and C{M} if available.
80 @raise TypeError: Invalid B{C{LatLon_and_kwds}} argument.
81 '''
82 kwds = _xkwds(LatLon_and_kwds, LatLon=LatLon, datum=self.datum)
83 return CartesianSphericalBase.toLatLon(self, **kwds)
86class LatLon(LatLonSphericalBase):
87 '''New point on a spherical earth model, based on trigonometry formulae.
88 '''
90 def _ab1_ab2_db5(self, other, wrap):
91 '''(INTERNAL) Helper for several methods.
92 '''
93 a1, b1 = self.philam
94 a2, b2 = self.others(other, up=2).philam
95 if wrap:
96 a2, b2 = _Wrap.philam(a2, b2)
97 db, b2 = unrollPI(b1, b2, wrap=wrap)
98 else: # unrollPI shortcut
99 db = b2 - b1
100 return a1, b1, a2, b2, db
102 def alongTrackDistanceTo(self, start, end, radius=R_M, wrap=False):
103 '''Compute the (signed) distance from the start to the closest
104 point on the great circle line defined by a start and an
105 end point.
107 That is, if a perpendicular is drawn from this point to the
108 great circle line, the along-track distance is the distance
109 from the start point to the point where the perpendicular
110 crosses the line.
112 @arg start: Start point of the great circle line (L{LatLon}).
113 @arg end: End point of the great circle line (L{LatLon}).
114 @kwarg radius: Mean earth radius (C{meter}) or C{None}.
115 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
116 the B{C{start}} and B{C{end}} point (C{bool}).
118 @return: Distance along the great circle line (C{radians}
119 if C{B{radius} is None} or C{meter}, same units
120 as B{C{radius}}), positive if I{after} the
121 B{C{start}} toward the B{C{end}} point of the
122 line, I{negative} if before or C{0} if at the
123 B{C{start}} point.
125 @raise TypeError: Invalid B{C{start}} or B{C{end}} point.
127 @raise ValueError: Invalid B{C{radius}}.
128 '''
129 r, x, b = self._a_x_b3(start, end, radius, wrap)
130 cx = cos(x)
131 return _0_0 if isnear0(cx) else \
132 _radians2m(copysign0(acos1(cos(r) / cx), cos(b)), radius)
134 def _a_x_b3(self, start, end, radius, wrap):
135 '''(INTERNAL) Helper for .along-/crossTrackDistanceTo.
136 '''
137 s = self.others(start=start)
138 e = self.others(end=end)
139 s, e, w = _unrollon3(self, s, e, wrap)
141 r = Radius_(radius)
142 r = s.distanceTo(self, r, wrap=w) / r
144 b = radians(s.initialBearingTo(self, wrap=w)
145 - s.initialBearingTo(e, wrap=w))
146 x = asin(sin(r) * sin(b))
147 return r, x, -b
149 @deprecated_method
150 def bearingTo(self, other, wrap=False, raiser=False): # PYCHOK no cover
151 '''DEPRECATED, use method L{initialBearingTo}.
152 '''
153 return self.initialBearingTo(other, wrap=wrap, raiser=raiser)
155 def crossingParallels(self, other, lat, wrap=False):
156 '''Return the pair of meridians at which a great circle defined
157 by this and an other point crosses the given latitude.
159 @arg other: The other point defining great circle (L{LatLon}).
160 @arg lat: Latitude at the crossing (C{degrees}).
161 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the
162 B{C{other}} point (C{bool}).
164 @return: 2-Tuple C{(lon1, lon2)}, both in C{degrees180} or
165 C{None} if the great circle doesn't reach B{C{lat}}.
166 '''
167 a1, b1, a2, b2, db = self._ab1_ab2_db5(other, wrap)
168 sa, ca, sa1, ca1, \
169 sa2, ca2, sdb, cdb = sincos2_(radians(lat), a1, a2, db)
170 sa1 *= ca2 * ca
172 x = sa1 * sdb
173 y = sa1 * cdb - ca1 * sa2 * ca
174 z = ca1 * sdb * ca2 * sa
176 h = hypot(x, y)
177 if h < EPS or fabs(z) > h: # PYCHOK no cover
178 return None # great circle doesn't reach latitude
180 m = atan2(-y, x) + b1 # longitude at max latitude
181 d = acos1(z / h) # delta longitude to intersections
182 return degrees180(m - d), degrees180(m + d)
184 def crossTrackDistanceTo(self, start, end, radius=R_M, wrap=False):
185 '''Compute the (signed) distance from this point to
186 the great circle defined by a start and an end point.
188 @arg start: Start point of the great circle line (L{LatLon}).
189 @arg end: End point of the great circle line (L{LatLon}).
190 @kwarg radius: Mean earth radius (C{meter}) or C{None}.
191 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
192 the B{C{start}} and B{C{end}} point (C{bool}).
194 @return: Distance to the great circle (C{radians} if
195 B{C{radius}} or C{meter}, same units as
196 B{C{radius}}), I{negative} if to the left or
197 I{positive} if to the right of the line.
199 @raise TypeError: If B{C{start}} or B{C{end}} is not L{LatLon}.
201 @raise ValueError: Invalid B{C{radius}}.
202 '''
203 _, x, _ = self._a_x_b3(start, end, radius, wrap)
204 return _radians2m(x, radius)
206 def destination(self, distance, bearing, radius=R_M, height=None):
207 '''Locate the destination from this point after having
208 travelled the given distance on the given initial bearing.
210 @arg distance: Distance travelled (C{meter}, same units as
211 B{C{radius}}).
212 @arg bearing: Bearing from this point (compass C{degrees360}).
213 @kwarg radius: Mean earth radius (C{meter}).
214 @kwarg height: Optional height at destination (C{meter}, same
215 units a B{C{radius}}).
217 @return: Destination point (L{LatLon}).
219 @raise ValueError: Invalid B{C{distance}}, B{C{bearing}},
220 B{C{radius}} or B{C{height}}.
221 '''
222 a, b = self.philam
223 r, t = _m2radians(distance, radius, low=None), Bearing_(bearing)
225 a, b = _destination2(a, b, r, t)
226 h = self._heigHt(height)
227 return self.classof(degrees90(a), degrees180(b), height=h)
229 def distanceTo(self, other, radius=R_M, wrap=False):
230 '''Compute the (angular) distance from this to an other point.
232 @arg other: The other point (L{LatLon}).
233 @kwarg radius: Mean earth radius (C{meter}) or C{None}.
234 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
235 the B{C{other}} point (C{bool}).
237 @return: Distance between this and the B{C{other}} point
238 (C{meter}, same units as B{C{radius}} or
239 C{radians} if B{C{radius}} is C{None}).
241 @raise TypeError: The B{C{other}} point is not L{LatLon}.
243 @raise ValueError: Invalid B{C{radius}}.
244 '''
245 a1, _, a2, _, db = self._ab1_ab2_db5(other, wrap)
246 return _radians2m(vincentys_(a2, a1, db), radius)
248# @Property_RO
249# def Ecef(self):
250# '''Get the ECEF I{class} (L{EcefVeness}), I{lazily}.
251# '''
252# return _MODS.ecef.EcefKarney
254 def greatCircle(self, bearing, Vector=Vector3d, **Vector_kwds):
255 '''Compute the vector normal to great circle obtained by heading
256 on the given initial bearing from this point.
258 Direction of vector is such that initial bearing vector
259 b = c × n, where n is an n-vector representing this point.
261 @arg bearing: Bearing from this point (compass C{degrees360}).
262 @kwarg Vector: Vector class to return the great circle,
263 overriding the default L{Vector3d}.
264 @kwarg Vector_kwds: Optional, additional keyword argunents
265 for B{C{Vector}}.
267 @return: Vector representing great circle (C{Vector}).
269 @raise ValueError: Invalid B{C{bearing}}.
270 '''
271 a, b = self.philam
272 sa, ca, sb, cb, st, ct = sincos2_(a, b, Bearing_(bearing))
274 return Vector(sb * ct - cb * sa * st,
275 -cb * ct - sb * sa * st,
276 ca * st, **Vector_kwds) # XXX .unit()?
278 def initialBearingTo(self, other, wrap=False, raiser=False):
279 '''Compute the initial bearing (forward azimuth) from this
280 to an other point.
282 @arg other: The other point (spherical L{LatLon}).
283 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
284 the B{C{other}} point (C{bool}).
285 @kwarg raiser: Optionally, raise L{CrossError} (C{bool}),
286 use C{B{raiser}=True} for behavior like
287 C{sphericalNvector.LatLon.initialBearingTo}.
289 @return: Initial bearing (compass C{degrees360}).
291 @raise CrossError: If this and the B{C{other}} point coincide,
292 provided both B{C{raiser}} is C{True} and
293 L{pygeodesy.crosserrors} is C{True}.
295 @raise TypeError: The B{C{other}} point is not L{LatLon}.
296 '''
297 a1, b1, a2, b2, db = self._ab1_ab2_db5(other, wrap)
298 # XXX behavior like sphericalNvector.LatLon.initialBearingTo
299 if raiser and crosserrors() and max(fabs(a2 - a1), fabs(db)) < EPS:
300 raise CrossError(_point_, self, other=other, wrap=wrap, txt=_coincident_)
302 return degrees(bearing_(a1, b1, a2, b2, final=False))
304 def intermediateTo(self, other, fraction, height=None, wrap=False):
305 '''Locate the point at given fraction between (or along) this
306 and an other point.
308 @arg other: The other point (L{LatLon}).
309 @arg fraction: Fraction between both points (C{scalar},
310 0.0 at this and 1.0 at the other point).
311 @kwarg height: Optional height, overriding the intermediate
312 height (C{meter}).
313 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the
314 B{C{other}} point (C{bool}).
316 @return: Intermediate point (L{LatLon}).
318 @raise TypeError: The B{C{other}} point is not L{LatLon}.
320 @raise ValueError: Invalid B{C{fraction}} or B{C{height}}.
322 @see: Methods C{midpointTo} and C{rhumbMidpointTo}.
323 '''
324 p = self
325 f = Scalar(fraction=fraction)
326 if not isnear0(f):
327 p = p.others(other)
328 if wrap:
329 p = _Wrap.point(p)
330 if not isnear1(f): # and not near0
331 a1, b1 = self.philam
332 a2, b2 = p.philam
333 db, b2 = unrollPI(b1, b2, wrap=wrap)
334 r = vincentys_(a2, a1, db)
335 sr = sin(r)
336 if isnon0(sr):
337 sa1, ca1, sa2, ca2, \
338 sb1, cb1, sb2, cb2 = sincos2_(a1, a2, b1, b2)
340 t = f * r
341 a = sin(r - t) # / sr superflous
342 b = sin( t) # / sr superflous
344 x = a * ca1 * cb1 + b * ca2 * cb2
345 y = a * ca1 * sb1 + b * ca2 * sb2
346 z = a * sa1 + b * sa2
348 a = atan1d(z, hypot(x, y))
349 b = atan2d(y, x)
351 else: # PYCHOK no cover
352 a = degrees90( favg(a1, a2, f=f)) # coincident
353 b = degrees180(favg(b1, b2, f=f))
355 h = self._havg(other, f=f, h=height)
356 p = self.classof(a, b, height=h)
357 return p
359 def intersection(self, end1, other, end2, height=None, wrap=False):
360 '''Compute the intersection point of two lines, each defined by
361 two points or a start point and bearing from North.
363 @arg end1: End point of this line (L{LatLon}) or the initial
364 bearing at this point (compass C{degrees360}).
365 @arg other: Start point of the other line (L{LatLon}).
366 @arg end2: End point of the other line (L{LatLon}) or the
367 initial bearing at the B{C{other}} point (compass
368 C{degrees360}).
369 @kwarg height: Optional height for intersection point,
370 overriding the mean height (C{meter}).
371 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
372 B{C{start2}} and both B{C{end*}} points (C{bool}).
374 @return: The intersection point (L{LatLon}). An alternate
375 intersection point might be the L{antipode} to
376 the returned result.
378 @raise IntersectionError: Ambiguous or infinite intersection
379 or colinear, parallel or otherwise
380 non-intersecting lines.
382 @raise TypeError: If B{C{other}} is not L{LatLon} or B{C{end1}}
383 or B{C{end2}} not C{scalar} nor L{LatLon}.
385 @raise ValueError: Invalid B{C{height}} or C{null} line.
386 '''
387 try:
388 s2 = self.others(other)
389 return _intersect(self, end1, s2, end2, height=height, wrap=wrap,
390 LatLon=self.classof)
391 except (TypeError, ValueError) as x:
392 raise _xError(x, start1=self, end1=end1,
393 other=other, end2=end2, wrap=wrap)
395 def intersections2(self, rad1, other, rad2, radius=R_M, eps=_0_0,
396 height=None, wrap=True):
397 '''Compute the intersection points of two circles, each defined
398 by a center point and radius.
400 @arg rad1: Radius of the this circle (C{meter} or C{radians},
401 see B{C{radius}}).
402 @arg other: Center point of the other circle (L{LatLon}).
403 @arg rad2: Radius of the other circle (C{meter} or C{radians},
404 see B{C{radius}}).
405 @kwarg radius: Mean earth radius (C{meter} or C{None} if B{C{rad1}},
406 B{C{rad2}} and B{C{eps}} are given in C{radians}).
407 @kwarg eps: Required overlap (C{meter} or C{radians}, see
408 B{C{radius}}).
409 @kwarg height: Optional height for the intersection points (C{meter},
410 conventionally) or C{None} for the I{"radical height"}
411 at the I{radical line} between both centers.
412 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the
413 B{C{other}} point (C{bool}).
415 @return: 2-Tuple of the intersection points, each a L{LatLon}
416 instance. For abutting circles, both intersection
417 points are the same instance, aka the I{radical center}.
419 @raise IntersectionError: Concentric, antipodal, invalid or
420 non-intersecting circles.
422 @raise TypeError: If B{C{other}} is not L{LatLon}.
424 @raise ValueError: Invalid B{C{rad1}}, B{C{rad2}}, B{C{radius}},
425 B{C{eps}} or B{C{height}}.
426 '''
427 try:
428 c2 = self.others(other)
429 return _intersects2(self, rad1, c2, rad2, radius=radius, eps=eps,
430 height=height, wrap=wrap,
431 LatLon=self.classof)
432 except (TypeError, ValueError) as x:
433 raise _xError(x, center=self, rad1=rad1,
434 other=other, rad2=rad2, wrap=wrap)
436 @deprecated_method
437 def isEnclosedBy(self, points): # PYCHOK no cover
438 '''DEPRECATED, use method C{isenclosedBy}.'''
439 return self.isenclosedBy(points)
441 def isenclosedBy(self, points, wrap=False):
442 '''Check whether a (convex) polygon or composite encloses this point.
444 @arg points: The polygon points or composite (L{LatLon}[],
445 L{BooleanFHP} or L{BooleanGH}).
446 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the
447 B{C{points}} (C{bool}).
449 @return: C{True} if this point is inside the polygon or
450 composite, C{False} otherwise.
452 @raise PointsError: Insufficient number of B{C{points}}.
454 @raise TypeError: Some B{C{points}} are not L{LatLon}.
456 @raise ValueError: Invalid B{C{points}}, non-convex polygon.
458 @see: Functions L{pygeodesy.isconvex}, L{pygeodesy.isenclosedBy}
459 and L{pygeodesy.ispolar} especially if the B{C{points}} may
460 enclose a pole or wrap around the earth I{longitudinally}.
461 '''
462 if _MODS.booleans.isBoolean(points):
463 return points._encloses(self.lat, self.lon, wrap=wrap)
465 Ps = self.PointsIter(points, loop=2, dedup=True, wrap=wrap)
466 n0 = self._N_vector
468 v2 = Ps[0]._N_vector
469 p1 = Ps[1]
470 v1 = p1._N_vector
471 # check whether this point on same side of all
472 # polygon edges (to the left or right depending
473 # on the anti-/clockwise polygon direction)
474 gc1 = v2.cross(v1)
475 t0 = gc1.angleTo(n0) > PI_2
476 s0 = None
477 # get great-circle vector for each edge
478 for i, p2 in Ps.enumerate(closed=True):
479 if wrap and not Ps.looped:
480 p2 = _unrollon(p1, p2)
481 p1 = p2
482 v2 = p2._N_vector
483 gc = v1.cross(v2)
484 t = gc.angleTo(n0) > PI_2
485 if t != t0: # different sides of edge i
486 return False # outside
488 # check for convex polygon: angle between
489 # gc vectors, signed by direction of n0
490 # (otherwise the test above is not reliable)
491 s = signOf(gc1.angleTo(gc, vSign=n0))
492 if s != s0:
493 if s0 is None:
494 s0 = s
495 else:
496 t = _Fmt.SQUARE(points=i)
497 raise _ValueError(t, p2, wrap=wrap, txt=_not_(_convex_))
498 gc1, v1 = gc, v2
500 return True # inside
502 def midpointTo(self, other, height=None, fraction=_0_5, wrap=False):
503 '''Find the midpoint between this and an other point.
505 @arg other: The other point (L{LatLon}).
506 @kwarg height: Optional height for midpoint, overriding
507 the mean height (C{meter}).
508 @kwarg fraction: Midpoint location from this point (C{scalar}),
509 may be negative or greater than 1.0.
510 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the
511 B{C{other}} point (C{bool}).
513 @return: Midpoint (L{LatLon}).
515 @raise TypeError: The B{C{other}} point is not L{LatLon}.
517 @raise ValueError: Invalid B{C{height}}.
519 @see: Methods C{intermediateTo} and C{rhumbMidpointTo}.
520 '''
521 if fraction is _0_5:
522 # see <https://MathForum.org/library/drmath/view/51822.html>
523 a1, b, a2, _, db = self._ab1_ab2_db5(other, wrap)
524 sa1, ca1, sa2, ca2, sdb, cdb = sincos2_(a1, a2, db)
526 x = ca2 * cdb + ca1
527 y = ca2 * sdb
529 a = atan1d(sa1 + sa2, hypot(x, y))
530 b = degrees180(b + atan2(y, x))
532 h = self._havg(other, h=height)
533 r = self.classof(a, b, height=h)
534 else:
535 r = self.intermediateTo(other, fraction, height=height, wrap=wrap)
536 return r
538 def nearestOn(self, point1, point2, radius=R_M, **wrap_adjust_limit):
539 '''Locate the point between two points closest to this point.
541 Distances are approximated by function L{pygeodesy.equirectangular_},
542 subject to the supplied B{C{options}}.
544 @arg point1: Start point (L{LatLon}).
545 @arg point2: End point (L{LatLon}).
546 @kwarg radius: Mean earth radius (C{meter}).
547 @kwarg wrap_adjust_limit: Optional keyword arguments for functions
548 L{sphericalTrigonometry.nearestOn3} and
549 L{pygeodesy.equirectangular_},
551 @return: Closest point on the great circle line (L{LatLon}).
553 @raise LimitError: Lat- and/or longitudinal delta exceeds B{C{limit}},
554 see function L{pygeodesy.equirectangular_}.
556 @raise NotImplementedError: Keyword argument C{B{within}=False}
557 is not (yet) supported.
559 @raise TypeError: Invalid B{C{point1}} or B{C{point2}}.
561 @raise ValueError: Invalid B{C{radius}} or B{C{options}}.
563 @see: Functions L{pygeodesy.equirectangular_} and L{pygeodesy.nearestOn5}
564 and method L{sphericalTrigonometry.LatLon.nearestOn3}.
565 '''
566 # remove kwarg B{C{within}} if present
567 w, kwds = _xkwds_pop2(wrap_adjust_limit, within=True)
568 if not w:
569 self._notImplemented(within=w)
571# # UNTESTED - handle C{B{within}=False} and C{B{within}=True}
572# wrap = _xkwds_get(options, wrap=False)
573# a = self.alongTrackDistanceTo(point1, point2, radius=radius, wrap=wrap)
574# if fabs(a) < EPS or (within and a < EPS):
575# return point1
576# d = point1.distanceTo(point2, radius=radius, wrap=wrap)
577# if isnear0(d):
578# return point1 # or point2
579# elif fabs(d - a) < EPS or (a + EPS) > d:
580# return point2
581# f = a / d
582# if within:
583# if f > EPS1:
584# return point2
585# elif f < EPS:
586# return point1
587# return point1.intermediateTo(point2, f, wrap=wrap)
589 # without kwarg B{C{within}}, use backward compatible .nearestOn3
590 return self.nearestOn3([point1, point2], closed=False, radius=radius,
591 **kwds)[0]
593 @deprecated_method
594 def nearestOn2(self, points, closed=False, radius=R_M, **options): # PYCHOK no cover
595 '''DEPRECATED, use method L{sphericalTrigonometry.LatLon.nearestOn3}.
597 @return: ... 2-Tuple C{(closest, distance)} of the closest
598 point (L{LatLon}) on the polygon and the distance
599 to that point from this point in C{meter}, same
600 units of B{C{radius}}.
601 '''
602 r = self.nearestOn3(points, closed=closed, radius=radius, **options)
603 return r.closest, r.distance
605 def nearestOn3(self, points, closed=False, radius=R_M, **wrap_adjust_limit):
606 '''Locate the point on a polygon closest to this point.
608 Distances are approximated by function L{pygeodesy.equirectangular_},
609 subject to the supplied B{C{options}}.
611 @arg points: The polygon points (L{LatLon}[]).
612 @kwarg closed: Optionally, close the polygon (C{bool}).
613 @kwarg radius: Mean earth radius (C{meter}).
614 @kwarg wrap_adjust_limit: Optional keyword arguments for function
615 L{sphericalTrigonometry.nearestOn3} and
616 L{pygeodesy.equirectangular_},
618 @return: A L{NearestOn3Tuple}C{(closest, distance, angle)} of the
619 C{closest} point (L{LatLon}), the L{pygeodesy.equirectangular_}
620 C{distance} between this and the C{closest} point converted to
621 C{meter}, same units as B{C{radius}}. The C{angle} from this
622 to the C{closest} point is in compass C{degrees360}, like
623 function L{pygeodesy.compassAngle}.
625 @raise LimitError: Lat- and/or longitudinal delta exceeds B{C{limit}},
626 see function L{pygeodesy.equirectangular_}.
628 @raise PointsError: Insufficient number of B{C{points}}.
630 @raise TypeError: Some B{C{points}} are not C{LatLon}.
632 @raise ValueError: Invalid B{C{radius}} or B{C{options}}.
634 @see: Functions L{pygeodesy.compassAngle}, L{pygeodesy.equirectangular_}
635 and L{pygeodesy.nearestOn5}.
636 '''
637 return nearestOn3(self, points, closed=closed, radius=radius,
638 LatLon=self.classof, **wrap_adjust_limit)
640 def toCartesian(self, **Cartesian_datum_kwds): # PYCHOK Cartesian=Cartesian, datum=None
641 '''Convert this point to C{Karney}-based cartesian (ECEF)
642 coordinates.
644 @kwarg Cartesian_datum_kwds: Optional L{Cartesian}, B{C{datum}}
645 and other keyword arguments, ignored
646 if C{B{Cartesian} is None}. Use
647 C{B{Cartesian}=...} to override
648 this L{Cartesian} class or specify
649 C{B{Cartesian}=None}.
651 @return: The cartesian point (L{Cartesian}) or if B{C{Cartesian}}
652 is C{None}, an L{Ecef9Tuple}C{(x, y, z, lat, lon, height,
653 C, M, datum)} with C{C} and C{M} if available.
655 @raise TypeError: Invalid B{C{Cartesian_datum_kwds}} argument.
656 '''
657 kwds = _xkwds(Cartesian_datum_kwds, Cartesian=Cartesian, datum=self.datum)
658 return LatLonSphericalBase.toCartesian(self, **kwds)
660 def triangle7(self, otherB, otherC, radius=R_M, wrap=False):
661 '''Compute the angles, sides and area of a spherical triangle.
663 @arg otherB: Second triangle point (C{LatLon}).
664 @arg otherC: Third triangle point (C{LatLon}).
665 @kwarg radius: Mean earth radius, ellipsoid or datum
666 (C{meter}, L{Ellipsoid}, L{Ellipsoid2},
667 L{Datum} or L{a_f2Tuple}) or C{None}.
668 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the
669 B{C{otherB}} and B{C{otherC}} points (C{bool}).
671 @return: L{Triangle7Tuple}C{(A, a, B, b, C, c, area)} or if
672 B{C{radius}} is C{None}, a L{Triangle8Tuple}C{(A,
673 a, B, b, C, c, D, E)}.
675 @see: Function L{triangle7} and U{Spherical trigonometry
676 <https://WikiPedia.org/wiki/Spherical_trigonometry>}.
677 '''
678 B = self.others(otherB=otherB)
679 C = self.others(otherC=otherC)
680 B, C, _ = _unrollon3(self, B, C, wrap)
682 r = self.philam + B.philam + C.philam
683 t = triangle8_(*r, wrap=wrap)
684 return self._xnamed(_t7Tuple(t, radius))
686 def triangulate(self, bearing1, other, bearing2, **height_wrap):
687 '''Locate a point given this, an other point and the (initial) bearing
688 at this and at the other point.
690 @arg bearing1: Bearing at this point (compass C{degrees360}).
691 @arg other: The other point (C{LatLon}).
692 @arg bearing2: Bearing at the other point (compass C{degrees360}).
693 @kwarg height_wrap_tol: Optional keyword arguments C{B{height}=None},
694 C{B{wrap}=False}, see method L{intersection}.
696 @return: Triangulated point (C{LatLon}).
698 @see: Method L{intersection} for further details.
699 '''
700 if _isDegrees(bearing1) and _isDegrees(bearing2):
701 return self.intersection(bearing1, other, bearing2, **height_wrap)
702 raise _TypeError(bearing1=bearing1, bearing2=bearing2, **height_wrap)
704 def trilaterate5(self, distance1, point2, distance2, point3, distance3,
705 area=True, eps=EPS1, radius=R_M, wrap=False):
706 '''Trilaterate three points by I{area overlap} or I{perimeter
707 intersection} of three corresponding circles.
709 @arg distance1: Distance to this point (C{meter}, same units
710 as B{C{radius}}).
711 @arg point2: Second center point (C{LatLon}).
712 @arg distance2: Distance to point2 (C{meter}, same units as
713 B{C{radius}}).
714 @arg point3: Third center point (C{LatLon}).
715 @arg distance3: Distance to point3 (C{meter}, same units as
716 B{C{radius}}).
717 @kwarg area: If C{True} compute the area overlap, otherwise the
718 perimeter intersection of the circles (C{bool}).
719 @kwarg eps: The required I{minimal overlap} for C{B{area}=True}
720 or the I{intersection margin} for C{B{area}=False}
721 (C{meter}, same units as B{C{radius}}).
722 @kwarg radius: Mean earth radius (C{meter}, conventionally).
723 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
724 B{C{point2}} and B{C{point3}} (C{bool}).
726 @return: A L{Trilaterate5Tuple}C{(min, minPoint, max, maxPoint, n)}
727 with C{min} and C{max} in C{meter}, same units as B{C{eps}},
728 the corresponding trilaterated points C{minPoint} and
729 C{maxPoint} as I{spherical} C{LatLon} and C{n}, the number
730 of trilatered points found for the given B{C{eps}}.
732 If only a single trilaterated point is found, C{min I{is}
733 max}, C{minPoint I{is} maxPoint} and C{n = 1}.
735 For C{B{area}=True}, C{min} and C{max} are the smallest
736 respectively largest I{radial} overlap found.
738 For C{B{area}=False}, C{min} and C{max} represent the
739 nearest respectively farthest intersection margin.
741 If C{B{area}=True} and all 3 circles are concentric, C{n =
742 0} and C{minPoint} and C{maxPoint} are both the B{C{point#}}
743 with the smallest B{C{distance#}} C{min} and C{max} the
744 largest B{C{distance#}}.
746 @raise IntersectionError: Trilateration failed for the given B{C{eps}},
747 insufficient overlap for C{B{area}=True} or
748 no intersection or all (near-)concentric for
749 C{B{area}=False}.
751 @raise TypeError: Invalid B{C{point2}} or B{C{point3}}.
753 @raise ValueError: Coincident B{C{point2}} or B{C{point3}} or invalid
754 B{C{distance1}}, B{C{distance2}}, B{C{distance3}}
755 or B{C{radius}}.
756 '''
757 return _trilaterate5(self, distance1,
758 self.others(point2=point2), distance2,
759 self.others(point3=point3), distance3,
760 area=area, radius=radius, eps=eps, wrap=wrap)
763_T00 = LatLon(0, 0, name='T00') # reference instance (L{LatLon})
766def areaOf(points, radius=R_M, wrap=False): # was=True
767 '''Calculate the area of a (spherical) polygon or composite
768 (with the pointsjoined by great circle arcs).
770 @arg points: The polygon points or clips (L{LatLon}[], L{BooleanFHP}
771 or L{BooleanGH}).
772 @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter},
773 L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple})
774 or C{None}.
775 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the B{C{points}}
776 (C{bool}).
778 @return: Polygon area (C{meter} I{quared}, same units as B{C{radius}}
779 or C{radians} if B{C{radius}} is C{None}).
781 @raise PointsError: Insufficient number of B{C{points}}.
783 @raise TypeError: Some B{C{points}} are not L{LatLon}.
785 @raise ValueError: Invalid B{C{radius}} or semi-circular polygon edge.
787 @note: The area is based on I{Karney}'s U{'Area of a spherical
788 polygon'<https://MathOverflow.net/questions/97711/
789 the-area-of-spherical-polygons>}, 3rd Answer.
791 @see: Functions L{pygeodesy.areaOf}, L{sphericalNvector.areaOf},
792 L{pygeodesy.excessKarney}, L{ellipsoidalExact.areaOf} and
793 L{ellipsoidalKarney.areaOf}.
794 '''
795 if _MODS.booleans.isBoolean(points):
796 return points._sum2(LatLon, areaOf, radius=radius, wrap=wrap)
798 _at2, _t_2 = atan2, tan_2
799 _un, _w180 = unrollPI, wrap180
801 Ps = _T00.PointsIter(points, loop=1, wrap=wrap)
802 p1 = p2 = Ps[0]
803 a1, b1 = p1.philam
804 ta1, z1 = _t_2(a1), None
806 A = Fsum() # mean phi
807 R = Fsum() # see L{pygeodesy.excessKarney_}
808 # ispolar: Summation of course deltas around pole is 0° rather than normally ±360°
809 # <https://blog.Element84.com/determining-if-a-spherical-polygon-contains-a-pole.html>
810 # XXX duplicate of function C{points.ispolar} to avoid copying all iterated points
811 D = Fsum()
812 for i, p2 in Ps.enumerate(closed=True):
813 a2, b2 = p2.philam
814 db, b2 = _un(b1, b2, wrap=wrap and not Ps.looped)
815 A += a2
816 ta2 = _t_2(a2)
817 tdb = _t_2(db, points=i)
818 R += _at2(tdb * (ta1 + ta2),
819 _1_0 + ta1 * ta2)
820 ta1, b1 = ta2, b2
822 if not p2.isequalTo(p1, eps=EPS):
823 z, z2 = _bearingTo2(p1, p2, wrap=wrap)
824 if z1 is not None:
825 D += _w180(z - z1) # (z - z1 + 540) ...
826 D += _w180(z2 - z) # (z2 - z + 540) % 360 - 180
827 p1, z1 = p2, z2
829 R = abs(R * _2_0)
830 if abs(D) < _90_0: # ispolar(points)
831 R = abs(R - PI2)
832 if radius:
833 a = degrees(A.fover(len(A))) # mean lat
834 R *= _mean_radius(radius, a)**2
835 return float(R)
838def _destination2(a, b, r, t):
839 '''(INTERNAL) Destination lat- and longitude in C{radians}.
841 @arg a: Latitude (C{radians}).
842 @arg b: Longitude (C{radians}).
843 @arg r: Angular distance (C{radians}).
844 @arg t: Bearing (compass C{radians}).
846 @return: 2-Tuple (phi, lam) of (C{radians}, C{radiansPI}).
847 '''
848 # see <https://www.EdWilliams.org/avform.htm#LL>
849 sa, ca, sr, cr, st, ct = sincos2_(a, r, t)
850 ca *= sr
852 a = asin1(ct * ca + cr * sa)
853 d = atan2(st * ca, cr - sa * sin(a))
854 # note, in EdWilliams.org/avform.htm W is + and E is -
855 return a, (b + d) # (mod(b + d + PI, PI2) - PI)
858def _int3d2(s, end, wrap, _i_, Vector, hs):
859 # see <https://www.EdWilliams.org/intersect.htm> (5) ff
860 # and similar logic in .ellipsoidalBaseDI._intersect3
861 a1, b1 = s.philam
863 if _isDegrees(end): # bearing, get pseudo-end point
864 a2, b2 = _destination2(a1, b1, PI_4, radians(end))
865 else: # must be a point
866 s.others(end, name=_end_ + _i_)
867 hs.append(end.height)
868 a2, b2 = end.philam
869 if wrap:
870 a2, b2 = _Wrap.philam(a2, b2)
872 db, b2 = unrollPI(b1, b2, wrap=wrap)
873 if max(fabs(db), fabs(a2 - a1)) < EPS:
874 raise _ValueError(_SPACE_(_line_ + _i_, _null_))
875 # note, in EdWilliams.org/avform.htm W is + and E is -
876 sb21, cb21, sb12, cb12 = sincos2_(db * _0_5,
877 -(b1 + b2) * _0_5)
878 cb21 *= sin(a1 - a2) # sa21
879 sb21 *= sin(a1 + a2) # sa12
880 x = Vector(sb12 * cb21 - cb12 * sb21,
881 cb12 * cb21 + sb12 * sb21,
882 cos(a1) * cos(a2) * sin(db)) # ll=start
883 return x.unit(), (db, (a2 - a1)) # negated d
886def _intdot(ds, a1, b1, a, b, wrap):
887 # compute dot product ds . (-b + b1, a - a1)
888 db, _ = unrollPI(b1, b, wrap=wrap)
889 return fdot(ds, db, a - a1)
892def intersecant2(center, circle, point, other, **radius_exact_height_wrap):
893 '''Compute the intersections of a circle and a (great circle) line given as
894 two points or as a point and bearing.
896 @arg center: Center of the circle (L{LatLon}).
897 @arg circle: Radius of the circle (C{meter}, same units as B{C{radius}})
898 or a point on the circle (L{LatLon}).
899 @arg point: A point on the (great circle) line (L{LatLon}).
900 @arg other: An other point on the (great circle) line (L{LatLon}) or
901 the bearing at the B{C{point}} (compass C{degrees360}).
902 @kwarg radius_exact_height_wrap: Optional keyword arguments, see
903 method L{LatLon.intersecant2} for further details.
905 @return: 2-Tuple of the intersection points (representing a chord), each
906 an instance of the B{C{point}} class. Both points are the same
907 instance if the (great circle) line is tangent to the circle.
909 @raise IntersectionError: The circle and line do not intersect.
911 @raise TypeError: If B{C{center}} or B{C{point}} not L{LatLon} or
912 B{C{circle}} or B{C{other}} invalid.
914 @raise UnitError: Invalid B{C{circle}}, B{C{other}}, B{C{radius}},
915 B{C{exact}}, B{C{height}} or B{C{napieradius}}.
916 '''
917 c = _T00.others(center=center)
918 p = _T00.others(point=point)
919 try:
920 return _intersecant2(c, circle, p, other, **radius_exact_height_wrap)
921 except (TypeError, ValueError) as x:
922 raise _xError(x, center=center, circle=circle, point=point, other=other,
923 **radius_exact_height_wrap)
926def _intersect(start1, end1, start2, end2, height=None, wrap=False, # in.ellipsoidalBaseDI._intersect3
927 LatLon=None, **LatLon_kwds):
928 # (INTERNAL) Intersect two (spherical) lines, see L{intersection}
929 # above, separated to allow callers to embellish any exceptions
931 s1, s2 = start1, start2
932 if wrap:
933 s2 = _Wrap.point(s2)
934 hs = [s1.height, s2.height]
936 a1, b1 = s1.philam
937 a2, b2 = s2.philam
938 db, b2 = unrollPI(b1, b2, wrap=wrap)
939 r12 = vincentys_(a2, a1, db)
940 if fabs(r12) < EPS: # [nearly] coincident points
941 a, b = favg(a1, a2), favg(b1, b2)
943 # see <https://www.EdWilliams.org/avform.htm#Intersection>
944 elif _isDegrees(end1) and _isDegrees(end2): # both bearings
945 sa1, ca1, sa2, ca2, sr12, cr12 = sincos2_(a1, a2, r12)
947 x1, x2 = (sr12 * ca1), (sr12 * ca2)
948 if isnear0(x1) or isnear0(x2):
949 raise IntersectionError(_parallel_)
950 # handle domain error for equivalent longitudes,
951 # see also functions asin_safe and acos_safe at
952 # <https://www.EdWilliams.org/avform.htm#Math>
953 t12, t13 = acos1((sa2 - sa1 * cr12) / x1), radiansPI2(end1)
954 t21, t23 = acos1((sa1 - sa2 * cr12) / x2), radiansPI2(end2)
955 if sin(db) > 0:
956 t21 = PI2 - t21
957 else:
958 t12 = PI2 - t12
959 sx1, cx1, sx2, cx2 = sincos2_(wrapPI(t13 - t12), # angle 2-1-3
960 wrapPI(t21 - t23)) # angle 1-2-3)
961 if isnear0(sx1) and isnear0(sx2):
962 raise IntersectionError(_infinite_)
963 sx3 = sx1 * sx2
964# XXX if sx3 < 0:
965# XXX raise ValueError(_ambiguous_)
966 x3 = acos1(cr12 * sx3 - cx2 * cx1)
967 r13 = atan2(sr12 * sx3, cx2 + cx1 * cos(x3))
969 a, b = _destination2(a1, b1, r13, t13)
970 # like .ellipsoidalBaseDI,_intersect3, if this intersection
971 # is "before" the first point, use the antipodal intersection
972 if opposing_(t13, bearing_(a1, b1, a, b, wrap=wrap)):
973 a, b = antipode_(a, b) # PYCHOK PhiLam2Tuple
975 else: # end point(s) or bearing(s)
976 _N_vector_ = _MODS.nvectorBase._N_vector_
978 x1, d1 = _int3d2(s1, end1, wrap, _1_, _N_vector_, hs)
979 x2, d2 = _int3d2(s2, end2, wrap, _2_, _N_vector_, hs)
980 x = x1.cross(x2)
981 if x.length < EPS: # [nearly] colinear or parallel lines
982 raise IntersectionError(_colinear_)
983 a, b = x.philam
984 # choose intersection similar to sphericalNvector
985 if not (_intdot(d1, a1, b1, a, b, wrap) *
986 _intdot(d2, a2, b2, a, b, wrap)) > 0:
987 a, b = antipode_(a, b) # PYCHOK PhiLam2Tuple
989 h = fmean(hs) if height is None else Height(height)
990 return _LL3Tuple(degrees90(a), degrees180(b), h,
991 intersection, LatLon, LatLon_kwds)
994def intersection(start1, end1, start2, end2, height=None, wrap=False,
995 LatLon=LatLon, **LatLon_kwds):
996 '''Compute the intersection point of two lines, each defined
997 by two points or a start point and bearing from North.
999 @arg start1: Start point of the first line (L{LatLon}).
1000 @arg end1: End point of the first line (L{LatLon}) or
1001 the initial bearing at the first start point
1002 (compass C{degrees360}).
1003 @arg start2: Start point of the second line (L{LatLon}).
1004 @arg end2: End point of the second line (L{LatLon}) or
1005 the initial bearing at the second start point
1006 (compass C{degrees360}).
1007 @kwarg height: Optional height for the intersection point,
1008 overriding the mean height (C{meter}).
1009 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll
1010 B{C{start2}} and both B{C{end*}} points (C{bool}).
1011 @kwarg LatLon: Optional class to return the intersection
1012 point (L{LatLon}) or C{None}.
1013 @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword
1014 arguments, ignored if C{B{LatLon} is None}.
1016 @return: The intersection point as a (B{C{LatLon}}) or if
1017 C{B{LatLon} is None} a L{LatLon3Tuple}C{(lat, lon,
1018 height)}. An alternate intersection point might
1019 be the L{antipode} to the returned result.
1021 @raise IntersectionError: Ambiguous or infinite intersection
1022 or colinear, parallel or otherwise
1023 non-intersecting lines.
1025 @raise TypeError: A B{C{start1}}, B{C{end1}}, B{C{start2}}
1026 or B{C{end2}} point not L{LatLon}.
1028 @raise ValueError: Invalid B{C{height}} or C{null} line.
1029 '''
1030 s1 = _T00.others(start1=start1)
1031 s2 = _T00.others(start2=start2)
1032 try:
1033 return _intersect(s1, end1, s2, end2, height=height, wrap=wrap,
1034 LatLon=LatLon, **LatLon_kwds)
1035 except (TypeError, ValueError) as x:
1036 raise _xError(x, start1=start1, end1=end1, start2=start2, end2=end2)
1039def intersections2(center1, rad1, center2, rad2, radius=R_M, eps=_0_0,
1040 height=None, wrap=False, # was=True
1041 LatLon=LatLon, **LatLon_kwds):
1042 '''Compute the intersection points of two circles each defined
1043 by a center point and a radius.
1045 @arg center1: Center of the first circle (L{LatLon}).
1046 @arg rad1: Radius of the first circle (C{meter} or C{radians},
1047 see B{C{radius}}).
1048 @arg center2: Center of the second circle (L{LatLon}).
1049 @arg rad2: Radius of the second circle (C{meter} or C{radians},
1050 see B{C{radius}}).
1051 @kwarg radius: Mean earth radius (C{meter} or C{None} if B{C{rad1}},
1052 B{C{rad2}} and B{C{eps}} are given in C{radians}).
1053 @kwarg eps: Required overlap (C{meter} or C{radians}, see
1054 B{C{radius}}).
1055 @kwarg height: Optional height for the intersection points (C{meter},
1056 conventionally) or C{None} for the I{"radical height"}
1057 at the I{radical line} between both centers.
1058 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll B{C{center2}}
1059 (C{bool}).
1060 @kwarg LatLon: Optional class to return the intersection
1061 points (L{LatLon}) or C{None}.
1062 @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword
1063 arguments, ignored if C{B{LatLon} is None}.
1065 @return: 2-Tuple of the intersection points, each a B{C{LatLon}}
1066 instance or if C{B{LatLon} is None} a L{LatLon3Tuple}C{(lat,
1067 lon, height)}. For abutting circles, both intersection
1068 points are the same instance, aka the I{radical center}.
1070 @raise IntersectionError: Concentric, antipodal, invalid or
1071 non-intersecting circles.
1073 @raise TypeError: If B{C{center1}} or B{C{center2}} not L{LatLon}.
1075 @raise ValueError: Invalid B{C{rad1}}, B{C{rad2}}, B{C{radius}},
1076 B{C{eps}} or B{C{height}}.
1078 @note: Courtesy of U{Samuel Čavoj<https://GitHub.com/mrJean1/PyGeodesy/issues/41>}.
1080 @see: This U{Answer<https://StackOverflow.com/questions/53324667/
1081 find-intersection-coordinates-of-two-circles-on-earth/53331953>}.
1082 '''
1083 c1 = _T00.others(center1=center1)
1084 c2 = _T00.others(center2=center2)
1085 try:
1086 return _intersects2(c1, rad1, c2, rad2, radius=radius, eps=eps,
1087 height=height, wrap=wrap,
1088 LatLon=LatLon, **LatLon_kwds)
1089 except (TypeError, ValueError) as x:
1090 raise _xError(x, center1=center1, rad1=rad1,
1091 center2=center2, rad2=rad2, wrap=wrap)
1094def _intersects2(c1, rad1, c2, rad2, radius=R_M, eps=_0_0, # in .ellipsoidalBaseDI._intersects2
1095 height=None, too_d=None, wrap=False, # was=True
1096 LatLon=LatLon, **LatLon_kwds):
1097 # (INTERNAL) Intersect two spherical circles, see L{intersections2}
1098 # above, separated to allow callers to embellish any exceptions
1100 def _dest3(bearing, h):
1101 a, b = _destination2(a1, b1, r1, bearing)
1102 return _LL3Tuple(degrees90(a), degrees180(b), h,
1103 intersections2, LatLon, LatLon_kwds)
1105 a1, b1 = c1.philam
1106 a2, b2 = c2.philam
1107 if wrap:
1108 a2, b2 = _Wrap.philam(a2, b2)
1110 r1, r2, f = _rads3(rad1, rad2, radius)
1111 if f: # swapped radii, swap centers
1112 a1, a2 = a2, a1 # PYCHOK swap!
1113 b1, b2 = b2, b1 # PYCHOK swap!
1115 db, b2 = unrollPI(b1, b2, wrap=wrap)
1116 d = vincentys_(a2, a1, db) # radians
1117 if d < max(r1 - r2, EPS):
1118 raise IntersectionError(_near_(_concentric_)) # XXX ConcentricError?
1120 r = eps if radius is None else (m2radians(
1121 eps, radius=radius) if eps else _0_0)
1122 if r < _0_0:
1123 raise _ValueError(eps=r)
1125 x = fsumf_(r1, r2, -d) # overlap
1126 if x > max(r, EPS):
1127 sd, cd, sr1, cr1, _, cr2 = sincos2_(d, r1, r2)
1128 x = sd * sr1
1129 if isnear0(x):
1130 raise _ValueError(_invalid_)
1131 x = acos1((cr2 - cd * cr1) / x) # 0 <= x <= PI
1133 elif x < r: # PYCHOK no cover
1134 t = (d * radius) if too_d is None else too_d
1135 raise IntersectionError(_too_(_Fmt.distant(t)))
1137 if height is None: # "radical height"
1138 f = _radical2(d, r1, r2).ratio
1139 h = Height(favg(c1.height, c2.height, f=f))
1140 else:
1141 h = Height(height)
1143 b = bearing_(a1, b1, a2, b2, final=False, wrap=wrap)
1144 if x < EPS4: # externally ...
1145 r = _dest3(b, h)
1146 elif x > _PI_EPS4: # internally ...
1147 r = _dest3(b + PI, h)
1148 else:
1149 return _dest3(b + x, h), _dest3(b - x, h)
1150 return r, r # ... abutting circles
1153@deprecated_function
1154def isPoleEnclosedBy(points, wrap=False): # PYCHOK no cover
1155 '''DEPRECATED, use function L{pygeodesy.ispolar}.
1156 '''
1157 return ispolar(points, wrap=wrap)
1160def _LL3Tuple(lat, lon, height, where, LatLon, LatLon_kwds):
1161 '''(INTERNAL) Helper for L{intersection}, L{intersections2} and L{meanOf}.
1162 '''
1163 n = where.__name__
1164 if LatLon is None:
1165 r = LatLon3Tuple(lat, lon, height, name=n)
1166 else:
1167 kwds = _xkwds(LatLon_kwds, height=height, name=n)
1168 r = LatLon(lat, lon, **kwds)
1169 return r
1172def meanOf(points, height=None, wrap=False, LatLon=LatLon, **LatLon_kwds):
1173 '''Compute the I{geographic} mean of several points.
1175 @arg points: Points to be averaged (L{LatLon}[]).
1176 @kwarg height: Optional height at mean point, overriding the mean
1177 height (C{meter}).
1178 @kwarg wrap: If C{True}, wrap or I{normalize} the B{C{points}}
1179 (C{bool}).
1180 @kwarg LatLon: Optional class to return the mean point (L{LatLon})
1181 or C{None}.
1182 @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword
1183 arguments, ignored if C{B{LatLon} is None}.
1185 @return: The geographic mean and height (B{C{LatLon}}) or a
1186 L{LatLon3Tuple}C{(lat, lon, height)} if B{C{LatLon}}
1187 is C{None}.
1189 @raise TypeError: Some B{C{points}} are not L{LatLon}.
1191 @raise ValueError: No B{C{points}} or invalid B{C{height}}.
1192 '''
1193 def _N_vs(ps, w):
1194 Ps = _T00.PointsIter(ps, wrap=w)
1195 for p in Ps.iterate(closed=False):
1196 yield p._N_vector
1198 m = _MODS.nvectorBase
1199 # geographic, vectorial mean
1200 n = m.sumOf(_N_vs(points, wrap), h=height, Vector=m.NvectorBase)
1201 lat, lon, h = n.latlonheight
1202 return _LL3Tuple(lat, lon, h, meanOf, LatLon, LatLon_kwds)
1205@deprecated_function
1206def nearestOn2(point, points, **closed_radius_LatLon_options): # PYCHOK no cover
1207 '''DEPRECATED, use function L{sphericalTrigonometry.nearestOn3}.
1209 @return: ... 2-tuple C{(closest, distance)} of the C{closest}
1210 point (L{LatLon}) on the polygon and the C{distance}
1211 between the C{closest} and the given B{C{point}}. The
1212 C{closest} is a B{C{LatLon}} or a L{LatLon2Tuple}C{(lat,
1213 lon)} if B{C{LatLon}} is C{None} ...
1214 '''
1215 ll, d, _ = nearestOn3(point, points, **closed_radius_LatLon_options) # PYCHOK 3-tuple
1216 if _xkwds_get(closed_radius_LatLon_options, LatLon=LatLon) is None:
1217 ll = LatLon2Tuple(ll.lat, ll.lon)
1218 return ll, d
1221def nearestOn3(point, points, closed=False, radius=R_M, wrap=False, adjust=True,
1222 limit=9, **LatLon_and_kwds):
1223 '''Locate the point on a path or polygon closest to a reference point.
1225 Distances are I{approximated} using function L{pygeodesy.equirectangular_},
1226 subject to the supplied B{C{options}}.
1228 @arg point: The reference point (L{LatLon}).
1229 @arg points: The path or polygon points (L{LatLon}[]).
1230 @kwarg closed: Optionally, close the polygon (C{bool}).
1231 @kwarg radius: Mean earth radius (C{meter}).
1232 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the
1233 B{C{points}} (C{bool}).
1234 @kwarg adjust: See function L{pygeodesy.equirectangular_} (C{bool}).
1235 @kwarg limit: See function L{pygeodesy.equirectangular_} (C{degrees}),
1236 default C{9 degrees} is about C{1,000 Kmeter} (for mean
1237 spherical earth radius L{R_KM}).
1238 @kwarg LatLon: Optional class to return the closest point (L{LatLon})
1239 or C{None}.
1240 @kwarg options: Optional keyword arguments for function
1241 L{pygeodesy.equirectangular_}.
1243 @return: A L{NearestOn3Tuple}C{(closest, distance, angle)} with the
1244 C{closest} point as B{C{LatLon}} or L{LatLon3Tuple}C{(lat,
1245 lon, height)} if B{C{LatLon}} is C{None}. The C{distance}
1246 is the L{pygeodesy.equirectangular_} distance between the
1247 C{closest} and the given B{C{point}} converted to C{meter},
1248 same units as B{C{radius}}. The C{angle} from the given
1249 B{C{point}} to the C{closest} is in compass C{degrees360},
1250 like function L{pygeodesy.compassAngle}. The C{height} is
1251 the (interpolated) height at the C{closest} point.
1253 @raise LimitError: Lat- and/or longitudinal delta exceeds the B{C{limit}},
1254 see function L{pygeodesy.equirectangular_}.
1256 @raise PointsError: Insufficient number of B{C{points}}.
1258 @raise TypeError: Some B{C{points}} are not C{LatLon}.
1260 @raise ValueError: Invalid B{C{radius}}.
1262 @see: Functions L{pygeodesy.equirectangular_} and L{pygeodesy.nearestOn5}.
1263 '''
1264 t = _nearestOn5(point, points, closed=closed, wrap=wrap,
1265 adjust=adjust, limit=limit)
1266 d = degrees2m(t.distance, radius=radius)
1267 h = t.height
1268 n = nearestOn3.__name__
1270 LL, kwds = _xkwds_pop2(LatLon_and_kwds, LatLon=LatLon)
1271 r = LatLon3Tuple(t.lat, t.lon, h, name=n) if LL is None else \
1272 LL(t.lat, t.lon, **_xkwds(kwds, height=h, name=n))
1273 return NearestOn3Tuple(r, d, t.angle, name=n)
1276def perimeterOf(points, closed=False, radius=R_M, wrap=True):
1277 '''Compute the perimeter of a (spherical) polygon or composite
1278 (with great circle arcs joining the points).
1280 @arg points: The polygon points or clips (L{LatLon}[], L{BooleanFHP}
1281 or L{BooleanGH}).
1282 @kwarg closed: Optionally, close the polygon (C{bool}).
1283 @kwarg radius: Mean earth radius (C{meter}) or C{None}.
1284 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll the
1285 B{C{points}} (C{bool}).
1287 @return: Polygon perimeter (C{meter}, same units as B{C{radius}}
1288 or C{radians} if B{C{radius}} is C{None}).
1290 @raise PointsError: Insufficient number of B{C{points}}.
1292 @raise TypeError: Some B{C{points}} are not L{LatLon}.
1294 @raise ValueError: Invalid B{C{radius}} or C{B{closed}=False} with
1295 C{B{points}} a composite.
1297 @note: Distances are based on function L{pygeodesy.vincentys_}.
1299 @see: Functions L{perimeterOf<pygeodesy.perimeterOf>},
1300 L{sphericalNvector.perimeterOf} and L{ellipsoidalKarney.perimeterOf}.
1301 '''
1302 def _rads(ps, c, w): # angular edge lengths in radians
1303 Ps = _T00.PointsIter(ps, loop=1, wrap=w)
1304 a1, b1 = Ps[0].philam
1305 for p in Ps.iterate(closed=c):
1306 a2, b2 = p.philam
1307 db, b2 = unrollPI(b1, b2, wrap=w and not (c and Ps.looped))
1308 yield vincentys_(a2, a1, db)
1309 a1, b1 = a2, b2
1311 if _MODS.booleans.isBoolean(points):
1312 if not closed:
1313 raise _ValueError(closed=closed, points=_composite_)
1314 r = points._sum2(LatLon, perimeterOf, closed=True, radius=radius, wrap=wrap)
1315 else:
1316 r = fsum(_rads(points, closed, wrap), floats=True)
1317 return _radians2m(r, radius)
1320def triangle7(latA, lonA, latB, lonB, latC, lonC, radius=R_M,
1321 excess=excessAbc_,
1322 wrap=False):
1323 '''Compute the angles, sides, and area of a (spherical) triangle.
1325 @arg latA: First corner latitude (C{degrees}).
1326 @arg lonA: First corner longitude (C{degrees}).
1327 @arg latB: Second corner latitude (C{degrees}).
1328 @arg lonB: Second corner longitude (C{degrees}).
1329 @arg latC: Third corner latitude (C{degrees}).
1330 @arg lonC: Third corner longitude (C{degrees}).
1331 @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter},
1332 L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple})
1333 or C{None}.
1334 @kwarg excess: I{Spherical excess} callable (L{excessAbc_},
1335 L{excessGirard_} or L{excessLHuilier_}).
1336 @kwarg wrap: If C{True}, wrap and L{pygeodesy.unroll180}
1337 longitudes (C{bool}).
1339 @return: A L{Triangle7Tuple}C{(A, a, B, b, C, c, area)} with
1340 spherical angles C{A}, C{B} and C{C}, angular sides
1341 C{a}, C{b} and C{c} all in C{degrees} and C{area}
1342 in I{square} C{meter} or same units as B{C{radius}}
1343 I{squared} or if C{B{radius}=0} or C{None}, a
1344 L{Triangle8Tuple}C{(A, a, B, b, C, c, D, E)} all in
1345 C{radians} with the I{spherical excess} C{E} as the
1346 C{unit area} in C{radians}.
1347 '''
1348 t = triangle8_(Phi_(latA=latA), Lam_(lonA=lonA),
1349 Phi_(latB=latB), Lam_(lonB=lonB),
1350 Phi_(latC=latC), Lam_(lonC=lonC),
1351 excess=excess, wrap=wrap)
1352 return _t7Tuple(t, radius)
1355def triangle8_(phiA, lamA, phiB, lamB, phiC, lamC, excess=excessAbc_,
1356 wrap=False):
1357 '''Compute the angles, sides, I{spherical deficit} and I{spherical
1358 excess} of a (spherical) triangle.
1360 @arg phiA: First corner latitude (C{radians}).
1361 @arg lamA: First corner longitude (C{radians}).
1362 @arg phiB: Second corner latitude (C{radians}).
1363 @arg lamB: Second corner longitude (C{radians}).
1364 @arg phiC: Third corner latitude (C{radians}).
1365 @arg lamC: Third corner longitude (C{radians}).
1366 @kwarg excess: I{Spherical excess} callable (L{excessAbc_},
1367 L{excessGirard_} or L{excessLHuilier_}).
1368 @kwarg wrap: If C{True}, L{pygeodesy.unrollPI} the
1369 longitudinal deltas (C{bool}).
1371 @return: A L{Triangle8Tuple}C{(A, a, B, b, C, c, D, E)} with
1372 spherical angles C{A}, C{B} and C{C}, angular sides
1373 C{a}, C{b} and C{c}, I{spherical deficit} C{D} and
1374 I{spherical excess} C{E}, all in C{radians}.
1375 '''
1376 def _a_r(w, phiA, lamA, phiB, lamB, phiC, lamC):
1377 d, _ = unrollPI(lamB, lamC, wrap=w)
1378 a = vincentys_(phiC, phiB, d)
1379 return a, (phiB, lamB, phiC, lamC, phiA, lamA) # rotate A, B, C
1381 def _A_r(a, sa, ca, sb, cb, sc, cc):
1382 s = sb * sc
1383 A = acos1((ca - cb * cc) / s) if isnon0(s) else a
1384 return A, (sb, cb, sc, cc, sa, ca) # rotate sincos2_'s
1386 # notation: side C{a} is oposite to corner C{A}, etc.
1387 a, r = _a_r(wrap, phiA, lamA, phiB, lamB, phiC, lamC)
1388 b, r = _a_r(wrap, *r)
1389 c, _ = _a_r(wrap, *r)
1391 A, r = _A_r(a, *sincos2_(a, b, c))
1392 B, r = _A_r(b, *r)
1393 C, _ = _A_r(c, *r)
1395 D = fsumf_(PI2, -a, -b, -c) # deficit aka defect
1396 E = excessGirard_(A, B, C) if excess in (excessGirard_, True) else (
1397 excessLHuilier_(a, b, c) if excess in (excessLHuilier_, False) else
1398 excessAbc_(*max((A, b, c), (B, c, a), (C, a, b))))
1400 return Triangle8Tuple(A, a, B, b, C, c, D, E)
1403def _t7Tuple(t, radius):
1404 '''(INTERNAL) Convert a L{Triangle8Tuple} to L{Triangle7Tuple}.
1405 '''
1406 if radius: # not in (None, _0_0)
1407 r = radius if _isRadius(radius) else \
1408 _ellipsoidal_datum(radius).ellipsoid.Rmean
1409 A, B, C = map1(degrees, t.A, t.B, t.C)
1410 t = Triangle7Tuple(A, (r * t.a),
1411 B, (r * t.b),
1412 C, (r * t.c), t.E * r**2)
1413 return t
1416__all__ += _ALL_OTHER(Cartesian, LatLon, # classes
1417 areaOf, # functions
1418 intersecant2, intersection, intersections2, ispolar,
1419 isPoleEnclosedBy, # DEPRECATED, use ispolar
1420 meanOf,
1421 nearestOn2, nearestOn3,
1422 perimeterOf,
1423 sumOf, # XXX == vector3d.sumOf
1424 triangle7, triangle8_)
1426# **) MIT License
1427#
1428# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved.
1429#
1430# Permission is hereby granted, free of charge, to any person obtaining a
1431# copy of this software and associated documentation files (the "Software"),
1432# to deal in the Software without restriction, including without limitation
1433# the rights to use, copy, modify, merge, publish, distribute, sublicense,
1434# and/or sell copies of the Software, and to permit persons to whom the
1435# Software is furnished to do so, subject to the following conditions:
1436#
1437# The above copyright notice and this permission notice shall be included
1438# in all copies or substantial portions of the Software.
1439#
1440# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
1441# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
1442# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
1443# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
1444# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
1445# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
1446# OTHER DEALINGS IN THE SOFTWARE.