# Coverage for pygeodesy/sphericalTrigonometry.py : 96%

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1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 | # -*- coding: utf-8 -*- u'''Trigonometric classes geodetic (lat-/longitude) L{LatLon} and geocentric (ECEF) L{Cartesian} and functions L{areaOf}, L{intersection}, L{intersections2}, L{isPoleEnclosedBy}, L{meanOf}, L{nearestOn3} and L{perimeterOf}, I{all spherical}. Pure Python implementation of geodetic (lat-/longitude) methods using spherical trigonometry, transcribed from JavaScript originals by I{(C) Chris Veness 2011-2016} published under the same MIT Licence**, see U{Latitude/Longitude<https://www.Movable-Type.co.UK/scripts/latlong.html>}. @newfield example: Example, Examples ''' # make sure int/int division yields float quotient, see .basics from __future__ import division from pygeodesy.basics import copysign, isscalar, map1 from pygeodesy.errors import _AssertionError, CrossError, crosserrors, \ IntersectionError, _ValueError, _xkwds, _xkwds_get from pygeodesy.fmath import favg, fdot, fmean, fsum, fsum_ from pygeodesy.formy import antipode_, bearing_, _radical2, vincentys_ from pygeodesy.interns import EPS, EPS1, PI, PI2, PI_2, PI_4, R_M, \ _EPS4, _coincident_, _colinear_, _convex_, \ _end_, _invalid_, _LatLon_, _near_concentric_, \ _not_, _points_, _too_, _1_, _2_, _0_0, _0_5 from pygeodesy.lazily import _ALL_LAZY, _ALL_OTHER from pygeodesy.named import notImplemented, _xnamed from pygeodesy.namedTuples import LatLon2Tuple, LatLon3Tuple, \ NearestOn3Tuple from pygeodesy.nvectorBase import NvectorBase as _Nvector from pygeodesy.points import _imdex2, ispolar, nearestOn5 as _nearestOn5 from pygeodesy.sphericalBase import _angular, CartesianSphericalBase, \ LatLonSphericalBase, _rads3, _trilaterate5 from pygeodesy.streprs import Fmt from pygeodesy.units import Bearing_, Height, Radius, Radius_, Scalar from pygeodesy.utily import acos1, asin1, degrees90, degrees180, degrees2m, \ iterNumpy2, m2radians, radiansPI2, sincos2, \ tan_2, unrollPI, wrapPI from pygeodesy.vector3d import sumOf, Vector3d from math import asin, atan2, cos, degrees, hypot, radians, sin __all__ = _ALL_LAZY.sphericalTrigonometry __version__ = '20.12.22' _PI_EPS4 = PI - _EPS4 if _PI_EPS4 >= PI: raise _AssertionError(EPS4=_EPS4, PI=PI, PI_EPS4=_PI_EPS4) def _destination2(a, b, r, t): '''(INTERNAL) Destination lat- and longitude in C{radians}. @arg a: Latitude (C{radians}). @arg b: Longitude (C{radians}). @arg r: Angular distance (C{radians}). @arg t: Bearing (compass C{radians}). @return: 2-Tuple (phi, lam) of (C{radians}, C{radiansPI}). ''' # see <https://www.EdWilliams.org/avform.htm#LL> sa, ca, sr, cr, st, ct = sincos2(a, r, t) a = asin1(ct * sr * ca + cr * sa) d = atan2(st * sr * ca, cr - sa * sin(a)) # note, in EdWilliams.org/avform.htm W is + and E is - return a, b + d class Cartesian(CartesianSphericalBase): '''Extended to convert geocentric, L{Cartesian} points to spherical, geodetic L{LatLon}. ''' def toLatLon(self, **LatLon_datum_kwds): # PYCHOK LatLon=LatLon '''Convert this cartesian point to an C{Nvector}-based geodetic point. @kwarg LatLon_datum_kwds: Optional L{LatLon}, B{C{datum}} and other keyword arguments, ignored if C{B{LatLon}=None}. Use B{C{LatLon=...}} to override this L{LatLon} class or specify C{B{LatLon}=None}. @return: The geodetic point (L{LatLon}) or if B{C{LatLon}} is C{None}, an L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)} with C{C} and C{M} if available. @raise TypeError: Invalid B{C{LatLon_datum_kwds}}. ''' kwds = _xkwds(LatLon_datum_kwds, LatLon=LatLon, datum=self.datum) return CartesianSphericalBase.toLatLon(self, **kwds) class LatLon(LatLonSphericalBase): '''New point on spherical model earth model. @example: >>> p = LatLon(52.205, 0.119) # height=0 ''' def _trackDistanceTo3(self, start, end, radius, wrap): '''(INTERNAL) Helper for .along-/crossTrackDistanceTo. ''' self.others(start=start) self.others(end=end) r = Radius_(radius) r = start.distanceTo(self, r, wrap=wrap) / r b = radians(start.initialBearingTo(self, wrap=wrap)) e = radians(start.initialBearingTo(end, wrap=wrap)) x = asin(sin(r) * sin(b - e)) return r, x, (e - b) def alongTrackDistanceTo(self, start, end, radius=R_M, wrap=False): '''Compute the (signed) distance from the start to the closest point on the great circle path defined by a start and an end point. That is, if a perpendicular is drawn from this point to the great circle path, the along-track distance is the distance from the start point to the point where the perpendicular crosses the path. @arg start: Start point of great circle path (L{LatLon}). @arg end: End point of great circle path (L{LatLon}). @kwarg radius: Mean earth radius (C{meter}). @kwarg wrap: Wrap and unroll longitudes (C{bool}). @return: Distance along the great circle path (C{meter}, same units as B{C{radius}}), positive if after the B{C{start}} toward the B{C{end}} point of the path or negative if before the B{C{start}} point. @raise TypeError: Invalid B{C{start}} or B{C{end}} point. @raise ValueError: Invalid B{C{radius}}. @example: >>> p = LatLon(53.2611, -0.7972) >>> s = LatLon(53.3206, -1.7297) >>> e = LatLon(53.1887, 0.1334) >>> d = p.alongTrackDistanceTo(s, e) # 62331.58 ''' r, x, b = self._trackDistanceTo3(start, end, radius, wrap) cx = cos(x) if abs(cx) > EPS: return copysign(acos1(cos(r) / cx), cos(b)) * radius else: return _0_0 def bearingTo(self, other, wrap=False, raiser=False): # PYCHOK no cover '''DEPRECATED, use method C{initialBearingTo}. ''' return self.initialBearingTo(other, wrap=wrap, raiser=raiser) def crossingParallels(self, other, lat, wrap=False): '''Return the pair of meridians at which a great circle defined by this and an other point crosses the given latitude. @arg other: The other point defining great circle (L{LatLon}). @arg lat: Latitude at the crossing (C{degrees}). @kwarg wrap: Wrap and unroll longitudes (C{bool}). @return: 2-Tuple C{(lon1, lon2)}, both in C{degrees180} or C{None} if the great circle doesn't reach B{C{lat}}. ''' self.others(other) a1, b1 = self.philam a2, b2 = other.philam a = radians(lat) db, b2 = unrollPI(b1, b2, wrap=wrap) sa, ca, sa1, ca1, \ sa2, ca2, sdb, cdb = sincos2(a, a1, a2, db) x = sa1 * ca2 * ca * sdb y = sa1 * ca2 * ca * cdb - ca1 * sa2 * ca z = ca1 * ca2 * sa * sdb h = hypot(x, y) if h < EPS or abs(z) > h: return None # great circle doesn't reach latitude m = atan2(-y, x) + b1 # longitude at max latitude d = acos1(z / h) # delta longitude to intersections return degrees180(m - d), degrees180(m + d) def crossTrackDistanceTo(self, start, end, radius=R_M, wrap=False): '''Compute the (signed) distance from this point to the great circle defined by a start and an end point. @arg start: Start point of great circle path (L{LatLon}). @arg end: End point of great circle path (L{LatLon}). @kwarg radius: Mean earth radius (C{meter}). @kwarg wrap: Wrap and unroll longitudes (C{bool}). @return: Distance to great circle (negative if to the left or positive if to the right of the path). @raise TypeError: The B{C{start}} or B{C{end}} point is not L{LatLon}. @raise ValueError: Invalid B{C{radius}}. @example: >>> p = LatLon(53.2611, -0.7972) >>> s = LatLon(53.3206, -1.7297) >>> e = LatLon(53.1887, 0.1334) >>> d = p.crossTrackDistanceTo(s, e) # -307.5 ''' _, x, _ = self._trackDistanceTo3(start, end, radius, wrap) return x * radius def destination(self, distance, bearing, radius=R_M, height=None): '''Locate the destination from this point after having travelled the given distance on the given initial bearing. @arg distance: Distance travelled (C{meter}, same units as B{C{radius}}). @arg bearing: Bearing from this point (compass C{degrees360}). @kwarg radius: Mean earth radius (C{meter}). @kwarg height: Optional height at destination (C{meter}, same units a B{C{radius}}). @return: Destination point (L{LatLon}). @raise ValueError: Invalid B{C{distance}}, B{C{bearing}}, B{C{radius}} or B{C{height}}. @example: >>> p1 = LatLon(51.4778, -0.0015) >>> p2 = p1.destination(7794, 300.7) >>> p2.toStr() # '51.5135°N, 000.0983°W' @JSname: I{destinationPoint}. ''' a, b = self.philam r, t = _angular(distance, radius), Bearing_(bearing) a, b = _destination2(a, b, r, t) h = self.height if height is None else Height(height) return self.classof(degrees90(a), degrees180(b), height=h) def distanceTo(self, other, radius=R_M, wrap=False): '''Compute the distance from this to an other point. @arg other: The other point (L{LatLon}). @kwarg radius: Mean earth radius (C{meter}). @kwarg wrap: Wrap and unroll longitudes (C{bool}). @return: Distance between this and the B{C{other}} point (C{meter}, same units as B{C{radius}}). @raise TypeError: The B{C{other}} point is not L{LatLon}. @raise ValueError: Invalid B{C{radius}}. @example: >>> p1 = LatLon(52.205, 0.119) >>> p2 = LatLon(48.857, 2.351); >>> d = p1.distanceTo(p2) # 404300 ''' self.others(other) a1, b1 = self.philam a2, b2 = other.philam db, _ = unrollPI(b1, b2, wrap=wrap) r = vincentys_(a2, a1, db) return r * (radius if radius is R_M else Radius(radius)) def greatCircle(self, bearing): '''Compute the vector normal to great circle obtained by heading on the given initial bearing from this point. Direction of vector is such that initial bearing vector b = c × n, where n is an n-vector representing this point. @arg bearing: Bearing from this point (compass C{degrees360}). @return: Vector representing great circle (L{Vector3d}). @raise ValueError: Invalid B{C{bearing}}. @example: >>> p = LatLon(53.3206, -1.7297) >>> g = p.greatCircle(96.0) >>> g.toStr() # (-0.794, 0.129, 0.594) ''' a, b = self.philam t = Bearing_(bearing) sa, ca, sb, cb, st, ct = sincos2(a, b, t) return Vector3d(sb * ct - cb * sa * st, -cb * ct - sb * sa * st, ca * st) # XXX .unit()? def initialBearingTo(self, other, wrap=False, raiser=False): '''Compute the initial bearing (forward azimuth) from this to an other point. @arg other: The other point (spherical L{LatLon}). @kwarg wrap: Wrap and unroll longitudes (C{bool}). @kwarg raiser: Optionally, raise L{CrossError} (C{bool}), use B{C{raiser}}=C{True} for behavior like C{sphericalNvector.LatLon.initialBearingTo}. @return: Initial bearing (compass C{degrees360}). @raise CrossError: If this and the B{C{other}} point coincide, provided B{C{raiser}} is C{True} and L{crosserrors} is C{True}. @raise TypeError: The B{C{other}} point is not L{LatLon}. @example: >>> p1 = LatLon(52.205, 0.119) >>> p2 = LatLon(48.857, 2.351) >>> b = p1.initialBearingTo(p2) # 156.2 @JSname: I{bearingTo}. ''' self.others(other) a1, b1 = self.philam a2, b2 = other.philam # XXX behavior like sphericalNvector.LatLon.initialBearingTo if raiser and crosserrors() and max(abs(a2 - a1), abs(b2 - b1)) < EPS: raise CrossError(_points_, self, txt=_coincident_) return degrees(bearing_(a1, b1, a2, b2, final=False, wrap=wrap)) def intermediateTo(self, other, fraction, height=None, wrap=False): '''Locate the point at given fraction between this and an other point. @arg other: The other point (L{LatLon}). @arg fraction: Fraction between both points (float, between 0.0 for this and 1.0 for the other point). @kwarg height: Optional height, overriding the fractional height (C{meter}). @kwarg wrap: Wrap and unroll longitudes (C{bool}). @return: Intermediate point (L{LatLon}). @raise TypeError: The B{C{other}} point is not L{LatLon}. @raise ValueError: Invalid B{C{fraction}} or B{C{height}}. @example: >>> p1 = LatLon(52.205, 0.119) >>> p2 = LatLon(48.857, 2.351) >>> p = p1.intermediateTo(p2, 0.25) # 51.3721°N, 000.7073°E @JSname: I{intermediatePointTo}. ''' self.others(other) f = Scalar(fraction=fraction) a1, b1 = self.philam a2, b2 = other.philam db, b2 = unrollPI(b1, b2, wrap=wrap) r = vincentys_(a2, a1, db) sr = sin(r) if abs(sr) > EPS: sa1, ca1, sa2, ca2, \ sb1, cb1, sb2, cb2 = sincos2(a1, a2, b1, b2) A = sin((1 - f) * r) / sr B = sin( f * r) / sr x = A * ca1 * cb1 + B * ca2 * cb2 y = A * ca1 * sb1 + B * ca2 * sb2 z = A * sa1 + B * sa2 a = atan2(z, hypot(x, y)) b = atan2(y, x) else: # points too close a = favg(a1, a2, f=f) b = favg(b1, b2, f=f) h = self._havg(other, f=f) if height is None else Height(height) return self.classof(degrees90(a), degrees180(b), height=h) def intersection(self, end1, other, end2, height=None, wrap=False): '''Compute the intersection point of two paths both defined by two points or a start point and bearing from North. @arg end1: End point of this path (L{LatLon}) or the initial bearing at this point (compass C{degrees360}). @arg other: Start point of the other path (L{LatLon}). @arg end2: End point of the other path (L{LatLon}) or the initial bearing at the other start point (compass C{degrees360}). @kwarg height: Optional height for intersection point, overriding the mean height (C{meter}). @kwarg wrap: Wrap and unroll longitudes (C{bool}). @return: The intersection point (L{LatLon}). An alternate intersection point might be the L{antipode} to the returned result. @raise IntersectionError: Intersection is ambiguous or infinite or the paths are coincident, colinear or parallel. @raise TypeError: If B{C{end1}}, B{C{other}} or B{C{end2}} is not L{LatLon}. @raise ValueError: Invalid B{C{height}}. @example: >>> p = LatLon(51.8853, 0.2545) >>> s = LatLon(49.0034, 2.5735) >>> i = p.intersection(108.547, s, 32.435) # '50.9078°N, 004.5084°E' ''' return intersection(self, end1, other, end2, height=height, wrap=wrap, LatLon=self.classof) def intersections2(self, rad1, other, rad2, radius=R_M, eps=_0_0, height=None, wrap=True): '''Compute the intersection points of two circles each defined by a center point and radius. @arg rad1: Radius of the this circle (C{meter} or C{radians}, see B{C{radius}}). @arg other: Center point of the other circle (L{LatLon}). @arg rad2: Radius of the other circle (C{meter} or C{radians}, see B{C{radius}}). @kwarg radius: Mean earth radius (C{meter} or C{None} if B{C{rad1}}, B{C{rad2}} and B{C{eps}} are given in C{radians}). @kwarg eps: Required overlap (C{meter} or C{radians}, see B{C{radius}}). @kwarg height: Optional height for the intersection points, overriding the "radical height" at the "radical line" between both centers (C{meter}) or C{None}. @kwarg wrap: Wrap and unroll longitudes (C{bool}). @return: 2-Tuple of the intersection points, each a L{LatLon} instance. For abutting circles, both intersection points are the same instance. @raise IntersectionError: Concentric, antipodal, invalid or non-intersecting circles. @raise TypeError: If B{C{other}} is not L{LatLon}. @raise ValueError: Invalid B{C{rad1}}, B{C{rad2}}, B{C{radius}}, B{C{eps}} or B{C{height}}. ''' c2 = self.others(other) try: return _intersects2(self, rad1, c2, rad2, radius=radius, eps=eps, height=height, wrap=wrap, LatLon=self.classof) except (TypeError, ValueError) as x: raise IntersectionError(center=self, rad1=rad1, other=other, rad2=rad2, txt=str(x)) def isenclosedBy(self, points): '''Check whether a (convex) polygon encloses this point. @arg points: The polygon points (L{LatLon}[]). @return: C{True} if the polygon encloses this point, C{False} otherwise. @raise PointsError: Insufficient number of B{C{points}}. @raise TypeError: Some B{C{points}} are not L{LatLon}. @raise ValueError: Invalid B{C{points}}, non-convex polygon. @example: >>> b = LatLon(45,1), LatLon(45,2), LatLon(46,2), LatLon(46,1) >>> p = LatLon(45,1, 1.1) >>> inside = p.isEnclosedBy(b) # True ''' n, points = self.points2(points, closed=True) n0 = self._N_vector if iterNumpy2(points): v1 = points[n-1]._N_vector v2 = points[n-2]._N_vector gc1 = v2.cross(v1) t0 = gc1.angleTo(n0) > PI_2 for i in range(n): v2 = points[i]._N_vector gc = v1.cross(v2) v1 = v2 ti = gc.angleTo(n0) > PI_2 if ti != t0: return False # outside if gc1.angleTo(gc, vSign=n0) < 0: ti = Fmt.SQUARE(points=i) raise _ValueError(ti, points[i], txt=_not_(_convex_)) gc1 = gc else: # get great-circle vector for each edge gc, v1 = [], points[n-1]._N_vector for i in range(n): v2 = points[i]._N_vector gc.append(v1.cross(v2)) v1 = v2 # check whether this point on same side of all # polygon edges (to the left or right depending # on anti-/clockwise polygon direction) t0 = gc[0].angleTo(n0) > PI_2 # True if on the right for i in range(1, n): ti = gc[i].angleTo(n0) > PI_2 if ti != t0: # different sides of edge i return False # outside # check for convex polygon (otherwise # the test above is not reliable) gc1 = gc[n-1] for i, gc2 in enumerate(gc): # angle between gc vectors, signed by direction of n0 if gc1.angleTo(gc2, vSign=n0) < 0: ti = Fmt.SQUARE(points=i) raise _ValueError(ti, points[i], txt=_not_(_convex_)) gc1 = gc2 return True # inside def isEnclosedBy(self, points): # PYCHOK no cover '''DEPRECATED, use method C{isenclosedBy}. ''' return self.isenclosedBy(points) def midpointTo(self, other, height=None, wrap=False): '''Find the midpoint between this and an other point. @arg other: The other point (L{LatLon}). @kwarg height: Optional height for midpoint, overriding the mean height (C{meter}). @kwarg wrap: Wrap and unroll longitudes (C{bool}). @return: Midpoint (L{LatLon}). @raise TypeError: The B{C{other}} point is not L{LatLon}. @raise ValueError: Invalid B{C{height}}. @example: >>> p1 = LatLon(52.205, 0.119) >>> p2 = LatLon(48.857, 2.351) >>> m = p1.midpointTo(p2) # '50.5363°N, 001.2746°E' ''' self.others(other) # see <https://MathForum.org/library/drmath/view/51822.html> a1, b1 = self.philam a2, b2 = other.philam db, b2 = unrollPI(b1, b2, wrap=wrap) sa1, ca1, sa2, ca2, sdb, cdb = sincos2(a1, a2, db) x = ca2 * cdb + ca1 y = ca2 * sdb a = atan2(sa1 + sa2, hypot(x, y)) b = atan2(y, x) + b1 if height is None: h = self._havg(other) else: h = Height(height) return self.classof(degrees90(a), degrees180(b), height=h) def nearestOn(self, point1, point2, radius=R_M, **options): '''Locate the point between two points closest to this point. Distances are approximated by function L{equirectangular_}, subject to the supplied B{C{options}}. @arg point1: Start point (L{LatLon}). @arg point2: End point (L{LatLon}). @kwarg radius: Mean earth radius (C{meter}). @kwarg options: Optional keyword arguments for function L{equirectangular_}. @return: Closest point on the great circle path (L{LatLon}). @raise LimitError: Lat- and/or longitudinal delta exceeds B{C{limit}}, see function L{equirectangular_}. @raise NotImplementedError: Keyword argument C{B{within}=False} is not (yet) supported. @raise TypeError: Invalid B{C{point1}} or B{C{point2}}. @raise ValueError: Invalid B{C{radius}} or B{C{options}}. @see: Functions L{equirectangular_} and L{nearestOn5} and method L{sphericalTrigonometry.LatLon.nearestOn3}. ''' try: # remove kwarg B{C{within}} if present within = options.pop('within') if not within: notImplemented(self, self.nearestOn, within=within) # # UNTESTED - handle C{B{within}=False} and B{C{within}=True} # wrap = options.get('wrap', False) # a = self.alongTrackDistanceTo(point1, point2, radius=radius, wrap=wrap) # if abs(a) < EPS or (within and a < EPS): # return point1 # d = point1.distanceTo(point2, radius=radius, wrap=wrap) # if abs(d) < EPS: # return point1 # or point2 # elif abs(d - a) < EPS or (a + EPS) > d: # return point2 # f = a / d # if within: # if f > EPS1: # return point2 # elif f < EPS: # return point1 # return point1.intermediateTo(point2, f, wrap=wrap) except KeyError: pass # without kwarg B{C{within}}, use backward compatible .nearestOn3 return self.nearestOn3([point1, point2], closed=False, radius=radius, **options)[0] def nearestOn2(self, points, closed=False, radius=R_M, **options): # PYCHOK no cover '''DEPRECATED, use method L{sphericalTrigonometry.LatLon.nearestOn3}. @return: ... 2-Tuple C{(closest, distance)} of the closest point (L{LatLon}) on the polygon and the distance to that point from this point in C{meter}, same units of B{C{radius}}. ''' r = self.nearestOn3(points, closed=closed, radius=radius, **options) return tuple(r[:2]) def nearestOn3(self, points, closed=False, radius=R_M, **options): '''Locate the point on a polygon closest to this point. Distances are approximated by function L{equirectangular_}, subject to the supplied B{C{options}}. @arg points: The polygon points (L{LatLon}[]). @kwarg closed: Optionally, close the polygon (C{bool}). @kwarg radius: Mean earth radius (C{meter}). @kwarg options: Optional keyword arguments for function L{equirectangular_}. @return: A L{NearestOn3Tuple}C{(closest, distance, angle)} of the C{closest} point (L{LatLon}), the L{equirectangular_} C{distance} between this and the C{closest} point in C{meter}, same units as B{C{radius}}. The C{angle} from this to the C{closest} point is in compass C{degrees360}, like function L{compassAngle}. @raise LimitError: Lat- and/or longitudinal delta exceeds B{C{limit}}, see function L{equirectangular_}. @raise PointsError: Insufficient number of B{C{points}}. @raise TypeError: Some B{C{points}} are not C{LatLon}. @raise ValueError: Invalid B{C{radius}} or B{C{options}}. @see: Functions L{compassAngle}, L{equirectangular_} and L{nearestOn5}. ''' if _LatLon_ in options: raise _ValueError(**options) lat, lon, d, c, h = _nearestOn5(self, points, closed=closed, **options) return NearestOn3Tuple(self.classof(lat, lon, height=h), degrees2m(d, radius=radius), c) def toCartesian(self, **Cartesian_datum_kwds): # PYCHOK Cartesian=Cartesian, datum=None '''Convert this point to C{Karney}-based cartesian (ECEF) coordinates. @kwarg Cartesian_datum_kwds: Optional L{Cartesian}, B{C{datum}} and other keyword arguments, ignored if B{C{Cartesian=None}}. Use B{C{Cartesian=...}} to override this L{Cartesian} class or specify B{C{Cartesian=None}}. @return: The cartesian point (L{Cartesian}) or if B{C{Cartesian}} is C{None}, an L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)} with C{C} and C{M} if available. @raise TypeError: Invalid B{C{Cartesian_datum_kwds}}. ''' kwds = _xkwds(Cartesian_datum_kwds, Cartesian=Cartesian, datum=self.datum) return LatLonSphericalBase.toCartesian(self, **kwds) def trilaterate5(self, distance1, point2, distance2, point3, distance3, area=True, eps=EPS1, radius=R_M, wrap=False): '''Trilaterate three points by area overlap or perimeter intersection of three corresponding circles. @arg distance1: Distance to this point (C{meter}, same units as B{C{radius}}). @arg point2: Second center point (C{LatLon}). @arg distance2: Distance to point2 (C{meter}, same units as B{C{radius}}). @arg point3: Third center point (C{LatLon}). @arg distance3: Distance to point3 (C{meter}, same units as B{C{radius}}). @kwarg area: If C{True} compute the area overlap, otherwise the perimeter intersection of the circles (C{bool}). @kwarg eps: The required I{minimal overlap} for C{B{area}=True} or the I{intersection margin} for C{B{area}=False} (C{meter}, same units as B{C{radius}}). @kwarg radius: Mean earth radius (C{meter}, conventionally). @kwarg wrap: Wrap/unroll angular distances (C{bool}). @return: A L{Trilaterate5Tuple}C{(min, minPoint, max, maxPoint, n)} with C{min} and C{max} in C{meter}, same units as B{C{eps}}, the corresponding trilaterated points C{minPoint} and C{maxPoint} as I{spherical} C{LatLon} and C{n}, the number of trilatered points found for the given B{C{eps}}. If only a single trilaterated point is found, C{min I{is} max}, C{minPoint I{is} maxPoint} and C{n = 1}. For C{B{area}=True}, C{min} and C{max} are the smallest respectively largest I{radial} overlap found. For C{B{area}=False}, C{min} and C{max} represent the nearest respectively farthest intersection margin. If C{B{area}=True} and all 3 circles are concentric, C{n = 0} and C{minPoint} and C{maxPoint} are both the B{C{point#}} with the smallest B{C{distance#}} C{min} and C{max} the largest B{C{distance#}}. @raise IntersectionError: Trilateration failed for the given B{C{eps}}, insufficient overlap for C{B{area}=True} or no intersection or all (near-)concentric for C{B{area}=False}. @raise TypeError: Invalid B{C{point2}} or B{C{point3}}. @raise ValueError: Some B{C{points}} coincide or invalid B{C{distance1}}, B{C{distance2}}, B{C{distance3}} or B{C{radius}}. ''' return _trilaterate5(self, distance1, self.others(point2=point2), distance2, self.others(point3=point3), distance3, area=area, radius=radius, eps=eps, wrap=wrap) _T00 = LatLon(0, 0) # reference instance (L{LatLon}) def areaOf(points, radius=R_M, wrap=True): '''Calculate the area of a (spherical) polygon (with great circle arcs joining the points). @arg points: The polygon points (L{LatLon}[]). @kwarg radius: Mean earth radius (C{meter}). @kwarg wrap: Wrap and unroll longitudes (C{bool}). @return: Polygon area (C{meter}, same units as B{C{radius}}, squared). @raise PointsError: Insufficient number of B{C{points}}. @raise TypeError: Some B{C{points}} are not L{LatLon}. @raise ValueError: Invalid B{C{radius}}. @note: The area is based on I{Karney}'s U{'Area of a spherical polygon' <https://OSGeo-org.1560.x6.nabble.com/ Area-of-a-spherical-polygon-td3841625.html>}. @see: L{pygeodesy.areaOf}, L{sphericalNvector.areaOf} and L{ellipsoidalKarney.areaOf}. @example: >>> b = LatLon(45, 1), LatLon(45, 2), LatLon(46, 2), LatLon(46, 1) >>> areaOf(b) # 8666058750.718977 >>> c = LatLon(0, 0), LatLon(1, 0), LatLon(0, 1) >>> areaOf(c) # 6.18e9 ''' n, points = _T00.points2(points, closed=True) # Area method due to Karney: for each edge of the polygon, # # tan(Δλ/2) · (tan(φ1/2) + tan(φ2/2)) # tan(E/2) = ------------------------------------ # 1 + tan(φ1/2) · tan(φ2/2) # # where E is the spherical excess of the trapezium obtained by # extending the edge to the equator-circle vector for each edge def _exs(n, points): # iterate over spherical edge excess a1, b1 = points[n-1].philam ta1 = tan_2(a1) for i in range(n): a2, b2 = points[i].philam db, b2 = unrollPI(b1, b2, wrap=wrap) ta2, tdb = map1(tan_2, a2, db) yield atan2(tdb * (ta1 + ta2), 1 + ta1 * ta2) ta1, b1 = ta2, b2 s = fsum(_exs(n, points)) * 2 if isPoleEnclosedBy(points): s = abs(s) - PI2 return abs(s * Radius(radius)**2) def _xb(a1, b1, end, a, b, wrap): # difference between the bearing to (a, b) and the given # bearing is negative if both are in opposite directions r = bearing_(a1, b1, a, b, wrap=wrap) return PI_2 - abs(wrapPI(r - radians(end))) def _xdot(d, a1, b1, a, b, wrap): # compute dot product d . (-b + b1, a - a1) db, _ = unrollPI(b1, b, wrap=wrap) return fdot(d, db, a - a1) def _x3d2(start, end, wrap, _i_, hs): # see <https://www.EdWilliams.org/intersect.htm> (5) ff a1, b1 = start.philam if isscalar(end): # bearing, make a point a2, b2 = _destination2(a1, b1, PI_4, radians(end)) else: # must be a point _T00.others(end, name=_end_ + _i_) hs.append(end.height) a2, b2 = end.philam db, b2 = unrollPI(b1, b2, wrap=wrap) if max(abs(db), abs(a2 - a1)) < EPS: raise IntersectionError(start=start, end=end, txt='null path' + _i_) # note, in EdWilliams.org/avform.htm W is + and E is - b21, b12 = db * _0_5, -(b1 + b2) * _0_5 sb21, cb21, sb12, cb12, \ sa21, _, sa12, _ = sincos2(b21, b12, a1 - a2, a1 + a2) x = _Nvector(sa21 * sb12 * cb21 - sa12 * cb12 * sb21, sa21 * cb12 * cb21 + sa12 * sb12 * sb21, cos(a1) * cos(a2) * sin(db)) # ll=start return x.unit(), (db, (a2 - a1)) # negated d def intersection(start1, end1, start2, end2, height=None, wrap=False, LatLon=LatLon, **LatLon_kwds): '''Compute the intersection point of two paths both defined by two points or a start point and bearing from North. @arg start1: Start point of the first path (L{LatLon}). @arg end1: End point ofthe first path (L{LatLon}) or the initial bearing at the first start point (compass C{degrees360}). @arg start2: Start point of the second path (L{LatLon}). @arg end2: End point of the second path (L{LatLon}) or the initial bearing at the second start point (compass C{degrees360}). @kwarg height: Optional height for the intersection point, overriding the mean height (C{meter}). @kwarg wrap: Wrap and unroll longitudes (C{bool}). @kwarg LatLon: Optional class to return the intersection point (L{LatLon}) or C{None}. @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword arguments, ignored if C{B{LatLon}=None}. @return: The intersection point (B{C{LatLon}}) or a L{LatLon3Tuple}C{(lat, lon, height)} if B{C{LatLon}} is C{None}. An alternate intersection point might be the L{antipode} to the returned result. @raise IntersectionError: Intersection is ambiguous or infinite or the paths are coincident, colinear or parallel. @raise TypeError: A B{C{start}} or B{C{end}} point not L{LatLon}. @raise ValueError: Invalid B{C{height}}. @example: >>> p = LatLon(51.8853, 0.2545) >>> s = LatLon(49.0034, 2.5735) >>> i = intersection(p, 108.547, s, 32.435) # '50.9078°N, 004.5084°E' ''' _T00.others(start1=start1) _T00.others(start2=start2) hs = [start1.height, start2.height] a1, b1 = start1.philam a2, b2 = start2.philam db, b2 = unrollPI(b1, b2, wrap=wrap) r12 = vincentys_(a2, a1, db) if abs(r12) < EPS: # [nearly] coincident points a, b = favg(a1, a2), favg(b1, b2) # see <https://www.EdWilliams.org/avform.htm#Intersection> elif isscalar(end1) and isscalar(end2): # both bearings sa1, ca1, sa2, ca2, sr12, cr12 = sincos2(a1, a2, r12) x1, x2 = (sr12 * ca1), (sr12 * ca2) if abs(x1) < EPS or abs(x2) < EPS: raise IntersectionError(start1=start1, end1=end1, start2=start2, end2=end2, txt='parallel') # handle domain error for equivalent longitudes, # see also functions asin_safe and acos_safe at # <https://www.EdWilliams.org/avform.htm#Math> t1, t2 = map1(acos1, (sa2 - sa1 * cr12) / x1, (sa1 - sa2 * cr12) / x2) if sin(db) > 0: t12, t21 = t1, PI2 - t2 else: t12, t21 = PI2 - t1, t2 t13, t23 = map1(radiansPI2, end1, end2) x1, x2 = map1(wrapPI, t13 - t12, # angle 2-1-3 t21 - t23) # angle 1-2-3 sx1, cx1, sx2, cx2 = sincos2(x1, x2) if sx1 == 0 and sx2 == 0: # max(abs(sx1), abs(sx2)) < EPS raise IntersectionError(start1=start1, end1=end1, start2=start2, end2=end2, txt='infinite') sx3 = sx1 * sx2 # if sx3 < 0: # raise IntersectionError(start1=start1, end1=end1, # start2=start2, end2=end2, txt=_ambiguous_) x3 = acos1(cr12 * sx3 - cx2 * cx1) r13 = atan2(sr12 * sx3, cx2 + cx1 * cos(x3)) a, b = _destination2(a1, b1, r13, t13) # choose antipode for opposing bearings if _xb(a1, b1, end1, a, b, wrap) < 0 or \ _xb(a2, b2, end2, a, b, wrap) < 0: a, b = antipode_(a, b) # PYCHOK PhiLam2Tuple else: # end point(s) or bearing(s) x1, d1 = _x3d2(start1, end1, wrap, _1_, hs) x2, d2 = _x3d2(start2, end2, wrap, _2_, hs) x = x1.cross(x2) if x.length < EPS: # [nearly] colinear or parallel paths raise IntersectionError(start1=start1, end1=end1, start2=start2, end2=end2, txt=_colinear_) a, b = x.philam # choose intersection similar to sphericalNvector d1 = _xdot(d1, a1, b1, a, b, wrap) if d1: d2 = _xdot(d2, a2, b2, a, b, wrap) if (d2 < 0 and d1 > 0) or (d2 > 0 and d1 < 0): a, b = antipode_(a, b) # PYCHOK PhiLam2Tuple h = fmean(hs) if height is None else Height(height) return _latlon3(degrees90(a), degrees180(b), h, intersection, LatLon, LatLon_kwds) def intersections2(center1, rad1, center2, rad2, radius=R_M, eps=_0_0, height=None, wrap=True, LatLon=LatLon, **LatLon_kwds): '''Compute the intersection points of two circles each defined by a center point and a radius. @arg center1: Center of the first circle (L{LatLon}). @arg rad1: Radius of the first circle (C{meter} or C{radians}, see B{C{radius}}). @arg center2: Center of the second circle (L{LatLon}). @arg rad2: Radius of the second circle (C{meter} or C{radians}, see B{C{radius}}). @kwarg radius: Mean earth radius (C{meter} or C{None} if B{C{rad1}}, B{C{rad2}} and B{C{eps}} are given in C{radians}). @kwarg eps: Required overlap (C{meter} or C{radians}, see B{C{radius}}). @kwarg height: Optional height for the intersection points, overriding the "radical height" at the "radcal line" between both centers (C{meter}) or C{None}. @kwarg wrap: Wrap and unroll longitudes (C{bool}). @kwarg LatLon: Optional class to return the intersection points (L{LatLon}) or C{None}. @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword arguments, ignored if C{B{LatLon}=None}. @return: 2-Tuple of the intersection points, each a B{C{LatLon}} instance or a L{LatLon3Tuple}C{(lat, lon, height)} if B{C{LatLon}} is C{None}. For abutting circles, both intersection points are the same instance. @raise IntersectionError: Concentric, antipodal, invalid or non-intersecting circles. @raise TypeError: If B{C{center1}} or B{C{center2}} not L{LatLon}. @raise ValueError: Invalid B{C{rad1}}, B{C{rad2}}, B{C{radius}}, B{C{eps}} or B{C{height}}. @note: Courtesy U{Samuel Čavoj<https://GitHub.com/mrJean1/PyGeodesy/issues/41>}. @see: This U{Answer<https://StackOverflow.com/questions/53324667/ find-intersection-coordinates-of-two-circles-on-earth/53331953>}. ''' c1 = _T00.others(center1=center1) c2 = _T00.others(center2=center2) try: return _intersects2(c1, rad1, c2, rad2, radius=radius, eps=eps, height=height, wrap=wrap, LatLon=LatLon, **LatLon_kwds) except (TypeError, ValueError) as x: raise IntersectionError(center1=center1, rad1=rad1, center2=center2, rad2=rad2, txt=str(x)) def _intersects2(c1, rad1, c2, rad2, radius=R_M, eps=_0_0, # in .ellipsoidalBase._intersects2 height=None, too_d=None, wrap=True, LatLon=LatLon, **LatLon_kwds): # (INTERNAL) Intersect two spherical circles, see L{intersections2} # above, separated to allow callers to embellish any exceptions def _dest3(bearing, h): a, b = _destination2(a1, b1, r1, bearing) return _latlon3(degrees90(a), degrees180(b), h, intersections2, LatLon, LatLon_kwds) r1, r2, f = _rads3(rad1, rad2, radius) if f: # swapped c1, c2 = c2, c1 # PYCHOK swap a1, b1 = c1.philam a2, b2 = c2.philam db, b2 = unrollPI(b1, b2, wrap=wrap) d = vincentys_(a2, a1, db) # radians if d < max(r1 - r2, EPS): raise ValueError(_near_concentric_) r = eps if radius is None else (m2radians(eps, radius=radius) if eps else _0_0) if r < _0_0: raise _ValueError(eps=r) x = fsum_(r1, r2, -d) # overlap if x > max(r, EPS): sd, cd, sr1, cr1, _, cr2 = sincos2(d, r1, r2) x = sd * sr1 if abs(x) < EPS: raise ValueError(_invalid_) x = acos1((cr2 - cd * cr1) / x) # 0 <= x <= PI elif x < r: t = (d * radius) if too_d is None else too_d raise ValueError(_too_(Fmt.distant(t))) if height is None: # "radical height" f = _radical2(d, r1, r2).ratio h = Height(favg(c1.height, c2.height, f=f)) else: h = Height(height) b = bearing_(a1, b1, a2, b2, final=False, wrap=wrap) if x < _EPS4: # externally ... r = _dest3(b, h) elif x > _PI_EPS4: # internally ... r = _dest3(b + PI, h) else: return _dest3(b + x, h), _dest3(b - x, h) return r, r # ... abutting circles def isPoleEnclosedBy(points, wrap=False): # PYCHOK no cover '''DEPRECATED, use function L{ispolar}. ''' return ispolar(points, wrap=wrap) def _latlon3(lat, lon, height, func, LatLon, LatLon_kwds): '''(INTERNAL) Helper for L{intersection}, L{intersections2} and L{meanof}. ''' if LatLon is None: r = LatLon3Tuple(lat, lon, height) else: kwds = _xkwds(LatLon_kwds, height=height) r = LatLon(lat, lon, **kwds) return _xnamed(r, func.__name__) def meanOf(points, height=None, LatLon=LatLon, **LatLon_kwds): '''Compute the geographic mean of several points. @arg points: Points to be averaged (L{LatLon}[]). @kwarg height: Optional height at mean point, overriding the mean height (C{meter}). @kwarg LatLon: Optional class to return the mean point (L{LatLon}) or C{None}. @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword arguments, ignored if C{B{LatLon}=None}. @return: Point at geographic mean and height (B{C{LatLon}}) or a L{LatLon3Tuple}C{(lat, lon, height)} if B{C{LatLon}} is C{None}. @raise TypeError: Some B{C{points}} are not L{LatLon}. @raise ValueError: No B{C{points}} or invalid B{C{height}}. ''' # geographic mean n, points = _T00.points2(points, closed=False) m = sumOf(points[i]._N_vector for i in range(n)) lat, lon = m._N_vector.latlon if height is None: h = fmean(points[i].height for i in range(n)) else: h = Height(height) return _latlon3(lat, lon, h, meanOf, LatLon, LatLon_kwds) def nearestOn2(point, points, **closed_radius_LatLon_options): # PYCHOK no cover '''DEPRECATED, use function L{sphericalTrigonometry.nearestOn3}. @return: ... 2-tuple C{(closest, distance)} of the C{closest} point (L{LatLon}) on the polygon and the C{distance} between the C{closest} and the given B{C{point}}. The C{closest} is a B{C{LatLon}} or a L{LatLon2Tuple}C{(lat, lon)} if B{C{LatLon}} is C{None} ... ''' ll, d, _ = nearestOn3(point, points, **closed_radius_LatLon_options) if _xkwds_get(closed_radius_LatLon_options, LatLon=LatLon) is None: ll = LatLon2Tuple(ll.lat, ll.lon) return ll, d def nearestOn3(point, points, closed=False, radius=R_M, LatLon=LatLon, **options): '''Locate the point on a polygon closest to an other, reference point. Distances are approximated by function L{equirectangular_}, subject to the supplied B{C{options}}. @arg point: The other, reference point (L{LatLon}). @arg points: The polygon points (L{LatLon}[]). @kwarg closed: Optionally, close the polygon (C{bool}). @kwarg radius: Mean earth radius (C{meter}). @kwarg LatLon: Optional class to return the closest point (L{LatLon}) or C{None}. @kwarg options: Optional keyword arguments for function L{equirectangular_}. @return: A L{NearestOn3Tuple}C{(closest, distance, angle)} with the C{closest} point as B{L{LatLon}} or L{LatLon3Tuple}C{(lat, lon, height)} if B{C{LatLon}} is C{None}. The C{distance} is the L{equirectangular_} distance between the C{closest} and the given B{C{point}} in C{meter}, same units as B{C{radius}}. The C{angle} from the given B{C{point}} to the C{closest} is in compass C{degrees360}, like function L{compassAngle}. The C{height} is the (interpolated) height at the C{closest} point. @raise LimitError: Lat- and/or longitudinal delta exceeds the B{C{limit}}, see function L{equirectangular_}. @raise PointsError: Insufficient number of B{C{points}}. @raise TypeError: Some B{C{points}} are not C{LatLon}. @raise ValueError: Invalid B{C{radius}}. @see: Functions L{equirectangular_} and L{nearestOn5}. ''' lat, lon, d, c, h = _nearestOn5(point, points, closed=closed, LatLon=None, **options) r = LatLon3Tuple(lat, lon, h) if LatLon is None else \ LatLon(lat, lon, height=h) r = NearestOn3Tuple(r, degrees2m(d, radius=radius), c) return _xnamed(r, nearestOn3.__name__) def perimeterOf(points, closed=False, radius=R_M, wrap=True): '''Compute the perimeter of a (spherical) polygon (with great circle arcs joining the points). @arg points: The polygon points (L{LatLon}[]). @kwarg closed: Optionally, close the polygon (C{bool}). @kwarg radius: Mean earth radius (C{meter}). @kwarg wrap: Wrap and unroll longitudes (C{bool}). @return: Polygon perimeter (C{meter}, same units as B{C{radius}}). @raise PointsError: Insufficient number of B{C{points}}. @raise TypeError: Some B{C{points}} are not L{LatLon}. @raise ValueError: Invalid B{C{radius}}. @note: This perimeter is based on the L{haversine} formula. @see: L{pygeodesy.perimeterOf}, L{sphericalNvector.perimeterOf} and L{ellipsoidalKarney.perimeterOf}. ''' n, points = _T00.points2(points, closed=closed) def _rads(n, points, closed): # angular edge lengths in radians i, m = _imdex2(closed, n) a1, b1 = points[i].philam for i in range(m, n): a2, b2 = points[i].philam db, b2 = unrollPI(b1, b2, wrap=wrap) yield vincentys_(a2, a1, db) a1, b1 = a2, b2 r = fsum(_rads(n, points, closed)) return r * Radius(radius) __all__ += _ALL_OTHER(Cartesian, LatLon, # classes areaOf, # functions intersection, intersections2, ispolar, isPoleEnclosedBy, # DEPRECATED, use ispolar meanOf, nearestOn2, nearestOn3, perimeterOf, sumOf) # == vector3d.sumOf # **) MIT License # # Copyright (C) 2016-2021 -- mrJean1 at Gmail -- All Rights Reserved. # # Permission is hereby granted, free of charge, to any person obtaining a # copy of this software and associated documentation files (the "Software"), # to deal in the Software without restriction, including without limitation # the rights to use, copy, modify, merge, publish, distribute, sublicense, # and/or sell copies of the Software, and to permit persons to whom the # Software is furnished to do so, subject to the following conditions: # # The above copyright notice and this permission notice shall be included # in all copies or substantial portions of the Software. # # THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS # OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, # FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL # THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR # OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, # ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR # OTHER DEALINGS IN THE SOFTWARE.