# Coverage for pygeodesy/geodesicx/gx.py : 93%

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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 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 | # -*- coding: utf-8 -*- u'''A Python version of I{Karney}'s C++ class U{GeodesicExact <https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1GeodesicExact.html>}. Classes L{GeodesicExact} and L{GeodesicLineExact} follow the naming, methods and return values from I{Karney}s' Python classes C{Geodesic} and C{GeodesicLine}, respectively. Copyright (C) Charles Karney (2012-2019) <Charles@Karney.com> and licensed under the MIT/X11 License. For more information, see U{GeographicLib<https://GeographicLib.SourceForge.io>}. ''' # make sure int/int division yields float quotient from __future__ import division # A copy of comments from Karney's C{GeodesicExact.cpp}: # # This is a reformulation of the geodesic problem. The # notation is as follows: # - at a general point (no suffix or 1 or 2 as suffix) # - phi = latitude # - beta = latitude on auxiliary sphere # - omega = longitude on auxiliary sphere # - lambda = longitude # - alpha = azimuth of great circle # - sigma = arc length along great circle # - s = distance # - tau = scaled distance (= sigma at multiples of PI/2) # - at northwards equator crossing # - beta = phi = 0 # - omega = lambda = 0 # - alpha = alpha0 # - sigma = s = 0 # - a 12 suffix means a difference, e.g., s12 = s2 - s1. # - s and c prefixes mean sin and cos from pygeodesy.basics import _xinstanceof, _xor, unsign0 from pygeodesy.datums import _ellipsoidal_datum from pygeodesy.ellipsoids import Ellipsoid2, Ellipsoids from pygeodesy.fmath import cbrt, fsum_, norm2 from pygeodesy.geodesicx.gxbases import _ALL_DOCS, Caps, _coSeries, \ _GeodesicBase, _polynomial, \ _sincos12, _TINY, _xnC4 from pygeodesy.geodesicx.gxline import _GeodesicLineExact, pairs, \ _update_glXs from pygeodesy.interns import EPS, MANT_DIG, NAN, NN, PI, PI_2, \ _COMMASPACE_, _convergence_, _EPSqrt, \ _no_, _0_0, _0_01, _0_5, _1_0, _2_0, \ _3_0, _4_0, _6_0, _8_0, _16_0, _90_0, \ _180_0, _1000_0 from pygeodesy.interns import _0_001, _0_1 # PYCHOK used! from pygeodesy.karney import _around, GDict, GeodesicError, \ _diff182, _fix90, _norm180 from pygeodesy.namedTuples import Destination3Tuple, Distance3Tuple from pygeodesy.props import Property, Property_RO # from pygeodesy.streprs import pairs # from .geodesicx.gxline from pygeodesy.utily import atan2d, sincos2, sincos2d, unroll180, wrap360 from math import atan2, copysign, cos, degrees, hypot, radians, sqrt __all__ = () __version__ = '21.06.02' _MAXIT1 = 20 _MAXIT2 = 10 + _MAXIT1 + MANT_DIG # MANT_DIG == C++ digits _SQRT2_2 = -0.7071 # negative sqrt(2) / 2 _1_75 = 1.75 # increased multiplier in defn of _TOL1 from 100 to 200 to fix Inverse # case 52.784459512564 0 -52.784459512563990912 179.634407464943777557 # which otherwise failed for Visual Studio 10 (Release and Debug) _TOL0 = EPS _TOL1 = _TOL0 * 200 _TOL2 = _EPSqrt # == sqrt(_TOL0) _TOLb = _TOL2 * _TOL0 # Check on bisection interval _THR1 = _TOL2 * _1000_0 + _1_0 _TINY3 = _TINY * _3_0 _TOL08 = _TOL0 * _8_0 _TOL016 = _TOL0 * _16_0 class _PDict(GDict): '''(INTERNAL) Parameters passed around in C{._GDictInverse} and optionally returned when C{GeodesicExact.debug} is C{True}. ''' def setsigs(self, ssig1, csig1, ssig2, csig2): '''Update the C{sig1} and C{sig2} parameters. ''' self.set_(ssig1=ssig1, csig1=csig1, sncndn1=(ssig1, csig1, self.dn1), # PYCHOK dn1 ssig2=ssig2, csig2=csig2, sncndn2=(ssig2, csig2, self.dn2)) # PYCHOK dn2 def toGDict(self): '''Return as C{GDict} without attrs C{sncndn1} and C{sncndn2}. ''' def _rest(sncndn1=None, sncndn2=None, **rest): # PYCHOK unused return GDict(rest) return _rest(**self) class GeodesicExact(_GeodesicBase): '''A pure Python version of I{Karney}'s C++ class U{GeodesicExact <https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1GeodesicExact.html>}, modeled after I{Karney}'s Python class U{Geodesic<https://GeographicLib.SourceForge.io/ html/python/code.html#module-geographiclib.geodesic>}. ''' _E = Ellipsoids.WGS84 _nC4 = 30 # default C4Order def __init__(self, a_ellipsoid=Ellipsoids.WGS84, f=None, name=NN, C4Order=None): '''New L{GeodesicExact} instance. @arg a_ellipsoid: An ellipsoid (L{Ellipsoid}) or datum (L{datum}) the equatorial radius of the ellipsoid (C{meter}), see B{C{f}}. @arg f: The flattening of the ellipsoid (C{scalar}) if B{C{a_ellipsoid}} is specified as C{meter}. @kwarg name: Optional name (C{str}). @kwarg C4Order: Optional series expansion order (C{int}), see property L{C4Order}, default C{30}. @raise GeodesicError: Invalid B{C{C4Order}}. ''' if a_ellipsoid in (GeodesicExact._E, None): pass # ignore f, default WGS84 elif f is None: self._E = _ellipsoidal_datum(a_ellipsoid, name=name, raiser=True).ellipsoid else: self._E = Ellipsoid2(a_ellipsoid, f, name=name) if name: self.name = name if C4Order: # XXX private copy, always? self.C4Order = C4Order @Property_RO def a(self): '''Get the I{equatorial} radius, semi-axis (C{meter}). ''' return self.ellipsoid.a def ArcDirect(self, lat1, lon1, azi1, a12, outmask=Caps.STANDARD): '''Solve the I{Direct} geodesic problem in terms of (spherical) arc length. @arg lat1: Latitude of the first point (C{degrees}). @arg lon1: Longitude of the first point (C{degrees}). @arg azi1: Azimuth at the first point (compass C{degrees}). @arg a12: Arc length between the points (C{degrees}), can be negative. @kwarg outmask: Bit-or'ed combination of L{Caps} values specifying the quantities to be returned. @return: A C{dict} with up to 12 items C{lat1, lon1, azi1, lat2, lon2, azi2, m12, a12, s12, M12, M21, S12} with C{lat1}, C{lon1}, C{azi1} and arc length C{a12} always included. @see: C++ U{GeodesicExact.ArcDirect <https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1GeodesicExact.html>} and Python U{Geodesic.ArcDirect<https://GeographicLib.SourceForge.io/html/python/code.html>}. ''' return self._GDictDirect(lat1, lon1, azi1, True, a12, outmask) def ArcDirectLine(self, lat1, lon1, azi1, a12, caps): '''Define an L{GeodesicLineExact} in terms of the I{direct} geodesic problem and as arc length. @arg lat1: Latitude of the first point (C{degrees}). @arg lon1: Longitude of the first point (C{degrees}). @arg azi1: Azimuth at the first point (compass C{degrees}). @arg a12: Arc length between the points (C{degrees}), can be negative. @kwarg caps: Bit-or'ed combination of L{Caps} values specifying the capabilities the L{GeodesicLineExact} instance should possess, i.e., which quantities can be returned by calls to L{GeodesicLineExact.Position} and L{GeodesicLineExact.ArcPosition}. @return: A L{GeodesicLineExact} instance. @note: The third point of the L{GeodesicLineExact} is set to correspond to the second point of the I{Inverse} geodesic problem. @note: Latitude B{C{lat1}} should in the range C{[-90, +90]}. @see: C++ U{GeodesicExact.ArcDirectLine <https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1GeodesicExact.html>} and Python U{Geodesic.ArcDirectLine<https://GeographicLib.SourceForge.io/html/python/code.html>}. ''' return self._GenDirectLine(lat1, lon1, azi1, True, a12, caps) def Area(self, polyline=False, name=NN): '''Set up an L{GeodesicAreaExact} to compute area and perimeter of a polygon. @kwarg polyline: If C{True} perimeter only, otherwise area and perimeter (C{bool}). @kwarg name: Optional name (C{str}). @return: A L{GeodesicAreaExact} instance. @note: The B{C{debug}} setting is passed as C{verbose} to the returned L{GeodesicAreaExact} instance. ''' from pygeodesy.geodesicx.gxarea import GeodesicAreaExact gaX = GeodesicAreaExact(self, polyline=polyline, name=name or self.name) if self.debug: gaX.verbose = True return gaX Polygon = Area # for C{geographiclib} compatibility @Property_RO def b(self): '''Get the ellipsoid's I{polar} radius, semi-axis (C{meter}). ''' return self.ellipsoid.b @Property_RO def c2x(self): '''Get the ellipsoid's I{authalic} earth radius I{squared} (C{meter**2}). ''' # The Geodesic class substitutes atanh(sqrt(e2)) for asinh(sqrt(ep2)) # in the definition of _c2. The latter is more accurate for very # oblate ellipsoids (which the Geodesic class does not handle). Of # course, the area calculation in GeodesicExact is still based on a # series and only holds for moderately oblate (or prolate) ellipsoids. return self.ellipsoid.c2x c2 = c2x # in this particular case def C4f(self, eps): '''Evaluate the C{C4x} coefficients for B{C{eps}}. @arg eps: Polynomial factor (C{float}). @return: C{C4Order}-Tuple of C{C4x(B{eps})} coefficients. ''' def _c4(nC4, C4x, _p_eps, eps): i, e = 0, _1_0 for r in range(nC4, 0, -1): j = i + r yield _p_eps(eps, C4x, i, j) * e e *= eps i = j # assert i == (nC4 * (nC4 + 1)) // 2 return tuple(_c4(self._nC4, self._C4x, _polynomial, eps)) def _C4f_k2(self, k2): # in ._GDictInverse and gxline._GeodesicExactLine._C4a '''(INTERNAL) Compute C{eps} from B{C{k2}} and invoke C{C4f}. ''' return self.C4f(k2 / fsum_(_2_0, sqrt(k2 + _1_0) * _2_0, k2)) @Property def C4Order(self): '''Get the series expansion order (C{int}, 24, 27 or 30). ''' return self._nC4 @C4Order.setter # PYCHOK .setter def C4Order(self, order): '''Set the series expansion order. @arg order: New order (C{int}, 24, 27 or 30). @raise GeodesicError: Invalid B{C{order}}. ''' _xnC4(C4Order=order) if self._nC4 != order: GeodesicExact._C4x._update(self) _update_glXs(self) self._nC4 = order @Property_RO def _C4x(self): '''Get this ellipsoid's C{C4} coefficients, I{cached} tuple. @see: Property L{nC4}. ''' # see C4coeff() in GeographicLib.src.GeodesicExactC4.cpp def _C4(nC4, coeffs, _p_n, n): i = 0 for r in range(nC4 + 1, 1, -1): for j in range(1, r): j = i + j # (j - i - 1) order of polynomial yield _p_n(n, coeffs, i, j) / coeffs[j] i = j + 1 # assert i == len(cs) == (nC4 * (nC4 + 1) * (nC4 + 5)) // 6 return tuple(_C4(self._nC4, self._coeffs(self._nC4), _polynomial, self.n)) # 3rd flattening def _coeffs(self, nC4): '''(INTERNAL) Get the series coefficients. ''' if nC4 == 30: from pygeodesy.geodesicx._C4_30 import _coeffs_30 as _coeffs elif nC4 == 27: from pygeodesy.geodesicx._C4_27 import _coeffs_27 as _coeffs elif nC4 == 24: from pygeodesy.geodesicx._C4_24 import _coeffs_24 as _coeffs else: # PYCHOK no cover _coeffs = () n = _xnC4(nC4=nC4) if len(_coeffs) != n: # double check raise GeodesicError(_coeffs=len(_coeffs), _xnC4=n, nC4=nC4) return _coeffs def Direct(self, lat1, lon1, azi1, s12, outmask=Caps.STANDARD): '''Solve the I{Direct} geodesic problem @arg lat1: Latitude of the first point (C{degrees}). @arg lon1: Longitude of the first point (C{degrees}). @arg azi1: Azimuth at the first point (compass C{degrees}). @arg s12: Distance between the points (C{meter}), can be negative. @kwarg outmask: Bit-or'ed combination of L{Caps} values specifying the quantities to be returned. @return: A C{dict} with up to 12 items C{lat1, lon1, azi1, lat2, lon2, azi2, m12, a12, s12, M12, M21, S12} with C{lat1}, C{lon1}, C{azi1} and distance C{s12} always included. @see: C++ U{GeodesicExact.Direct <https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1GeodesicExact.html>} and Python U{Geodesic.Direct<https://GeographicLib.SourceForge.io/html/python/code.html>}. ''' return self._GDictDirect(lat1, lon1, azi1, False, s12, outmask) def Direct3(self, lat1, lon1, azi1, s12): # PYCHOK outmask '''Return the destination lat, lon and reverse azimuth (final bearing) in C{degrees}. @return: L{Destination3Tuple}C{(lat, lon, final)}. ''' r = self._GDictDirect(lat1, lon1, azi1, False, s12, Caps._AZIMUTH_LATITUDE_LONGITUDE) return Destination3Tuple(r.lat2, r.lon2, r.azi2) def DirectLine(self, lat1, lon1, azi1, s12, caps=Caps.STANDARD): '''Define a {GeodesicLineExact} in terms of the I{direct} geodesic problem and as distance. @arg lat1: Latitude of the first point (C{degrees}). @arg lon1: Longitude of the first point (C{degrees}). @arg azi1: Azimuth at the first point (compass C{degrees}). @arg s12: Distance between the points (C{meter}), can be negative. @kwarg caps: Bit-or'ed combination of L{Caps} values specifying the capabilities the L{GeodesicLineExact} instance should possess, i.e., which quantities can be returned by calls to L{GeodesicLineExact.Position}. @return: A L{GeodesicLineExact} instance. @note: The third point of the L{GeodesicLineExact} is set to correspond to the second point of the I{Inverse} geodesic problem. @note: Latitude B{C{lat1}} should in the range C{[-90, +90]}. @see: C++ U{GeodesicExact.DirectLine <https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1GeodesicExact.html>} and Python U{Geodesic.DirectLine<https://GeographicLib.SourceForge.io/html/python/code.html>}. ''' return self._GenDirectLine(lat1, lon1, azi1, False, s12, caps) def _dn(self, sbet, cbet): # in gxline._GeodesicLineExact.__init__ '''(INTERNAL) Helper. ''' return sqrt(_1_0 + self.ep2 * sbet**2) if self.f >= 0 else ( sqrt(_1_0 - self.e2 * cbet**2) / self.f1) @Property_RO def e2(self): '''Get the ellipsoid's I{(1st) eccentricity squared} (C{float}), M{f * (2 - f)}. ''' return self.ellipsoid.e2 @Property_RO def _e2a2(self): '''(INTERNAL) Cache M{E.e2 * E.a2}. ''' return self.e2 * self.ellipsoid.a2 @Property_RO def _e2_f1(self): '''(INTERNAL) Cache M{E.e2 * E.f1}. ''' return self.e2 / self.f1 @Property_RO def _eF(self): '''(INTERNAL) Get the elliptic function, aka C{.E}. ''' from pygeodesy.elliptic import Elliptic return Elliptic(k2=-self.ep2) def _eF_reset_e2_f1_cH(self, x, y): '''(INTERNAL) Reset elliptic function and return M{e2 / f1 * cH * ...}. ''' self._eF_reset_k2(x) return self._e2_f1 * y * self._eF.cH def _eF_reset_k2(self, x): '''(INTERNAL) Reset elliptic function and return C{k2}. ''' ep2 = self.ep2 k2 = x**2 * ep2 self._eF.reset(k2=-k2, alpha2=-ep2, kp2=k2 + _1_0, alphap2=ep2 + _1_0) # defaults _update_glXs(self) # zap cached/memoized _GeodesicLineExact attrs return k2 @Property_RO def ellipsoid(self): '''Get the ellipsoid (C{Ellipsoid}). ''' return self._E @Property_RO def ep2(self): # aka .e22 '''Get the ellipsoid's I{2nd eccentricity squared} (C{float}), M{e2 / (1 - e2)}. ''' return self.ellipsoid.e22 # == self.e2 / self.f1**2 @Property_RO def _eTOL2(self): '''(INTERNAL) The si12 threshold for "really short". ''' # Using the auxiliary sphere solution with dnm computed at # (bet1 + bet2) / 2, the relative error in the azimuth # consistency check is sig12^2 * abs(f) * min(1, 1-f/2) / 2. # (Error measured for 1/100 < b/a < 100 and abs(f) >= 1/1000. # For a given f and sig12, the max error occurs for lines # near the pole. If the old rule for computing dnm = (dn1 # + dn2)/2 is used, then the error increases by a factor of # 2.) Setting this equal to epsilon gives sig12 = etol2. # Here 0.1 is a safety factor (error decreased by 100) and # max(0.001, abs(f)) stops etol2 getting too large in the # nearly spherical case. f = self.f return _0_1 * _TOL2 / sqrt(min(_1_0, _1_0 - f * _0_5) * max(_0_001, abs(f)) * _0_5) @Property_RO def f(self): '''Get the ellipsoid's I{flattening} (C{float}), M{(a - b) / a}, C{0} for spherical, negative for prolate. ''' return self.ellipsoid.f flattening = f @Property_RO def f1(self): # aka .b_a '''Get the ellipsoid's ratio I{polar} over I{equatorial} radius (C{float}), M{b / a} == M{1 - f}. ''' return self.ellipsoid.b_a # _1_0 - self.f @Property_RO def _f_180(self): '''(INTERNAL) Cached/memoized. ''' return self.f * _180_0 def _GDictDirect(self, lat1, lon1, azi1, arcmode, s12_a12, outmask=Caps.STANDARD): '''(INTERNAL) As C{_GenDirect}, but returning a C{dict}. @return: A C{dict} ... ''' if not arcmode: # supply DISTANCE_IN outmask |= Caps.DISTANCE_IN glX = self._Line(lat1, lon1, azi1, outmask) return glX._GDictPosition(arcmode, s12_a12, outmask) def _GDictInverse(self, lat1, lon1, lat2, lon2, outmask=Caps.STANDARD): # MCCABE 32, 41 vars '''(INTERNAL) As C{_GenInverse}, but returning a C{dict}. @return: A C{dict} ... ''' if self.debug: # PYCHOK no cover outmask |= Caps._DEBUG_INVERSE & self._debug outmask &= Caps._OUTMASK # incl. _SALPs_CALPs and _DEBUG_ # compute longitude difference carefully (with _diff182): # result is in [-180, +180] but -180 is only for west-going # geodesics, +180 is for east-going and meridional geodesics lon12, lon12s = _diff182(lon1, lon2) # see C{result} from geographiclib.geodesic.Inverse if (outmask & Caps.LONG_UNROLL): # == (lon1 + lon12) + lon12s r = GDict(lon1=lon1, lon2=fsum_(lon1, lon12, lon12s)) else: r = GDict(lon1=_norm180(lon1), lon2=_norm180(lon2)) # make longitude difference positive and if very close to # being on the same half-meridian, then make it so. if lon12 < 0: lon_, lon12 = True, -_around(lon12) lon12s = _around((_180_0 - lon12) + lon12s) else: lon_, lon12 = False, _around(lon12) lon12s = _around((_180_0 - lon12) - lon12s) lam12 = radians(lon12) if lon12 > _90_0: slam12, clam12 = sincos2d(lon12s) clam12 = -clam12 else: slam12, clam12 = sincos2d(lon12) # If really close to the equator, treat as on equator. lat1 = _around(_fix90(lat1)) lat2 = _around(_fix90(lat2)) r.set_(lat1=lat1, lat2=lat2) # Swap points so that point with higher (abs) latitude is # point 1. If one latitude is a NAN, then it becomes lat1. swap_ = abs(lat1) < abs(lat2) if swap_: lat1, lat2 = lat2, lat1 lon_ = not lon_ if lat1 < 0: lat_ = False else: # make lat1 <= 0 lat_ = True lat1, lat2 = -lat1, -lat2 # Now 0 <= lon12 <= 180, -90 <= lat1 <= 0 and lat1 <= lat2 <= -lat1 # and lat_, lon_, swap_ register the transformation to bring the # coordinates to this canonical form, where False means no change # made. We make these transformations so that there are few cases # to check, e.g., on verifying quadrants in atan2. In addition, # this enforces some symmetries in the results returned. # Initialize for the meridian. No longitude calculation is # done in this case to let the parameter default to 0. sbet1, cbet1 = self._sincos2f1(lat1) sbet2, cbet2 = self._sincos2f1(lat2) # If cbet1 < -sbet1, then cbet2 - cbet1 is a sensitive measure # of the |bet1| - |bet2|. Alternatively (cbet1 >= -sbet1), # abs(sbet2) + sbet1 is a better measure. This logic is used # in assigning calp2 in _Lambda10. Sometimes these quantities # vanish and in that case we force bet2 = +/- bet1 exactly. An # example where is is necessary is the inverse problem # 48.522876735459 0 -48.52287673545898293 179.599720456223079643 # which failed with Visual Studio 10 (Release and Debug) if cbet1 < -sbet1: if cbet2 == cbet1: sbet2 = sbet1 if sbet2 < 0 else -sbet1 elif abs(sbet2) == -sbet1: cbet2 = cbet1 p = _PDict(sbet1=sbet1, cbet1=cbet1, dn1=self._dn(sbet1, cbet1), sbet2=sbet2, cbet2=cbet2, dn2=self._dn(sbet2, cbet2)) _meridian = True # i.e. meridian = False if lat1 == -90 or slam12 == 0: # Endpoints are on a single full meridian, # so the geodesic might lie on a meridian. salp1, calp1 = slam12, clam12 # head to target lon salp2, calp2 = _0_0, _1_0 # then head north # tan(bet) = tan(sig) * cos(alp) p.setsigs(sbet1, calp1 * cbet1, sbet2, calp2 * cbet2) # sig12 = sig2 - sig1 sig12 = atan2(*_sincos12(sbet1, p.csig1, sbet2, p.csig2, noneg=True)) # PYCHOK csig* s12x, m12x, _, \ M12, M21 = self._Lengths5(sig12, outmask | Caps.REDUCEDLENGTH, p) # Add the check for sig12 since zero length geodesics # might yield m12 < 0. Test case was # echo 20.001 0 20.001 0 | GeodSolve -i # In fact, we will have sig12 > PI/2 for meridional # geodesic which is not a shortest path. if m12x >= 0 or sig12 < _1_0: # Need at least 2, to handle 90 0 90 180 if sig12 < _TINY3: sig12 = m12x = s12x = a12 = _0_0 else: m12x *= self.b s12x *= self.b a12 = degrees(sig12) _meridian = False C = 1 # elif m12 < 0: # prolate and too close to anti-podal # _meridian = True if _meridian: if sbet1 == 0 and ( # sbet2 == 0 and self.f <= 0 or lon12s >= self._f_180): # Geodesic runs along equator calp1 = calp2 = _0_0 salp1 = salp2 = _1_0 sig12 = lam12 / self.f1 # == omg12 somg12, comg12 = sincos2(sig12) m12x = self.b * somg12 s12x = self.a * lam12 if (outmask & Caps.GEODESICSCALE): r.set_(M12=comg12, M21=comg12) a12 = lon12 / self.f1 C = 2 else: # Now point1 and point2 belong within a hemisphere bounded by a # meridian and geodesic is neither meridional or equatorial. p.set_(slam12=slam12, clam12=clam12) # Figure a starting point for Newton's method sig12, salp1, calp1, \ salp2, calp2, dnm = self._InverseStart6(lam12, p) if sig12 is None: # use Newton's method # pre-compute the constant _Lambda10 term, once p.set_(bet12=None if cbet2 == cbet1 and abs(sbet2) == -sbet1 else ( ((cbet1 + cbet2) * (cbet2 - cbet1)) if cbet1 < -sbet1 else ((sbet1 + sbet2) * (sbet1 - sbet2)))) sig12, salp1, calp1, salp2, calp2, domg12, \ s12x, m12x, M12, M21 = self._Newton10(salp1, calp1, outmask, p) if (outmask & Caps.AREA): # omg12 = lam12 - domg12 s, c = sincos2(domg12) somg12, comg12 = _sincos12(s, c, slam12, clam12) if (outmask & Caps.GEODESICSCALE): if swap_: M12, M21 = M21, M12 r.set_(M12=unsign0(M12), M21=unsign0(M21)) C = 3 # Newton else: C = 4 # Short lines, _Inverse6 set salp2, calp2, dnm s, c = sincos2(sig12 / dnm) m12x = s * dnm**2 s12x = sig12 * dnm if (outmask & Caps.GEODESICSCALE): r.set_(M12=c, M21=c) if (outmask & Caps.AREA): somg12, comg12 = sincos2(lam12 / (self.f1 * dnm)) a12 = degrees(sig12) m12x *= self.b s12x *= self.b else: # _meridian is False somg12 = comg12 = NAN if (outmask & Caps.DISTANCE): r.set_(s12=unsign0(s12x)) if (outmask & Caps.REDUCEDLENGTH): r.set_(m12=unsign0(m12x)) if (outmask & Caps.AREA): S12 = self._InverseArea(_meridian, salp1, calp1, salp2, calp2, somg12, comg12, p) if _xor(swap_, lat_, lon_): S12 = -S12 r.set_(S12=unsign0(S12)) if swap_: salp1, salp2 = salp2, salp1 calp1, calp2 = calp2, calp1 if _xor(swap_, lon_): salp1, salp2 = -salp1, -salp2 if _xor(swap_, lat_): calp1, calp2 = -calp1, -calp2 r.set_(a12=a12) # in [0, 180] if (outmask & Caps.AZIMUTH): r.set_(azi1=atan2d(salp1, calp1), azi2=atan2d(salp2, calp2, reverse=outmask & Caps.REVERSE2)) if (outmask & Caps._SALPs_CALPs): r.set_(salp1=salp1, calp1=calp1, salp2=salp2, calp2=calp2) if (outmask & Caps._DEBUG_INVERSE): # PYCHOK no cover eF = self._eF p.set_(C=C, a=self.a, f=self.f, f1=self.f1, e=self.ellipsoid.e, e2=self.e2, ep2=self.ep2, c2x=self.c2x, c2=self.ellipsoid.c2, eFcD=eF.cD, eFcE=eF.cE, eFcH=eF.cH, eFk2=eF.k2, eFa2=eF.alpha2) p.update(r) # r overrides r = p.toGDict() return r def _GenDirect(self, lat1, lon1, azi1, arcmode, s12_a12, outmask): '''(INTERNAL) The general I{Inverse} geodesic calculation. @return: L{Direct9Tuple}C{(a12, lat2, lon2, azi2, s12, m12, M12, M21, S12)}. ''' r = self._GDictDirect(lat1, lon1, azi1, arcmode, s12_a12, outmask) return r.toDirect9Tuple() def _GenDirectLine(self, lat1, lon1, azi1, arcmode, s12_a12, caps): '''(INTERNAL) Helper for C{ArcDirectLine} and C{DirectLine}. @return: A L{GeodesicLineExact} instance. ''' azi1 = _norm180(azi1) # guard against underflow in salp0. Also -0 is converted to +0. s, c = sincos2d(_around(azi1)) if not arcmode: # supply DISTANCE_IN caps |= Caps.DISTANCE_IN return _GeodesicLineExact(self, lat1, lon1, azi1, caps, self._debug, s, c)._GenSet(arcmode, s12_a12) def _GenInverse(self, lat1, lon1, lat2, lon2, outmask): '''(INTERNAL) The general I{Inverse} geodesic calculation. @return: L{Inverse10Tuple}C{(a12, s12, salp1, calp1, salp2, calp2, m12, M12, M21, S12)}. ''' r = self._GDictInverse(lat1, lon1, lat2, lon2, outmask | Caps._SALPs_CALPs) return r.toInverse10Tuple() def Inverse(self, lat1, lon1, lat2, lon2, outmask=Caps.STANDARD): '''Perform the I{Inverse} geodesic calculation. @arg lat1: Latitude of the first point (C{degrees}). @arg lon1: Longitude of the first point (C{degrees}). @arg lat2: Latitude of the second point (C{degrees}). @arg lon2: Longitude of the second point (C{degrees}). @kwarg outmask: Bit-or'ed combination of L{Caps} values specifying the quantities to be returned. @return: A C{dict} with up to 12 items C{lat1, lon1, azi1, lat2, lon2, azi2, m12, a12, s12, M12, M21, S12} with C{lat1}, C{lon1}, C{azi1} and distance C{s12} always included. @note: The third point of the L{GeodesicLineExact} is set to correspond to the second point of the I{Inverse} geodesic problem. @note: Both B{C{lat1}} and B{C{lat2}} should in the range C{[-90, +90]}. @see: C++ U{GeodesicExact.InverseLine <https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1GeodesicExact.html>} and Python U{Geodesic.InverseLine<https://GeographicLib.SourceForge.io/html/python/code.html>}. ''' return self._GDictInverse(lat1, lon1, lat2, lon2, outmask) def Inverse1(self, lat1, lon1, lat2, lon2, wrap=False): '''Return the non-negative, I{angular} distance in C{degrees}. ''' # see .FrechetKarney.distance, .HausdorffKarney._distance # and .HeightIDWkarney._distances _, lon2 = unroll180(lon1, lon2, wrap=wrap) # self.LONG_UNROLL return abs(self._GDictInverse(lat1, lon1, lat2, lon2, Caps.STANDARD).a12) def Inverse3(self, lat1, lon1, lat2, lon2): # PYCHOK outmask '''Return the distance in C{meter} and the forward and reverse azimuths (initial and final bearing) in C{degrees}. @return: L{Distance3Tuple}C{(distance, initial, final)}. ''' r = self._GDictInverse(lat1, lon1, lat2, lon2, Caps._AZIMUTH_DISTANCE) return Distance3Tuple(r.s12, wrap360(r.azi1), wrap360(r.azi2)) def InverseLine(self, lat1, lon1, lat2, lon2, caps=Caps.STANDARD): '''Define a {GeodesicLineExact} in terms of the I{Inverse} geodesic problem. @arg lat1: Latitude of the first point (C{degrees}). @arg lon1: Longitude of the first point (C{degrees}). @arg lat2: Latitude of the second point (C{degrees}). @arg lon2: Longitude of the second point (C{degrees}). @kwarg caps: Bit-or'ed combination of L{Caps} values specifying the capabilities the L{GeodesicLineExact} instance should possess, i.e., which quantities can be returned by calls to L{GeodesicLineExact.Position} and L{GeodesicLineExact.ArcPosition}. @return: A L{GeodesicLineExact} instance. @note: The third point of the L{GeodesicLineExact} is set to correspond to the second point of the I{Inverse} geodesic problem. @note: Both B{C{lat1}} and B{C{lat2}} should in the range C{[-90, +90]}. @see: C++ U{GeodesicExact.InverseLine <https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1GeodesicExact.html>} and Python U{Geodesic.InverseLine<https://GeographicLib.SourceForge.io/html/python/code.html>}. ''' r = self._GDictInverse(lat1, lon1, lat2, lon2, Caps._SALPs_CALPs) # No need for AZIMUTH azi1 = atan2d(r.salp1, r.calp1) if (caps & (Caps._OUTMASK & Caps.DISTANCE_IN)): caps |= Caps.DISTANCE # ensure that a12 is a distance return _GeodesicLineExact(self, lat1, lon1, azi1, caps, self._debug, r.salp1, r.calp1)._GenSet(True, r.a12) def _InverseArea(self, _meridian, salp1, calp1, # PYCHOK 9 args salp2, calp2, somg12, comg12, p): '''(INTERNAL) Split off from C{_GDictInverse} to reduce complexity/length. @return: Area C{S12}. ''' # from _Lambda10: sin(alp1) * cos(bet1) = sin(alp0) salp0 = salp1 * p.cbet1 calp0 = hypot(calp1, salp1 * p.sbet1) # calp0 > 0 if calp0 and salp0: # from _Lambda10: tan(bet) = tan(sig) * cos(alp) k2 = calp0**2 * self.ep2 C4a = self._C4f_k2(k2) B41 = _coSeries(C4a, *norm2(p.sbet1, calp1 * p.cbet1)) B42 = _coSeries(C4a, *norm2(p.sbet2, calp2 * p.cbet2)) # multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0) A4 = self._e2a2 * calp0 * salp0 S12 = A4 * (B42 - B41) p.set_(A4=A4, B41=B41, B42=B42, k2=k2) else: # avoid problems with indeterminate sig1, sig2 on equator S12 = _0_0 if (_meridian and # omg12 < 3/4 * PI comg12 > _SQRT2_2 and # lon diff not too big (p.sbet2 - p.sbet1) < _1_75): # lat diff not too big # use tan(Gamma/2) = tan(omg12/2) * # (tan(bet1/2) + tan(bet2/2)) / # (tan(bet1/2) * tan(bet2/2) + 1)) # with tan(x/2) = sin(x) / (1 + cos(x)) cbet1 = _1_0 + p.cbet1 cbet2 = _1_0 + p.cbet2 salp12 = (p.sbet1 * cbet2 + cbet1 * p.sbet2) * somg12 calp12 = (p.sbet1 * p.sbet2 + cbet1 * cbet2) * (comg12 + _1_0) alp12 = _2_0 * atan2(salp12, calp12) else: # alp12 = alp2 - alp1, used in atan2, no need to normalize salp12, calp12 = _sincos12(salp1, calp1, salp2, calp2) # The right thing appears to happen if alp1 = +/-180 and # alp2 = 0, viz salp12 = -0 and alp12 = -180. However, # this depends on the sign being attached to 0 correctly. # Following ensures the correct behavior. if salp12 == 0 and calp12 < 0: alp12 = copysign(PI, calp1) else: alp12 = atan2(salp12, calp12) p.set_(alp12=alp12) return S12 + self.c2x * alp12 def _InverseStart6(self, lam12, p): '''(INTERNAL) Return a starting point for Newton's method in C{salp1} and C{calp1} indicated by C{sig12=None}. If Newton's method doesn't need to be used, return also C{salp2}, C{calp2}, C{dnm} and C{sig12} non-C{None}. @return: 6-Tuple C{(sig12, salp1, calp1, salp2, calp2, dnm)} and updated C{p.setsigs} if C{sig12==None}. ''' sig12 = None # use Newton salp1 = calp1 = salp2 = calp2 = dnm = NAN # bet12 = bet2 - bet1 in [0, PI) sbet12, cbet12 = _sincos12(p.sbet1, p.cbet1, p.sbet2, p.cbet2) shortline = cbet12 >= 0 and sbet12 < _0_5 and (p.cbet2 * lam12) < _0_5 if shortline: # sin((bet1 + bet2)/2)^2 = (sbet1 + sbet2)^2 / ( # (cbet1 + cbet2)^2 + (sbet1 + sbet2)^2) t = (p.sbet1 + p.sbet2)**2 s = t / ((p.cbet1 + p.cbet2)**2 + t) dnm = sqrt(_1_0 + self.ep2 * s) somg12, comg12 = sincos2(lam12 / (self.f1 * dnm)) else: somg12, comg12 = p.slam12, p.clam12 # bet12a = bet2 + bet1 in (-PI, 0], note -sbet1 sbet12a, cbet12a = _sincos12(-p.sbet1, p.cbet1, p.sbet2, p.cbet2) s = somg12**2 / (abs(comg12) + _1_0) t = p.cbet2 * p.sbet1 * s salp1 = p.cbet2 * somg12 calp1 = (sbet12a - t) if comg12 < 0 else (sbet12 + t) ssig12 = hypot(salp1, calp1) csig12 = p.sbet1 * p.sbet2 + p.cbet1 * p.cbet2 * comg12 if shortline and ssig12 < self._eTOL2: # really short lines if comg12 < 0: s = _1_0 - comg12 salp2, calp2 = norm2(somg12 * p.cbet1, sbet12 - p.cbet1 * p.sbet2 * s) sig12 = atan2(ssig12, csig12) # return value elif (self._n_0_1 or # Skip astroid calc if too eccentric csig12 >= 0 or ssig12 >= (self._n_6PI * p.cbet1**2)): pass # nothing to do, 0th order spherical approximation OK else: # Scale lam12 and bet2 to x, y coordinate system where antipodal # point is at origin and singular point is at y = 0, x = -1 lam12x = atan2(-p.slam12, -p.clam12) # lam12 - PI if self.f < 0: # x = dlat, y = dlon # ssig1=sbet1, csig1=-cbet1, ssig2=sbet2, csig2=cbet2 p.setsigs(p.sbet1, -p.cbet1, p.sbet2, p.cbet2) # if lon12 = 180, this repeats a calculation made in Inverse _, m12b, m0, _, _ = self._Lengths5(atan2(sbet12a, cbet12a) + PI, Caps.REDUCEDLENGTH, p) x = m12b / (p.cbet1 * p.cbet2 * m0 * PI) - _1_0 sca = ((sbet12a / x) if x < -_0_01 else (-self.f * p.cbet1**2 * PI)) / p.cbet1 y = lam12x / sca else: # _f >= 0, in fact f == 0 does not get here sca = self._eF_reset_e2_f1_cH(p.sbet1, p.cbet1 * _2_0) x = lam12x / sca # dlon y = sbet12a / (sca * p.cbet1) # dlat if y > -_TOL1 and x > -_THR1: # strip near cut if self.f < 0: calp1 = max((_0_0 if x > -_TOL1 else -_1_0), x) salp1 = sqrt(_1_0 - calp1**2) else: salp1 = min(_1_0, -x) calp1 = -sqrt(_1_0 - salp1**2) else: # Estimate alp1, by solving the astroid problem. # # Could estimate alpha1 = theta + PI/2, directly, i.e., # calp1 = y/k; salp1 = -x/(1+k); for _f >= 0 # calp1 = x/(1+k); salp1 = -y/k; for _f < 0 (need to check) # # However, it's better to estimate omg12 from astroid and use # spherical formula to compute alp1. This reduces the mean # number of Newton iterations for astroid cases from 2.24 # (min 0, max 6) to 2.12 (min 0 max 5). The changes in the # number of iterations are as follows: # # change percent # 1 5 # 0 78 # -1 16 # -2 0.6 # -3 0.04 # -4 0.002 # # The histogram of iterations is (m = number of iterations # estimating alp1 directly, n = number of iterations # estimating via omg12, total number of trials = 148605): # # iter m n # 0 148 186 # 1 13046 13845 # 2 93315 102225 # 3 36189 32341 # 4 5396 7 # 5 455 1 # 6 56 0 # # omg12 is near PI, estimate work with omg12a = PI - omg12 k = _Astroid(x, y) k1 = k + _1_0 sca *= (y * k1 / k) if self.f < 0 else (x * k / k1) s, c = sincos2(-sca) # update spherical estimate of alp1 using omg12 instead of lam12 salp1 = p.cbet2 * s calp1 = sbet12a - p.cbet2 * p.sbet1 * s**2 / (_1_0 + c) # sanity check on starting guess. Backwards check allows NaN through. salp1, calp1 = norm2(salp1, calp1) if salp1 > 0 else (_1_0, _0_0) return sig12, salp1, calp1, salp2, calp2, dnm def _Lambda5(self, salp1, calp1, diffp, p): '''(INTERNAL) Helper. @return: 5-Tuple C{(lam12, sig12, salp2, calp2, dlam12)} and updated C{p.setsigs} and C{p.domg12}. ''' eF = self._eF f1 = self.f1 if p.sbet1 == 0 and calp1 == 0: # Break degeneracy of equatorial line calp1 = -_TINY # sin(alp1) * cos(bet1) = sin(alp0) salp0 = salp1 * p.cbet1 calp0 = hypot(calp1, salp1 * p.sbet1) # calp0 > 0 # tan(bet1) = tan(sig1) * cos(alp1) # tan(omg1) = sin(alp0) * tan(sig1) # = sin(bet1) * tan(alp1) somg1 = salp0 * p.sbet1 comg1 = calp1 * p.cbet1 ssig1, csig1 = norm2(p.sbet1, comg1) # Without normalization we have schi1 = somg1 cchi1 = f1 * p.dn1 * comg1 # Enforce symmetries in the case abs(bet2) = -bet1. # Need to be careful about this case, since this can # yield singularities in the Newton iteration. # sin(alp2) * cos(bet2) = sin(alp0) salp2 = (salp0 / p.cbet2) if p.cbet2 != p.cbet1 else salp1 # calp2 = sqrt(1 - sq(salp2)) # = sqrt(sq(calp0) - sq(sbet2)) / cbet2 # and subst for calp0 and rearrange to give (choose # positive sqrt to give alp2 in [0, PI/2]). calp2 = abs(calp1) if p.bet12 is None else ( sqrt(p.bet12 + (calp1 * p.cbet1)**2) / p.cbet2) # tan(bet2) = tan(sig2) * cos(alp2) # tan(omg2) = sin(alp0) * tan(sig2). somg2 = salp0 * p.sbet2 comg2 = calp2 * p.cbet2 ssig2, csig2 = norm2(p.sbet2, comg2) # without normalization we have schi2 = somg2 cchi2 = f1 * p.dn2 * comg2 # omg12 = omg2 - omg1, limit to [0, PI] somg12, comg12 = _sincos12(somg1, comg1, somg2, comg2, noneg=True) # chi12 = chi2 - chi1, limit to [0, PI] schi12, cchi12 = _sincos12(somg1, cchi1, somg2, cchi2, noneg=True) p.setsigs(ssig1, csig1, ssig2, csig2) # sig12 = sig2 - sig1, limit to [0, PI] sig12 = atan2(*_sincos12(ssig1, csig1, ssig2, csig2, noneg=True)) eta12 = self._eF_reset_e2_f1_cH(calp0, salp0) / PI_2 # then ... eta12 *= fsum_(eF.deltaH(*p.sncndn2), -eF.deltaH(*p.sncndn1), sig12) # eta = chi12 - lam12 lam12 = atan2(*_sincos12(p.slam12, p.clam12, schi12, cchi12)) - eta12 # domg12 = chi12 - omg12 - deta12 domg12 = atan2(*_sincos12( somg12, comg12, schi12, cchi12)) - eta12 if not diffp: dlam12 = NAN # dv > 0 in ._Newton9 elif calp2: _, dlam12, _, _, _ = self._Lengths5(sig12, Caps.REDUCEDLENGTH, p) dlam12 *= f1 / (calp2 * p.cbet2) else: dlam12 = -f1 * _2_0 * p.dn1 / p.sbet1 # p.set_(deta12=-eta12, lam12=lam12) return lam12, sig12, salp2, calp2, domg12, dlam12 def _Lengths5(self, sig12, outmask, p): '''(INTERNAL) Return M{m12b = (reduced length) / self.b} and calculate M{s12b = distance / self.b} and M{m0}, the coefficient of secular term in expression for reduced length. @return: 5-Tuple C{(s12b, m12b, m0, M12, M21)}. ''' s12b = m12b = m0 = M12 = M21 = NAN eF = self._eF # outmask &= Caps._OUTMASK if (outmask & Caps.DISTANCE): # Missing a factor of self.b s12b = eF.cE / PI_2 * fsum_(eF.deltaE(*p.sncndn2), -eF.deltaE(*p.sncndn1), sig12) if (outmask & Caps._REDUCEDLENGTH_GEODESICSCALE): m0x = -eF.k2 * eF.cD / PI_2 J12 = -m0x * fsum_(eF.deltaD(*p.sncndn2), -eF.deltaD(*p.sncndn1), sig12) c12 = p.csig1 * p.csig2 if (outmask & Caps.REDUCEDLENGTH): m0 = m0x # Missing a factor of self.b. Add parens around # (csig1 * ssig2) and (ssig1 * csig2) to ensure # accurate cancellation for coincident points. m12b = fsum_(p.dn2 * (p.csig1 * p.ssig2), -p.dn1 * (p.ssig1 * p.csig2), J12 * c12) if (outmask & Caps.GEODESICSCALE): M12 = M21 = p.ssig1 * p.ssig2 + c12 t = (p.cbet1 - p.cbet2) * self.ep2 * \ (p.cbet1 + p.cbet2) / (p.dn1 + p.dn2) M12 += (p.ssig2 * t + p.csig2 * J12) * p.ssig1 / p.dn1 M21 -= (p.ssig1 * t + p.csig1 * J12) * p.ssig2 / p.dn2 return s12b, m12b, m0, M12, M21 def Line(self, lat1, lon1, azi1, caps=Caps.ALL): '''Set up an L{GeodesicLineExact} to compute several points on a single geodesic. @arg lat1: Latitude of the first point (C{degrees}). @arg lon1: Longitude of the first point (C{degrees}). @arg azi1: Azimuth at the first point (compass C{degrees}). @kwarg caps: Bit-or'ed combination of L{Caps} values specifying the capabilities the L{GeodesicLineExact} instance should possess, i.e., which quantities can be returnedby calls to L{GeodesicLineExact.Position} and L{GeodesicLineExact.ArcPosition}. @return: A L{GeodesicLineExact} instance. @note: If the point is at a pole, the azimuth is defined by keeping B{C{lon1}} fixed, writing C{B{lat1} = ±(90 − ε)}, and taking the limit C{ε → 0+}. @see: C++ U{GeodesicExact.Line <https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1GeodesicExact.html>} and Python U{Geodesic.Line<https://GeographicLib.SourceForge.io/html/python/code.html>}. ''' return _GeodesicLineExact(self, lat1, lon1, azi1, caps, self._debug) def _Line(self, lat1, lon1, azi1, caps): # for .azimuthal._GnomonicBase.reverse '''(INTERNAL) Get a temporary L{GeodesicLineExact} instance. ''' glX = _GeodesicLineExact(self, lat1, lon1, azi1, caps, self._debug) _update_glXs(glX) # remove glX from .gxline._glXs list return glX @Property_RO def n(self): '''Get the ellipsoid's I{3rd flattening} (C{float}), M{f / (2 - f) == (a - b) / (a + b)}. ''' return self.ellipsoid.n @Property_RO def _n_0_1(self): '''(INTERNAL) Cached once. ''' return abs(self.n) > _0_1 @Property_RO def _n_6PI(self): '''(INTERNAL) Cached once. ''' return abs(self.n) * _6_0 * PI def _Newton10(self, salp1, calp1, outmask, p): '''(INTERNAL) Split off from C{_GDictInverse} to reduce complexity/length. @return: 10-Tuple C{(sig12, salp1, calp1, salp2, calp2, domg12, s12x, m12x, M12, M21)}. ''' # This is a straightforward solution of f(alp1) = lambda12(alp1) - # lam12 = 0 with one wrinkle. f(alp) has exactly one root in the # interval (0, PI) and its derivative is positive at the root. # Thus f(alp) is positive for alp > alp1 and negative for alp < alp1. # During the course of the iteration, a range (alp1a, alp1b) is # maintained which brackets the root and with each evaluation of # f(alp) the range is shrunk, if possible. Newton's method is # restarted whenever the derivative of f is negative (because the # new value of alp1 is then further from the solution) or if the # new estimate of alp1 lies outside (0,PI); in this case, the new # starting guess is taken to be (alp1a + alp1b) / 2. salp1a = salp1b = _TINY calp1a, calp1b = _1_0, -_1_0 tripb, TOLv = False, _TOL0 for it in range(_MAXIT2): # 1/4 meridian = 10e6 meter and random input, # estimated max error in nm (nano meter, by # checking Inverse problem by Direct). # # max iterations # log2(b/a) error mean sd # -7 387 5.33 3.68 # -6 345 5.19 3.43 # -5 269 5.00 3.05 # -4 210 4.76 2.44 # -3 115 4.55 1.87 # -2 69 4.35 1.38 # -1 36 4.05 1.03 # 0 15 0.01 0.13 # 1 25 5.10 1.53 # 2 96 5.61 2.09 # 3 318 6.02 2.74 # 4 985 6.24 3.22 # 5 2352 6.32 3.44 # 6 6008 6.30 3.45 # 7 19024 6.19 3.30 v, sig12, salp2, calp2, \ domg12, dv = self._Lambda5(salp1, calp1, it < _MAXIT1, p) # reversed test to allow escape with NaNs if tripb or abs(v) < TOLv: break # update bracketing values if v > 0 and (it > _MAXIT1 or (calp1 / salp1) > (calp1b / salp1b)): salp1b, calp1b = salp1, calp1 elif v < 0 and (it > _MAXIT1 or (calp1 / salp1) < (calp1a / salp1a)): salp1a, calp1a = salp1, calp1 if it < _MAXIT1 and dv > 0: dalp1 = -v / dv if abs(dalp1) < PI: s, c = sincos2(dalp1) # nalp1 = alp1 + dalp1 s, c = _sincos12(-s, c, salp1, calp1) if s > 0: salp1, calp1 = norm2(s, c) # in some regimes we don't get quadratic convergence # because slope -> 0. So use convergence conditions # based on epsilon instead of sqrt(epsilon) TOLv = _TOL0 if abs(v) > _TOL016 else _TOL08 continue # Either dv was not positive or updated value was outside # legal range. Use the midpoint of the bracket as the next # estimate. This mechanism is not needed for the WGS84 # ellipsoid, but it does catch problems with more eccentric # ellipsoids. Its efficacy is such for the WGS84 test set # with the starting guess set to alp1 = 90deg: the WGS84 # test set: mean = 5.21, sd = 3.93, max = 24 WGS84 and # random input: mean = 4.74, sd = 0.99 salp1, calp1 = norm2((salp1a + salp1b) * _0_5, (calp1a + calp1b) * _0_5) tripb = fsum_(calp1a, -calp1, abs(salp1a - salp1)) < _TOLb or \ fsum_(calp1b, -calp1, abs(salp1b - salp1)) < _TOLb TOLv = _TOL0 else: raise GeodesicError(_no_(_convergence_), _MAXIT2, txt=repr(self)) # self.toRepr() s12x, m12x, _, M12, M21 = self._Lengths5(sig12, outmask, p) p.set_(iter=it, trip=tripb) return (sig12, salp1, calp1, salp2, calp2, domg12, s12x, m12x, M12, M21) def _sincos2f1(self, lat): '''(INTERNAL) Helper. ''' sbet, cbet = sincos2d(lat) # ensure cbet1 = +epsilon at poles; doing the fix on beta means # that sig12 will be <= 2*tiny for two points at the same pole sbet, cbet = norm2(sbet * self.f1, cbet) return sbet, max(_TINY, cbet) def toStr(self, prec=6, sep=_COMMASPACE_, **unused): # PYCHOK signature '''Return this C{GeodesicExact} as string. @kwarg prec: The C{float} precision, number of decimal digits (0..9). Trailing zero decimals are stripped for B{C{prec}} values of 1 and above, but kept for negative B{C{prec}} values. @kwarg sep: Optional separator to join (C{str}). @return: Tuple items (C{str}). ''' d = dict(ellipsoid=self.ellipsoid, C4Order=self.C4Order) return sep.join(pairs(d, prec=prec)) class GeodesicLineExact(_GeodesicLineExact): '''A pure Python version of I{Karney}'s C++ class U{GeodesicLineExact <https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1GeodesicLineExact.html>}, modeled after I{Karney}'s Python class U{GeodesicLine<https://GeographicLib.SourceForge.io/ html/python/code.html#module-geographiclib.geodesicline>}. ''' def __init__(self, geodesic, lat1, lon1, azi1, caps=Caps.STANDARD, name=NN): '''New L{GeodesicLineExact} instance, allowing points to be found along a geodesic starting at C{(B{lat1}, B{lon1})} with azimuth B{C{azi1}}. @arg geodesic: The geodesic to use (L{GeodesicExact}). @arg lat1: Latitude of the first point (C{degrees}). @arg lon1: Longitude of the first point (C{degrees}). @arg azi1: Azimuth at the first points (compass C{degrees}). @kwarg caps: Bit-or'ed combination of L{Caps} values specifying the capabilities the L{GeodesicLineExact} instance should possess, i.e., which quantities can be returned by calls to L{GeodesicLineExact.Position} and L{GeodesicLineExact.ArcPosition}. @kwarg name: Optional name (C{str}). @raise TypeError: Invalid B{C{geodesic}}. ''' _xinstanceof(GeodesicExact, geodesic=geodesic) _GeodesicLineExact.__init__(self, geodesic, lat1, lon1, azi1, caps, 0, name=name) def _Astroid(x, y): '''(INTERNAL) Solve M{k^4 + 2 * k^3 - (x^2 + y^2 - 1 ) * k^2 - (2 * k + 1) * y^2 = 0} for positive root k. ''' p = x**2 q = y**2 r = q + p - _1_0 if q or r > 0: r = r / _6_0 # /= chokes PyChecker # avoid possible division by zero when r = 0 # by multiplying s and t by r^3 and r, resp. S = p * q / _4_0 # S = r^3 * s r3 = r**3 T3 = r3 + S # discriminant of the quadratic equation for T3 is # zero on the evolute curve p^(1/3)+q^(1/3) = 1 d = S * (S + _2_0 * r3) if d < 0: # T is complex, but u is defined for a real result a = atan2(sqrt(-d), -T3) # There are three possible cube roots, choose the one which # avoids cancellation. Note d < 0 implies that r < 0. u = r * (_1_0 + _2_0 * cos(a / _3_0)) else: # pick the sign on the sqrt to maximize abs(T3) to # minimize loss of precision due to cancellation. if d: T3 += copysign(sqrt(d), T3) # T3 = (r * t)^3 # cbrt always returns the real root, cbrt(-8) = -2 u = cbrt(T3) # T = r * t if u: # T can be zero; but then r2 / T -> 0 u += r**2 / u u += r v = sqrt(u**2 + q) # avoid loss of accuracy when u < 0 uv = (q / (v - u)) if u < 0 else (u + v) w = (uv - q) / (_2_0 * v) # positive? # rearrange expression for k to avoid loss of accuracy due to # subtraction, division by 0 impossible because uv > 0, w >= 0 k = uv / (sqrt(w**2 + uv) + w) # guaranteed positive else: # q == 0 && r <= 0 # y = 0 with |x| <= 1. Handle this case directly, for # y small, positive root is k = abs(y) / sqrt(1 - x^2) k = _0_0 return k __all__ += _ALL_DOCS(GeodesicExact, GeodesicLineExact) # **) 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.