Coverage for pygeodesy/rhumbx.py : 98%

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# -*- coding: utf-8 -*-
<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Rhumb.html>} and U{RhumbLine <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1RhumbLine.html>}.
Class L{RhumbLine} has been enhanced with methods C{intersection2} and C{nearestOn4} to find the intersection of two rhumb lines, respectively the nearest point on a rumb line.
For more details, see the C++ U{GeographicLib<https://GeographicLib.SourceForge.io/C++/doc/index.html>} documentation, especially the U{Class List<https://GeographicLib.SourceForge.io/C++/doc/annotated.html>}, the background information on U{Rhumb lines<https://GeographicLib.SourceForge.io/C++/doc/rhumb.html>}, the utily U{RhumbSolve<https://GeographicLib.SourceForge.io/C++/doc/RhumbSolve.1.html>} and U{Online rhumb line calculations<https://GeographicLib.SourceForge.io/cgi-bin/RhumbSolve>}.
Copyright (C) U{Charles Karney<mailto:Charles@Karney.com>} (2014-2022) and licensed under the MIT/X11 License. For more information, see the U{GeographicLib<https://GeographicLib.SourceForge.io>} documentation. ''' # make sure int/int division yields float quotient
_0_0, _0_5, _1_0, _2_0, _4_0, _90_0, _180_0, _720_0 # from pygeodesy.datums import _spherical_datum # in Rhumb.ellipsoid.setter _xdatum, _xkwds # from pygeodesy.etm import ExactTransverseMercator # in ._RhumbLine.xTM # from pygeodesy.fsums import fsum1_ # from .fmath _intersection_, _lat1_, _lat2_, _lon1_, _lon2_, \ _no_, _s12_, _S12_, _under_name _diff182, Direct9Tuple, _EWGS84, _fix90, GDict, \ _GTuple, Inverse10Tuple, _norm180 _AlpCoeffs, _BetCoeffs # PYCHOK used! _update_all Meter as _M, Meter2 as _M2
'''(INTERNAL) Latitude B{C{lat}}. '''
'''(INTERNAL) Longitude B{C{lon}}. '''
'''(INTERNAL) Zap cached/memoized C{Property[_RO]}s of any L{RhumbLine} instances tied to the given L{Rhumb} instance B{C{r}}. '''
'''Class to solve of the I{direct} and I{inverse rhumb} problems, accurately.
@see: The U{Detailed Description<https://GeographicLib.SourceForge.io/C++/doc/ classGeographicLib_1_1Rhumb.html>} of I{Karney}'s C++ C{Rhumb Class}. '''
'''New L{Rhumb}.
@kwarg a_earth: This rhumb's earth (L{Ellipsoid}, L{Ellipsoid2}, L{a_f2Tuple}, L{Datum}, 2-tuple C{(a, f)}) or the (equatorial) radius (C{scalar}). @kwarg f: The ellipsoid's flattening (C{scalar}), iff B{C{a_earth}} is a C{scalar}, ignored otherwise. @kwarg exact: If C{True}, use an addition theorem for elliptic integrals to compute I{Divided differences}, otherwise use the Krüger series expansion (C{bool}), see also property C{exact}. @kwarg name: Optional name (C{str}). @kwarg RA_TMorder: Optional keyword arguments B{C{RAorder}} and B{C{TMorder}} to set the respective C{order}, see properties C{RAorder} and C{TMorder} and method C{orders}.
@raise RhumbError: Invalid B{C{a_earth}}, B{C{f}} or B{C{RA_TMorder}}. ''' self.orders(**RA_TMorder)
'''Solve the I{direct rhumb} problem, optionally with the area.
@arg lat1: Latitude of the first point (C{degrees90}). @arg lon1: Longitude of the first point (C{degrees180}). @arg azi12: Azimuth of the rhumb line (compass C{degrees}). @arg s12: Distance along the rhumb line from the given to the destination point (C{meter}), can be negative.
@return: L{GDict} with 2 up to 8 items C{lat2, lon2, a12, S12, lat1, lon1, azi12, s12} with the destination point's latitude C{lat2} and longitude C{lon2} in C{degrees}, the rhumb angle C{a12} in C{degrees} and area C{S12} under the rhumb line in C{meter} I{squared}.
@note: If B{C{s12}} is large enough that the rhumb line crosses a pole, the longitude of the second point is indeterminate and C{NAN} is returned for C{lon2} and area C{S12}.
If the given point is a pole, the cosine of its latitude is taken to be C{epsilon}**-2 (where C{epsilon} is 2.0**-52. This position is extremely close to the actual pole and allows the calculation to be carried out in finite terms. ''' name=self.name)
'''DEPRECATED, use method L{Rhumb.Direct8}.
@return: A I{DEPRECATED} L{Rhumb7Tuple}. '''
'''Like method L{Rhumb.Direct} but returning a L{Rhumb8Tuple} with area C{S12}. '''
'''Define a L{RhumbLine} in terms of the I{direct} rhumb problem.
@arg lat1: Latitude of the first point (C{degrees90}). @arg lon1: Longitude of the first point (C{degrees180}). @arg azi12: Azimuth of the rhumb line (compass C{degrees}). @kwarg caps: Optional C{caps}, see L{RhumbLine} C{B{caps}}.
@return: A L{RhumbLine} instance and invoke its method L{RhumbLine.Position} to compute each point.
@note: Updates to this rhumb are reflected in the returned rhumb line. ''' name=name or self.name, **caps)
_Dsin(phix, phiy) * _DeatanhE(sin(phix), sin(phiy), E)
self._DIsometrict(phix, phiy, tphix, tphiy, t) else:
* _DfEt( tbetx, tbety, self._eF) * _Datan(tbetx, tbety)) / E.L
self._DRectifyingt( tphix, tphiy, t) else: _Dgdinv(E.es_taupf(tphix), E.es_taupf(tphiy))
'''(INTERNAL) Get the ellipsoid's elliptic function. ''' # .k2 = 0.006739496742276434
'''Get this rhumb's ellipsoid (L{Ellipsoid}). '''
'''Set this rhumb's ellipsoid (L{Ellipsoid}, L{Ellipsoid2}, L{Datum}, L{a_f2Tuple}, 2-tuple C{(a, f)}) or the (equatorial) radius (C{scalar}). '''
'''Get the C{ellipsoid}'s equatorial radius, semi-axis (C{meter}). ''' return self.ellipsoid.a
'''Get the I{exact} option (C{bool}). '''
'''Set the I{exact} option (C{bool}). If C{True}, use I{exact} rhumb calculations, if C{False} results are less precise for more oblate or more prolate ellipsoids with M{abs(flattening) > 0.01} (C{bool}).
@see: Option U{B{-s}<https://GeographicLib.SourceForge.io/C++/doc/RhumbSolve.1.html>} and U{ACCURACY<https://GeographicLib.SourceForge.io/C++/doc/RhumbSolve.1.html#ACCURACY>}. '''
'''Get the C{ellipsoid}'s flattening (C{float}). ''' return self.ellipsoid.f
'''Solve the I{inverse rhumb} problem.
@arg lat1: Latitude of the first point (C{degrees90}). @arg lon1: Longitude of the first point (C{degrees180}). @arg lat2: Latitude of the second point (C{degrees90}). @arg lon2: Longitude of the second point (C{degrees180}).
@return: L{GDict} with 5 to 8 items C{azi12, s12, a12, S12, lat1, lon1, lat2, lon2}, the rhumb line's azimuth C{azi12} in compass C{degrees} between C{-180} and C{+180}, the distance C{s12} and rhumb angle C{a12} between both points in C{meter} respectively C{degrees} and the area C{S12} under the rhumb line in C{meter} I{squared}.
@note: The shortest rhumb line is found. If the end points are on opposite meridians, there are two shortest rhumb lines and the East-going one is chosen.
If either point is a pole, the cosine of its latitude is taken to be C{epsilon}**-2 (where C{epsilon} is 2.0**-52). This position is extremely close to the actual pole and allows the calculation to be carried out in finite terms. ''' if ((outmask | self._debug) & Caps._DEBUG_INVERSE): # PYCHOK no cover r.set_(a=E.a, f=E.f, f1=E.f1, L=E.L, b=E.b, e=E.e, e2=E.e2, k2=self._eF.k2, lon12=lon12, psi1=psi1, exact=self.exact, psi12=psi12, psi2=psi2)
# 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.Inverse(lat1, lon1, lat2, lon2) # return Distance3Tuple(r.s12, r.azi12, r.azi12)
'''DEPRECATED, use method L{Rhumb.Inverse8}.
@return: A I{DEPRECATED} L{Rhumb7Tuple}. '''
'''Like method L{Rhumb.Inverse} but returning a L{Rhumb8Tuple} with area C{S12}. '''
'''Define a L{RhumbLine} in terms of the I{inverse} rhumb problem.
@arg lat1: Latitude of the first point (C{degrees90}). @arg lon1: Longitude of the first point (C{degrees180}). @arg lat2: Latitude of the second point (C{degrees90}). @arg lon2: Longitude of the second point (C{degrees180}). @kwarg caps: Optional C{caps}, see L{RhumbLine} C{B{caps}}.
@return: A L{RhumbLine} instance and invoke its method L{RhumbLine.Position} to compute each point.
@note: Updates to this rhumb are reflected in the returned rhumb line. ''' name=name or self.name, **caps)
'''Get and set the I{RAorder} and/or I{TMorder}.
@kwarg RAorder: I{Rhumb Area} order (C{int}, 4, 5, 6, 7 or 8). @kwarg TMorder: I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8).
@return: L{RhumbOrder2Tuple}C{(RAorder, TMorder)} with the previous C{RAorder} and C{TMorder} setting. '''
# for WGS84: (0, -0.0005583633519275459, -3.743803759172812e-07, -4.633682270824446e-10, # RAorder 6: -7.709197397676237e-13, -1.5323287106694307e-15, -3.462875359099873e-18)
'''Get the I{Rhumb Area} order (C{int}, 4, 5, 6, 7 or 8). '''
'''Set the I{Rhumb Area} order (C{int}, 4, 5, 6, 7 or 8). '''
'''(INTERNAL) Compute the area C{S12}. ''' self.ellipsoid.area) * lon12 / _720_0
'''Get the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8). '''
'''Set the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8).
@note: Setting C{TMorder} turns property C{exact} off. '''
'''Return this C{Rhumb} 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: Separator to join (C{str}).
@return: Tuple items (C{str}). ''' exact=self.exact, TMorder=self.TMorder)
'''Raised for an L{Rhumb} or L{RhumbLine} issue. '''
'''(INTERNAL) Class L{RhumbLine} ''' # _lat1 = _0_0 # _lon1 = _0_0
'''New C{RhumbLine}. '''
try: # PYCHOK no cover _rls.remove(self) except (TypeError, ValueError): pass # _update_all(self) # throws TypeError during Python 2 cleanup
'''Get this rhumb line's I{azimuth} (compass C{degrees}). '''
'''Set this rhumb line's I{azimuth} (compass C{degrees}). '''
'''Return the distance and (initial) bearing of a point to this rhumb line's start point.
@arg lat: Latitude of the point (C{degrees}). @arg lon: Longitude of the points (C{degrees}).
@return: A L{Distance2Tuple}C{(distance, initial)} with the C{distance} in C{meter} and C{initial} bearing in C{degrees}.
@see: Methods L{RhumbLine.intersection2} and L{RhumbLine.nearestOn4}. ''' # outmask=Caps.AZIMUTH_DISTANCE)
'''Get this rhumb line's ellipsoid (L{Ellipsoid}). '''
'''Get this rhumb line's I{exact} option (C{bool}). '''
'''Iteratively find the intersection of this and an other rhumb line.
@arg other: The other rhumb line ({RhumbLine}). @kwarg tol: Tolerance for longitudinal convergence (C{degrees}). @kwarg eps: Tolerance for L{intersection3d3} (C{EPS}).
@return: A L{LatLon2Tuple}{(lat, lon)} with the C{lat}- and C{lon}gitude of the intersection point.
@raise IntersectionError: No convergence for this B{C{tol}} or no intersection for an other reason.
@see: Methods L{RhumbLine.distance2} and L{RhumbLine.nearestOn4} and function L{pygeodesy.intersection3d3}.
@note: Each iteration involves a round trip to this rhumb line's L{ExactTransverseMercator} or L{KTransverseMercator} projection and invoking function L{intersection3d3} in that domain. ''' raise ValueError(_coincident_) # make globals and invariants locals # use halfway point as initial estimate favg(self.lon1, other.lon1)) _o_3d(p), o_az, useZ=False, **eps)[0] name=self.intersection2.__name__) except Exception as x: raise IntersectionError(_no_(_intersection_), txt=str(x)) t = unstr(self.intersection2.__name__, tol=tol, **eps) raise IntersectionError(Fmt.no_convergence(d), txt=t)
'''Get this rhumb line's latitude (C{degrees90}). '''
'''Get this rhumb line's longitude (C{degrees180}). '''
'''Get this rhumb line's lat- and longitude (L{LatLon2Tuple}C{(lat, lon)}). ''' return LatLon2Tuple(self.lat1, self.lon1)
'''(INTERNAL) Get the I{rectifying auxiliary} latitude C{mu} (C{degrees}). '''
'''Iteratively locate the point on this rhumb line nearest to the given point.
@arg lat: Latitude of the point (C{degrees}). @arg lon: Longitude of the point (C{degrees}). @kwarg tol: Longitudinal convergence tolerance (C{degrees}). @kwarg eps: Tolerance for L{intersection3d3} (C{EPS}).
@return: A L{NearestOn4Tuple}C{(lat, lon, distance, normal)} with the C{lat}- and C{lon}gitude of the nearest point on and the C{distance} in C{meter} to this rhumb line and with the azimuth of the C{normal}, perpendicular to this rhumb line.
@raise IntersectionError: No convergence for this B{C{eps}} or no intersection for an other reason.
@see: Methods L{RhumbLine.distance2} and L{RhumbLine.intersection2} and function L{intersection3d3}. ''' iteration=p.iteration)
'''(INTERNAL) Get the I{isometric auxiliary} latitude C{psi} (C{degrees}). '''
'''Get this rhumb line's I{Rhumb Area} order (C{int}, 4, 5, 6, 7 or 8). ''' return self.rhumb.RAorder
def _r1rad(self): # PYCHOK no cover '''(INTERNAL) Get this rhumb line's parallel I{circle radius} (C{meter}). ''' return radians(self.ellipsoid.circle4(self.lat1).radius)
'''Get this rhumb line's rhumb (L{Rhumb}). '''
'''Compute a position at a distance on this rhumb line.
@arg s12: The distance along this rhumb between its point and the other point (C{meters}), can be negative. @kwarg outmask: Bit-or'ed combination of L{Caps} values specifying the quantities to be returned.
@return: L{GDict} with 4 to 8 items C{azi12, a12, s12, S12, lat2, lon2, lat1, lon1} with latitude C{lat2} and longitude C{lon2} of the other point in C{degrees}, the rhumb angle C{a12} between both points in C{degrees} and the area C{S12} under the rhumb line in C{meter} I{squared}.
@note: If B{C{s12}} is large enough that the rhumb line crosses a pole, the longitude of the second point is indeterminate and C{NAN} is returned for C{lon2} and area C{S12}.
If the first point is a pole, the cosine of its latitude is taken to be C{epsilon}**-2 (where C{epsilon} is 2**-52). This position is extremely close to the actual pole and allows the calculation to be carried out in finite terms. ''' if fabs(mu2) > 90: # PYCHOK no cover mu2 = _norm180(mu2) # reduce to [-180, 180) if fabs(mu2) > 90: # point on anti-meridian mu2 = _norm180(_180_0 - mu2) lat2x = E.auxRectifying(mu2, inverse=True) lon2x = NAN if (outmask & Caps.AREA): r.set_(S12=NAN) else: self._mu1) * mu12 else: # PYCHOK no cover lat2x = self.lat1 lon2x = self._salp * s12 / self._r1rad lon2x += self.lon1 else: if ((outmask | self._debug) & Caps._DEBUG_DIRECT_LINE): # PYCHOK no cover r.set_(a=E.a, f=E.f, f1=E.f1, L=E.L, exact=R.exact, b=E.b, e=E.e, e2=E.e2, k2=R._eF.k2, calp=self._calp, mu1 =self._mu1, mu12=mu12, salp=self._salp, psi1=self._psi1, mu2=mu2)
'''Return this C{RhumbLine} 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: Separator to join (C{str}).
@return: C{RhumbLine} (C{str}). ''' azi12=self.azi12, exact=self.exact, TMorder=self.TMorder, xTM=self.xTM)
'''Get this rhumb line's I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8). '''
'''Get this rhumb line's I{Transverse Mercator} projection (L{ExactTransverseMercator} if I{exact} and I{ellipsoidal}, otherwise L{KTransverseMercator}). ''' # ExactTransverseMercator doesn't handle spherical earth models KTransverseMercator(E, TMorder=self.TMorder)
'''(INTERNAL) C{xTM.forward} this C{latlon1} to C{V3d} with B{C{latlon0}} as current intersection estimate and central meridian. '''
'''Compute one or several points on a single rhumb line.
Class L{RhumbLine} facilitates the determination of points on a single rhumb line. The starting point (C{lat1}, C{lon1}) and the azimuth C{azi12} are specified once.
Method L{RhumbLine.Position} returns the location of an other point and optionally the distance C{s12} along the corresponding area C{S12} under the rhumb line.
Method L{RhumbLine.intersection2} finds the intersection between two rhumb lines.
Method L{RhumbLine.nearestOn4} computes the nearest point on and its distance to a rhumb line. ''' '''New L{RhumbLine}.
@arg rhumb: The rhumb reference (L{Rhumb}). @kwarg lat1: Latitude of the start point (C{degrees90}). @kwarg lon1: Longitude of the start point (C{degrees180}). @kwarg azi12: Azimuth of this rhumb line (compass C{degrees}). @kwarg caps: Bit-or'ed combination of L{Caps} values specifying the capabilities. Include C{Caps.LINE_OFF} if updates to B{C{rhumb}} should I{not} be reflected in this rhumb line. @kwarg name: Optional name (C{str}). ''' rhumb = rhumb.copy(deep=False, name=_under_name(rhumb.name))
'''2-Tuple C{(RAorder, TMorder)} with a I{Rhumb Area} and I{Transverse Mercator} order, both C{int}. '''
'''8-Tuple C{(lat1, lon1, lat2, lon2, azi12, s12, S12, a12)} with lat- C{lat1}, C{lat2} and longitudes C{lon1}, C{lon2} of both points, the azimuth of the rhumb line C{azi12}, the distance C{s12}, the area C{S12} under the rhumb line and the angular distance C{a12} between both points. '''
'''Convert this L{Rhumb8Tuple} result to a 9-tuple, like I{Karney}'s method C{geographiclib.geodesic.Geodesic._GenDirect}.
@kwarg dflt: Default value for missing items (C{any}). @kwarg a12_azi1_azi2_m12_M12_M21: Optional keyword arguments to specify or override L{Inverse10Tuple} items.
@return: L{Direct9Tuple}C{(a12, lat2, lon2, azi2, s12, m12, M12, M21, S12)} ''' azi2=self.azi12, M21=_1_0) # PYCHOK attr d.update(a12_azi1_azi2_m12_M12_M21)
'''Convert this L{Rhumb8Tuple} to a 10-tuple, like I{Karney}'s method C{geographiclib.geodesic.Geodesic._GenInverse}.
@kwarg dflt: Default value for missing items (C{any}). @kwarg a12_m12_M12_M21_salp1_calp1_salp2_calp2: Optional keyword arguments to specify or override L{Inverse10Tuple} items.
@return: L{Inverse10Tuple}C{(a12, s12, salp1, calp1, salp2, calp2, m12, M12, M21, S12)}. ''' salp2=s, calp2=c, M21=_1_0) d.update(a12_m12_M12_M21_salp1_calp1_salp2_calp2)
'''(INTERNAL) Convert this C{Rhumb8Tuple} to an B{C{nTuple}}. '''
'''DEPRECATED, do not use! '''
# Use I{Divided Differences} to determine (mu2 - mu1) / (psi2 - psi1) accurately. # Definition: _Df(x,y,d) = (f(x) - f(y)) / (x - y), @see W. M. Kahan & R. J. # Fateman, "Symbolic computation of Divided Differences", SIGSAM Bull. 33(3), # 7-28 (1999). U{ACM<https://DL.ACM.org/doi/pdf/10.1145/334714.334716>, @see # U{UCB<https://www.CS.Berkeley.edu/~fateman/papers/divdiff.pdf>}, Dec 8, 1999.
else:
else:
# Deatanhe(x, y) = eatanhe((x - y) / (1 - e^2 * x * y)) / (x - y) # assert not isnear0(e)
# eF = Elliptic(-E.e12) # -E.e2 / (1 - E.e2) # See U{DLMF<https://DLMF.NIST.gov/19.11>}: 19.11.2 and 19.11.4 # letting theta -> x, phi -> -y, psi -> z # (E(x) - E(y)) / d = E(z)/d - k2 * sin(x) * sin(y) * sin(z)/d # tan(z/2) = (sin(x)*Delta(y) - sin(y)*Delta(x)) / (cos(x) + cos(y)) # = d * Dsin(x,y) * (sin(x) + sin(y))/(cos(x) + cos(y)) / # (sin(x)*Delta(y) + sin(y)*Delta(x)) # = t = d * Dt # sin(z) = 2*t/(1+t^2); cos(z) = (1-t^2)/(1+t^2) # Alt (this only works for |z| <= pi/2 -- however, this conditions # holds if x*y > 0): # sin(z) = d * Dsin(x,y) * (sin(x) + sin(y)) / # (sin(x)*cos(y)*Delta(y) + sin(y)*cos(x)*Delta(x)) # cos(z) = sqrt((1-sin(z))*(1+sin(z))) eF.fDelta(sx, cx) * sy)
# Changed atanh(t / (x + y)) to asinh(t / (2 * sqrt(x*y))) to # avoid taking atanh(1) when x is large and y is 1. This also # fixes bogus results being returned for the area when an endpoint # is at a pole. N.B. this routine is invoked with positive x # and y, so the sqrt is always taken of a positive quantity.
def _Dtan(x, y): # PYCHOK no cover return _Dtant(x - y, tan(x), tan(y))
# get inverse auxiliary lats in radians and tangents
# N.B. C[] has n+1 elements of which # C[0] is ignored and n >= 0 # Use Clenshaw summation to evaluate # m = (g(x) + g(y)) / 2 -- mean value # s = (g(x) - g(y)) / (x - y) -- average slope # where # g(x) = sum(C[j] * SC(2 * j * x), j = 1..n) # SC = sinp ? sin : cos # CS = sinp ? cos : sin # ... # 2x2 matrices in row-major order # b1 = a * b2 - b1 + C[j] * I _fsum1_(a0 * m1, a1 * m3, n1, floats=True), _fsum1_(a2 * m0, a3 * m2, n2, floats=True), _fsum1_(a2 * m1, a3 * m3, n3, Cj, floats=True)) # Here are the full expressions for m and s # f01, f02, f11, f12 = (0, 0, cd * sp, 2 * sd * cp) if sinp else \ # (1, 0, cd * cp, -2 * sd * sp) # m = -b2[1] * f02 + (C[0] - b2[0]) * f01 + b1[0] * f11 + b1[1] * f12 # s = -b2[2] * f01 + (C[0] - b2[3]) * f02 + b1[2] * f11 + b1[3] * f12 _fsum1_(cd * cp, _neg(sd * sp), _neg(b2[2]), floats=True)
4: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 4 691, 7860, -20160, 18900, 0, 56700, # R[0]/n^0, polynomial(n), order 4 1772, -5340, 6930, -4725, 14175, # R[1]/n^1, polynomial(n), order 3 -1747, 1590, -630, 4725, # PYCHOK R[2]/n^2, polynomial(n), order 2 104, -31, 315, # R[3]/n^3, polynomial(n), order 1 -41, 420), # PYCHOK R[4]/n^4, polynomial(n), order 0, count = 20 5: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 5 -79036, 22803, 259380, -665280, 623700, 0, 1871100, # PYCHOK R[0]/n^0, polynomial(n), order 5 41662, 58476, -176220, 228690, -155925, 467775, # PYCHOK R[1]/n^1, polynomial(n), order 4 18118, -57651, 52470, -20790, 155925, # PYCHOK R[2]/n^2, polynomial(n), order 3 -23011, 17160, -5115, 51975, # PYCHOK R[3]/n^3, polynomial(n), order 2 5480, -1353, 13860, # PYCHOK R[4]/n^4, polynomial(n), order 1 -668, 5775), # PYCHOK R[5]/n^5, polynomial(n), order 0, count = 27 6: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 6 128346268, -107884140, 31126095, 354053700, -908107200, 851350500, 0, 2554051500, # R[0]/n^0, polynomial(n), order 6 -114456994, 56868630, 79819740, -240540300, 312161850, -212837625, 638512875, # PYCHOK R[1]/n^1, polynomial(n), order 5 51304574, 24731070, -78693615, 71621550, -28378350, 212837625, # R[2]/n^2, polynomial(n), order 4 1554472, -6282003, 4684680, -1396395, 14189175, # R[3]/n^3, polynomial(n), order 3 -4913956, 3205800, -791505, 8108100, # PYCHOK R[4]/n^4, polynomial(n), order 2 1092376, -234468, 2027025, # R[5]/n^5, polynomial(n), order 1 -313076, 2027025), # PYCHOK R[6]/n^6, polynomial(n), order 0, count = 35 7: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 7 -317195588, 385038804, -323652420, 93378285, 1062161100, -2724321600, 2554051500, 0, 7662154500, # PYCHOK R[0]/n^0, polynomial(n), order 7 258618446, -343370982, 170605890, 239459220, -721620900, 936485550, -638512875, 1915538625, # PYCHOK R[1]/n^1, polynomial(n), order 6 -248174686, 153913722, 74193210, -236080845, 214864650, -85135050, 638512875, # PYCHOK R[2]/n^2, polynomial(n), order 5 114450437, 23317080, -94230045, 70270200, -20945925, 212837625, # PYCHOK R[3]/n^3, polynomial(n), order 4 15445736, -103193076, 67321800, -16621605, 170270100, # PYCHOK R[4]/n^4, polynomial(n), order 3 -27766753, 16385640, -3517020, 30405375, # PYCHOK R[4]/n^4, polynomial(n), order 3 4892722, -939228, 6081075, # PYCHOK R[4]/n^4, polynomial(n), order 3 -3189007, 14189175), # PYCHOK R[7]/n^7, polynomial(n), order 0, count = 44 8: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 8 71374704821, -161769749880, 196369790040, -165062734200, 47622925350, 541702161000, -1389404016000, 1302566265000, 0, 3907698795000, # R[0]/n^0, polynomial(n), order 8 -13691187484, 65947703730, -87559600410, 43504501950, 61062101100, -184013329500, 238803815250, -162820783125, 488462349375, # PYCHOK R[1]/n^1, polynomial(n), order 7 30802104839, -63284544930, 39247999110, 18919268550, -60200615475, 54790485750, -21709437750, 162820783125, # R[2]/n^2, polynomial(n), order 6 -8934064508, 5836972287, 1189171080, -4805732295, 3583780200, -1068242175, 10854718875, # PYCHOK R[3]/n^3, polynomial(n), order 5 50072287748, 3938662680, -26314234380, 17167059000, -4238509275, 43418875500, # R[4]/n^4, polynomial(n), order 4 359094172, -9912730821, 5849673480, -1255576140, 10854718875, # R[5]/n^5, polynomial(n), order 3 -16053944387, 8733508770, -1676521980, 10854718875, # PYCHOK R[6]/n^6, polynomial(n), order 2 930092876, -162639357, 723647925, # R[7]/n^7, polynomial(n), order 1 -673429061, 1929727800) # PYCHOK R[8]/n^8, polynomial(n), order 0, count = 54 }
if __name__ == '__main__':
def _re(fmt, r3, x3): e3 = [] for r, x in _zip(r3, x3): # strict=True e = abs(r - x) / abs(x) e3.append('%.g' % (e,)) print((fmt % r3) + ' rel errors: ' + ', '.join(e3))
# <https://GeographicLib.SourceForge.io/cgi-bin/RhumbSolve> rhumb = Rhumb(exact=True) # WGS84 default print('# %r\n' % rhumb) r = rhumb.Direct8(40.6, -73.8, 51, 5.5e6) # from JFK about NE _re('# JFK NE lat2=%.8f, lon2=%.8f, S12=%.1f', (r.lat2, r.lon2, r.S12), (71.68889988, 0.25551982, 44095641862956.148438)) r = rhumb.Inverse8(40.6, -73.8, 51.6, -0.5) # JFK to LHR _re('# JFK-LHR azi12=%.8f, s12=%.3f S12=%.1f', (r.azi12, r.s12, r.S12), (77.76838971, 5771083.383328, 37395209100030.367188)) r = rhumb.Inverse8(40.6, -73.8, 35.8, 140.3) # JFK to Tokyo Narita _re('# JFK-NRT azi12=%.8f, s12=%.3f S12=%.1f', (r.azi12, r.s12, r.S12), (-92.388887981699639, 12782581.0676841792, -63760642939072.492))
# % python3 -m pygeodesy.rhumbx
# Rhumb(RAorder=6, TMorder=6, ellipsoid=Ellipsoid(name='WGS84', a=6378137, b=6356752.31424518, f_=298.25722356, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181919, e2=0.00669438, e22=0.0067395, e32=0.00335843, A=6367449.14582341, L=10001965.72931272, R1=6371008.77141506, R2=6371007.18091847, R3=6371000.79000916, Rbiaxial=6367453.63451633, Rtriaxial=6372797.5559594), exact=True)
# JFK NE lat2=71.68889988, lon2=0.25551982, S12=44095641862956.1 rel errors: 4e-11, 2e-08, 2e-15 # JFK-LHR azi12=77.76838971, s12=5771083.383 S12=37395209100030.4 rel errors: 3e-12, 5e-15, 2e-16 # JFK-NRT azi12=-92.38888798, s12=12782581.068 S12=-63760642939072.5 rel errors: 2e-16, 3e-16, 0
# **) MIT License # # Copyright (C) 2022-2022 -- 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. |