# Coverage for pygeodesy/rhumbx.py : 98%

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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 from __future__ import division as _; del _ # PYCHOK semicolon from pygeodesy.basics import copysign0, isnan, _xinstanceof, _zip from pygeodesy.errors import IntersectionError, _ValueError, _xdatum, _xkwds # from pygeodesy.etm import ExactTransverseMercator # in ._eTM below from pygeodesy.fmath import euclid, favg, hypot, hypot1 from pygeodesy.fsums import Fmt, fsum1_, pairs from pygeodesy.interns import INT0, NAN, NN, PI_2, _azi12_, _coincident_, \ _COMMASPACE_, _convergence_, _ellipsoidal_, \ _EPSqrt as _TOL, _intersection_, _lat1_, _lat2_, \ _lon1_, _lon2_, _no_, _not_, _s12_, _S12_, \ _under_name, _0_0, _0_5, _1_0, _2_0, _4_0, \ _90_0, _180_0, _720_0 # PYCHOK used! from pygeodesy.karney import _a12_, _atan2d, Caps, _CapsBase as _RhumbBase, \ _diff182, Direct9Tuple, _EWGS84, _fix90, GDict, \ _GTuple, Inverse10Tuple, _norm180, _tand from pygeodesy.ktm import _Xellipsoid, _Xorder, _Xs, \ _AlpCoeffs, _BetCoeffs # PYCHOK used! from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY, _ALL_MODS as _MODS from pygeodesy.namedTuples import Distance2Tuple, LatLon2Tuple, NearestOn4Tuple from pygeodesy.props import deprecated_method, Property, Property_RO, property_RO, \ _update_all # from pygeodesy.streprs import Fmt, pairs # from .fsums from pygeodesy.units import Bearing as _Azi, Degrees as _Deg, Int, Lat, Lon, \ Meter as _M, Meter2 as _M2 from pygeodesy.utily import sincos2_, sincos2d # from pygeodesy.vector3d import intersection3d3, Vector3d # in ._eTM3d, ... below from math import asinh, atan, cos, cosh, fabs, radians, sin, sinh, sqrt __all__ = _ALL_LAZY.rhumbx __version__ = '22.07.01' _rls = [] # instances of C{RbumbLine} to be updated _TRIPS = 65 # .intersection2, 18+ class _Lat(Lat): '''(INTERNAL) Latitude B{C{lat}}. ''' def __init__(self, *lat, **Error_name): kwds = _xkwds(Error_name, clip=0, Error=RhumbError) Lat.__new__(_Lat, *lat, **kwds) class _Lon(Lon): '''(INTERNAL) Longitude B{C{lon}}. ''' def __init__(self, *lon, **Error_name): kwds = _xkwds(Error_name, clip=0, Error=RhumbError) Lon.__new__(_Lon, *lon, **kwds) def _update_all_rls(r): '''(INTERNAL) Zap cached/memoized C{Property[_RO]}s of any L{RhumbLine} instances tied to the given L{Rhumb} instance B{C{r}}. ''' _xinstanceof(r, Rhumb) _update_all(r) for rl in _rls: # PYCHOK use weakref? if rl._rhumb is r: _update_all(rl) class Rhumb(_RhumbBase): '''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}. ''' _E = _EWGS84 _exact = True _mRA = 6 _mTM = 6 def __init__(self, a_earth=_EWGS84, f=None, exact=True, name=NN, **RA_TMorder): '''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}}. ''' if f is not None: self.ellipsoid = a_earth, f elif a_earth not in (_EWGS84, None): self.ellipsoid = a_earth if not exact: self._exact = False if name: self.name = name if RA_TMorder: self.orders(**RA_TMorder) def _DConformal2Rectifying(self, chix, chiy): return _1_0 + _sincosSeries(True, chix, chiy, *self._DRectifying2ConformalA2) @Property_RO def _DRectifying2ConformalA2(self): m = self.TMorder return _Xs(_AlpCoeffs, m, self.ellipsoid), m def Direct(self, lat1, lon1, azi12, s12, outmask=Caps.LATITUDE_LONGITUDE): '''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. ''' rl = _RhumbLine(self, lat1, lon1, azi12, caps=Caps.LINE_OFF, name=self.name) return rl.Position(s12, outmask | self._debug) # lat2, lon2, S12 @deprecated_method def Direct7(self, lat1, lon1, azi12, s12, outmask=Caps.LATITUDE_LONGITUDE_AREA): '''DEPRECATED, use method L{Rhumb.Direct8}. @return: A I{DEPRECATED} L{Rhumb7Tuple}. ''' return self.Direct8(lat1, lon1, azi12, s12, outmask=outmask)._to7Tuple() def Direct8(self, lat1, lon1, azi12, s12, outmask=Caps.LATITUDE_LONGITUDE_AREA): '''Like method L{Rhumb.Direct} but returning a L{Rhumb8Tuple} with area C{S12}. ''' return self.Direct(lat1, lon1, azi12, s12, outmask=outmask).toRhumb8Tuple() def DirectLine(self, lat1, lon1, azi12, name=NN, **caps): # caps=Caps.STANDARD '''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. ''' return RhumbLine(self, lat1=lat1, lon1=lon1, azi12=azi12, name=name or self.name, **caps) def _DIsometric(self, latx, laty): # degrees phix = radians(latx) phiy = radians(laty) return (_Dtand(latx, laty) * _Dasinh(_tand(latx), _tand(laty)) - _Dsin(phix, phiy) * _DeatanhE(sin(phix), sin(phiy), self._E)) def _DIsometric2Rectifying(self, psix, psiy): # degrees if self.exact: E = self.ellipsoid latx = E.auxIsometric(psix, inverse=True) laty = E.auxIsometric(psiy, inverse=True) r = self._DRectifying(latx, laty) / self._DIsometric(latx, laty) else: x, y = radians(psix), radians(psiy) r = self._DConformal2Rectifying(_gd(x), _gd(y)) * _Dgd(x, y) return r def _DRectifying(self, latx, laty): # degrees E, eF = self._E, self._eF tbetx = E.f1 * _tand(latx) tbety = E.f1 * _tand(laty) return (E.f1 * _DfEt(tbetx, tbety, eF) * E.b * _Dtand(latx, laty) * PI_2 * _Datan(tbetx, tbety)) / E.L def _DRectifying2Conformal(self, mux, muy): # radians return _1_0 - _sincosSeries(True, mux, muy, *self._DRectifying2ConformalB2) @Property_RO def _DRectifying2ConformalB2(self): m = self.TMorder return _Xs(_BetCoeffs, m, self.ellipsoid), m def _DRectifying2Isometric(self, mux, muy): # degrees E = self.ellipsoid latx = E.auxRectifying(mux, inverse=True) laty = E.auxRectifying(muy, inverse=True) if self.exact: r = self._DIsometric(latx, laty) / self._DRectifying(latx, laty) else: x, y = radians(mux), radians(muy) r = self._DRectifying2Conformal(x, y) * \ _Dgdinv(E.es_taupf(_tand(latx)), E.es_taupf(_tand(laty))) return r @Property_RO def _eF(self): '''(INTERNAL) Get the ellipsoid's elliptic function. ''' # .k2 = 0.006739496742276434 return self._E._elliptic_e12 # _MODS.elliptic.Elliptic(-self._E._e12) @Property def ellipsoid(self): '''Get this rhumb's ellipsoid (L{Ellipsoid}). ''' return self._E @ellipsoid.setter # PYCHOK setter! def ellipsoid(self, a_earth_f): '''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}). ''' E = _Xellipsoid(a_earth_f, RhumbError) if self._E != E: _update_all_rls(self) self._E = E @property_RO def equatoradius(self): '''Get the C{ellipsoid}'s equatorial radius, semi-axis (C{meter}). ''' return self.ellipsoid.a a = equatoradius @Property def exact(self): '''Get the I{exact} option (C{bool}). ''' return self._exact @exact.setter # PYCHOK setter! def exact(self, exact): '''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>}. ''' x = bool(exact) if self._exact != x: _update_all_rls(self) self._exact = x def flattening(self): '''Get the C{ellipsoid}'s flattening (C{float}). ''' return self.ellipsoid.f f = flattening def Inverse(self, lat1, lon1, lat2, lon2, outmask=Caps.AZIMUTH_DISTANCE): '''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. ''' r = GDict(name=self.name) if (outmask & Caps.AZIMUTH_DISTANCE_AREA): r.set_(lat1=lat1, lon1=lon1, lat2=lat2, lon2=lon2) E = self.ellipsoid psi1 = E.auxIsometric(lat1) psi2 = E.auxIsometric(lat2) psi12 = psi2 - psi1 lon12, _ = _diff182(lon1, lon2) if (outmask & Caps.AZIMUTH): r.set_(azi12=_atan2d(lon12, psi12)) if (outmask & Caps.DISTANCE): a12 = hypot(lon12, psi12) * self._DIsometric2Rectifying(psi2, psi1) s12 = a12 * E._L_90 r.set_(s12=s12, a12=copysign0(a12, s12)) if (outmask & Caps.AREA): r.set_(S12=self._S12d(lon12, psi2, psi1)) 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) return r # 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_method def Inverse7(self, lat1, lon1, azi12, s12, outmask=Caps.AZIMUTH_DISTANCE_AREA): '''DEPRECATED, use method L{Rhumb.Inverse8}. @return: A I{DEPRECATED} L{Rhumb7Tuple}. ''' return self.Inverse8(lat1, lon1, azi12, s12, outmask=outmask)._to7Tuple() def Inverse8(self, lat1, lon1, azi12, s12, outmask=Caps.AZIMUTH_DISTANCE_AREA): '''Like method L{Rhumb.Inverse} but returning a L{Rhumb8Tuple} with area C{S12}. ''' return self.Inverse(lat1, lon1, azi12, s12, outmask=outmask).toRhumb8Tuple() def InverseLine(self, lat1, lon1, lat2, lon2, name=NN, **caps): # caps=Caps.STANDARD '''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. ''' r = self.Inverse(lat1, lon1, lat2, lon2, outmask=Caps.AZIMUTH) return RhumbLine(self, lat1=lat1, lon1=lon1, azi12=r.azi12, name=name or self.name, **caps) Line = DirectLine # synonyms def _MeanSinXid(self, psix, psiy): # degrees x, y = radians(psix), radians(psiy) t = _sincosSeries(False, _gd(x), _gd(y), *self._RA2) return _Dlog(cosh(x), cosh(y)) * _Dcosh(x, y) + t * _Dgd(x, y) def orders(self, RAorder=None, TMorder=None): '''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. ''' t = RhumbOrder2Tuple(self.RAorder, self.TMorder) if RAorder not in (None, self._mRA): self.RAorder = RAorder if TMorder not in (None, self._mTM): self.TMorder = TMorder return t @Property_RO def _RA2(self): # for WGS84: (0, -0.0005583633519275459, -3.743803759172812e-07, -4.633682270824446e-10, # RAorder 6: -7.709197397676237e-13, -1.5323287106694307e-15, -3.462875359099873e-18) m = self.RAorder return _Xs(_RACoeffs, m, self.ellipsoid, RA=True), m @Property def RAorder(self): '''Get the I{Rhumb Area} order (C{int}, 4, 5, 6, 7 or 8). ''' return self._mRA @RAorder.setter # PYCHOK setter! def RAorder(self, order): '''Set the I{Rhumb Area} order (C{int}, 4, 5, 6, 7 or 8). ''' n = _Xorder(_RACoeffs, RhumbError, RAorder=order) if self._mRA != n: _update_all_rls(self) self._mRA = n def _S12d(self, lon12, psi2, psi1): # degrees '''(INTERNAL) Compute the area C{S12}. ''' r = self._E.areax if self.exact else self._E.area # for WGS84: r = 510065621724088.44 # a = 6378137.0, f = 0.0033528106647474805 # b = 6356752.314245179, c2 = 40589732499314.76 r *= lon12 / _720_0 return r * self._MeanSinXid(psi2, psi1) @Property def TMorder(self): '''Get the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8). ''' return self._mTM @TMorder.setter # PYCHOK setter! def TMorder(self, order): '''Set the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8). @note: Setting C{TMorder} turns C{exact} off. ''' n = _Xorder(_AlpCoeffs, RhumbError, TMorder=order) if self._mTM != n: _update_all_rls(self) self._mTM = n self.exact = False def toStr(self, prec=6, sep=_COMMASPACE_, **unused): # PYCHOK signature '''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}). ''' d = dict(ellipsoid=self.ellipsoid, RAorder=self.RAorder, exact=self.exact, TMorder=self.TMorder) return sep.join(pairs(d, prec=prec)) class RhumbError(_ValueError): '''Raised for an L{Rhumb} or L{RhumbLine} issue. ''' pass class _RhumbLine(_RhumbBase): '''(INTERNAL) Class L{RhumbLine} ''' _azi12 = _0_0 # _lat1 = _0_0 # _lon1 = _0_0 _salp = _0_0 _calp = _1_0 _rhumb = None # L{Rhumb} instance def __init__(self, rhumb, lat1, lon1, azi12, caps=0, name=NN): # case=Caps.? '''New C{RhumbLine}. ''' _xinstanceof(Rhumb, rhumb=rhumb) self._lat1 = _Lat(lat1=_fix90(lat1)) self._lon1 = _Lon(lon1= lon1) self._debug |= (caps | rhumb._debug) & Caps._DEBUG_DIRECT_LINE if azi12: # non-zero self.azi12 = azi12 self._caps = caps if not (caps & Caps.LINE_OFF): _rls.append(self) n = name or rhumb.name if n: self.name=n self._rhumb = rhumb # last def __del__(self): # XXX use weakref? if _rls: # may be empty or None try: # PYCHOK no cover _rls.remove(self) except (TypeError, ValueError): pass self._rhumb = None # _update_all(self) # throws TypeError during Python 2 cleanup @Property def azi12(self): '''Get this rhumb line's I{azimuth} (compass C{degrees}). ''' return self._azi12 @azi12.setter # PYCHOK setter! def azi12(self, azi12): '''Set this rhumb line's I{azimuth} (compass C{degrees}). ''' z = _norm180(azi12) if z != self._azi12: if self._rhumb: _update_all(self) self._azi12 = z self._salp, self._calp = sincos2d(z) # no NEG0 def distance2(self, lat, lon): '''Return the distance and (initial) bearing of a 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}. ''' r = self.rhumb.Inverse(self.lat1, self.lon1, lat, lon, Caps.AZIMUTH_DISTANCE) return Distance2Tuple(r.s12, r.azi12) @Property_RO def ellipsoid(self): '''Get this rhumb line's ellipsoid (L{Ellipsoid}). ''' return self.rhumb._E @property_RO def exact(self): '''Get this rhumb line's I{exact} option (C{bool}). ''' return self.rhumb.exact def intersection2(self, other, tol=_TOL, **eps): '''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. ''' _xinstanceof(other, RhumbLine) _xdatum(self.rhumb, other.rhumb, Error=RhumbError) try: if other is self: raise ValueError(_coincident_) # set initial origin and halfway point p = LatLon2Tuple(favg(self.lat1, other.lat1), favg(self.lon1, other.lon1)) _xTM4 = self.xTM.reverse # ellipsoidal or spherical _i3d3 = _MODS.vector3d._intersect3d3 # NOT .intersection3d3! for i in range(1, _TRIPS): v = _i3d3(self._xTM3d(p), self.azi12, other._xTM3d(p), other.azi12, useZ=False, **eps)[0] t = _xTM4(v.x, v.y, lon0=p.lon) # PYCHOK Reverse4Tuple d = euclid(t.lon - p.lon, t.lat) # PYCHOK t.lat + p.lat - p.lat p = LatLon2Tuple(t.lat + p.lat, t.lon) # PYCHOK p. if d < tol: return LatLon2Tuple(p.lat, p.lon, iteration=i, # PYCHOK p. name=self.intersection2.__name__) except Exception as x: raise IntersectionError(_no_(_intersection_), txt=str(x)) t = _no_(_convergence_, Fmt.PAREN(d)) raise IntersectionError(interation=i, tol=tol, txt=t) @property_RO def lat1(self): '''Get this rhumb line's latitude (C{degrees90}). ''' return self._lat1 @property_RO def lon1(self): '''Get this rhumb line's longitude (C{degrees180}). ''' return self._lon1 @Property_RO def latlon1(self): '''Get this rhumb line's lat- and longitude (L{LatLon2Tuple}C{(lat, lon)}). ''' return LatLon2Tuple(self.lat1, self.lon1) @Property_RO def _mu1(self): '''(INTERNAL) Get the I{rectifying auxiliary} latitude C{mu} (C{degrees}).. ''' return self.ellipsoid.auxRectifying(self.lat1) def nearestOn4(self, lat, lon, tol=_TOL, **eps): '''Find the point on this rhumb line nearest to a 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}. ''' z = _norm180(self.azi12 + _90_0) # perpendicular r = _RhumbLine(self.rhumb, lat, lon, z, caps=Caps.LINE_OFF) p = self.intersection2(r, tol=tol, **eps) t = r.distance2(p.lat, p.lon) return NearestOn4Tuple(p.lat, p.lon, t.distance, z, iteration=p.iteration) @Property_RO def _psi1(self): '''(INTERNAL) Get the I{isometric auxiliary} latitude C{psi} (C{degrees}). ''' return self.ellipsoid.auxIsometric(self.lat1) @property_RO def RAorder(self): '''Get this rhumb line's I{Rhumb Area} order (C{int}, 4, 5, 6, 7 or 8). ''' return self.rhumb.RAorder @Property_RO def _r1rad(self): # PYCHOK no cover '''(INTERNAL) Get this rhumb line's I{icircle radius} (C{meter}). ''' return radians(self.ellipsoid.circle4(self.lat1).radius) @Property_RO def rhumb(self): '''Get this rhumb line's rhumb (L{Rhumb}). ''' return self._rhumb def Position(self, s12, outmask=Caps.LATITUDE_LONGITUDE): '''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. ''' r = GDict(name=self.name) if (outmask & Caps.LATITUDE_LONGITUDE_AREA): E, R = self.ellipsoid, self.rhumb mu12 = s12 * self._calp / E._L_90 mu2 = mu12 + self._mu1 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: psi2 = self._psi1 if self._calp: lat2x = E.auxRectifying(mu2, inverse=True) psi12 = R._DRectifying2Isometric(mu2, self._mu1) * mu12 lon2x = psi12 * self._salp / self._calp psi2 += psi12 else: # PYCHOK no cover lat2x = self.lat1 lon2x = self._salp * s12 / self._r1rad # psi12 = _0_0 # a12 = copysign0(hypot(psi12, lon2x - self.lon1), s12) if (outmask & Caps.AREA): r.set_(S12=R._S12d(lon2x, self._psi1, psi2)) r.set_(s12=s12, azi12=self.azi12, a12=s12 / E._L_90) if (outmask & Caps.LATITUDE): r.set_(lat2=lat2x, lat1=self.lat1) if (outmask & Caps.LONGITUDE): if (outmask & Caps.LONG_UNROLL) and not isnan(lat2x): lon2x += self.lon1 else: lon2x = _norm180(_norm180(self.lon1) + lon2x) r.set_(lon2=lon2x, lon1=self.lon1) 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 r def toStr(self, prec=6, sep=_COMMASPACE_, **unused): # PYCHOK signature '''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}). ''' d = dict(rhumb=self.rhumb, lat1=self.lat1, lon1=self.lon1, azi12=self.azi12, exact=self.exact, TMorder=self.TMorder, xTM=self.xTM) return sep.join(pairs(d, prec=prec)) @property_RO def TMorder(self): '''Get this rhumb line's I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8). ''' return self.rhumb.TMorder @Property_RO def xTM(self): '''Get this rhumb line's I{Transverse Mercator} projection (L{ExactTransverseMercator} if I{exact} and I{ellipsoidal}, L{KTransverseMercator} otherwise). ''' E = self.ellipsoid return _MODS.etm.ExactTransverseMercator(E) if E.isEllipsoidal and self.exact else \ _MODS.ktm.KTransverseMercator(E, TMorder=self.TMorder) def _xTM3d(self, latlon0, z=INT0): '''(INTERNAL) Convert this C{latlon1 - latlon0} to C{Vector3d}. ''' t = self.xTM.forward(self.lat1 - latlon0.lat, self.lon1, lon0=latlon0.lon) return _MODS.vector3d.Vector3d(t.easting, t.northing, z) class RhumbLine(_RhumbLine): '''Compute one or more 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 rhumb line and the corresponding area C{S12} under the rhumb line. The intersection of two rhumb lines can be found with method L{RhumbLine.intersection2} and method L{RhumbLine.nearestOn4} computes the nearest point on a rhumb line. ''' def __init__(self, rhumb, lat1=0, lon1=0, azi12=None, caps=0, name=NN): # case=Caps.? '''New L{RhumbLine}. @arg rhumb: The rhumb reference (L{Rhumb}). @kwarg lat1: The latitude of the starting point (C{degrees90}). @kwarg lon1: The longitude of the starting point (C{degrees180}). @kwarg azi12: The azimuth of this rhumb line (compass C{degrees}). @kwarg caps: Bit-or'ed combination of L{Caps} values specifying the capabilities. Use C{Caps.LINE_OFF} if updates to the B{C{rhumb}} should I{not} be reflected in this rhumb line. @kwarg name: Optional name (C{str}). ''' if (caps & Caps.LINE_OFF): # copy to avoid updates rhumb = rhumb.copy(deep=False, name=_under_name(rhumb.name)) _RhumbLine.__init__(self, rhumb, lat1, lon1, azi12, caps=caps, name=name) class RhumbOrder2Tuple(_GTuple): '''2-Tuple C{(RAorder, TMorder)} with a I{Rhumb Area} and I{Transverse Mercator} order, both C{int}. ''' _Names_ = (Rhumb.RAorder.name, Rhumb.TMorder.name) _Units_ = ( Int, Int) class Rhumb8Tuple(_GTuple): '''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. ''' _Names_ = (_lat1_, _lon1_, _lat2_, _lon2_, _azi12_, _s12_, _S12_, _a12_) _Units_ = (_Lat, _Lon, _Lat, _Lon, _Azi, _M, _M2, _Deg) def toDirect9Tuple(self, dflt=NAN, **a12_azi1_azi2_m12_M12_M21): '''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)} ''' d = dict(azi1=self.azi12, azi2=self.azi12, m12=self.s12, M12=_1_0, M21=_1_0) # PYCHOK attr d.update(a12_azi1_azi2_m12_M12_M21) return self._toTuple(Direct9Tuple, dflt, d) def toInverse10Tuple(self, dflt=NAN, **a12_m12_M12_M21_salp1_calp1_salp2_calp2): '''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)}. ''' s, c = sincos2d(self.azi12) # PYCHOK attr d = dict(m12=self.s12, M12=_1_0, M21=_1_0, # PYCHOK attr salp1=s, calp1=c, salp2=s, calp2=c) d.update(a12_m12_M12_M21_salp1_calp1_salp2_calp2) return self._toTuple(Inverse10Tuple, dflt, d) def _toTuple(self, nTuple, dflt, updates={}): '''(INTERNAL) Convert this C{Rhumb8Tuple} to an B{C{nTuple}}. ''' r = self.toGDict(**updates) t = tuple(r.get(n, dflt) for n in nTuple._Names_) return nTuple(t, name=self.name) @deprecated_method def _to7Tuple(self): '''DEPRECATED, do not use! ''' return _MODS.deprecated.Rhumb7Tuple(self[:-1]) # 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. def _Datan(x, y): xy = x * y r = xy + _1_0 d = x - y if d: # 2 * xy > -1 == 2 * xy + 1 > 0 == xy + r > 0 == xy > -r r = (atan(d / r) if xy > -r else (atan(x) - atan(y))) / d else: r = _1_0 / r return r def _Dasinh(x, y): hx = hypot1(x) d = x - y if d: hx *= y hy = x * hypot1(y) t = (d * (x + y) / (hy + hx)) if (x * y) > 0 else (hy - hx) r = asinh(t) / d else: r = _1_0 / hx return r def _Dcosh(x, y): return _Dsincos(x, y, sinh, sinh) def _DeatanhE(x, y, E): # Deatanhe(x, y) = eatanhe((x - y) / (1 - e^2 * x * y)) / (x - y) e = _1_0 - E.e2 * x * y # assert not isnear0(e) d = x - y return (E._es_atanh(d / e) / d) if d else (E.e2 / e) def _DfEt(x, y, eF): # x, y are tangents # eF = Elliptic(-E.e12) # -E.e2 / (1 - E.e2) x, y, r = atan(x), atan(y), _1_0 d = x - y if (x * y) > 0: # 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))) sx, cx, sy, cy = sincos2_(x, y) D = (cx + cy) * (eF.fDelta(sy, cy) * sx + eF.fDelta(sx, cx) * sy) D = (sx + sy) * _Dsin(x, y) / D t = D * d t2 = t**2 + _1_0 D *= _2_0 / t2 s = D * d if s: c = (t + _1_0) * (_1_0 - t) / t2 r = eF.fE(s, c, eF.fDelta(s, c)) / s r = D * (r - eF.k2 * sx * sy) elif d: r = (eF.fE(x) - eF.fE(y)) / d return r def _Dgd(x, y): return _Datan(sinh(x), sinh(y)) * _Dsinh(x, y) def _Dgdinv(x, y): # x, y are tangents return _Dasinh(x, y) / _Datan(x, y) def _Dlog(x, y): d = x - y # 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. N.B. this routine is # invoked with positive x and y, so no need to guard against taking # the sqrt of a negative quantity. This also fixes bogus results # being returned for the area when an endpoint is at a pole. return (asinh(d / (sqrt(x * y) * _2_0)) * _2_0 / d) if d else (_1_0 / x) def _Dsin(x, y): return _Dsincos(x, y, sin, cos) def _Dsincos(x, y, s_, c_): r = c_((x + y) * _0_5) d = (x - y) * _0_5 if d: r *= s_(d) / d return r def _Dsinh(x, y): return _Dsincos(x, y, sinh, cosh) def _Dtand(x, y): # x, y in degrees tx, ty = _tand(x), _tand(y) txy = tx * ty r = txy + _1_0 d = x - y if d: # 2 * txy > -1 == 2 * txy + 1 > 0 == txy + r > 0 == txy > -r r = ((_tand(d) * r) if txy > -r else (tx - ty)) / radians(d) return r def _gd(x): return atan(sinh(x)) def _sincosSeries(sinp, x, y, C, n): # 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 # ... d = x - y sp, cp, sd, cd = sincos2_(x + y, d) sd = (sd / d) if d else _1_0 m = cp * cd * _2_0 s = sp * sd # 2x2 matrices in row-major order A0, A1 = m, (-s * d**2) A2, A3 = (-s * _4_0), m b2 = b1 = (_0_0,) * 4 if n > 0: b1 = C[n], _0_0, _0_0, C[n] for j in range(n - 1, 0, -1): b1, b2, Cj = b2, b1, C[j] # C[0] unused # b1 = A * b2 - b1 + C[j] * I B0, B1, B2, B3 = b2 b1 = (fsum1_(A0 * B0, A1 * B2, -b1[0], Cj), fsum1_(A0 * B1, A1 * B3, -b1[1]), fsum1_(A2 * B0, A3 * B2, -b1[2]), fsum1_(A2 * B1, A3 * B3, -b1[3], Cj)) # 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 s = fsum1_(b1[2] * cd * sp, b1[3] * sd * cp * _2_0) if sinp else \ fsum1_(b1[2] * cd * cp, -b1[3] * sd * sp * _2_0, -b2[2]) return s _RACoeffs = { # Generated by Maxima on 2015-05-15 08:24:04-04:00 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 } __all__ += _ALL_DOCS(Caps, _RhumbLine) 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) + ' 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('# 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('# 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('# 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) # lat2=71.68889988, lon2=0.25551982, S12=44095641862956.1 errors: 4e-11, 2e-08, 2e-15 # azi12=77.76838971, s12=5771083.383 S12=37395209100030.4 errors: 3e-12, 5e-15, 2e-16 # azi12=-92.38888798, s12=12782581.068 S12=-63760642939072.5 errors: 2e-16, 0, 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.