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Henry Winkel
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/**
* \file Geodesic30.hpp
* \brief Header for GeographicLib::Geodesic30 class
*
* Copyright (c) Charles Karney (2009-2022) <charles@karney.com> and licensed
* under the MIT/X11 License. For more information, see
* https://geographiclib.sourceforge.io/
**********************************************************************/
#if !defined(GEOGRAPHICLIB_GEODESICEXACT_HPP)
#define GEOGRAPHICLIB_GEODESICEXACT_HPP 1
#include <GeographicLib/Constants.hpp>
#if !defined(GEOGRAPHICLIB_GEODESICEXACT_ORDER)
/**
* The order of the expansions used by Geodesic30.
**********************************************************************/
# define GEOGRAPHICLIB_GEODESICEXACT_ORDER 30
#endif
namespace GeographicLib {
template<typename real> class GeodesicLine30;
/**
* \brief %Geodesic calculations
*
* The shortest path between two points on an ellipsoid at (\e lat1, \e lon1)
* and (\e lat2, \e lon2) is called the geodesic. Its length is \e s12 and
* the geodesic from point 1 to point 2 has azimuths \e azi1 and \e azi2 at
* the two end points. (The azimuth is the heading measured clockwise from
* north. \e azi2 is the "forward" azimuth, i.e., the heading that takes you
* beyond point 2 not back to point 1.)
*
* Given \e lat1, \e lon1, \e azi1, and \e s12, we can determine \e lat2, \e
* lon2, and \e azi2. This is the \e direct geodesic problem and its
* solution is given by the function Geodesic30::Direct. (If \e s12 is
* sufficiently large that the geodesic wraps more than halfway around the
* earth, there will be another geodesic between the points with a smaller \e
* s12.)
*
* Given \e lat1, \e lon1, \e lat2, and \e lon2, we can determine \e azi1, \e
* azi2, and \e s12. This is the \e inverse geodesic problem, whose solution
* is given by Geodesic30::Inverse. Usually, the solution to the inverse
* problem is unique. In cases where there are multiple solutions (all with
* the same \e s12, of course), all the solutions can be easily generated
* once a particular solution is provided.
*
* The standard way of specifying the direct problem is the specify the
* distance \e s12 to the second point. However it is sometimes useful
* instead to specify the arc length \e a12 (in degrees) on the auxiliary
* sphere. This is a mathematical construct used in solving the geodesic
* problems. The solution of the direct problem in this form is provided by
* Geodesic30::ArcDirect. An arc length in excess of 180&deg; indicates that
* the geodesic is not a shortest path. In addition, the arc length between
* an equatorial crossing and the next extremum of latitude for a geodesic is
* 90&deg;.
*
* This class can also calculate several other quantities related to
* geodesics. These are:
* - <i>reduced length</i>. If we fix the first point and increase \e azi1
* by \e dazi1 (radians), the second point is displaced \e m12 \e dazi1 in
* the direction \e azi2 + 90&deg;. The quantity \e m12 is called
* the "reduced length" and is symmetric under interchange of the two
* points. On a curved surface the reduced length obeys a symmetry
* relation, \e m12 + \e m21 = 0. On a flat surface, we have \e m12 = \e
* s12. The ratio <i>s12</i>/\e m12 gives the azimuthal scale for an
* azimuthal equidistant projection.
* - <i>geodesic scale</i>. Consider a reference geodesic and a second
* geodesic parallel to this one at point 1 and separated by a small
* distance \e dt. The separation of the two geodesics at point 2 is \e
* M12 \e dt where \e M12 is called the "geodesic scale". \e M21 is
* defined similarly (with the geodesics being parallel at point 2). On a
* flat surface, we have \e M12 = \e M21 = 1. The quantity 1/\e M12 gives
* the scale of the Cassini-Soldner projection.
* - <i>area</i>. Consider the quadrilateral bounded by the following lines:
* the geodesic from point 1 to point 2, the meridian from point 2 to the
* equator, the equator from \e lon2 to \e lon1, the meridian from the
* equator to point 1. The area of this quadrilateral is represented by \e
* S12 with a clockwise traversal of the perimeter counting as a positive
* area and it can be used to compute the area of any simple geodesic
* polygon.
*
* Overloaded versions of Geodesic30::Direct, Geodesic30::ArcDirect,
* and Geodesic30::Inverse allow these quantities to be returned. In
* addition there are general functions Geodesic30::GenDirect, and
* Geodesic30::GenInverse which allow an arbitrary set of results to be
* computed. The quantities \e m12, \e M12, \e M21 which all specify the
* behavior of nearby geodesics obey addition rules. Let points 1, 2, and 3
* all lie on a single geodesic, then
* - \e m13 = \e m12 \e M23 + \e m23 \e M21
* - \e M13 = \e M12 \e M23 &minus; (1 &minus; \e M12 \e M21) \e m23 / \e m12
* - \e M31 = \e M32 \e M21 &minus; (1 &minus; \e M23 \e M32) \e m12 / \e m23
*
* Additional functionality is provided by the GeodesicLine30 class, which
* allows a sequence of points along a geodesic to be computed.
*
* The calculations are accurate to better than 15 nm (15 nanometers). See
* Sec. 9 of
* <a href="https://arxiv.org/abs/1102.1215v1">arXiv:1102.1215v1</a>
* for details.
*
* The algorithms are described in
* - C. F. F. Karney,
* <a href="https://doi.org/10.1007/s00190-012-0578-z">
* Algorithms for geodesics</a>,
* J. Geodesy <b>87</b>, 43--55 (2013);
* DOI: <a href="https://doi.org/10.1007/s00190-012-0578-z">
* 10.1007/s00190-012-0578-z</a>;
* addenda: <a href="https://geographiclib.sourceforge.io/geod-addenda.html">
* geod-addenda.html</a>.
* .
* For more information on geodesics see \ref geodesic.
**********************************************************************/
template<typename real>
class Geodesic30 {
private:
friend class GeodesicLine30<real>;
static const int nA1_ = GEOGRAPHICLIB_GEODESICEXACT_ORDER;
static const int nC1_ = GEOGRAPHICLIB_GEODESICEXACT_ORDER;
static const int nC1p_ = GEOGRAPHICLIB_GEODESICEXACT_ORDER;
static const int nA2_ = GEOGRAPHICLIB_GEODESICEXACT_ORDER;
static const int nC2_ = GEOGRAPHICLIB_GEODESICEXACT_ORDER;
static const int nA3_ = GEOGRAPHICLIB_GEODESICEXACT_ORDER;
static const int nA3x_ = nA3_;
static const int nC3_ = GEOGRAPHICLIB_GEODESICEXACT_ORDER;
static const int nC3x_ = (nC3_ * (nC3_ - 1)) / 2;
static const int nC4_ = GEOGRAPHICLIB_GEODESICEXACT_ORDER;
static const int nC4x_ = (nC4_ * (nC4_ + 1)) / 2;
static const unsigned maxit_ = 50;
static const real tiny_;
static const real tol0_;
static const real tol1_;
static const real tol2_;
static const real xthresh_;
enum captype {
CAP_NONE = 0U,
CAP_C1 = 1U<<0,
CAP_C1p = 1U<<1,
CAP_C2 = 1U<<2,
CAP_C3 = 1U<<3,
CAP_C4 = 1U<<4,
CAP_ALL = 0x1FU,
OUT_ALL = 0x7F80U,
};
static real SinCosSeries(bool sinp,
real sinx, real cosx, const real c[], int n)
;
static inline real AngRound(real x) {
// The makes the smallest gap in x = 1/16 - nextafter(1/16, 0) = 1/2^57
// for reals = 0.7 pm on the earth if x is an angle in degrees. (This
// is about 1000 times more resolution than we get with angles around 90
// degrees.) We use this to avoid having to deal with near singular
// cases when x is non-zero but tiny (e.g., 1.0e-200).
const real z = real(0.0625); // 1/16
volatile real y = std::abs(x);
// The compiler mustn't "simplify" z - (z - y) to y
y = y < z ? z - (z - y) : y;
return x < 0 ? -y : y;
}
static inline void SinCosNorm(real& sinx, real& cosx) {
using std::hypot;
real r = hypot(sinx, cosx);
sinx /= r;
cosx /= r;
}
static real Astroid(real x, real y);
real _a, _f, _f1, _e2, _ep2, _n, _b, _c2, _etol2;
real _A3x[nA3x_], _C3x[nC3x_], _C4x[nC4x_];
void Lengths(real eps, real sig12,
real ssig1, real csig1, real ssig2, real csig2,
real cbet1, real cbet2,
real& s12s, real& m12a, real& m0,
bool scalep, real& M12, real& M21,
real C1a[], real C2a[]) const;
real InverseStart(real sbet1, real cbet1, real sbet2, real cbet2,
real lam12,
real& salp1, real& calp1,
real& salp2, real& calp2,
real C1a[], real C2a[]) const;
real Lambda12(real sbet1, real cbet1, real sbet2, real cbet2,
real salp1, real calp1,
real& salp2, real& calp2, real& sig12,
real& ssig1, real& csig1, real& ssig2, real& csig2,
real& eps, real& domg12, bool diffp, real& dlam12,
real C1a[], real C2a[], real C3a[])
const;
// These are Maxima generated functions to provide series approximations to
// the integrals for the ellipsoidal geodesic.
static real A1m1f(real eps);
static void C1f(real eps, real c[]);
static void C1pf(real eps, real c[]);
static real A2m1f(real eps);
static void C2f(real eps, real c[]);
void A3coeff();
real A3f(real eps) const;
void C3coeff();
void C3f(real eps, real c[]) const;
void C4coeff();
void C4f(real k2, real c[]) const;
public:
/**
* Bit masks for what calculations to do. These masks do double duty.
* They signify to the GeodesicLine30::GeodesicLine30 constructor and
* to Geodesic30::Line what capabilities should be included in the
* GeodesicLine30 object. They also specify which results to return in
* the general routines Geodesic30::GenDirect and
* Geodesic30::GenInverse routines. GeodesicLine30::mask is a
* duplication of this enum.
**********************************************************************/
enum mask {
/**
* No capabilities, no output.
* @hideinitializer
**********************************************************************/
NONE = 0U,
/**
* Calculate latitude \e lat2. (It's not necessary to include this as a
* capability to GeodesicLine30 because this is included by default.)
* @hideinitializer
**********************************************************************/
LATITUDE = 1U<<7 | CAP_NONE,
/**
* Calculate longitude \e lon2.
* @hideinitializer
**********************************************************************/
LONGITUDE = 1U<<8 | CAP_C3,
/**
* Calculate azimuths \e azi1 and \e azi2. (It's not necessary to
* include this as a capability to GeodesicLine30 because this is
* included by default.)
* @hideinitializer
**********************************************************************/
AZIMUTH = 1U<<9 | CAP_NONE,
/**
* Calculate distance \e s12.
* @hideinitializer
**********************************************************************/
DISTANCE = 1U<<10 | CAP_C1,
/**
* Allow distance \e s12 to be used as input in the direct geodesic
* problem.
* @hideinitializer
**********************************************************************/
DISTANCE_IN = 1U<<11 | CAP_C1 | CAP_C1p,
/**
* Calculate reduced length \e m12.
* @hideinitializer
**********************************************************************/
REDUCEDLENGTH = 1U<<12 | CAP_C1 | CAP_C2,
/**
* Calculate geodesic scales \e M12 and \e M21.
* @hideinitializer
**********************************************************************/
GEODESICSCALE = 1U<<13 | CAP_C1 | CAP_C2,
/**
* Calculate area \e S12.
* @hideinitializer
**********************************************************************/
AREA = 1U<<14 | CAP_C4,
/**
* All capabilities, calculate everything.
* @hideinitializer
**********************************************************************/
ALL = OUT_ALL| CAP_ALL,
};
/** \name Constructor
**********************************************************************/
///@{
/**
* Constructor for an ellipsoid with
*
* @param[in] a equatorial radius (meters).
* @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere.
* Negative \e f gives a prolate ellipsoid. If \e f > 1, set flattening
* to 1/\e f.
* @exception GeographicErr if \e a or (1 &minus; \e f ) \e a is not
* positive.
**********************************************************************/
Geodesic30(real a, real f);
///@}
/** \name Direct geodesic problem specified in terms of distance.
**********************************************************************/
///@{
/**
* Perform the direct geodesic calculation where the length of the geodesic
* is specified in terms of distance.
*
* @param[in] lat1 latitude of point 1 (degrees).
* @param[in] lon1 longitude of point 1 (degrees).
* @param[in] azi1 azimuth at point 1 (degrees).
* @param[in] s12 distance between point 1 and point 2 (meters); it can be
* signed.
* @param[out] lat2 latitude of point 2 (degrees).
* @param[out] lon2 longitude of point 2 (degrees).
* @param[out] azi2 (forward) azimuth at point 2 (degrees).
* @param[out] m12 reduced length of geodesic (meters).
* @param[out] M12 geodesic scale of point 2 relative to point 1
* (dimensionless).
* @param[out] M21 geodesic scale of point 1 relative to point 2
* (dimensionless).
* @param[out] S12 area under the geodesic (meters<sup>2</sup>).
* @return \e a12 arc length of between point 1 and point 2 (degrees).
*
* \e lat1 should be in the range [&minus;90&deg;, 90&deg;]; \e lon1 and \e
* azi1 should be in the range [&minus;540&deg;, 540&deg;). The values of
* \e lon2 and \e azi2 returned are in the range [&minus;180&deg;,
* 180&deg;).
*
* If either point is at a pole, the azimuth is defined by keeping the
* longitude fixed and writing \e lat = 90&deg; &minus; &epsilon; or
* &minus;90&deg; + &epsilon; and taking the limit &epsilon; &rarr; 0 from
* above. An arc length greater that 180&deg; signifies a geodesic which
* is not a shortest path. (For a prolate ellipsoid, an additional
* condition is necessary for a shortest path: the longitudinal extent must
* not exceed of 180&deg;.)
*
* The following functions are overloaded versions of Geodesic30::Direct
* which omit some of the output parameters. Note, however, that the arc
* length is always computed and returned as the function value.
**********************************************************************/
real Direct(real lat1, real lon1, real azi1, real s12,
real& lat2, real& lon2, real& azi2,
real& m12, real& M12, real& M21, real& S12)
const {
real t;
return GenDirect(lat1, lon1, azi1, false, s12,
LATITUDE | LONGITUDE | AZIMUTH |
REDUCEDLENGTH | GEODESICSCALE | AREA,
lat2, lon2, azi2, t, m12, M12, M21, S12);
}
/**
* See the documentation for Geodesic30::Direct.
**********************************************************************/
real Direct(real lat1, real lon1, real azi1, real s12,
real& lat2, real& lon2)
const {
real t;
return GenDirect(lat1, lon1, azi1, false, s12,
LATITUDE | LONGITUDE,
lat2, lon2, t, t, t, t, t, t);
}
/**
* See the documentation for Geodesic30::Direct.
**********************************************************************/
real Direct(real lat1, real lon1, real azi1, real s12,
real& lat2, real& lon2, real& azi2)
const {
real t;
return GenDirect(lat1, lon1, azi1, false, s12,
LATITUDE | LONGITUDE | AZIMUTH,
lat2, lon2, azi2, t, t, t, t, t);
}
/**
* See the documentation for Geodesic30::Direct.
**********************************************************************/
real Direct(real lat1, real lon1, real azi1, real s12,
real& lat2, real& lon2, real& azi2, real& m12)
const {
real t;
return GenDirect(lat1, lon1, azi1, false, s12,
LATITUDE | LONGITUDE | AZIMUTH | REDUCEDLENGTH,
lat2, lon2, azi2, t, m12, t, t, t);
}
/**
* See the documentation for Geodesic30::Direct.
**********************************************************************/
real Direct(real lat1, real lon1, real azi1, real s12,
real& lat2, real& lon2, real& azi2,
real& M12, real& M21)
const {
real t;
return GenDirect(lat1, lon1, azi1, false, s12,
LATITUDE | LONGITUDE | AZIMUTH | GEODESICSCALE,
lat2, lon2, azi2, t, t, M12, M21, t);
}
/**
* See the documentation for Geodesic30::Direct.
**********************************************************************/
real Direct(real lat1, real lon1, real azi1, real s12,
real& lat2, real& lon2, real& azi2,
real& m12, real& M12, real& M21)
const {
real t;
return GenDirect(lat1, lon1, azi1, false, s12,
LATITUDE | LONGITUDE | AZIMUTH |
REDUCEDLENGTH | GEODESICSCALE,
lat2, lon2, azi2, t, m12, M12, M21, t);
}
///@}
/** \name Direct geodesic problem specified in terms of arc length.
**********************************************************************/
///@{
/**
* Perform the direct geodesic calculation where the length of the geodesic
* is specified in terms of arc length.
*
* @param[in] lat1 latitude of point 1 (degrees).
* @param[in] lon1 longitude of point 1 (degrees).
* @param[in] azi1 azimuth at point 1 (degrees).
* @param[in] a12 arc length between point 1 and point 2 (degrees); it can
* be signed.
* @param[out] lat2 latitude of point 2 (degrees).
* @param[out] lon2 longitude of point 2 (degrees).
* @param[out] azi2 (forward) azimuth at point 2 (degrees).
* @param[out] s12 distance between point 1 and point 2 (meters).
* @param[out] m12 reduced length of geodesic (meters).
* @param[out] M12 geodesic scale of point 2 relative to point 1
* (dimensionless).
* @param[out] M21 geodesic scale of point 1 relative to point 2
* (dimensionless).
* @param[out] S12 area under the geodesic (meters<sup>2</sup>).
*
* \e lat1 should be in the range [&minus;90&deg;, 90&deg;]; \e lon1 and \e
* azi1 should be in the range [&minus;540&deg;, 540&deg;). The values of
* \e lon2 and \e azi2 returned are in the range [&minus;180&deg;,
* 180&deg;).
*
* If either point is at a pole, the azimuth is defined by keeping the
* longitude fixed and writing \e lat = 90&deg; &minus; &epsilon; or
* &minus;90&deg; + &epsilon; and taking the limit &epsilon; &rarr; 0 from
* above. An arc length greater that 180&deg; signifies a geodesic which
* is not a shortest path. (For a prolate ellipsoid, an additional
* condition is necessary for a shortest path: the longitudinal extent must
* not exceed of 180&deg;.)
*
* The following functions are overloaded versions of Geodesic30::Direct
* which omit some of the output parameters.
**********************************************************************/
void ArcDirect(real lat1, real lon1, real azi1, real a12,
real& lat2, real& lon2, real& azi2, real& s12,
real& m12, real& M12, real& M21, real& S12)
const {
GenDirect(lat1, lon1, azi1, true, a12,
LATITUDE | LONGITUDE | AZIMUTH | DISTANCE |
REDUCEDLENGTH | GEODESICSCALE | AREA,
lat2, lon2, azi2, s12, m12, M12, M21, S12);
}
/**
* See the documentation for Geodesic30::ArcDirect.
**********************************************************************/
void ArcDirect(real lat1, real lon1, real azi1, real a12,
real& lat2, real& lon2) const {
real t;
GenDirect(lat1, lon1, azi1, true, a12,
LATITUDE | LONGITUDE,
lat2, lon2, t, t, t, t, t, t);
}
/**
* See the documentation for Geodesic30::ArcDirect.
**********************************************************************/
void ArcDirect(real lat1, real lon1, real azi1, real a12,
real& lat2, real& lon2, real& azi2) const {
real t;
GenDirect(lat1, lon1, azi1, true, a12,
LATITUDE | LONGITUDE | AZIMUTH,
lat2, lon2, azi2, t, t, t, t, t);
}
/**
* See the documentation for Geodesic30::ArcDirect.
**********************************************************************/
void ArcDirect(real lat1, real lon1, real azi1, real a12,
real& lat2, real& lon2, real& azi2, real& s12)
const {
real t;
GenDirect(lat1, lon1, azi1, true, a12,
LATITUDE | LONGITUDE | AZIMUTH | DISTANCE,
lat2, lon2, azi2, s12, t, t, t, t);
}
/**
* See the documentation for Geodesic30::ArcDirect.
**********************************************************************/
void ArcDirect(real lat1, real lon1, real azi1, real a12,
real& lat2, real& lon2, real& azi2,
real& s12, real& m12) const {
real t;
GenDirect(lat1, lon1, azi1, true, a12,
LATITUDE | LONGITUDE | AZIMUTH | DISTANCE |
REDUCEDLENGTH,
lat2, lon2, azi2, s12, m12, t, t, t);
}
/**
* See the documentation for Geodesic30::ArcDirect.
**********************************************************************/
void ArcDirect(real lat1, real lon1, real azi1, real a12,
real& lat2, real& lon2, real& azi2, real& s12,
real& M12, real& M21) const {
real t;
GenDirect(lat1, lon1, azi1, true, a12,
LATITUDE | LONGITUDE | AZIMUTH | DISTANCE |
GEODESICSCALE,
lat2, lon2, azi2, s12, t, M12, M21, t);
}
/**
* See the documentation for Geodesic30::ArcDirect.
**********************************************************************/
void ArcDirect(real lat1, real lon1, real azi1, real a12,
real& lat2, real& lon2, real& azi2, real& s12,
real& m12, real& M12, real& M21) const {
real t;
GenDirect(lat1, lon1, azi1, true, a12,
LATITUDE | LONGITUDE | AZIMUTH | DISTANCE |
REDUCEDLENGTH | GEODESICSCALE,
lat2, lon2, azi2, s12, m12, M12, M21, t);
}
///@}
/** \name General version of the direct geodesic solution.
**********************************************************************/
///@{
/**
* The general direct geodesic calculation. Geodesic30::Direct and
* Geodesic30::ArcDirect are defined in terms of this function.
*
* @param[in] lat1 latitude of point 1 (degrees).
* @param[in] lon1 longitude of point 1 (degrees).
* @param[in] azi1 azimuth at point 1 (degrees).
* @param[in] arcmode boolean flag determining the meaning of the second
* parameter.
* @param[in] s12_a12 if \e arcmode is false, this is the distance between
* point 1 and point 2 (meters); otherwise it is the arc length between
* point 1 and point 2 (degrees); it can be signed.
* @param[in] outmask a bitor'ed combination of Geodesic30::mask values
* specifying which of the following parameters should be set.
* @param[out] lat2 latitude of point 2 (degrees).
* @param[out] lon2 longitude of point 2 (degrees).
* @param[out] azi2 (forward) azimuth at point 2 (degrees).
* @param[out] s12 distance between point 1 and point 2 (meters).
* @param[out] m12 reduced length of geodesic (meters).
* @param[out] M12 geodesic scale of point 2 relative to point 1
* (dimensionless).
* @param[out] M21 geodesic scale of point 1 relative to point 2
* (dimensionless).
* @param[out] S12 area under the geodesic (meters<sup>2</sup>).
* @return \e a12 arc length of between point 1 and point 2 (degrees).
*
* The Geodesic30::mask values possible for \e outmask are
* - \e outmask |= Geodesic30::LATITUDE for the latitude \e lat2.
* - \e outmask |= Geodesic30::LONGITUDE for the latitude \e lon2.
* - \e outmask |= Geodesic30::AZIMUTH for the latitude \e azi2.
* - \e outmask |= Geodesic30::DISTANCE for the distance \e s12.
* - \e outmask |= Geodesic30::REDUCEDLENGTH for the reduced length \e
* m12.
* - \e outmask |= Geodesic30::GEODESICSCALE for the geodesic scales \e
* M12 and \e M21.
* - \e outmask |= Geodesic30::AREA for the area \e S12.
* .
* The function value \e a12 is always computed and returned and this
* equals \e s12_a12 is \e arcmode is true. If \e outmask includes
* Geodesic30::DISTANCE and \e arcmode is false, then \e s12 = \e
* s12_a12. It is not necessary to include Geodesic30::DISTANCE_IN in
* \e outmask; this is automatically included is \e arcmode is false.
**********************************************************************/
real GenDirect(real lat1, real lon1, real azi1,
bool arcmode, real s12_a12, unsigned outmask,
real& lat2, real& lon2, real& azi2,
real& s12, real& m12, real& M12, real& M21,
real& S12) const;
///@}
/** \name Inverse geodesic problem.
**********************************************************************/
///@{
/**
* Perform the inverse geodesic calculation.
*
* @param[in] lat1 latitude of point 1 (degrees).
* @param[in] lon1 longitude of point 1 (degrees).
* @param[in] lat2 latitude of point 2 (degrees).
* @param[in] lon2 longitude of point 2 (degrees).
* @param[out] s12 distance between point 1 and point 2 (meters).
* @param[out] azi1 azimuth at point 1 (degrees).
* @param[out] azi2 (forward) azimuth at point 2 (degrees).
* @param[out] m12 reduced length of geodesic (meters).
* @param[out] M12 geodesic scale of point 2 relative to point 1
* (dimensionless).
* @param[out] M21 geodesic scale of point 1 relative to point 2
* (dimensionless).
* @param[out] S12 area under the geodesic (meters<sup>2</sup>).
* @return \e a12 arc length of between point 1 and point 2 (degrees).
*
* \e lat1 and \e lat2 should be in the range [&minus;90&deg;, 90&deg;]; \e
* lon1 and \e lon2 should be in the range [&minus;540&deg;, 540&deg;).
* The values of \e azi1 and \e azi2 returned are in the range
* [&minus;180&deg;, 180&deg;).
*
* If either point is at a pole, the azimuth is defined by keeping the
* longitude fixed and writing \e lat = 90&deg; &minus; &epsilon; or
* &minus;90&deg; + &epsilon; and taking the limit &epsilon; &rarr; 0 from
* above. If the routine fails to converge, then all the requested outputs
* are set to Math::NaN(). (Test for such results with Math::isnan.) This
* is not expected to happen with ellipsoidal models of the earth; please
* report all cases where this occurs.
*
* The following functions are overloaded versions of Geodesic30::Inverse
* which omit some of the output parameters. Note, however, that the arc
* length is always computed and returned as the function value.
**********************************************************************/
real Inverse(real lat1, real lon1, real lat2, real lon2,
real& s12, real& azi1, real& azi2, real& m12,
real& M12, real& M21, real& S12) const {
return GenInverse(lat1, lon1, lat2, lon2,
DISTANCE | AZIMUTH |
REDUCEDLENGTH | GEODESICSCALE | AREA,
s12, azi1, azi2, m12, M12, M21, S12);
}
/**
* See the documentation for Geodesic30::Inverse.
**********************************************************************/
real Inverse(real lat1, real lon1, real lat2, real lon2,
real& s12) const {
real t;
return GenInverse(lat1, lon1, lat2, lon2,
DISTANCE,
s12, t, t, t, t, t, t);
}
/**
* See the documentation for Geodesic30::Inverse.
**********************************************************************/
real Inverse(real lat1, real lon1, real lat2, real lon2,
real& azi1, real& azi2) const {
real t;
return GenInverse(lat1, lon1, lat2, lon2,
AZIMUTH,
t, azi1, azi2, t, t, t, t);
}
/**
* See the documentation for Geodesic30::Inverse.
**********************************************************************/
real Inverse(real lat1, real lon1, real lat2, real lon2,
real& s12, real& azi1, real& azi2)
const {
real t;
return GenInverse(lat1, lon1, lat2, lon2,
DISTANCE | AZIMUTH,
s12, azi1, azi2, t, t, t, t);
}
/**
* See the documentation for Geodesic30::Inverse.
**********************************************************************/
real Inverse(real lat1, real lon1, real lat2, real lon2,
real& s12, real& azi1, real& azi2, real& m12)
const {
real t;
return GenInverse(lat1, lon1, lat2, lon2,
DISTANCE | AZIMUTH | REDUCEDLENGTH,
s12, azi1, azi2, m12, t, t, t);
}
/**
* See the documentation for Geodesic30::Inverse.
**********************************************************************/
real Inverse(real lat1, real lon1, real lat2, real lon2,
real& s12, real& azi1, real& azi2,
real& M12, real& M21) const {
real t;
return GenInverse(lat1, lon1, lat2, lon2,
DISTANCE | AZIMUTH | GEODESICSCALE,
s12, azi1, azi2, t, M12, M21, t);
}
/**
* See the documentation for Geodesic30::Inverse.
**********************************************************************/
real Inverse(real lat1, real lon1, real lat2, real lon2,
real& s12, real& azi1, real& azi2, real& m12,
real& M12, real& M21) const {
real t;
return GenInverse(lat1, lon1, lat2, lon2,
DISTANCE | AZIMUTH |
REDUCEDLENGTH | GEODESICSCALE,
s12, azi1, azi2, m12, M12, M21, t);
}
///@}
/** \name General version of inverse geodesic solution.
**********************************************************************/
///@{
/**
* The general inverse geodesic calculation. Geodesic30::Inverse is
* defined in terms of this function.
*
* @param[in] lat1 latitude of point 1 (degrees).
* @param[in] lon1 longitude of point 1 (degrees).
* @param[in] lat2 latitude of point 2 (degrees).
* @param[in] lon2 longitude of point 2 (degrees).
* @param[in] outmask a bitor'ed combination of Geodesic30::mask values
* specifying which of the following parameters should be set.
* @param[out] s12 distance between point 1 and point 2 (meters).
* @param[out] azi1 azimuth at point 1 (degrees).
* @param[out] azi2 (forward) azimuth at point 2 (degrees).
* @param[out] m12 reduced length of geodesic (meters).
* @param[out] M12 geodesic scale of point 2 relative to point 1
* (dimensionless).
* @param[out] M21 geodesic scale of point 1 relative to point 2
* (dimensionless).
* @param[out] S12 area under the geodesic (meters<sup>2</sup>).
* @return \e a12 arc length of between point 1 and point 2 (degrees).
*
* The Geodesic30::mask values possible for \e outmask are
* - \e outmask |= Geodesic30::DISTANCE for the distance \e s12.
* - \e outmask |= Geodesic30::AZIMUTH for the latitude \e azi2.
* - \e outmask |= Geodesic30::REDUCEDLENGTH for the reduced length \e
* m12.
* - \e outmask |= Geodesic30::GEODESICSCALE for the geodesic scales \e
* M12 and \e M21.
* - \e outmask |= Geodesic30::AREA for the area \e S12.
* .
* The arc length is always computed and returned as the function value.
**********************************************************************/
real GenInverse(real lat1, real lon1, real lat2, real lon2,
unsigned outmask,
real& s12, real& azi1, real& azi2,
real& m12, real& M12, real& M21, real& S12)
const;
///@}
/** \name Interface to GeodesicLine30.
**********************************************************************/
///@{
/**
* Set up to compute several points on a single geodesic.
*
* @param[in] lat1 latitude of point 1 (degrees).
* @param[in] lon1 longitude of point 1 (degrees).
* @param[in] azi1 azimuth at point 1 (degrees).
* @param[in] caps bitor'ed combination of Geodesic30::mask values
* specifying the capabilities the GeodesicLine30 object should
* possess, i.e., which quantities can be returned in calls to
* GeodesicLine::Position.
*
* \e lat1 should be in the range [&minus;90&deg;, 90&deg;]; \e lon1 and \e
* azi1 should be in the range [&minus;540&deg;, 540&deg;).
*
* The Geodesic30::mask values are
* - \e caps |= Geodesic30::LATITUDE for the latitude \e lat2; this is
* added automatically
* - \e caps |= Geodesic30::LONGITUDE for the latitude \e lon2
* - \e caps |= Geodesic30::AZIMUTH for the latitude \e azi2; this is
* added automatically
* - \e caps |= Geodesic30::DISTANCE for the distance \e s12
* - \e caps |= Geodesic30::REDUCEDLENGTH for the reduced length \e m12
* - \e caps |= Geodesic30::GEODESICSCALE for the geodesic scales \e M12
* and \e M21
* - \e caps |= Geodesic30::AREA for the area \e S12
* - \e caps |= Geodesic30::DISTANCE_IN permits the length of the
* geodesic to be given in terms of \e s12; without this capability the
* length can only be specified in terms of arc length.
* .
* The default value of \e caps is Geodesic30::ALL which turns on all
* the capabilities.
*
* If the point is at a pole, the azimuth is defined by keeping the \e lon1
* fixed and writing \e lat1 = &plusmn;(90&deg; &minus; &epsilon;) and
* taking the limit &epsilon; &rarr; 0+.
**********************************************************************/
GeodesicLine30<real>
Line(real lat1, real lon1, real azi1, unsigned caps = ALL)
const;
///@}
/** \name Inspector functions.
**********************************************************************/
///@{
/**
* @return \e a the equatorial radius of the ellipsoid (meters). This is
* the value used in the constructor.
**********************************************************************/
real EquatorialRadius() const { return _a; }
/**
* @return \e f the flattening of the ellipsoid. This is the
* value used in the constructor.
**********************************************************************/
real Flattening() const { return _f; }
/// \cond SKIP
/**
* <b>DEPRECATED</b>
* @return \e r the inverse flattening of the ellipsoid.
**********************************************************************/
real InverseFlattening() const { return 1/_f; }
/// \endcond
/**
* @return total area of ellipsoid in meters<sup>2</sup>. The area of a
* polygon encircling a pole can be found by adding
* Geodesic30::EllipsoidArea()/2 to the sum of \e S12 for each side of
* the polygon.
**********************************************************************/
real EllipsoidArea() const
{ return 4 * Math::pi<real>() * _c2; }
///@}
/**
* A global instantiation of Geodesic30 with the parameters for the WGS84
* ellipsoid.
**********************************************************************/
static const Geodesic30 WGS84;
};
} // namespace GeographicLib
#endif // GEOGRAPHICLIB_GEODESICEXACT_HPP