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Geodesic.hpp
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/**
* \file Geodesic.hpp
* \brief Header for GeographicLib::Geodesic class
*
* Copyright (c) Charles Karney (2009-2024) <karney@alum.mit.edu> and licensed
* under the MIT/X11 License. For more information, see
* https://geographiclib.sourceforge.io/
**********************************************************************/
#if !defined(GEOGRAPHICLIB_GEODESIC_HPP)
#define GEOGRAPHICLIB_GEODESIC_HPP 1
#include <GeographicLib/Constants.hpp>
#include <GeographicLib/GeodesicExact.hpp>
#if !defined(GEOGRAPHICLIB_GEODESIC_ORDER)
/**
* The order of the expansions used by Geodesic.
* GEOGRAPHICLIB_GEODESIC_ORDER can be set to any integer in [3, 8].
**********************************************************************/
# define GEOGRAPHICLIB_GEODESIC_ORDER \
(GEOGRAPHICLIB_PRECISION == 2 ? 6 : \
(GEOGRAPHICLIB_PRECISION == 1 ? 3 : \
(GEOGRAPHICLIB_PRECISION == 3 ? 7 : 8)))
#endif
namespace GeographicLib {
class GeodesicLine;
/**
* \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.) In the figure below, latitude is
* labeled &phi;, longitude &lambda; (with &lambda;<sub>12</sub> =
* &lambda;<sub>2</sub> &minus; &lambda;<sub>1</sub>), and azimuth &alpha;.
*
* <img src="https://upload.wikimedia.org/wikipedia/commons/c/cb/Geodesic_problem_on_an_ellipsoid.svg" width=250 alt="spheroidal triangle">
*
* 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 Geodesic::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 Geodesic::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
* Geodesic::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>. The area between the geodesic from point 1 to point 2 and
* the equation is represented by \e S12; it is the area, measured
* counterclockwise, of the geodesic quadrilateral with corners
* (<i>lat1</i>,<i>lon1</i>), (0,<i>lon1</i>), (0,<i>lon2</i>), and
* (<i>lat2</i>,<i>lon2</i>). It can be used to compute the area of any
* geodesic polygon.
*
* Overloaded versions of Geodesic::Direct, Geodesic::ArcDirect, and
* Geodesic::Inverse allow these quantities to be returned. In addition
* there are general functions Geodesic::GenDirect, and Geodesic::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. If points 1, 2, and 3 all lie on a single geodesic,
* then the following rules hold:
* - \e s13 = \e s12 + \e s23
* - \e a13 = \e a12 + \e a23
* - \e S13 = \e S12 + \e S23
* - \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 GeodesicLine class, which
* allows a sequence of points along a geodesic to be computed.
*
* The shortest distance returned by the solution of the inverse problem is
* (obviously) uniquely defined. However, in a few special cases there are
* multiple azimuths which yield the same shortest distance. Here is a
* catalog of those cases:
* - \e lat1 = &minus;\e lat2 (with neither point at a pole). If \e azi1 =
* \e azi2, the geodesic is unique. Otherwise there are two geodesics and
* the second one is obtained by setting [\e azi1, \e azi2] &rarr; [\e
* azi2, \e azi1], [\e M12, \e M21] &rarr; [\e M21, \e M12], \e S12 &rarr;
* &minus;\e S12. (This occurs when the longitude difference is near
* &plusmn;180&deg; for oblate ellipsoids.)
* - \e lon2 = \e lon1 &plusmn; 180&deg; (with neither point at a pole). If
* \e azi1 = 0&deg; or &plusmn;180&deg;, the geodesic is unique. Otherwise
* there are two geodesics and the second one is obtained by setting [\e
* azi1, \e azi2] &rarr; [&minus;\e azi1, &minus;\e azi2], \e S12 &rarr;
* &minus;\e S12. (This occurs when \e lat2 is near &minus;\e lat1 for
* prolate ellipsoids.)
* - Points 1 and 2 at opposite poles. There are infinitely many geodesics
* which can be generated by setting [\e azi1, \e azi2] &rarr; [\e azi1, \e
* azi2] + [\e d, &minus;\e d], for arbitrary \e d. (For spheres, this
* prescription applies when points 1 and 2 are antipodal.)
* - \e s12 = 0 (coincident points). There are infinitely many geodesics
* which can be generated by setting [\e azi1, \e azi2] &rarr;
* [\e azi1, \e azi2] + [\e d, \e d], for arbitrary \e d.
*
* The calculations are accurate to better than 15 nm (15 nanometers) for the
* WGS84 ellipsoid. See Sec. 9 of
* <a href="https://arxiv.org/abs/1102.1215v1">arXiv:1102.1215v1</a> for
* details. With \e exact = false (the default) in the constructor,
* the algorithms used by this class are based on series expansions
* using the flattening \e f as a small parameter. These are only accurate
* for |<i>f</i>| &lt; 0.02; however reasonably accurate results will be
* obtained for |<i>f</i>| &lt; 0.2. Here is a table of the approximate
* maximum error (expressed as a distance) for an ellipsoid with the same
* equatorial radius as the WGS84 ellipsoid and different values of the
* flattening.<pre>
* |f| error
* 0.01 25 nm
* 0.02 30 nm
* 0.05 10 um
* 0.1 1.5 mm
* 0.2 300 mm
* </pre>
* For very eccentric ellipsoids, set \e exact to true in the constructor;
* this will delegate the calculations to the GeodesicExact class.
*
* The algorithms are described in
* - C. F. F. Karney,
* <a href="https://doi.org/10.1007/s00190-012-0578-z">
* <i>Algorithms for geodesics</i></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.
*
* Example of use:
* \include example-Geodesic.cpp
*
* <a href="GeodSolve.1.html">GeodSolve</a> is a command-line utility
* providing access to the functionality of Geodesic and GeodesicLine.
**********************************************************************/
class GEOGRAPHICLIB_EXPORT Geodesic {
private:
typedef Math::real real;
friend class GeodesicLine;
static const int nA1_ = GEOGRAPHICLIB_GEODESIC_ORDER;
static const int nC1_ = GEOGRAPHICLIB_GEODESIC_ORDER;
static const int nC1p_ = GEOGRAPHICLIB_GEODESIC_ORDER;
static const int nA2_ = GEOGRAPHICLIB_GEODESIC_ORDER;
static const int nC2_ = GEOGRAPHICLIB_GEODESIC_ORDER;
static const int nA3_ = GEOGRAPHICLIB_GEODESIC_ORDER;
static const int nA3x_ = nA3_;
static const int nC3_ = GEOGRAPHICLIB_GEODESIC_ORDER;
static const int nC3x_ = (nC3_ * (nC3_ - 1)) / 2;
static const int nC4_ = GEOGRAPHICLIB_GEODESIC_ORDER;
static const int nC4x_ = (nC4_ * (nC4_ + 1)) / 2;
// Size for temporary array
// nC = max(max(nC1_, nC1p_, nC2_) + 1, max(nC3_, nC4_))
static const int nC_ = GEOGRAPHICLIB_GEODESIC_ORDER + 1;
static const unsigned maxit1_ = 20;
unsigned maxit2_;
real tiny_, tol0_, tol1_, tol2_, tolb_, xthresh_;
static constexpr unsigned CAP_NONE = 0U;
static constexpr unsigned CAP_C1 = 1U<<0;
static constexpr unsigned CAP_C1p = 1U<<1;
static constexpr unsigned CAP_C2 = 1U<<2;
static constexpr unsigned CAP_C3 = 1U<<3;
static constexpr unsigned CAP_C4 = 1U<<4;
static constexpr unsigned CAP_ALL = 0x1FU;
static constexpr unsigned CAP_MASK = CAP_ALL;
static constexpr unsigned OUT_ALL = 0x7F80U;
static constexpr unsigned OUT_MASK = 0xFF80U; // Includes LONG_UNROLL
static real SinCosSeries(bool sinp,
real sinx, real cosx, const real c[], int n);
static real Astroid(real x, real y);
real _a, _f;
bool _exact;
real _f1, _e2, _ep2, _n, _b, _c2, _etol2;
real _aA3x[nA3x_], _cC3x[nC3x_], _cC4x[nC4x_];
GeodesicExact _geodexact;
void Lengths(real eps, real sig12,
real ssig1, real csig1, real dn1,
real ssig2, real csig2, real dn2,
real cbet1, real cbet2, unsigned outmask,
real& s12s, real& m12a, real& m0,
real& M12, real& M21, real Ca[]) const;
real InverseStart(real sbet1, real cbet1, real dn1,
real sbet2, real cbet2, real dn2,
real lam12, real slam12, real clam12,
real& salp1, real& calp1,
real& salp2, real& calp2, real& dnm,
real Ca[]) const;
real Lambda12(real sbet1, real cbet1, real dn1,
real sbet2, real cbet2, real dn2,
real salp1, real calp1, real slam120, real clam120,
real& salp2, real& calp2, real& sig12,
real& ssig1, real& csig1, real& ssig2, real& csig2,
real& eps, real& domg12,
bool diffp, real& dlam12, real Ca[]) const;
real GenInverse(real lat1, real lon1, real lat2, real lon2,
unsigned outmask, real& s12,
real& salp1, real& calp1, real& salp2, real& calp2,
real& m12, real& M12, real& M21, real& S12) 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 GeodesicLine constructor and to
* Geodesic::Line what capabilities should be included in the GeodesicLine
* object. They also specify which results to return in the general
* routines Geodesic::GenDirect and Geodesic::GenInverse routines.
* GeodesicLine::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 GeodesicLine 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 GeodesicLine because this is included
* by default.)
* @hideinitializer
**********************************************************************/
AZIMUTH = 1U<<9 | CAP_NONE,
/**
* Calculate distance \e s12.
* @hideinitializer
**********************************************************************/
DISTANCE = 1U<<10 | CAP_C1,
/**
* A combination of the common capabilities: Geodesic::LATITUDE,
* Geodesic::LONGITUDE, Geodesic::AZIMUTH, Geodesic::DISTANCE.
* @hideinitializer
**********************************************************************/
STANDARD = LATITUDE | LONGITUDE | AZIMUTH | DISTANCE,
/**
* 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,
/**
* Unroll \e lon2 in the direct calculation.
* @hideinitializer
**********************************************************************/
LONG_UNROLL = 1U<<15,
/**
* All capabilities, calculate everything. (Geodesic::LONG_UNROLL is not
* included in this mask.)
* @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.
* @param[in] exact if true use exact formulation in terms of elliptic
* integrals instead of series expansions (default false).
* @exception GeographicErr if \e a or (1 &minus; \e f) \e a is not
* positive.
*
* With \e exact = true, this class delegates the calculations to the
* GeodesicExact and GeodesicLineExact classes which solve the geodesic
* problems in terms of elliptic integrals.
**********************************************************************/
Geodesic(real a, real f, bool exact = false);
///@}
/** \name Direct geodesic problem specified in terms of distance.
**********************************************************************/
///@{
/**
* Solve the direct geodesic problem 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
* negative.
* @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;]. 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, writing \e lat = &plusmn;(90&deg; &minus; &epsilon;),
* and taking the limit &epsilon; &rarr; 0+. 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 Geodesic::Direct
* which omit some of the output parameters. Note, however, that the arc
* length is always computed and returned as the function value.
**********************************************************************/
Math::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 Geodesic::Direct.
**********************************************************************/
Math::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 Geodesic::Direct.
**********************************************************************/
Math::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 Geodesic::Direct.
**********************************************************************/
Math::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 Geodesic::Direct.
**********************************************************************/
Math::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 Geodesic::Direct.
**********************************************************************/
Math::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.
**********************************************************************/
///@{
/**
* Solve the direct geodesic problem 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 negative.
* @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;]. 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, writing \e lat = &plusmn;(90&deg; &minus; &epsilon;),
* and taking the limit &epsilon; &rarr; 0+. 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 Geodesic::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 Geodesic::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 Geodesic::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 Geodesic::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 Geodesic::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 Geodesic::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 Geodesic::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 problem. Geodesic::Direct and
* Geodesic::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 \e
* s12_a12.
* @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 negative.
* @param[in] outmask a bitor'ed combination of Geodesic::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 Geodesic::mask values possible for \e outmask are
* - \e outmask |= Geodesic::LATITUDE for the latitude \e lat2;
* - \e outmask |= Geodesic::LONGITUDE for the latitude \e lon2;
* - \e outmask |= Geodesic::AZIMUTH for the latitude \e azi2;
* - \e outmask |= Geodesic::DISTANCE for the distance \e s12;
* - \e outmask |= Geodesic::REDUCEDLENGTH for the reduced length \e
* m12;
* - \e outmask |= Geodesic::GEODESICSCALE for the geodesic scales \e
* M12 and \e M21;
* - \e outmask |= Geodesic::AREA for the area \e S12;
* - \e outmask |= Geodesic::ALL for all of the above;
* - \e outmask |= Geodesic::LONG_UNROLL to unroll \e lon2 instead of
* wrapping it into the range [&minus;180&deg;, 180&deg;].
* .
* 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
* Geodesic::DISTANCE and \e arcmode is false, then \e s12 = \e s12_a12.
* It is not necessary to include Geodesic::DISTANCE_IN in \e outmask; this
* is automatically included is \e arcmode is false.
*
* With the Geodesic::LONG_UNROLL bit set, the quantity \e lon2 &minus; \e
* lon1 indicates how many times and in what sense the geodesic encircles
* the ellipsoid.
**********************************************************************/
Math::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.
**********************************************************************/
///@{
/**
* Solve the inverse geodesic problem.
*
* @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;].
* 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, writing \e lat = &plusmn;(90&deg; &minus; &epsilon;),
* and taking the limit &epsilon; &rarr; 0+.
*
* The solution to the inverse problem is found using Newton's method. If
* this fails to converge (this is very unlikely in geodetic applications
* but does occur for very eccentric ellipsoids), then the bisection method
* is used to refine the solution.
*
* The following functions are overloaded versions of Geodesic::Inverse
* which omit some of the output parameters. Note, however, that the arc
* length is always computed and returned as the function value.
**********************************************************************/
Math::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 Geodesic::Inverse.
**********************************************************************/
Math::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 Geodesic::Inverse.
**********************************************************************/
Math::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 Geodesic::Inverse.
**********************************************************************/
Math::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 Geodesic::Inverse.
**********************************************************************/
Math::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 Geodesic::Inverse.
**********************************************************************/
Math::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 Geodesic::Inverse.
**********************************************************************/
Math::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. Geodesic::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 Geodesic::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 Geodesic::mask values possible for \e outmask are
* - \e outmask |= Geodesic::DISTANCE for the distance \e s12;
* - \e outmask |= Geodesic::AZIMUTH for the latitude \e azi2;
* - \e outmask |= Geodesic::REDUCEDLENGTH for the reduced length \e
* m12;
* - \e outmask |= Geodesic::GEODESICSCALE for the geodesic scales \e
* M12 and \e M21;
* - \e outmask |= Geodesic::AREA for the area \e S12;
* - \e outmask |= Geodesic::ALL for all of the above.
* .
* The arc length is always computed and returned as the function value.
**********************************************************************/
Math::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 GeodesicLine.
**********************************************************************/
///@{
/**
* Typedef for the class for computing multiple points on a geodesic.
**********************************************************************/
typedef GeodesicLine LineClass;
/**
* 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 Geodesic::mask values
* specifying the capabilities the GeodesicLine object should possess,
* i.e., which quantities can be returned in calls to
* GeodesicLine::Position.
* @return a GeodesicLine object.
*
* \e lat1 should be in the range [&minus;90&deg;, 90&deg;].
*
* The Geodesic::mask values are
* - \e caps |= Geodesic::LATITUDE for the latitude \e lat2; this is
* added automatically;
* - \e caps |= Geodesic::LONGITUDE for the latitude \e lon2;
* - \e caps |= Geodesic::AZIMUTH for the azimuth \e azi2; this is
* added automatically;
* - \e caps |= Geodesic::DISTANCE for the distance \e s12;
* - \e caps |= Geodesic::REDUCEDLENGTH for the reduced length \e m12;
* - \e caps |= Geodesic::GEODESICSCALE for the geodesic scales \e M12
* and \e M21;
* - \e caps |= Geodesic::AREA for the area \e S12;
* - \e caps |= Geodesic::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;
* - \e caps |= Geodesic::ALL for all of the above.
* .
* The default value of \e caps is Geodesic::ALL.
*
* If the point is at a pole, the azimuth is defined by keeping \e lon1
* fixed, writing \e lat1 = &plusmn;(90 &minus; &epsilon;), and taking the
* limit &epsilon; &rarr; 0+.
**********************************************************************/
GeodesicLine Line(real lat1, real lon1, real azi1, unsigned caps = ALL)
const;
/**
* Define a GeodesicLine in terms of the inverse geodesic problem.
*
* @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] caps bitor'ed combination of Geodesic::mask values
* specifying the capabilities the GeodesicLine object should possess,
* i.e., which quantities can be returned in calls to
* GeodesicLine::Position.
* @return a GeodesicLine object.
*
* This function sets point 3 of the GeodesicLine to correspond to point 2
* of the inverse geodesic problem.
*
* \e lat1 and \e lat2 should be in the range [&minus;90&deg;, 90&deg;].
**********************************************************************/
GeodesicLine InverseLine(real lat1, real lon1, real lat2, real lon2,
unsigned caps = ALL) const;
/**
* Define a GeodesicLine in terms of the direct geodesic problem 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
* negative.
* @param[in] caps bitor'ed combination of Geodesic::mask values
* specifying the capabilities the GeodesicLine object should possess,
* i.e., which quantities can be returned in calls to
* GeodesicLine::Position.
* @return a GeodesicLine object.
*
* This function sets point 3 of the GeodesicLine to correspond to point 2
* of the direct geodesic problem.
*
* \e lat1 should be in the range [&minus;90&deg;, 90&deg;].
**********************************************************************/
GeodesicLine DirectLine(real lat1, real lon1, real azi1, real s12,
unsigned caps = ALL) const;
/**
* Define a GeodesicLine in terms of the direct geodesic problem 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 negative.
* @param[in] caps bitor'ed combination of Geodesic::mask values
* specifying the capabilities the GeodesicLine object should possess,
* i.e., which quantities can be returned in calls to
* GeodesicLine::Position.
* @return a GeodesicLine object.
*
* This function sets point 3 of the GeodesicLine to correspond to point 2
* of the direct geodesic problem.
*
* \e lat1 should be in the range [&minus;90&deg;, 90&deg;].
**********************************************************************/
GeodesicLine ArcDirectLine(real lat1, real lon1, real azi1, real a12,
unsigned caps = ALL) const;
/**
* Define a GeodesicLine in terms of the direct geodesic problem specified
* in terms of either distance or 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] arcmode boolean flag determining the meaning of the \e
* s12_a12.
* @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 negative.
* @param[in] caps bitor'ed combination of Geodesic::mask values
* specifying the capabilities the GeodesicLine object should possess,
* i.e., which quantities can be returned in calls to
* GeodesicLine::Position.
* @return a GeodesicLine object.
*
* This function sets point 3 of the GeodesicLine to correspond to point 2
* of the direct geodesic problem.
*
* \e lat1 should be in the range [&minus;90&deg;, 90&deg;].
**********************************************************************/
GeodesicLine GenDirectLine(real lat1, real lon1, real azi1,
bool arcmode, real s12_a12,
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.
**********************************************************************/
Math::real EquatorialRadius() const { return _a; }
/**
* @return \e f the flattening of the ellipsoid. This is the
* value used in the constructor.
**********************************************************************/
Math::real Flattening() const { return _f; }
/**
* @return \e exact whether the exact formulation is used. This is the
* value used in the constructor.
**********************************************************************/
bool Exact() const { return _exact; }
/**
* @return total area of ellipsoid in meters<sup>2</sup>. The area of a
* polygon encircling a pole can be found by adding
* Geodesic::EllipsoidArea()/2 to the sum of \e S12 for each side of the
* polygon.
**********************************************************************/
Math::real EllipsoidArea() const
{ return 4 * Math::pi() * _c2; }
///@}
/**
* A global instantiation of Geodesic with the parameters for the WGS84
* ellipsoid.
**********************************************************************/
static const Geodesic& WGS84();
};
} // namespace GeographicLib
#endif // GEOGRAPHICLIB_GEODESIC_HPP
Constants.hpp
Header for GeographicLib::Constants class.
GEOGRAPHICLIB_EXPORT
#define GEOGRAPHICLIB_EXPORT
Definition Constants.hpp:59
real
GeographicLib::Math::real real
Definition Geod3Solve.cpp:25
GeodesicExact.hpp
Header for GeographicLib::GeodesicExact class.
GEOGRAPHICLIB_GEODESIC_ORDER
#define GEOGRAPHICLIB_GEODESIC_ORDER
Definition Geodesic.hpp:21
GeographicLib::GeodesicExact
Exact geodesic calculations.
Definition GeodesicExact.hpp:88
GeographicLib::Geodesic
Geodesic calculations
Definition Geodesic.hpp:175
GeographicLib::Geodesic::Direct
Math::real Direct(real lat1, real lon1, real azi1, real s12, real &lat2, real &lon2, real &azi2, real &m12) const
Definition Geodesic.hpp:432
GeographicLib::Geodesic::Direct
Math::real Direct(real lat1, real lon1, real azi1, real s12, real &lat2, real &lon2, real &azi2) const
Definition Geodesic.hpp:420
GeographicLib::Geodesic::ArcDirect
void ArcDirect(real lat1, real lon1, real azi1, real a12, real &lat2, real &lon2, real &azi2, real &s12, real &M12, real &M21) const
Definition Geodesic.hpp:566
GeographicLib::Geodesic::Inverse
Math::real Inverse(real lat1, real lon1, real lat2, real lon2, real &s12, real &azi1, real &azi2, real &m12, real &M12, real &M21) const
Definition Geodesic.hpp:759
GeographicLib::Geodesic::Flattening
Math::real Flattening() const
Definition Geodesic.hpp:969
GeographicLib::Geodesic::ArcDirect
void ArcDirect(real lat1, real lon1, real azi1, real a12, real &lat2, real &lon2) const
Definition Geodesic.hpp:519
GeographicLib::Geodesic::Inverse
Math::real Inverse(real lat1, real lon1, real lat2, real lon2, real &s12, real &azi1, real &azi2, real &M12, real &M21) const
Definition Geodesic.hpp:747
GeographicLib::Geodesic::InverseLine
GeodesicLine InverseLine(real lat1, real lon1, real lat2, real lon2, unsigned caps=ALL) const
Definition Geodesic.cpp:523
GeographicLib::Geodesic::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
Definition Geodesic.hpp:579
GeographicLib::Geodesic::LineClass
GeodesicLine LineClass
Definition Geodesic.hpp:819
GeographicLib::Geodesic::Direct
Math::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
Definition Geodesic.hpp:394
GeographicLib::Geodesic::ArcDirectLine
GeodesicLine ArcDirectLine(real lat1, real lon1, real azi1, real a12, unsigned caps=ALL) const
Definition Geodesic.cpp:163
GeographicLib::Geodesic::EquatorialRadius
Math::real EquatorialRadius() const
Definition Geodesic.hpp:963
GeographicLib::Geodesic::Inverse
Math::real Inverse(real lat1, real lon1, real lat2, real lon2, real &s12, real &azi1, real &azi2, real &m12) const
Definition Geodesic.hpp:735
GeographicLib::Geodesic::ArcDirect
void ArcDirect(real lat1, real lon1, real azi1, real a12, real &lat2, real &lon2, real &azi2, real &s12) const
Definition Geodesic.hpp:541
GeographicLib::Geodesic::Exact
bool Exact() const
Definition Geodesic.hpp:975
GeographicLib::Geodesic::Line
GeodesicLine Line(real lat1, real lon1, real azi1, unsigned caps=ALL) const
Definition Geodesic.cpp:123
GeographicLib::Geodesic::GenDirectLine
GeodesicLine GenDirectLine(real lat1, real lon1, real azi1, bool arcmode, real s12_a12, unsigned caps=ALL) const
Definition Geodesic.cpp:145
GeographicLib::Geodesic::Direct
Math::real Direct(real lat1, real lon1, real azi1, real s12, real &lat2, real &lon2, real &azi2, real &M12, real &M21) const
Definition Geodesic.hpp:444
GeographicLib::Geodesic::Inverse
Math::real Inverse(real lat1, real lon1, real lat2, real lon2, real &s12, real &azi1, real &azi2) const
Definition Geodesic.hpp:723
GeographicLib::Geodesic::Direct
Math::real Direct(real lat1, real lon1, real azi1, real s12, real &lat2, real &lon2) const
Definition Geodesic.hpp:408
GeographicLib::Geodesic::mask
mask
Definition Geodesic.hpp:266
GeographicLib::Geodesic::DISTANCE
@ DISTANCE
Definition Geodesic.hpp:294
GeographicLib::Geodesic::ALL
@ ALL
Definition Geodesic.hpp:332
GeographicLib::Geodesic::LONG_UNROLL
@ LONG_UNROLL
Definition Geodesic.hpp:326
GeographicLib::Geodesic::REDUCEDLENGTH
@ REDUCEDLENGTH
Definition Geodesic.hpp:311
GeographicLib::Geodesic::LATITUDE
@ LATITUDE
Definition Geodesic.hpp:277
GeographicLib::Geodesic::STANDARD
@ STANDARD
Definition Geodesic.hpp:300
GeographicLib::Geodesic::AREA
@ AREA
Definition Geodesic.hpp:321
GeographicLib::Geodesic::LONGITUDE
@ LONGITUDE
Definition Geodesic.hpp:282
GeographicLib::Geodesic::AZIMUTH
@ AZIMUTH
Definition Geodesic.hpp:289
GeographicLib::Geodesic::NONE
@ NONE
Definition Geodesic.hpp:271
GeographicLib::Geodesic::DISTANCE_IN
@ DISTANCE_IN
Definition Geodesic.hpp:306
GeographicLib::Geodesic::GEODESICSCALE
@ GEODESICSCALE
Definition Geodesic.hpp:316
GeographicLib::Geodesic::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, real &S12) const
Definition Geodesic.hpp:506
GeographicLib::Geodesic::GeodesicLine
friend class GeodesicLine
Definition Geodesic.hpp:178
GeographicLib::Geodesic::Inverse
Math::real Inverse(real lat1, real lon1, real lat2, real lon2, real &azi1, real &azi2) const
Definition Geodesic.hpp:712
GeographicLib::Geodesic::Inverse
Math::real Inverse(real lat1, real lon1, real lat2, real lon2, real &s12) const
Definition Geodesic.hpp:701
GeographicLib::Geodesic::GenDirect
Math::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
Definition Geodesic.cpp:128
GeographicLib::Geodesic::DirectLine
GeodesicLine DirectLine(real lat1, real lon1, real azi1, real s12, unsigned caps=ALL) const
Definition Geodesic.cpp:158
GeographicLib::Geodesic::ArcDirect
void ArcDirect(real lat1, real lon1, real azi1, real a12, real &lat2, real &lon2, real &azi2, real &s12, real &m12) const
Definition Geodesic.hpp:553
GeographicLib::Geodesic::EllipsoidArea
Math::real EllipsoidArea() const
Definition Geodesic.hpp:983
GeographicLib::Geodesic::Geodesic
Geodesic(real a, real f, bool exact=false)
Definition Geodesic.cpp:41
GeographicLib::Geodesic::Direct
Math::real Direct(real lat1, real lon1, real azi1, real s12, real &lat2, real &lon2, real &azi2, real &m12, real &M12, real &M21) const
Definition Geodesic.hpp:457
GeographicLib::Geodesic::ArcDirect
void ArcDirect(real lat1, real lon1, real azi1, real a12, real &lat2, real &lon2, real &azi2) const
Definition Geodesic.hpp:530
GeographicLib::Geodesic::Inverse
Math::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
Definition Geodesic.hpp:689
GeographicLib::Math::pi
static T pi()
Definition Math.hpp:187
GeographicLib::Math::real
double real
Definition Math.hpp:115
GeographicLib
Namespace for GeographicLib.
Definition Accumulator.cpp:12