GeographicLib
1.48

Geodesic calculations More...
#include <GeographicLib/Geodesic.hpp>
Public Types  
enum  mask { NONE, LATITUDE, LONGITUDE, AZIMUTH, DISTANCE, DISTANCE_IN, REDUCEDLENGTH, GEODESICSCALE, AREA, LONG_UNROLL, ALL } 
Public Member Functions  
Constructor  
Geodesic (real a, real f)  
Direct geodesic problem specified in terms of distance.  
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 
Math::real  Direct (real lat1, real lon1, real azi1, real s12, real &lat2, real &lon2) const 
Math::real  Direct (real lat1, real lon1, real azi1, real s12, real &lat2, real &lon2, real &azi2) const 
Math::real  Direct (real lat1, real lon1, real azi1, real s12, real &lat2, real &lon2, real &azi2, real &m12) const 
Math::real  Direct (real lat1, real lon1, real azi1, real s12, real &lat2, real &lon2, real &azi2, real &M12, real &M21) const 
Math::real  Direct (real lat1, real lon1, real azi1, real s12, real &lat2, real &lon2, real &azi2, real &m12, real &M12, real &M21) const 
Direct geodesic problem specified in terms of arc length.  
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 
void  ArcDirect (real lat1, real lon1, real azi1, real a12, real &lat2, real &lon2) const 
void  ArcDirect (real lat1, real lon1, real azi1, real a12, real &lat2, real &lon2, real &azi2) const 
void  ArcDirect (real lat1, real lon1, real azi1, real a12, real &lat2, real &lon2, real &azi2, real &s12) const 
void  ArcDirect (real lat1, real lon1, real azi1, real a12, real &lat2, real &lon2, real &azi2, real &s12, real &m12) const 
void  ArcDirect (real lat1, real lon1, real azi1, real a12, real &lat2, real &lon2, real &azi2, real &s12, real &M12, real &M21) const 
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 
General version of the direct geodesic solution.  
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 
Inverse geodesic problem.  
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 
Math::real  Inverse (real lat1, real lon1, real lat2, real lon2, real &s12) const 
Math::real  Inverse (real lat1, real lon1, real lat2, real lon2, real &azi1, real &azi2) const 
Math::real  Inverse (real lat1, real lon1, real lat2, real lon2, real &s12, real &azi1, real &azi2) const 
Math::real  Inverse (real lat1, real lon1, real lat2, real lon2, real &s12, real &azi1, real &azi2, real &m12) const 
Math::real  Inverse (real lat1, real lon1, real lat2, real lon2, real &s12, real &azi1, real &azi2, real &M12, real &M21) const 
Math::real  Inverse (real lat1, real lon1, real lat2, real lon2, real &s12, real &azi1, real &azi2, real &m12, real &M12, real &M21) const 
General version of inverse geodesic solution.  
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 
Interface to GeodesicLine.  
GeodesicLine  Line (real lat1, real lon1, real azi1, unsigned caps=ALL) const 
GeodesicLine  InverseLine (real lat1, real lon1, real lat2, real lon2, unsigned caps=ALL) const 
GeodesicLine  DirectLine (real lat1, real lon1, real azi1, real s12, unsigned caps=ALL) const 
GeodesicLine  ArcDirectLine (real lat1, real lon1, real azi1, real a12, unsigned caps=ALL) const 
GeodesicLine  GenDirectLine (real lat1, real lon1, real azi1, bool arcmode, real s12_a12, unsigned caps=ALL) const 
Inspector functions.  
Math::real  MajorRadius () const 
Math::real  Flattening () const 
Math::real  EllipsoidArea () const 
Static Public Member Functions  
static const Geodesic &  WGS84 () 
Friends  
class  GeodesicLine 
Geodesic calculations
The shortest path between two points on a ellipsoid at (lat1, lon1) and (lat2, lon2) is called the geodesic. Its length is s12 and the geodesic from point 1 to point 2 has azimuths azi1 and azi2 at the two end points. (The azimuth is the heading measured clockwise from north. 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 if labeled φ, longitude λ (with λ_{12} = λ_{2} − λ_{1}), and azimuth α.
Given lat1, lon1, azi1, and s12, we can determine lat2, lon2, and azi2. This is the direct geodesic problem and its solution is given by the function Geodesic::Direct. (If 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 s12.)
Given lat1, lon1, lat2, and lon2, we can determine azi1, azi2, and s12. This is the 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 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 s12 to the second point. However it is sometimes useful instead to specify the arc length 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° 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°.
This class can also calculate several other quantities related to geodesics. These are:
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 m12, M12, 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:
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:
The calculations are accurate to better than 15 nm (15 nanometers) for the WGS84 ellipsoid. See Sec. 9 of arXiv:1102.1215v1 for details. The algorithms used by this class are based on series expansions using the flattening f as a small parameter. These are only accurate for f < 0.02; however reasonably accurate results will be obtained for f < 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.
f error 0.01 25 nm 0.02 30 nm 0.05 10 um 0.1 1.5 mm 0.2 300 mm
For very eccentric ellipsoids, use GeodesicExact instead.
The algorithms are described in
For more information on geodesics see Geodesics on an ellipsoid of revolution.
Example of use:
GeodSolve is a commandline utility providing access to the functionality of Geodesic and GeodesicLine.
Definition at line 171 of file Geodesic.hpp.
Bit masks for what calculations to do. These masks do double duty. They signify to the GeodesicLine::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.
Enumerator  

NONE  No capabilities, no output. 
LATITUDE  Calculate latitude lat2. (It's not necessary to include this as a capability to GeodesicLine because this is included by default.) 
LONGITUDE  Calculate longitude lon2. 
AZIMUTH  Calculate azimuths azi1 and azi2. (It's not necessary to include this as a capability to GeodesicLine because this is included by default.) 
DISTANCE  Calculate distance s12. 
DISTANCE_IN  Allow distance s12 to be used as input in the direct geodesic problem. 
REDUCEDLENGTH  Calculate reduced length m12. 
GEODESICSCALE  Calculate geodesic scales M12 and M21. 
AREA  Calculate area S12. 
LONG_UNROLL  Unroll lon2 in the direct calculation. 
ALL  All capabilities, calculate everything. (LONG_UNROLL is not included in this mask.) 
Definition at line 262 of file Geodesic.hpp.
GeographicLib::Geodesic::Geodesic  (  real  a, 
real  f  
) 
Constructor for a ellipsoid with
[in]  a  equatorial radius (meters). 
[in]  f  flattening of ellipsoid. Setting f = 0 gives a sphere. Negative f gives a prolate ellipsoid. 
GeographicErr  if a or (1 − f) a is not positive. 
Definition at line 42 of file Geodesic.cpp.
References GeographicLib::Math::isfinite().

inline 
Solve the direct geodesic problem where the length of the geodesic is specified in terms of distance.
[in]  lat1  latitude of point 1 (degrees). 
[in]  lon1  longitude of point 1 (degrees). 
[in]  azi1  azimuth at point 1 (degrees). 
[in]  s12  distance between point 1 and point 2 (meters); it can be negative. 
[out]  lat2  latitude of point 2 (degrees). 
[out]  lon2  longitude of point 2 (degrees). 
[out]  azi2  (forward) azimuth at point 2 (degrees). 
[out]  m12  reduced length of geodesic (meters). 
[out]  M12  geodesic scale of point 2 relative to point 1 (dimensionless). 
[out]  M21  geodesic scale of point 1 relative to point 2 (dimensionless). 
[out]  S12  area under the geodesic (meters^{2}). 
lat1 should be in the range [−90°, 90°]. The values of lon2 and azi2 returned are in the range [−180°, 180°].
If either point is at a pole, the azimuth is defined by keeping the longitude fixed, writing lat = ±(90° − ε), and taking the limit ε → 0+. An arc length greater that 180° 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°.)
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.
Definition at line 378 of file Geodesic.hpp.
Referenced by GeographicLib::AzimuthalEquidistant::Reverse(), and GeographicLib::CassiniSoldner::Reverse().

inline 
See the documentation for Geodesic::Direct.
Definition at line 392 of file Geodesic.hpp.

inline 
See the documentation for Geodesic::Direct.
Definition at line 404 of file Geodesic.hpp.

inline 
See the documentation for Geodesic::Direct.
Definition at line 416 of file Geodesic.hpp.

inline 
See the documentation for Geodesic::Direct.
Definition at line 428 of file Geodesic.hpp.

inline 
See the documentation for Geodesic::Direct.
Definition at line 441 of file Geodesic.hpp.

inline 
Solve the direct geodesic problem where the length of the geodesic is specified in terms of arc length.
[in]  lat1  latitude of point 1 (degrees). 
[in]  lon1  longitude of point 1 (degrees). 
[in]  azi1  azimuth at point 1 (degrees). 
[in]  a12  arc length between point 1 and point 2 (degrees); it can be negative. 
[out]  lat2  latitude of point 2 (degrees). 
[out]  lon2  longitude of point 2 (degrees). 
[out]  azi2  (forward) azimuth at point 2 (degrees). 
[out]  s12  distance between point 1 and point 2 (meters). 
[out]  m12  reduced length of geodesic (meters). 
[out]  M12  geodesic scale of point 2 relative to point 1 (dimensionless). 
[out]  M21  geodesic scale of point 1 relative to point 2 (dimensionless). 
[out]  S12  area under the geodesic (meters^{2}). 
lat1 should be in the range [−90°, 90°]. The values of lon2 and azi2 returned are in the range [−180°, 180°].
If either point is at a pole, the azimuth is defined by keeping the longitude fixed, writing lat = ±(90° − ε), and taking the limit ε → 0+. An arc length greater that 180° 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°.)
The following functions are overloaded versions of Geodesic::Direct which omit some of the output parameters.
Definition at line 490 of file Geodesic.hpp.

inline 
See the documentation for Geodesic::ArcDirect.
Definition at line 503 of file Geodesic.hpp.

inline 
See the documentation for Geodesic::ArcDirect.
Definition at line 514 of file Geodesic.hpp.

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See the documentation for Geodesic::ArcDirect.
Definition at line 525 of file Geodesic.hpp.

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See the documentation for Geodesic::ArcDirect.
Definition at line 537 of file Geodesic.hpp.

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See the documentation for Geodesic::ArcDirect.
Definition at line 550 of file Geodesic.hpp.

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See the documentation for Geodesic::ArcDirect.
Definition at line 563 of file Geodesic.hpp.
Math::real GeographicLib::Geodesic::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 
The general direct geodesic problem. Geodesic::Direct and Geodesic::ArcDirect are defined in terms of this function.
[in]  lat1  latitude of point 1 (degrees). 
[in]  lon1  longitude of point 1 (degrees). 
[in]  azi1  azimuth at point 1 (degrees). 
[in]  arcmode  boolean flag determining the meaning of the s12_a12. 
[in]  s12_a12  if 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. 
[in]  outmask  a bitor'ed combination of Geodesic::mask values specifying which of the following parameters should be set. 
[out]  lat2  latitude of point 2 (degrees). 
[out]  lon2  longitude of point 2 (degrees). 
[out]  azi2  (forward) azimuth at point 2 (degrees). 
[out]  s12  distance between point 1 and point 2 (meters). 
[out]  m12  reduced length of geodesic (meters). 
[out]  M12  geodesic scale of point 2 relative to point 1 (dimensionless). 
[out]  M21  geodesic scale of point 1 relative to point 2 (dimensionless). 
[out]  S12  area under the geodesic (meters^{2}). 
The Geodesic::mask values possible for outmask are
The function value a12 is always computed and returned and this equals s12_a12 is arcmode is true. If outmask includes Geodesic::DISTANCE and arcmode is false, then s12 = s12_a12. It is not necessary to include Geodesic::DISTANCE_IN in outmask; this is automatically included is arcmode is false.
With the Geodesic::LONG_UNROLL bit set, the quantity lon2 − lon1 indicates how many times and in what sense the geodesic encircles the ellipsoid.
Definition at line 123 of file Geodesic.cpp.
References DISTANCE_IN, and GeodesicLine.

inline 
Solve the inverse geodesic problem.
[in]  lat1  latitude of point 1 (degrees). 
[in]  lon1  longitude of point 1 (degrees). 
[in]  lat2  latitude of point 2 (degrees). 
[in]  lon2  longitude of point 2 (degrees). 
[out]  s12  distance between point 1 and point 2 (meters). 
[out]  azi1  azimuth at point 1 (degrees). 
[out]  azi2  (forward) azimuth at point 2 (degrees). 
[out]  m12  reduced length of geodesic (meters). 
[out]  M12  geodesic scale of point 2 relative to point 1 (dimensionless). 
[out]  M21  geodesic scale of point 1 relative to point 2 (dimensionless). 
[out]  S12  area under the geodesic (meters^{2}). 
lat1 and lat2 should be in the range [−90°, 90°]. The values of azi1 and azi2 returned are in the range [−180°, 180°].
If either point is at a pole, the azimuth is defined by keeping the longitude fixed, writing lat = ±(90° − ε), and taking the limit ε → 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.
Definition at line 673 of file Geodesic.hpp.
Referenced by GeographicLib::AzimuthalEquidistant::Forward(), and GeographicLib::CassiniSoldner::Forward().

inline 
See the documentation for Geodesic::Inverse.
Definition at line 685 of file Geodesic.hpp.

inline 
See the documentation for Geodesic::Inverse.
Definition at line 696 of file Geodesic.hpp.

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See the documentation for Geodesic::Inverse.
Definition at line 707 of file Geodesic.hpp.

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See the documentation for Geodesic::Inverse.
Definition at line 719 of file Geodesic.hpp.

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See the documentation for Geodesic::Inverse.
Definition at line 731 of file Geodesic.hpp.

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See the documentation for Geodesic::Inverse.
Definition at line 743 of file Geodesic.hpp.
Math::real GeographicLib::Geodesic::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 
The general inverse geodesic calculation. Geodesic::Inverse is defined in terms of this function.
[in]  lat1  latitude of point 1 (degrees). 
[in]  lon1  longitude of point 1 (degrees). 
[in]  lat2  latitude of point 2 (degrees). 
[in]  lon2  longitude of point 2 (degrees). 
[in]  outmask  a bitor'ed combination of Geodesic::mask values specifying which of the following parameters should be set. 
[out]  s12  distance between point 1 and point 2 (meters). 
[out]  azi1  azimuth at point 1 (degrees). 
[out]  azi2  (forward) azimuth at point 2 (degrees). 
[out]  m12  reduced length of geodesic (meters). 
[out]  M12  geodesic scale of point 2 relative to point 1 (dimensionless). 
[out]  M21  geodesic scale of point 1 relative to point 2 (dimensionless). 
[out]  S12  area under the geodesic (meters^{2}). 
The Geodesic::mask values possible for outmask are
The arc length is always computed and returned as the function value.
Definition at line 493 of file Geodesic.cpp.
References GeographicLib::Math::atan2d(), and AZIMUTH.
GeodesicLine GeographicLib::Geodesic::Line  (  real  lat1, 
real  lon1,  
real  azi1,  
unsigned  caps = ALL 

)  const 
Set up to compute several points on a single geodesic.
[in]  lat1  latitude of point 1 (degrees). 
[in]  lon1  longitude of point 1 (degrees). 
[in]  azi1  azimuth at point 1 (degrees). 
[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. 
lat1 should be in the range [−90°, 90°].
The Geodesic::mask values are
The default value of caps is Geodesic::ALL.
If the point is at a pole, the azimuth is defined by keeping lon1 fixed, writing lat1 = ±(90 − ε), and taking the limit ε → 0+.
Definition at line 118 of file Geodesic.cpp.
References GeodesicLine.
Referenced by GeographicLib::CassiniSoldner::Forward(), GeographicLib::CassiniSoldner::Reset(), and GeographicLib::Gnomonic::Reverse().
GeodesicLine GeographicLib::Geodesic::InverseLine  (  real  lat1, 
real  lon1,  
real  lat2,  
real  lon2,  
unsigned  caps = ALL 

)  const 
Define a GeodesicLine in terms of the inverse geodesic problem.
[in]  lat1  latitude of point 1 (degrees). 
[in]  lon1  longitude of point 1 (degrees). 
[in]  lat2  latitude of point 2 (degrees). 
[in]  lon2  longitude of point 2 (degrees). 
[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. 
This function sets point 3 of the GeodesicLine to correspond to point 2 of the inverse geodesic problem.
lat1 and lat2 should be in the range [−90°, 90°].
Definition at line 510 of file Geodesic.cpp.
References GeographicLib::Math::atan2d(), GeographicLib::Math::cbrt(), DISTANCE, DISTANCE_IN, GeodesicLine, GEODESICSCALE, GEOGRAPHICLIB_VOLATILE, GeographicLib::Math::hypot(), GeographicLib::Math::norm(), GeographicLib::Math::pi(), GeographicLib::Math::polyval(), REDUCEDLENGTH, and GeographicLib::Math::sq().
GeodesicLine GeographicLib::Geodesic::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 distance.
[in]  lat1  latitude of point 1 (degrees). 
[in]  lon1  longitude of point 1 (degrees). 
[in]  azi1  azimuth at point 1 (degrees). 
[in]  s12  distance between point 1 and point 2 (meters); it can be negative. 
[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. 
This function sets point 3 of the GeodesicLine to correspond to point 2 of the direct geodesic problem.
lat1 should be in the range [−90°, 90°].
Definition at line 149 of file Geodesic.cpp.
References GenDirectLine().
GeodesicLine GeographicLib::Geodesic::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 arc length.
[in]  lat1  latitude of point 1 (degrees). 
[in]  lon1  longitude of point 1 (degrees). 
[in]  azi1  azimuth at point 1 (degrees). 
[in]  a12  arc length between point 1 and point 2 (degrees); it can be negative. 
[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. 
This function sets point 3 of the GeodesicLine to correspond to point 2 of the direct geodesic problem.
lat1 should be in the range [−90°, 90°].
Definition at line 154 of file Geodesic.cpp.
References GeographicLib::Math::AngDiff(), GeographicLib::Math::AngRound(), AREA, GeographicLib::Math::degree(), DISTANCE, GenDirectLine(), GEODESICSCALE, GEOGRAPHICLIB_PANIC, GeographicLib::Math::hypot(), GeographicLib::Math::LatFix(), NONE, GeographicLib::Math::norm(), GeographicLib::Math::pi(), REDUCEDLENGTH, GeographicLib::Math::sincosd(), GeographicLib::Math::sq(), and std::swap().
GeodesicLine GeographicLib::Geodesic::GenDirectLine  (  real  lat1, 
real  lon1,  
real  azi1,  
bool  arcmode,  
real  s12_a12,  
unsigned  caps = ALL 

)  const 
Define a GeodesicLine in terms of the direct geodesic problem specified in terms of either distance or arc length.
[in]  lat1  latitude of point 1 (degrees). 
[in]  lon1  longitude of point 1 (degrees). 
[in]  azi1  azimuth at point 1 (degrees). 
[in]  arcmode  boolean flag determining the meaning of the s12_a12. 
[in]  s12_a12  if 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. 
[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. 
This function sets point 3 of the GeodesicLine to correspond to point 2 of the direct geodesic problem.
lat1 should be in the range [−90°, 90°].
Definition at line 136 of file Geodesic.cpp.
References GeographicLib::Math::AngNormalize(), GeographicLib::Math::AngRound(), DISTANCE_IN, GeodesicLine, and GeographicLib::Math::sincosd().
Referenced by ArcDirectLine(), and DirectLine().

inline 
Definition at line 942 of file Geodesic.hpp.
Referenced by GeographicLib::AzimuthalEquidistant::MajorRadius(), GeographicLib::CassiniSoldner::MajorRadius(), and GeographicLib::Gnomonic::MajorRadius().

inline 
Definition at line 948 of file Geodesic.hpp.
Referenced by GeographicLib::AzimuthalEquidistant::Flattening(), GeographicLib::CassiniSoldner::Flattening(), GeographicLib::Gnomonic::Flattening(), and GeographicLib::CassiniSoldner::Reset().

inline 
Definition at line 956 of file Geodesic.hpp.
References GeographicLib::Math::pi().

static 
A global instantiation of Geodesic with the parameters for the WGS84 ellipsoid.
Definition at line 89 of file Geodesic.cpp.
References GeographicLib::Constants::WGS84_a(), and GeographicLib::Constants::WGS84_f().

friend 
Definition at line 174 of file Geodesic.hpp.
Referenced by GenDirect(), GenDirectLine(), InverseLine(), and Line().