GeographicLib
1.35
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Geodesic calculations More...
#include <GeographicLib/Geodesic.hpp>
Public Types | |
enum | mask { NONE, LATITUDE, LONGITUDE, AZIMUTH, DISTANCE, DISTANCE_IN, REDUCEDLENGTH, GEODESICSCALE, AREA, 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 throw () |
Math::real | Direct (real lat1, real lon1, real azi1, real s12, real &lat2, real &lon2) const throw () |
Math::real | Direct (real lat1, real lon1, real azi1, real s12, real &lat2, real &lon2, real &azi2) const throw () |
Math::real | Direct (real lat1, real lon1, real azi1, real s12, real &lat2, real &lon2, real &azi2, real &m12) const throw () |
Math::real | Direct (real lat1, real lon1, real azi1, real s12, real &lat2, real &lon2, real &azi2, real &M12, real &M21) const throw () |
Math::real | Direct (real lat1, real lon1, real azi1, real s12, real &lat2, real &lon2, real &azi2, real &m12, real &M12, real &M21) const throw () |
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 throw () |
void | ArcDirect (real lat1, real lon1, real azi1, real a12, real &lat2, real &lon2) const throw () |
void | ArcDirect (real lat1, real lon1, real azi1, real a12, real &lat2, real &lon2, real &azi2) const throw () |
void | ArcDirect (real lat1, real lon1, real azi1, real a12, real &lat2, real &lon2, real &azi2, real &s12) const throw () |
void | ArcDirect (real lat1, real lon1, real azi1, real a12, real &lat2, real &lon2, real &azi2, real &s12, real &m12) const throw () |
void | ArcDirect (real lat1, real lon1, real azi1, real a12, real &lat2, real &lon2, real &azi2, real &s12, real &M12, real &M21) const throw () |
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 throw () |
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 throw () |
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 throw () |
Math::real | Inverse (real lat1, real lon1, real lat2, real lon2, real &s12) const throw () |
Math::real | Inverse (real lat1, real lon1, real lat2, real lon2, real &azi1, real &azi2) const throw () |
Math::real | Inverse (real lat1, real lon1, real lat2, real lon2, real &s12, real &azi1, real &azi2) const throw () |
Math::real | Inverse (real lat1, real lon1, real lat2, real lon2, real &s12, real &azi1, real &azi2, real &m12) const throw () |
Math::real | Inverse (real lat1, real lon1, real lat2, real lon2, real &s12, real &azi1, real &azi2, real &M12, real &M21) const throw () |
Math::real | Inverse (real lat1, real lon1, real lat2, real lon2, real &s12, real &azi1, real &azi2, real &m12, real &M12, real &M21) const throw () |
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 throw () |
Interface to GeodesicLine. | |
GeodesicLine | Line (real lat1, real lon1, real azi1, unsigned caps=ALL) const throw () |
Inspector functions. | |
Math::real | MajorRadius () const throw () |
Math::real | Flattening () const throw () |
Math::real | EllipsoidArea () const throw () |
Static Public Attributes | |
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 λ12 = λ2 − λ1), 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 major 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 geodesic.
Example of use:
GeodSolve is a command-line utility providing access to the functionality of Geodesic and GeodesicLine.
Definition at line 169 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 | |
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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. |
ALL |
All capabilities, calculate everything. |
Definition at line 278 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. If f > 1, set flattening to 1/f. |
GeographicErr | if a or (1 − f ) a is not positive. |
Definition at line 55 of file Geodesic.cpp.
References GeographicLib::Math::isfinite().
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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 (meters2). |
lat1 should be in the range [−90°, 90°]; lon1 and azi1 should be in the range [−540°, 540°). 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 390 of file Geodesic.hpp.
Referenced by main().
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See the documentation for Geodesic::Direct.
Definition at line 404 of file Geodesic.hpp.
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See the documentation for Geodesic::Direct.
Definition at line 416 of file Geodesic.hpp.
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See the documentation for Geodesic::Direct.
Definition at line 428 of file Geodesic.hpp.
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See the documentation for Geodesic::Direct.
Definition at line 440 of file Geodesic.hpp.
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See the documentation for Geodesic::Direct.
Definition at line 453 of file Geodesic.hpp.
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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 (meters2). |
lat1 should be in the range [−90°, 90°]; lon1 and azi1 should be in the range [−540°, 540°). 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 503 of file Geodesic.hpp.
Referenced by main().
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See the documentation for Geodesic::ArcDirect.
Definition at line 516 of file Geodesic.hpp.
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See the documentation for Geodesic::ArcDirect.
Definition at line 527 of file Geodesic.hpp.
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See the documentation for Geodesic::ArcDirect.
Definition at line 538 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.
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See the documentation for Geodesic::ArcDirect.
Definition at line 576 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 | ||
throw | ( | ||
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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 (meters2). |
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.
Definition at line 121 of file Geodesic.cpp.
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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 (meters2). |
lat1 and lat2 should be in the range [−90°, 90°]; lon1 and lon2 should be in the range [−540°, 540°). 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 681 of file Geodesic.hpp.
Referenced by main().
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See the documentation for Geodesic::Inverse.
Definition at line 693 of file Geodesic.hpp.
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See the documentation for Geodesic::Inverse.
Definition at line 704 of file Geodesic.hpp.
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See the documentation for Geodesic::Inverse.
Definition at line 715 of file Geodesic.hpp.
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See the documentation for Geodesic::Inverse.
Definition at line 727 of file Geodesic.hpp.
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See the documentation for Geodesic::Inverse.
Definition at line 739 of file Geodesic.hpp.
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See the documentation for Geodesic::Inverse.
Definition at line 751 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 | ||
throw | ( | ||
) |
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 (meters2). |
The Geodesic::mask values possible for outmask are
The arc length is always computed and returned as the function value.
Definition at line 134 of file Geodesic.cpp.
References GeographicLib::Math::AngDiff(), GeographicLib::Math::AngNormalize(), GeographicLib::Math::hypot(), and GeographicLib::Math::sq().
GeodesicLine GeographicLib::Geodesic::Line | ( | real | lat1, |
real | lon1, | ||
real | azi1, | ||
unsigned | caps = ALL |
||
) | const | ||
throw | ( | ||
) |
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°]; lon1 and azi1 should be in the range [−540°, 540°).
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 116 of file Geodesic.cpp.
Referenced by main().
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Definition at line 859 of file Geodesic.hpp.
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Definition at line 865 of file Geodesic.hpp.
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Definition at line 881 of file Geodesic.hpp.
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Definition at line 172 of file Geodesic.hpp.
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A global instantiation of Geodesic with the parameters for the WGS84 ellipsoid.
Definition at line 889 of file Geodesic.hpp.