Package com.irurueta.navigation.geodesic

Class Geodesic

java.lang.Object

com.irurueta.navigation.geodesic.Geodesic

public class Geodesic
extends Object

Geodesic calculations. The shortest path between two points on an 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.). 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 th function direct(double, double, double, double). (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 inverse(double, double, double, double). 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 to 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 arcDirect(double, double, double, double). An arc length in excess of 180° indicates that the geodesic is not the 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:

• reduced length. If we fix the first point and increase azi1 by dazi1 (radians), the second point is displaced m12 dazi1 in the direction azi2 + 90°. The quantity 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, m12 + m21 = 0. On a flat surface, we have m12 = s12. The ratio s12/m12 gives the azimuthal scale for an azimuthal equidistant projection.
• geodesic scale. Consider a reference geodesic and a second geodesic parallel to this one at point 1 and separated by a small distance dt. The separation of the two geodesics at point 2 is M12dt where M12 is called the "geodesic scale". M21 is defined similarly (with the geodesics being parallel at point 2). On a flat surface, we have M12 = M21 = 1. The quantity 1/M12 gives the scale of the Cassini-Soldner projection.
• area. The area between the geodesic from point 1 to point 2 and the equation is represented by S12; it is the area, measured counter-clockwise, of the geodesic quadrilateral with corners (lat1, lon1), (0, lon1), (0, lon2), and (lat2, lon2). It can be used to compute the area of any single geodesic polygon.

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:

• s13 = s12 + s23
• a13 = a12 + a23
• S13 = S12 + S23
• m13 = m12 M23 + m23 M21
• M13 = M12 M23 − (1 − M12 M21) m23 / m12
• M31 = M32 M21 − (1 − M23 M32) m12 / m23

The results of the geodesic calculations are bundled up into a GeodesicData object which includes the input parameters and all the computed results, i.e. lat1, lon1, azi1, lat2, lon2, azi2, s12, a12, m12, M12, M21, S12. The functions direct(double, double, double, double, int), arcDirect(double, double, double, double, int) and inverse(double, double, double, double, int) include an optional final argument outmask which allows you to specify which results should be computed and returned. If you omit outmask, then the "standard" geodesic results are computed (latitudes, longitudes, azimuths, and distance). outmask is bitor'ed combination of GeodesicMask values. For example, if you wish just to compute the distance between two points you would call, e.g.,

     
     GeodesicData g = Geodesic.WGS84.inverse(lat1, lon1, lat2, lon2, GeodesicMask.DISTANCE);
     
 

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:

• lat1 = −lat2 (with neither point at a pole). If azi1 = azi2, the geodesic is unique. Otherwise there are two geodesics and the second one is obtained by setting [azi1, azi2] → [azi2, azi1], [M12, M21] → [M21, M12], S12 → −S12. (This occurs when the longitude difference is near ±180° for oblate ellipsoids.)
• lon2 = lon1 + 180° (with neither point at pole). If azi1 = 0° or ±180°, the geodesic is unique. Otherwise there are two geodesics and the second one is obtained by setting [azi1, azi2] → [−azi1, −azi2], S12 → − S12. (This occurs when lat2 is near −lat1 for prolate ellipsoids.)
• Points 1 and 2 at opposite poles. There are infinitely many geodesics which can be generated by setting [azi1, azi2] → [azi1, azi2] + [d, −d], for arbitrary d. (For spheres, this prescription applies when points 1 and 2 are antipodal.)
• s12 = 0 (coincident points). There are infinitely many geodesics which can be generated by setting [azi1, azi2] → [azi1, azi2] + [d, d], for arbitrary d.

The calculations are accurate to better than 15nm (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 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
 

The algorithms are described in

• C.F.F Karney, Algorithms for geodesics, J. Geodesy 87, 43–55 (2013)

Example of use:

     
     //Solve the direct geodesic problem.

     //This program reads in lines with lat1, lon1, azi1, s12 and prints out lines with lat2, lon2, azi2
     //(for the WGS84 ellipsoid).

     import java.util.*;
     import com.irurueta.navigation.geodesic.*;

     public class Direct {
         public static void main(String[] args) {
             try {
                 Scanner in = new Scanner(System.in);
                 double lat1, lon1, azi1, s12;
                 while (true) {
                     lat1 = in.nextDouble();
                     lon1 = in.nextDouble();
                     azi1 = in.nextDouble();
                     s12 = in.nextDouble();

                     GeodesicData g = Geodesic.WGS84.direct(lat1, lon1, azi1, s12);
                     System.out.println(g.lat2 + " " + g.lon2 + " " + g.azi2);
                 }
             } catch (Exception e) { }
         }
     }
     
 

Nested Class Summary

Nested Classes

Modifier and Type	Class	Description
private static class	Geodesic.InverseData
private static class	Geodesic.InverseStartV
private static class	Geodesic.Lambda12V
private static class	Geodesic.LengthsV

Field Summary

Fields

Modifier and Type	Field	Description
protected final double	a
private final double[]	a3x
protected final double	b
protected final double	c2
private final double[]	c3x
private final double[]	c4x
protected final double	e2
protected final double	ep2
private final double	etol2
protected final double	f
protected final double	f1
protected static final int	GEODESIC_ORDER	The order of the expansions used.
private static final int	MAXIT1
private static final int	MAXIT2
private final double	n
protected static final int	NA1
protected static final int	NA2
protected static final int	NA3
protected static final int	NA3X
protected static final int	NC1
protected static final int	NC1P
protected static final int	NC2
protected static final int	NC3
protected static final int	NC3X
protected static final int	NC4
protected static final int	NC4X
protected static final double	TINY	Underflow guard.
private static final double	TOL0
private static final double	TOL1	Increase multiplier in defn of TOl1 from 100 to 200 to fix inverse case 52.784459412564 0 -52.784459512563990912 179.634407464943777557
private static final double	TOL2
private static final double	TOLB	Check on bisection interval.
static final Geodesic	WGS84	A global instantiation of Geodesic with the parameters for the WGS84 ellipsoid.
private static final double	XTHRESH

Constructor Summary

Constructors

Constructor	Description
Geodesic(double a, double f)	Constructor for an ellipsoid with.

Method Summary

Modifier and Type	Method	Description
protected static double	a1m1f(double eps)
protected static double	a2m1f(double eps)
private void	a3coeff()
protected double	a3f(double eps)
GeodesicData	arcDirect(double lat1, double lon1, double azi1, double a12)	Solve the direct geodesic problem where the length of the geodesic is specified in terms of arc length.
GeodesicData	arcDirect(double lat1, double lon1, double azi1, double a12, int outmask)	Solve the direct geodesic problem where the length of the geodesic is specified in terms of arc length and with a subset of the geodesic results returned.
GeodesicLine	arcDirectLine(double lat1, double lon1, double azi1, double a12)	Define a GeodesicLine in terms of the direct geodesic problem specified in terms of arc length with all capabilities included.
GeodesicLine	arcDirectLine(double lat1, double lon1, double azi1, double a12, int caps)	Define a GeodesicLine in terms of the direct geodesic problem specified in terms of arc length with a subset of the capabilities included.
private static double	astroid(double x, double y)
protected static void	c1f(double eps, double[] c)
protected static void	c1pf(double eps, double[] c)
protected static void	c2f(double eps, double[] c)
private void	c3coeff()
protected void	c3f(double eps, double[] c)
private void	c4coeff()
protected void	c4f(double eps, double[] c)
GeodesicData	direct(double lat1, double lon1, double azi1, boolean arcmode, double s12A12, int outmask)	The general direct geodesic problem.
GeodesicData	direct(double lat1, double lon1, double azi1, double s12)	Solve the direct geodesic problem where the length of the geodesic is specified in terms of distance.
GeodesicData	direct(double lat1, double lon1, double azi1, double s12, int outmask)	Solve the direct geodesic problem where the length of the geodesic is specified in terms of distance with a subset of the geodesic results returned.
GeodesicLine	directLine(double lat1, double lon1, double azi1, double s12)	Define a GeodesicLine in terms of the direct geodesic problem specified in terms of distance with all capabilities included.
GeodesicLine	directLine(double lat1, double lon1, double azi1, double s12, int caps)	Define a GeodesicLine in terms of the direct geodesic problem specified in terms of distance with a subset of the capabilities included.
GeodesicLine	genDirectLine(double lat1, double lon1, double azi1, boolean arcmode, double s12A12, int caps)	Define a GeodesicLine in terms of the direct geodesic problem specified in terms of either distance or arc length with a subset of the capabilities included.
double	getEllipsoidArea()	Total area of ellipsoid in meters2.
double	getFlattening()	Gets the flattening of the ellipsoid.
double	getMajorRadius()	Gets the equatorial radius of the ellipsoid (meters).
GeodesicData	inverse(double lat1, double lon1, double lat2, double lon2)	Solve the inverse geodesic problem.
GeodesicData	inverse(double lat1, double lon1, double lat2, double lon2, int outmask)	Solve the inverse geodesic problem with a subset of the geodesic results returned.
private Geodesic.InverseData	inverseInt(double lat1, double lon1, double lat2, double lon2, int outmask)
GeodesicLine	inverseLine(double lat1, double lon1, double lat2, double lon2)	Define a GeodesicLine in terms of the inverse geodesic problem with all capabilities included.
GeodesicLine	inverseLine(double lat1, double lon1, double lat2, double lon2, int caps)	Define a GeodesicLine in terms of the inverse geodesic problem with a subet of the capabilities included.
private Geodesic.InverseStartV	inverseStart(double sbet1, double cbet1, double dn1, double sbet2, double cbet2, double dn2, double lam12, double slam12, double clam12, double[] c1a, double[] c2a)
private Geodesic.Lambda12V	lambda12(double sbet1, double cbet1, double dn1, double sbet2, double cbet2, double dn2, double salp1, double calp1, double slam120, double clam120, boolean diffp, double[] c1a, double[] c2a, double[] c3a)
private Geodesic.LengthsV	lengths(double eps, double sig12, double ssig1, double csig1, double dn1, double ssig2, double csig2, double dn2, double cbet1, double cbet2, int outmask, double[] c1a, double[] c2a)
GeodesicLine	line(double lat1, double lon1, double azi1)	Set up to compute several points on a single geodesic with all capabilities included.
GeodesicLine	line(double lat1, double lon1, double azi1, int caps)	Set up to compute several points on a single geodesic with a subset of the capabilities included.
private static Geodesic	safeInstance(double a, double f)	Safely creates an ellipsoid with.
protected static double	sinCosSeries(boolean sinp, double sinx, double cosx, double[] c)	This is a reformulation of the geodesic problem.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Details

WGS84

public static final Geodesic WGS84

A global instantiation of Geodesic with the parameters for the WGS84 ellipsoid.

GEODESIC_ORDER

protected static final int GEODESIC_ORDER

The order of the expansions used.

See Also:

• Constant Field Values

NA1

protected static final int NA1

See Also:

• Constant Field Values

NC1

protected static final int NC1

See Also:

• Constant Field Values

NC1P

protected static final int NC1P

See Also:

• Constant Field Values

NA2

protected static final int NA2

See Also:

• Constant Field Values

NC2

protected static final int NC2

See Also:

• Constant Field Values

NA3

protected static final int NA3

See Also:

• Constant Field Values

NA3X

protected static final int NA3X

See Also:

• Constant Field Values

NC3

protected static final int NC3

See Also:

• Constant Field Values

NC3X

protected static final int NC3X

See Also:

• Constant Field Values

NC4

protected static final int NC4

See Also:

• Constant Field Values

NC4X

protected static final int NC4X

See Also:

• Constant Field Values

TINY

protected static final double TINY

Underflow guard. We require TINY * epsilon() < 0 and TINY + epsilon() == epsilon()

MAXIT1

private static final int MAXIT1

See Also:

• Constant Field Values

MAXIT2

private static final int MAXIT2

See Also:

• Constant Field Values

TOL0

private static final double TOL0

TOL1

private static final double TOL1

Increase multiplier in defn of TOl1 from 100 to 200 to fix inverse case 52.784459412564 0 -52.784459512563990912 179.634407464943777557

TOL2

private static final double TOL2

TOLB

private static final double TOLB

Check on bisection interval.

XTHRESH

private static final double XTHRESH

a

protected final double a

f

protected final double f

f1

protected final double f1

e2

protected final double e2

ep2

protected final double ep2

b

protected final double b

c2

protected final double c2

n

private final double n

etol2

private final double etol2

a3x

private final double[] a3x

c3x

private final double[] c3x

c4x

private final double[] c4x

Constructor Details

Geodesic

public Geodesic(double a,
 double f)
         throws GeodesicException

Constructor for an ellipsoid with.

Parameters:

a - equatorial radius (meters).

f - flattening of ellipsoid. Setting f = 0 gives a sphere. Negative f gives a prolate ellipsoid.

Throws:

GeodesicException - if a or (1 − f) a is not positive.

Method Details

direct

public GeodesicData direct(double lat1,
 double lon1,
 double azi1,
 double s12)

Solve the direct geodesic problem where the length of the geodesic is specified in terms of distance. 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 than 180° signifies a geodesic which is not the shortest path. (For a prolate ellipsoid, an additional condition is necessary for the shortest path: the longitudinal extent must not exceed of 180°.)

Parameters:

lat1 - latitude of point 1 (degrees). lat1 should be in the range [−90°, 90°].

lon1 - longitude of point 1 (degrees).

azi1 - azimuth at point 1 (degrees).

s12 - distance between point 1 and point 2 (meters); it can be negative.

Returns:

a GeodesicData object with the following fields: lat1, lon1, azi1, lat2, lon2, azi2, s12, a12. The values of lon2 and azi2 returned are in the range [−180°, 180°].

direct

public GeodesicData direct(double lat1,
 double lon1,
 double azi1,
 double s12,
 int outmask)

Solve the direct geodesic problem where the length of the geodesic is specified in terms of distance with a subset of the geodesic results returned.

Parameters:

lat1 - latitude of point 1 (degrees).

lon1 - longitude of point 1 (degrees).

azi1 - azimuth at point 1 (degrees).

s12 - distance between point 1 and point 2 (meters); it can be negative.

outmask - a bitor'ed combination of GeodesicMask values specifying which results should be returned.

Returns:

a GeodesicData object with the fields specified by outmask computed. lat1, lon1, azi1, s12, and a12 are always included in the returned result. The value of lon2 returned is in the range [−180°, 180°], unless the outmask includes the GeodesicMask.LONG_UNROLL flag.

arcDirect

public GeodesicData arcDirect(double lat1,
 double lon1,
 double azi1,
 double a12)

Solve the direct geodesic problem where the length of the geodesic is specified in terms of arc length. 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 than 180° signifies a geodesic which is not the shortest path. (For a prolate ellipsoid, an additional condition is necessary for the shortest path: the longitudinal extent must not exceed of 180°.)

Parameters:

lat1 - latitude of point 1 (degrees). lat1 should be in the range [−90°, 90°].

lon1 - longitude of point 1 (degrees).

azi1 - azimuth at point 1 (degrees).

a12 - arc length between point 1 and point 2 (degrees); it can be negative.

Returns:

a GeodesicData object with the following fields: lat1, lon1, azi1, lat2, lon2, azi2, s12, a12. The values of lon2 and azi2 returned are in the range [−180°, 180°].

arcDirect

public GeodesicData arcDirect(double lat1,
 double lon1,
 double azi1,
 double a12,
 int outmask)

Solve the direct geodesic problem where the length of the geodesic is specified in terms of arc length and with a subset of the geodesic results returned.

Parameters:

lat1 - latitude of point 1 (degrees).

lon1 - longitude of point 1 (degrees).

azi1 - azimuth at point 1 (degrees).

a12 - arc length between point 1 and point 2 (degrees); it can be negative.

outmask - a bitor'ed combination of GeodesicMask values specifying which results should be returned.

Returns:

a GeodesicData object with the fields specified by outmask computed. lat1, lon1, azi1, and a12 are always included in the returned result. The value of lon2 returned is in the range [−180°, 180°], unless the outmask includes the GeodesicMask.LONG_UNROLL flag.

direct

public GeodesicData direct(double lat1,
 double lon1,
 double azi1,
 boolean arcmode,
 double s12A12,
 int outmask)

The general direct geodesic problem. direct(double, double, double, double) and arcDirect(double, double, double, double) are defined in terms of this function. The GeodesicMask values possible for outmask are

• outmask |= GeodesicMask.LATITUDE for the latitude lat2;
• outmask |= GeodesicMask.LONGITUDE for the longitude lon2;
• outmask |= GeodesicMask.AZIMUTH for the azimuth azi2;
• outmask |= GeodesicMask.DISTANCE for the distance s12;
• outmask |= GeodesicMask.REDUCED_LENGTH for the reduced length m12;
• outmask |= GeodesicMask.GEODESIC_SCALE for the geodesic scales M12 and M21;
• outmask |= GeodesicMask.AREA for the area S12;
• outmask |= GeodesicMask.LONG_UNROLL, if set then lon1 is unchanged and lon2 − lon1 indicates how many times and in what sense the geodesic encircles the ellipsoid. Otherwise lon1 and lon2 are both reduced to the range [−180°, 180°].

The function value a12 is always computed and returned and this equals s12A12 is arcmode is true. If outmask includes GeodesicMask.DISTANCE and arcmode is false, then s12 = s12A12. It is not necessary to include GeodesicMask.DISTANCE_IN in outmask; this is automatically included if arcmode is false.

Parameters:

lat1 - latitude of point 1 (degrees).

lon1 - longitude of point 1 (degrees).

azi1 - azimuth at point 1 (degrees).

arcmode - boolean flag determining the meaning of the s12A12.

s12A12 - 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.

outmask - a bitor'ed combination of GeodesicMask values specifying which results should be returned.

Returns:

a GeodesicData object with the fields specified by outmask computed.

directLine

public GeodesicLine directLine(double lat1,
 double lon1,
 double azi1,
 double s12)

Define a GeodesicLine in terms of the direct geodesic problem specified in terms of distance with all capabilities included. This function sets point 3 of the GeodesicLine to correspond to point 2 of the direct geodesic problem.

Parameters:

lat1 - latitude of point 1 (degrees). lat1 should be in the range [−90°, 90°].

lon1 - longitude of point 1 (degrees).

azi1 - azimuth at point 1 (degrees).

s12 - distance between point 1 and point 2 (meters); it can be negative.

Returns:

a GeodesicLine object.

directLine

public GeodesicLine directLine(double lat1,
 double lon1,
 double azi1,
 double s12,
 int caps)

Define a GeodesicLine in terms of the direct geodesic problem specified in terms of distance with a subset of the capabilities included. This function sets point 3 of the GeodesicLine to correspond to point 2 of the direct geodesic problem.

Parameters:

lat1 - latitude of point 1 (Degrees). lat1 should be in the range [−90°, 90°].

lon1 - longitude of point 1 (degrees).

azi1 - azimuth at point 1 (degrees).

s12 - distance between point 1 and point 2 (meters); it can be negative.

caps - bitor'ed combination of GeodesicMask values specifying the capabilities the GeodesicLine object should possess, i.e., which quantities can be returned in calls to GeodesicLine.position(double).

Returns:

a GeodesicLine object.

arcDirectLine

public GeodesicLine arcDirectLine(double lat1,
 double lon1,
 double azi1,
 double a12)

Define a GeodesicLine in terms of the direct geodesic problem specified in terms of arc length with all capabilities included. This function sets point 3 of the GeodesicLine to correspond to point 2 of the direct geodesic problem.

Parameters:

lat1 - latitude of point 1 (degrees). lat1 should be in the range [−90°, 90°].

lon1 - longitude of point 1 (degrees).

azi1 - azimuth at point 1 (degrees).

a12 - arc length between point 1 and point 2 (degrees); it can be negative.

Returns:

a GeodesicLine object.

arcDirectLine

public GeodesicLine arcDirectLine(double lat1,
 double lon1,
 double azi1,
 double a12,
 int caps)

Define a GeodesicLine in terms of the direct geodesic problem specified in terms of arc length with a subset of the capabilities included. This function sets point 3 of the GeodesicLine to correspond to point 2 of the direct geodesic problem.

Parameters:

lat1 - latitude of point 1 (degrees). lat1 should be in the range [−90°, 90°].

lon1 - longitude of point 1 (degrees).

azi1 - azimuth at point 1 (degrees).

a12 - arc length between point 1 and point 2 (degrees); it can be negative.

caps - bitor'ed combination of GeodesicMask values specifying the capabilities the GeodesicLine object should possess, i.e., which quantities can be returned in calls to GeodesicLine.position(double).

Returns:

a GeodesicLine object.

genDirectLine

public GeodesicLine genDirectLine(double lat1,
 double lon1,
 double azi1,
 boolean arcmode,
 double s12A12,
 int caps)

Define a GeodesicLine in terms of the direct geodesic problem specified in terms of either distance or arc length with a subset of the capabilities included. This function sets point 3 of the GeodesicLine to correspond to point 2 of the direct geodesic problem.

Parameters:

lat1 - latitude of point 1 (degrees). lat1 should be in the range [−90°, 90°].

lon1 - longitude of point 1 (degrees).

azi1 - azimuth at point 1 (degrees).

arcmode - boolean flag determining the meaning of the s12A12.

s12A12 - 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.

caps - bitor'ed combination of GeodesicMask values specifying the capabilities the GeodesicLine object should possess, i.e., which quantities can be returned in calls to GeodesicLine.position(double).

Returns:

a GeodesicLine object.

inverse

public GeodesicData inverse(double lat1,
 double lon1,
 double lat2,
 double lon2)

Solve the inverse geodesic problem. 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), 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.

Parameters:

lat1 - latitude of point 1 (degrees).

lon1 - longitude of point 1 (degrees).

lat2 - latitude of point 2 (degrees).

lon2 - longitude of point 2 (degrees).

Returns:

a GeodesicData object with the following fields: lat1, lon1, azi1, lat2, lon2, azi2, s12, a12.

inverse

public GeodesicData inverse(double lat1,
 double lon1,
 double lat2,
 double lon2,
 int outmask)

Solve the inverse geodesic problem with a subset of the geodesic results returned. The GeodesicMask values possible for outmask are

• outmask |= GeodesicMask.DISTANCE for the distance s12;
• outmask |= GeodesicMask.AZIMUTH for the latitude azi2.
• outmask |= GeodesicMask.REDUCED_LENGTH for the reduced length m12;
• outmask |= GeodesicMask.GEODESIC_SCALE for the geodesic scales M12 and M21;
• outmask |= GeodesicMask.AREA for the area S12;
• outmask |= GeodesicMask.ALL for all of the above.
• outmask |= GeodesicMask.LONG_UNROLL, if set then lon1 is unchanged and lon2 − lon1 indicates whether the geodesic is east going or west going. Otherwise lon1 and lon2 are both reduced to the range [−180°, 180°].

Parameters:

lat1 - latitude of point 1 (degrees).

lon1 - longitude of point 1 (degrees).

lat2 - latitude of point 2 (degrees).

lon2 - longitude of point 2 (degrees)

outmask - a bitor'ed combination of GeodesicMask values specifying which results should be returned.

Returns:

a GeodesicData object with the fields specified by outmask computed. lat1, lon1, lat2, lon2, and a12 are always included in the returned result.

inverseLine

public GeodesicLine inverseLine(double lat1,
 double lon1,
 double lat2,
 double lon2)

Define a GeodesicLine in terms of the inverse geodesic problem with all capabilities included. This function sets point 3 of the GeodesicLine to correspond to point 2 of the inverse geodesic problem. lat2 and lat2 should be in the range [−90°, 90°].

Parameters:

lat1 - latitude of point 1 (degrees).

lon1 - longitude of point 1 (degrees).

lat2 - latitude of point 2 (degrees).

lon2 - longitude of point 2 (degrees).

Returns:

a GeodesicLine object.

inverseLine

public GeodesicLine inverseLine(double lat1,
 double lon1,
 double lat2,
 double lon2,
 int caps)

Define a GeodesicLine in terms of the inverse geodesic problem with a subet of the capabilities included. 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°].

Parameters:

lat1 - latitude of point 1 (degrees).

lon1 - longitude of point 1 (degrees).

lat2 - latitude of point 2 (degrees).

lon2 - longitude of point 2 (degrees).

caps - bitor'ed combination of GeodesicMask values specifying the capabilities the GeodesicLine object should possess, i.e., which quantities can be returned in calls to GeodesicLine.position(double).

Returns:

a GeodesicLine object.

line

public GeodesicLine line(double lat1,
 double lon1,
 double azi1)

Set up to compute several points on a single geodesic with all capabilities included. If the point is at a pole, the azimuth is defined by keeping the lon1 fixed, writing lat1 = ± (90 − ε), taking the limit ε → 0+.

Parameters:

lat1 - latitude of point 1 (degrees). lat1 should be in the range [−90°, 90°].

lon1 - longitude of point 1 (degrees).

azi1 - azimuth at point 1 (degrees).

Returns:

a GeodesicLine object. The full set of capabilities is included.

line

public GeodesicLine line(double lat1,
 double lon1,
 double azi1,
 int caps)

Set up to compute several points on a single geodesic with a subset of the capabilities included. The GeodesicMask values are:

• caps |= GeodesicMask.LATITUDE for the latitude lat2; this is added automatically;
• caps |= GeodesicMask.LONGITUDE for the longitude lon2;
• caps |= GeodesicMask.AZIMUTH for the azimuth azi2; this is added automatically;
• caps |= GeodesicMask.DISTANCE for the distance s12;
• caps |= GeodesicMask.REDUCED_LENGTH for the reduced length m12;
• caps |= GeodesicMask.GEODESIC_SCALE for the geodesic scales M12 and M21;
• caps |= GeodesicMask.AREA for the area S12;
• caps |= GeodesicMask.DISTANCE_IN permits the length of the geodesic to be given in terms of s12; without this capability the length can only be specified in terms of arc length;
• caps |= GeodesicMask.ALL for all of the above.

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+.

Parameters:

lat1 - latitude of point 1 (degrees).

lon1 - longitude of point 1 (degrees).

azi1 - azimuth at point 1 (degrees).

caps - bitor'ed combination of GeodesicMask values specifying which quantities can be returned in calls to GeodesicLine.position(double).

Returns:

a GeodesicLine object.

getMajorRadius

public double getMajorRadius()

Gets the equatorial radius of the ellipsoid (meters). This is the value used in the constructor.

Returns:

a the equatorial radius of the ellipsoid (meters).

getFlattening

public double getFlattening()

Gets the flattening of the ellipsoid. This is the value used in the constructor.

Returns:

f the flattening of the ellipsoid.

getEllipsoidArea

public double getEllipsoidArea()

Total area of ellipsoid in meters². The area of a polygon encircling a pole can be found by adding ellipsoidArea()/2 to the sum of S12 for each side of the polygon.

Returns:

total area of ellipsoid in meters².

sinCosSeries

protected static double sinCosSeries(boolean sinp,
 double sinx,
 double cosx,
 double[] c)

This is a reformulation of the geodesic problem. The notation is as follows: - at a general point (no suffix or 1 or 2 as suffix) - phi = latitude - beta = latitude on auxiliary sphere - omega = longitude on auxiliary sphere - lambda = longitude - alpha = azimuth of great circle - sigma = arc length along great circle - s = distance - tau = scaled distance (= sigma at multiples of pi/2) - at northwards equator crossing - beta = phi = 0 - omega = lambda = 0 - alpha = alpha0 - sigma = s = 0 - a 12 suggix means a difference, e.g., s12 = s2 - s1. - s and c prefixes mean sin and cos

Parameters:

sinp - sinus of p

sinx - sinus of x

cosx - cosinus of x

c - c

Returns:

sin cos series.

a3f

protected double a3f(double eps)

c3f

protected void c3f(double eps,
 double[] c)

c4f

protected void c4f(double eps,
 double[] c)

a1m1f

protected static double a1m1f(double eps)

c1f

protected static void c1f(double eps,
 double[] c)

c1pf

protected static void c1pf(double eps,
 double[] c)

a2m1f

protected static double a2m1f(double eps)

c2f

protected static void c2f(double eps,
 double[] c)

a3coeff

private void a3coeff()

c3coeff

private void c3coeff()

c4coeff

private void c4coeff()

inverseInt

private Geodesic.InverseData inverseInt(double lat1,
 double lon1,
 double lat2,
 double lon2,
 int outmask)

safeInstance

private static Geodesic safeInstance(double a,
 double f)

Safely creates an ellipsoid with.

Parameters:

a - equatorial radius (meters).

f - flattening of ellipsoid. Setting f = 0 gives a sphere. Negative f gives a prolate ellipsoid.

Returns:

a new Geodesic instance or null if something fails.

lengths

private Geodesic.LengthsV lengths(double eps,
 double sig12,
 double ssig1,
 double csig1,
 double dn1,
 double ssig2,
 double csig2,
 double dn2,
 double cbet1,
 double cbet2,
 int outmask,
 double[] c1a,
 double[] c2a)

astroid

private static double astroid(double x,
 double y)

inverseStart

private Geodesic.InverseStartV inverseStart(double sbet1,
 double cbet1,
 double dn1,
 double sbet2,
 double cbet2,
 double dn2,
 double lam12,
 double slam12,
 double clam12,
 double[] c1a,
 double[] c2a)

lambda12

private Geodesic.Lambda12V lambda12(double sbet1,
 double cbet1,
 double dn1,
 double sbet2,
 double cbet2,
 double dn2,
 double salp1,
 double calp1,
 double slam120,
 double clam120,
 boolean diffp,
 double[] c1a,
 double[] c2a,
 double[] c3a)