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 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:

    • NA1

      protected static final int NA1
      See Also:

    • NC1

      protected static final int NC1
      See Also:

    • NC1P

      protected static final int NC1P
      See Also:

    • NA2

      protected static final int NA2
      See Also:

    • NC2

      protected static final int NC2
      See Also:

    • NA3

      protected static final int NA3
      See Also:

    • NA3X

      protected static final int NA3X
      See Also:

    • NC3

      protected static final int NC3
      See Also:

    • NC3X

      protected static final int NC3X
      See Also:

    • NC4

      protected static final int NC4
      See Also:

    • NC4X

      protected static final int NC4X
      See Also:

    • 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:

    • MAXIT2

      private static final int MAXIT2
      See Also:

    • 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 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
      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: 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)