OneAxisEllipsoid.java

  1. /* Copyright 2002-2024 CS GROUP
  2.  * Licensed to CS GROUP (CS) under one or more
  3.  * contributor license agreements.  See the NOTICE file distributed with
  4.  * this work for additional information regarding copyright ownership.
  5.  * CS licenses this file to You under the Apache License, Version 2.0
  6.  * (the "License"); you may not use this file except in compliance with
  7.  * the License.  You may obtain a copy of the License at
  8.  *
  9.  *   http://www.apache.org/licenses/LICENSE-2.0
  10.  *
  11.  * Unless required by applicable law or agreed to in writing, software
  12.  * distributed under the License is distributed on an "AS IS" BASIS,
  13.  * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  14.  * See the License for the specific language governing permissions and
  15.  * limitations under the License.
  16.  */
  17. package org.orekit.bodies;

  19. import java.io.Serializable;
  20. import java.util.function.DoubleFunction;

  22. import org.hipparchus.CalculusFieldElement;
  23. import org.hipparchus.Field;
  24. import org.hipparchus.analysis.differentiation.DerivativeStructure;
  25. import org.hipparchus.analysis.solvers.BracketingNthOrderBrentSolver;
  26. import org.hipparchus.geometry.euclidean.threed.FieldLine;
  27. import org.hipparchus.geometry.euclidean.threed.FieldVector3D;
  28. import org.hipparchus.geometry.euclidean.threed.Line;
  29. import org.hipparchus.geometry.euclidean.threed.Vector3D;
  30. import org.hipparchus.geometry.euclidean.twod.Vector2D;
  31. import org.hipparchus.util.FastMath;
  32. import org.hipparchus.util.FieldSinCos;
  33. import org.hipparchus.util.MathArrays;
  34. import org.hipparchus.util.MathUtils;
  35. import org.hipparchus.util.SinCos;
  36. import org.orekit.frames.FieldStaticTransform;
  37. import org.orekit.frames.Frame;
  38. import org.orekit.frames.StaticTransform;
  39. import org.orekit.frames.Transform;
  40. import org.orekit.time.AbsoluteDate;
  41. import org.orekit.time.FieldAbsoluteDate;
  42. import org.orekit.utils.PVCoordinates;
  43. import org.orekit.utils.TimeStampedPVCoordinates;


  46. /** Modeling of a one-axis ellipsoid.

  48.  * <p>One-axis ellipsoids is a good approximate model for most planet-size
  49.  * and larger natural bodies. It is the equilibrium shape reached by
  50.  * a fluid body under its own gravity field when it rotates. The symmetry
  51.  * axis is the rotation or polar axis.</p>

  53.  * @author Luc Maisonobe
  54.  * @author Guylaine Prat
  55.  */
  56. public class OneAxisEllipsoid extends Ellipsoid implements BodyShape {

  58.     /** Serializable UID. */
  59.     private static final long serialVersionUID = 20130518L;

  61.     /** Threshold for polar and equatorial points detection. */
  62.     private static final double ANGULAR_THRESHOLD = 1.0e-4;

  64.     /** Body frame related to body shape. */
  65.     private final Frame bodyFrame;

  67.     /** Equatorial radius power 2. */
  68.     private final double ae2;

  70.     /** Polar radius power 2. */
  71.     private final double ap2;

  73.     /** Flattening. */
  74.     private final double f;

  76.     /** Eccentricity. */
  77.     private final double e;

  79.     /** Eccentricity squared. */
  80.     private final double e2;

  82.     /** 1 minus flatness. */
  83.     private final double g;

  85.     /** g squared. */
  86.     private final double g2;

  88.     /** Convergence limit. */
  89.     private double angularThreshold;

  91.     /** Simple constructor.
  92.      * <p>Standard values for Earth models can be found in the {@link org.orekit.utils.Constants Constants} class:</p>
  93.      * <table border="1" style="background-color:#f5f5dc;">
  94.      * <caption>Ellipsoid Models</caption>
  95.      * <tr style="background-color:#c9d5c9;"><th>model</th><th>a<sub>e</sub> (m)</th> <th>f</th></tr>
  96.      * <tr><td style="background-color:#c9d5c9; padding:5px">GRS 80</td>
  97.      *     <td>{@link org.orekit.utils.Constants#GRS80_EARTH_EQUATORIAL_RADIUS Constants.GRS80_EARTH_EQUATORIAL_RADIUS}</td>
  98.      *     <td>{@link org.orekit.utils.Constants#GRS80_EARTH_FLATTENING Constants.GRS80_EARTH_FLATTENING}</td></tr>
  99.      * <tr><td style="background-color:#c9d5c9; padding:5px">WGS84</td>
  100.      *     <td>{@link org.orekit.utils.Constants#WGS84_EARTH_EQUATORIAL_RADIUS Constants.WGS84_EARTH_EQUATORIAL_RADIUS}</td>
  101.      *     <td>{@link org.orekit.utils.Constants#WGS84_EARTH_FLATTENING Constants.WGS84_EARTH_FLATTENING}</td></tr>
  102.      * <tr><td style="background-color:#c9d5c9; padding:5px">IERS96</td>
  103.      *     <td>{@link org.orekit.utils.Constants#IERS96_EARTH_EQUATORIAL_RADIUS Constants.IERS96_EARTH_EQUATORIAL_RADIUS}</td>
  104.      *     <td>{@link org.orekit.utils.Constants#IERS96_EARTH_FLATTENING Constants.IERS96_EARTH_FLATTENING}</td></tr>
  105.      * <tr><td style="background-color:#c9d5c9; padding:5px">IERS2003</td>
  106.      *     <td>{@link org.orekit.utils.Constants#IERS2003_EARTH_EQUATORIAL_RADIUS Constants.IERS2003_EARTH_EQUATORIAL_RADIUS}</td>
  107.      *     <td>{@link org.orekit.utils.Constants#IERS2003_EARTH_FLATTENING Constants.IERS2003_EARTH_FLATTENING}</td></tr>
  108.      * <tr><td style="background-color:#c9d5c9; padding:5px">IERS2010</td>
  109.      *     <td>{@link org.orekit.utils.Constants#IERS2010_EARTH_EQUATORIAL_RADIUS Constants.IERS2010_EARTH_EQUATORIAL_RADIUS}</td>
  110.      *     <td>{@link org.orekit.utils.Constants#IERS2010_EARTH_FLATTENING Constants.IERS2010_EARTH_FLATTENING}</td></tr>
  111.      * </table>
  112.      * @param ae equatorial radius
  113.      * @param f the flattening (f = (a-b)/a)
  114.      * @param bodyFrame body frame related to body shape
  115.      * @see org.orekit.frames.FramesFactory#getITRF(org.orekit.utils.IERSConventions, boolean)
  116.      */
  117.     public OneAxisEllipsoid(final double ae, final double f,
  118.                             final Frame bodyFrame) {
  119.         super(bodyFrame, ae, ae, ae * (1.0 - f));
  120.         this.f    = f;
  121.         this.ae2  = ae * ae;
  122.         this.e2   = f * (2.0 - f);
  123.         this.e    = FastMath.sqrt(e2);
  124.         this.g    = 1.0 - f;
  125.         this.g2   = g * g;
  126.         this.ap2  = ae2 * g2;
  127.         setAngularThreshold(1.0e-12);
  128.         this.bodyFrame = bodyFrame;
  129.     }

  131.     /** Set the angular convergence threshold.
  132.      * <p>The angular threshold is used both to identify points close to
  133.      * the ellipse axes and as the convergence threshold used to
  134.      * stop the iterations in the {@link #transform(Vector3D, Frame,
  135.      * AbsoluteDate)} method.</p>
  136.      * <p>If this method is not called, the default value is set to
  137.      * 10<sup>-12</sup>.</p>
  138.      * @param angularThreshold angular convergence threshold (rad)
  139.      */
  140.     public void setAngularThreshold(final double angularThreshold) {
  141.         this.angularThreshold = angularThreshold;
  142.     }

  144.     /** Get the equatorial radius of the body.
  145.      * @return equatorial radius of the body (m)
  146.      */
  147.     public double getEquatorialRadius() {
  148.         return getA();
  149.     }

  151.     /** Get the flattening of the body: f = (a-b)/a.
  152.      * @return the flattening
  153.      */
  154.     public double getFlattening() {
  155.         return f;
  156.     }

  158.     /** Get the first eccentricity squared of the ellipsoid: e^2 = f * (2.0 - f).
  159.      * @return the eccentricity squared
  160.      */
  161.     public double getEccentricitySquared() {
  162.         return e2;
  163.     }

  165.     /** Get the first eccentricity of the ellipsoid: e = sqrt(f * (2.0 - f)).
  166.      * @return the eccentricity
  167.      */
  168.     public double getEccentricity() {
  169.         return e;
  170.     }

  172.     /** {@inheritDoc} */
  173.     public Frame getBodyFrame() {
  174.         return bodyFrame;
  175.     }

  177.     /** Get the intersection point of a line with the surface of the body.
  178.      * <p>A line may have several intersection points with a closed
  179.      * surface (we consider the one point case as a degenerated two
  180.      * points case). The close parameter is used to select which of
  181.      * these points should be returned. The selected point is the one
  182.      * that is closest to the close point.</p>
  183.      * @param line test line (may intersect the body or not)
  184.      * @param close point used for intersections selection
  185.      * @param frame frame in which line is expressed
  186.      * @param date date of the line in given frame
  187.      * @return intersection point at altitude zero or null if the line does
  188.      * not intersect the surface
  189.      * @since 9.3
  190.      */
  191.     public Vector3D getCartesianIntersectionPoint(final Line line, final Vector3D close,
  192.                                                   final Frame frame, final AbsoluteDate date) {

  194.         // transform line and close to body frame
  195.         final StaticTransform frameToBodyFrame =
  196.                 frame.getStaticTransformTo(bodyFrame, date);
  197.         final Line lineInBodyFrame = frameToBodyFrame.transformLine(line);

  199.         // compute some miscellaneous variables
  200.         final Vector3D point    = lineInBodyFrame.getOrigin();
  201.         final double x          = point.getX();
  202.         final double y          = point.getY();
  203.         final double z          = point.getZ();
  204.         final double z2         = z * z;
  205.         final double r2         = x * x + y * y;

  207.         final Vector3D direction = lineInBodyFrame.getDirection();
  208.         final double dx         = direction.getX();
  209.         final double dy         = direction.getY();
  210.         final double dz         = direction.getZ();
  211.         final double cz2        = dx * dx + dy * dy;

  213.         // abscissa of the intersection as a root of a 2nd degree polynomial :
  214.         // a k^2 - 2 b k + c = 0
  215.         final double a  = 1.0 - e2 * cz2;
  216.         final double b  = -(g2 * (x * dx + y * dy) + z * dz);
  217.         final double c  = g2 * (r2 - ae2) + z2;
  218.         final double b2 = b * b;
  219.         final double ac = a * c;
  220.         if (b2 < ac) {
  221.             return null;
  222.         }
  223.         final double s  = FastMath.sqrt(b2 - ac);
  224.         final double k1 = (b < 0) ? (b - s) / a : c / (b + s);
  225.         final double k2 = c / (a * k1);

  227.         // select the right point
  228.         final Vector3D closeInBodyFrame = frameToBodyFrame.transformPosition(close);
  229.         final double   closeAbscissa    = lineInBodyFrame.getAbscissa(closeInBodyFrame);
  230.         final double k =
  231.             (FastMath.abs(k1 - closeAbscissa) < FastMath.abs(k2 - closeAbscissa)) ? k1 : k2;
  232.         return lineInBodyFrame.pointAt(k);

  234.     }

  236.     /** {@inheritDoc} */
  237.     public GeodeticPoint getIntersectionPoint(final Line line, final Vector3D close,
  238.                                               final Frame frame, final AbsoluteDate date) {

  240.         final Vector3D intersection = getCartesianIntersectionPoint(line, close, frame, date);
  241.         if (intersection == null) {
  242.             return null;
  243.         }
  244.         final double ix = intersection.getX();
  245.         final double iy = intersection.getY();
  246.         final double iz = intersection.getZ();

  248.         final double lambda = FastMath.atan2(iy, ix);
  249.         final double phi    = FastMath.atan2(iz, g2 * FastMath.sqrt(ix * ix + iy * iy));
  250.         return new GeodeticPoint(phi, lambda, 0.0);

  252.     }

  254.     /** Get the intersection point of a line with the surface of the body.
  255.      * <p>A line may have several intersection points with a closed
  256.      * surface (we consider the one point case as a degenerated two
  257.      * points case). The close parameter is used to select which of
  258.      * these points should be returned. The selected point is the one
  259.      * that is closest to the close point.</p>
  260.      * @param line test line (may intersect the body or not)
  261.      * @param close point used for intersections selection
  262.      * @param frame frame in which line is expressed
  263.      * @param date date of the line in given frame
  264.      * @param <T> type of the field elements
  265.      * @return intersection point at altitude zero or null if the line does
  266.      * not intersect the surface
  267.      * @since 9.3
  268.      */
  269.     public <T extends CalculusFieldElement<T>> FieldVector3D<T> getCartesianIntersectionPoint(final FieldLine<T> line,
  270.                                                                                           final FieldVector3D<T> close,
  271.                                                                                           final Frame frame,
  272.                                                                                           final FieldAbsoluteDate<T> date) {

  274.         // transform line and close to body frame
  275.         final FieldStaticTransform<T> frameToBodyFrame = frame.getStaticTransformTo(bodyFrame, date);
  276.         final FieldLine<T>            lineInBodyFrame  = frameToBodyFrame.transformLine(line);

  278.         // compute some miscellaneous variables
  279.         final FieldVector3D<T> point = lineInBodyFrame.getOrigin();
  280.         final T x  = point.getX();
  281.         final T y  = point.getY();
  282.         final T z  = point.getZ();
  283.         final T z2 = z.square();
  284.         final T r2 = x.square().add(y.square());

  286.         final FieldVector3D<T> direction = lineInBodyFrame.getDirection();
  287.         final T dx  = direction.getX();
  288.         final T dy  = direction.getY();
  289.         final T dz  = direction.getZ();
  290.         final T cz2 = dx.square().add(dy.square());

  292.         // abscissa of the intersection as a root of a 2nd degree polynomial :
  293.         // a k^2 - 2 b k + c = 0
  294.         final T a  = cz2.multiply(e2).subtract(1.0).negate();
  295.         final T b  = x.multiply(dx).add(y.multiply(dy)).multiply(g2).add(z.multiply(dz)).negate();
  296.         final T c  = r2.subtract(ae2).multiply(g2).add(z2);
  297.         final T b2 = b.square();
  298.         final T ac = a.multiply(c);
  299.         if (b2.getReal() < ac.getReal()) {
  300.             return null;
  301.         }
  302.         final T s  = b2.subtract(ac).sqrt();
  303.         final T k1 = (b.getReal() < 0) ? b.subtract(s).divide(a) : c.divide(b.add(s));
  304.         final T k2 = c.divide(a.multiply(k1));

  306.         // select the right point
  307.         final FieldVector3D<T>  closeInBodyFrame = frameToBodyFrame.transformPosition(close);
  308.         final T                 closeAbscissa    = lineInBodyFrame.getAbscissa(closeInBodyFrame);
  309.         final T k = (FastMath.abs(k1.getReal() - closeAbscissa.getReal()) < FastMath.abs(k2.getReal() - closeAbscissa.getReal())) ?
  310.                     k1 : k2;
  311.         return lineInBodyFrame.pointAt(k);
  312.     }

  314.     /** {@inheritDoc} */
  315.     public <T extends CalculusFieldElement<T>> FieldGeodeticPoint<T> getIntersectionPoint(final FieldLine<T> line,
  316.                                                                                       final FieldVector3D<T> close,
  317.                                                                                       final Frame frame,
  318.                                                                                       final FieldAbsoluteDate<T> date) {

  320.         final FieldVector3D<T> intersection = getCartesianIntersectionPoint(line, close, frame, date);
  321.         if (intersection == null) {
  322.             return null;
  323.         }
  324.         final T ix = intersection.getX();
  325.         final T iy = intersection.getY();
  326.         final T iz = intersection.getZ();

  328.         final T lambda = iy.atan2(ix);
  329.         final T phi    = iz.atan2(ix.multiply(ix).add(iy.multiply(iy)).sqrt().multiply(g2));
  330.         return new FieldGeodeticPoint<>(phi, lambda, phi.getField().getZero());

  332.     }

  334.     /** {@inheritDoc} */
  335.     public Vector3D transform(final GeodeticPoint point) {
  336.         final double longitude = point.getLongitude();
  337.         final SinCos scLambda  = FastMath.sinCos(longitude);
  338.         final double latitude  = point.getLatitude();
  339.         final SinCos scPhi     = FastMath.sinCos(latitude);
  340.         final double h         = point.getAltitude();
  341.         final double n         = getA() / FastMath.sqrt(1.0 - e2 * scPhi.sin() * scPhi.sin());
  342.         final double r         = (n + h) * scPhi.cos();
  343.         return new Vector3D(r * scLambda.cos(), r * scLambda.sin(), (g2 * n + h) * scPhi.sin());
  344.     }

  346.     /** {@inheritDoc} */
  347.     public <T extends CalculusFieldElement<T>> FieldVector3D<T> transform(final FieldGeodeticPoint<T> point) {

  349.         final T latitude  = point.getLatitude();
  350.         final T longitude = point.getLongitude();
  351.         final T altitude  = point.getAltitude();

  353.         final FieldSinCos<T> scLambda = FastMath.sinCos(longitude);
  354.         final FieldSinCos<T> scPhi    = FastMath.sinCos(latitude);
  355.         final T cLambda = scLambda.cos();
  356.         final T sLambda = scLambda.sin();
  357.         final T cPhi    = scPhi.cos();
  358.         final T sPhi    = scPhi.sin();
  359.         final T n       = sPhi.multiply(sPhi).multiply(e2).subtract(1.0).negate().sqrt().reciprocal().multiply(getA());
  360.         final T r       = n.add(altitude).multiply(cPhi);

  362.         return new FieldVector3D<>(r.multiply(cLambda),
  363.                                    r.multiply(sLambda),
  364.                                    sPhi.multiply(altitude.add(n.multiply(g2))));
  365.     }

  367.     /** {@inheritDoc} */
  368.     public Vector3D projectToGround(final Vector3D point, final AbsoluteDate date, final Frame frame) {

  370.         // transform point to body frame
  371.         final StaticTransform toBody = frame.getStaticTransformTo(bodyFrame, date);
  372.         final Vector3D   p         = toBody.transformPosition(point);
  373.         final double     z         = p.getZ();
  374.         final double     r         = FastMath.hypot(p.getX(), p.getY());

  376.         // set up the 2D meridian ellipse
  377.         final Ellipse meridian = new Ellipse(Vector3D.ZERO,
  378.                                              r == 0 ? Vector3D.PLUS_I : new Vector3D(p.getX() / r, p.getY() / r, 0),
  379.                                              Vector3D.PLUS_K,
  380.                                              getA(), getC(), bodyFrame);

  382.         // find the closest point in the meridian plane
  383.         final Vector3D groundPoint = meridian.toSpace(meridian.projectToEllipse(new Vector2D(r, z)));

  385.         // transform point back to initial frame
  386.         return toBody.getInverse().transformPosition(groundPoint);

  388.     }

  390.     /** {@inheritDoc} */
  391.     public TimeStampedPVCoordinates projectToGround(final TimeStampedPVCoordinates pv, final Frame frame) {

  393.         // transform point to body frame
  394.         final Transform                toBody        = frame.getTransformTo(bodyFrame, pv.getDate());
  395.         final TimeStampedPVCoordinates pvInBodyFrame = toBody.transformPVCoordinates(pv);
  396.         final Vector3D                 p             = pvInBodyFrame.getPosition();
  397.         final double                   r             = FastMath.hypot(p.getX(), p.getY());

  399.         // set up the 2D ellipse corresponding to first principal curvature along meridian
  400.         final Vector3D meridian = r == 0 ? Vector3D.PLUS_I : new Vector3D(p.getX() / r, p.getY() / r, 0);
  401.         final Ellipse firstPrincipalCurvature =
  402.                 new Ellipse(Vector3D.ZERO, meridian, Vector3D.PLUS_K, getA(), getC(), bodyFrame);

  404.         // project coordinates in the meridian plane
  405.         final TimeStampedPVCoordinates gpFirst = firstPrincipalCurvature.projectToEllipse(pvInBodyFrame);
  406.         final Vector3D                 gpP     = gpFirst.getPosition();
  407.         final double                   gr      = MathArrays.linearCombination(gpP.getX(), meridian.getX(),
  408.                                                                               gpP.getY(), meridian.getY());
  409.         final double                   gz      = gpP.getZ();

  411.         // topocentric frame
  412.         final Vector3D east   = new Vector3D(-meridian.getY(), meridian.getX(), 0);
  413.         final Vector3D zenith = new Vector3D(gr * getC() / getA(), meridian, gz * getA() / getC(), Vector3D.PLUS_K).normalize();
  414.         final Vector3D north  = Vector3D.crossProduct(zenith, east);

  416.         // set up the ellipse corresponding to second principal curvature in the zenith/east plane
  417.         final Ellipse secondPrincipalCurvature  = getPlaneSection(gpP, north);
  418.         final TimeStampedPVCoordinates gpSecond = secondPrincipalCurvature.projectToEllipse(pvInBodyFrame);

  420.         final Vector3D gpV = gpFirst.getVelocity().add(gpSecond.getVelocity());
  421.         final Vector3D gpA = gpFirst.getAcceleration().add(gpSecond.getAcceleration());

  423.         // moving projected point
  424.         final TimeStampedPVCoordinates groundPV =
  425.                 new TimeStampedPVCoordinates(pv.getDate(), gpP, gpV, gpA);

  427.         // transform moving projected point back to initial frame
  428.         return toBody.getInverse().transformPVCoordinates(groundPV);

  430.     }

  432.     /** {@inheritDoc}
  433.      * <p>
  434.      * This method is based on Toshio Fukushima's algorithm which uses Halley's method.
  435.      * <a href="https://www.researchgate.net/publication/227215135_Transformation_from_Cartesian_to_Geodetic_Coordinates_Accelerated_by_Halley's_Method">
  436.      * transformation from Cartesian to Geodetic Coordinates Accelerated by Halley's Method</a>,
  437.      * Toshio Fukushima, Journal of Geodesy 9(12):689-693, February 2006
  438.      * </p>
  439.      * <p>
  440.      * Some changes have been added to the original method:
  441.      * </p>
  442.      * <ul>
  443.      *   <li>in order to handle more accurately corner cases near the pole</li>
  444.      *   <li>in order to handle properly corner cases near the equatorial plane, even far inside the ellipsoid</li>
  445.      *   <li>in order to handle very flat ellipsoids</li>
  446.      * </ul>
  447.      * <p>
  448.      * In some rare cases (for example very flat ellipsoid, or points close to ellipsoid center), the loop
  449.      * may fail to converge. As this seems to happen only in degenerate cases, a design choice was to return
  450.      * an approximate point corresponding to last iteration. This point may be incorrect and fail to give the
  451.      * initial point back if doing roundtrip by calling {@link #transform(GeodeticPoint)}. This design choice
  452.      * was made to avoid NaNs appearing for example in inter-satellites visibility checks when two satellites
  453.      * are almost on opposite sides of Earth. The intermediate points far within the Earth should not prevent
  454.      * the detection algorithm to find visibility start/end.
  455.      * </p>
  456.      */
  457.     public GeodeticPoint transform(final Vector3D point, final Frame frame, final AbsoluteDate date) {

  459.         // transform point to body frame
  460.         final Vector3D pointInBodyFrame = frame.getStaticTransformTo(bodyFrame, date)
  461.                 .transformPosition(point);
  462.         final double   r2               = pointInBodyFrame.getX() * pointInBodyFrame.getX() +
  463.                                           pointInBodyFrame.getY() * pointInBodyFrame.getY();
  464.         final double   r                = FastMath.sqrt(r2);
  465.         final double   z                = pointInBodyFrame.getZ();

  467.         final double   lambda           = FastMath.atan2(pointInBodyFrame.getY(), pointInBodyFrame.getX());

  469.         double h;
  470.         double phi;
  471.         if (r <= ANGULAR_THRESHOLD * FastMath.abs(z)) {
  472.             // the point is almost on the polar axis, approximate the ellipsoid with
  473.             // the osculating sphere whose center is at evolute cusp along polar axis
  474.             final double osculatingRadius = ae2 / getC();
  475.             final double evoluteCuspZ     = FastMath.copySign(getA() * e2 / g, -z);
  476.             final double deltaZ           = z - evoluteCuspZ;
  477.             // we use Ï€/2 - atan(r/Δz) instead of atan(Δz/r) for accuracy purposes, as r is much smaller than Δz
  478.             phi = FastMath.copySign(0.5 * FastMath.PI - FastMath.atan(r / FastMath.abs(deltaZ)), deltaZ);
  479.             h   = FastMath.hypot(deltaZ, r) - osculatingRadius;
  480.         } else if (FastMath.abs(z) <= ANGULAR_THRESHOLD * r) {
  481.             // the point is almost on the major axis

  483.             final double osculatingRadius = ap2 / getA();
  484.             final double evoluteCuspR     = getA() * e2;
  485.             final double deltaR           = r - evoluteCuspR;
  486.             if (deltaR >= 0) {
  487.                 // the point is outside of the ellipse evolute, approximate the ellipse
  488.                 // with the osculating circle whose center is at evolute cusp along major axis
  489.                 phi = (deltaR == 0) ? 0.0 : FastMath.atan(z / deltaR);
  490.                 h   = FastMath.hypot(deltaR, z) - osculatingRadius;
  491.             } else {
  492.                 // the point is on the part of the major axis within ellipse evolute
  493.                 // we can compute the closest ellipse point analytically, and it is NOT near the equator
  494.                 final double rClose = r / e2;
  495.                 final double zClose = FastMath.copySign(g * FastMath.sqrt(ae2 - rClose * rClose), z);
  496.                 phi = FastMath.atan((zClose - z) / (rClose - r));
  497.                 h   = -FastMath.hypot(r - rClose, z - zClose);
  498.             }

  500.         } else {
  501.             // use Toshio Fukushima method, with several iterations
  502.             final double epsPhi = 1.0e-15;
  503.             final double epsH   = 1.0e-14 * FastMath.max(getA(), FastMath.sqrt(r2 + z * z));
  504.             final double c     = getA() * e2;
  505.             final double absZ  = FastMath.abs(z);
  506.             final double zc    = g * absZ;
  507.             double sn  = absZ;
  508.             double sn2 = sn * sn;
  509.             double cn  = g * r;
  510.             double cn2 = cn * cn;
  511.             double an2 = cn2 + sn2;
  512.             double an  = FastMath.sqrt(an2);
  513.             double bn  = 0;
  514.             phi = Double.POSITIVE_INFINITY;
  515.             h   = Double.POSITIVE_INFINITY;
  516.             for (int i = 0; i < 1000; ++i) {
  517.                 // this usually converges in 2 iterations, but in rare cases it can take much more
  518.                 // see https://gitlab.orekit.org/orekit/orekit/-/issues/1224 for examples
  519.                 // with points near Earth center which need 137 iterations for the first example
  520.                 // and 1150 iterations for the second example
  521.                 final double oldSn  = sn;
  522.                 final double oldCn  = cn;
  523.                 final double oldPhi = phi;
  524.                 final double oldH   = h;
  525.                 final double an3    = an2 * an;
  526.                 final double csncn  = c * sn * cn;
  527.                 bn    = 1.5 * csncn * ((r * sn - zc * cn) * an - csncn);
  528.                 sn    = (zc * an3 + c * sn2 * sn) * an3 - bn * sn;
  529.                 cn    = (r  * an3 - c * cn2 * cn) * an3 - bn * cn;
  530.                 if (sn * oldSn < 0 || cn < 0) {
  531.                     // the Halley iteration went too far, we restrict it and iterate again
  532.                     while (sn * oldSn < 0 || cn < 0) {
  533.                         sn = (sn + oldSn) / 2;
  534.                         cn = (cn + oldCn) / 2;
  535.                     }
  536.                 } else {

  538.                     // rescale components to avoid overflow when several iterations are used
  539.                     final int exp = (FastMath.getExponent(sn) + FastMath.getExponent(cn)) / 2;
  540.                     sn = FastMath.scalb(sn, -exp);
  541.                     cn = FastMath.scalb(cn, -exp);

  543.                     sn2 = sn * sn;
  544.                     cn2 = cn * cn;
  545.                     an2 = cn2 + sn2;
  546.                     an  = FastMath.sqrt(an2);

  548.                     final double cc = g * cn;
  549.                     h = (r * cc + absZ * sn - getA() * g * an) / FastMath.sqrt(an2 - e2 * cn2);
  550.                     if (FastMath.abs(oldH   - h)   < epsH) {
  551.                         phi = FastMath.copySign(FastMath.atan(sn / cc), z);
  552.                         if (FastMath.abs(oldPhi - phi) < epsPhi) {
  553.                             break;
  554.                         }
  555.                     }

  557.                 }

  559.             }

  561.             if (Double.isInfinite(phi)) {
  562.                 // we did not converge, the point is probably within the ellipsoid
  563.                 // we just compute the "best" phi we can to avoid NaN
  564.                 phi = FastMath.copySign(FastMath.atan(sn / (g * cn)), z);
  565.             }

  567.         }

  569.         return new GeodeticPoint(phi, lambda, h);

  571.     }

  573.     /** {@inheritDoc}
  574.      * <p>
  575.      * This method is based on Toshio Fukushima's algorithm which uses Halley's method.
  576.      * <a href="https://www.researchgate.net/publication/227215135_Transformation_from_Cartesian_to_Geodetic_Coordinates_Accelerated_by_Halley's_Method">
  577.      * transformation from Cartesian to Geodetic Coordinates Accelerated by Halley's Method</a>,
  578.      * Toshio Fukushima, Journal of Geodesy 9(12):689-693, February 2006
  579.      * </p>
  580.      * <p>
  581.      * Some changes have been added to the original method:
  582.      * <ul>
  583.      *   <li>in order to handle more accurately corner cases near the pole</li>
  584.      *   <li>in order to handle properly corner cases near the equatorial plane, even far inside the ellipsoid</li>
  585.      *   <li>in order to handle very flat ellipsoids</li>
  586.      * </ul>
  587.      * <p>
  588.      * In some rare cases (for example very flat ellipsoid, or points close to ellipsoid center), the loop
  589.      * may fail to converge. As this seems to happen only in degenerate cases, a design choice was to return
  590.      * an approximate point corresponding to last iteration. This point may be incorrect and fail to give the
  591.      * initial point back if doing roundtrip by calling {@link #transform(GeodeticPoint)}. This design choice
  592.      * was made to avoid NaNs appearing for example in inter-satellites visibility checks when two satellites
  593.      * are almost on opposite sides of Earth. The intermediate points far within the Earth should not prevent
  594.      * the detection algorithm to find visibility start/end.
  595.      * </p>
  596.      */
  597.     public <T extends CalculusFieldElement<T>> FieldGeodeticPoint<T> transform(final FieldVector3D<T> point,
  598.                                                                            final Frame frame,
  599.                                                                            final FieldAbsoluteDate<T> date) {

  601.         // transform point to body frame
  602.         final FieldVector3D<T> pointInBodyFrame = (frame == bodyFrame) ?
  603.                                                   point :
  604.                                                   frame.getStaticTransformTo(bodyFrame, date).transformPosition(point);
  605.         final T   r2                            = pointInBodyFrame.getX().multiply(pointInBodyFrame.getX()).
  606.                                                   add(pointInBodyFrame.getY().multiply(pointInBodyFrame.getY()));
  607.         final T   r                             = r2.sqrt();
  608.         final T   z                             = pointInBodyFrame.getZ();

  610.         final T   lambda                        = pointInBodyFrame.getY().atan2(pointInBodyFrame.getX());

  612.         T h;
  613.         T phi;
  614.         if (r.getReal() <= ANGULAR_THRESHOLD * FastMath.abs(z.getReal())) {
  615.             // the point is almost on the polar axis, approximate the ellipsoid with
  616.             // the osculating sphere whose center is at evolute cusp along polar axis
  617.             final double osculatingRadius = ae2 / getC();
  618.             final double evoluteCuspZ     = FastMath.copySign(getA() * e2 / g, -z.getReal());
  619.             final T      deltaZ           = z.subtract(evoluteCuspZ);
  620.             // we use Ï€/2 - atan(r/Δz) instead of atan(Δz/r) for accuracy purposes, as r is much smaller than Δz
  621.             phi = r.divide(deltaZ.abs()).atan().negate().add(r.getPi().multiply(0.5)).copySign(deltaZ);
  622.             h   = deltaZ.hypot(r).subtract(osculatingRadius);
  623.         } else if (FastMath.abs(z.getReal()) <= ANGULAR_THRESHOLD * r.getReal()) {
  624.             // the point is almost on the major axis

  626.             final double osculatingRadius = ap2 / getA();
  627.             final double evoluteCuspR     = getA() * e2;
  628.             final T      deltaR           = r.subtract(evoluteCuspR);
  629.             if (deltaR.getReal() >= 0) {
  630.                 // the point is outside of the ellipse evolute, approximate the ellipse
  631.                 // with the osculating circle whose center is at evolute cusp along major axis
  632.                 phi = (deltaR.getReal() == 0) ? z.getField().getZero() : z.divide(deltaR).atan();
  633.                 h   = deltaR.hypot(z).subtract(osculatingRadius);
  634.             } else {
  635.                 // the point is on the part of the major axis within ellipse evolute
  636.                 // we can compute the closest ellipse point analytically, and it is NOT near the equator
  637.                 final T rClose = r.divide(e2);
  638.                 final T zClose = rClose.multiply(rClose).negate().add(ae2).sqrt().multiply(g).copySign(z);
  639.                 phi = zClose.subtract(z).divide(rClose.subtract(r)).atan();
  640.                 h   = r.subtract(rClose).hypot(z.subtract(zClose)).negate();
  641.             }

  643.         } else {
  644.             // use Toshio Fukushima method, with several iterations
  645.             final double epsPhi = 1.0e-15;
  646.             final double epsH   = 1.0e-14 * getA();
  647.             final double c      = getA() * e2;
  648.             final T      absZ   = z.abs();
  649.             final T      zc     = absZ.multiply(g);
  650.             T            sn     = absZ;
  651.             T            sn2    = sn.multiply(sn);
  652.             T            cn     = r.multiply(g);
  653.             T            cn2    = cn.multiply(cn);
  654.             T            an2    = cn2.add(sn2);
  655.             T            an     = an2.sqrt();
  656.             T            bn     = an.getField().getZero();
  657.             phi = an.getField().getZero().add(Double.POSITIVE_INFINITY);
  658.             h   = an.getField().getZero().add(Double.POSITIVE_INFINITY);
  659.             for (int i = 0; i < 1000; ++i) {
  660.                 // this usually converges in 2 iterations, but in rare cases it can take much more
  661.                 // see https://gitlab.orekit.org/orekit/orekit/-/issues/1224 for examples
  662.                 // with points near Earth center which need 137 iterations for the first example
  663.                 // and 1150 iterations for the second example
  664.                 final T oldSn  = sn;
  665.                 final T oldCn  = cn;
  666.                 final T oldPhi = phi;
  667.                 final T oldH   = h;
  668.                 final T an3    = an2.multiply(an);
  669.                 final T csncn  = sn.multiply(cn).multiply(c);
  670.                 bn    = csncn.multiply(1.5).multiply((r.multiply(sn).subtract(zc.multiply(cn))).multiply(an).subtract(csncn));
  671.                 sn    = zc.multiply(an3).add(sn2.multiply(sn).multiply(c)).multiply(an3).subtract(bn.multiply(sn));
  672.                 cn    = r.multiply(an3).subtract(cn2.multiply(cn).multiply(c)).multiply(an3).subtract(bn.multiply(cn));
  673.                 if (sn.getReal() * oldSn.getReal() < 0 || cn.getReal() < 0) {
  674.                     // the Halley iteration went too far, we restrict it and iterate again
  675.                     while (sn.getReal() * oldSn.getReal() < 0 || cn.getReal() < 0) {
  676.                         sn = sn.add(oldSn).multiply(0.5);
  677.                         cn = cn.add(oldCn).multiply(0.5);
  678.                     }
  679.                 } else {

  681.                     // rescale components to avoid overflow when several iterations are used
  682.                     final int exp = (FastMath.getExponent(sn.getReal()) + FastMath.getExponent(cn.getReal())) / 2;
  683.                     sn = sn.scalb(-exp);
  684.                     cn = cn.scalb(-exp);

  686.                     sn2 = sn.square();
  687.                     cn2 = cn.square();
  688.                     an2 = cn2.add(sn2);
  689.                     an  = an2.sqrt();

  691.                     final T cc = cn.multiply(g);
  692.                     h = r.multiply(cc).add(absZ.multiply(sn)).subtract(an.multiply(getA() * g)).divide(an2.subtract(cn2.multiply(e2)).sqrt());
  693.                     if (FastMath.abs(oldH.getReal()  - h.getReal())   < epsH) {
  694.                         phi = sn.divide(cc).atan().copySign(z);
  695.                         if (FastMath.abs(oldPhi.getReal() - phi.getReal()) < epsPhi) {
  696.                             break;
  697.                         }
  698.                     }

  700.                 }

  702.             }

  704.             if (Double.isInfinite(phi.getReal())) {
  705.                 // we did not converge, the point is probably within the ellipsoid
  706.                 // we just compute the "best" phi we can to avoid NaN
  707.                 phi = sn.divide(cn.multiply(g)).atan().copySign(z);
  708.             }

  710.         }

  712.         return new FieldGeodeticPoint<>(phi, lambda, h);

  714.     }

  716.     /** Transform a Cartesian point to a surface-relative point.
  717.      * @param point Cartesian point
  718.      * @param frame frame in which Cartesian point is expressed
  719.      * @param date date of the computation (used for frames conversions)
  720.      * @return point at the same location but as a surface-relative point,
  721.      * using time as the single derivation parameter
  722.      */
  723.     public FieldGeodeticPoint<DerivativeStructure> transform(final PVCoordinates point,
  724.                                                              final Frame frame, final AbsoluteDate date) {

  726.         // transform point to body frame
  727.         final Transform toBody = frame.getTransformTo(bodyFrame, date);
  728.         final PVCoordinates pointInBodyFrame = toBody.transformPVCoordinates(point);
  729.         final FieldVector3D<DerivativeStructure> p = pointInBodyFrame.toDerivativeStructureVector(2);
  730.         final DerivativeStructure   pr2 = p.getX().square().add(p.getY().square());
  731.         final DerivativeStructure   pr  = pr2.sqrt();
  732.         final DerivativeStructure   pz  = p.getZ();

  734.         // project point on the ellipsoid surface
  735.         final TimeStampedPVCoordinates groundPoint = projectToGround(new TimeStampedPVCoordinates(date, pointInBodyFrame),
  736.                                                                      bodyFrame);
  737.         final FieldVector3D<DerivativeStructure> gp = groundPoint.toDerivativeStructureVector(2);
  738.         final DerivativeStructure   gpr2 = gp.getX().square().add(gp.getY().square());
  739.         final DerivativeStructure   gpr  = gpr2.sqrt();
  740.         final DerivativeStructure   gpz  = gp.getZ();

  742.         // relative position of test point with respect to its ellipse sub-point
  743.         final DerivativeStructure dr  = pr.subtract(gpr);
  744.         final DerivativeStructure dz  = pz.subtract(gpz);
  745.         final double insideIfNegative = g2 * (pr2.getReal() - ae2) + pz.getReal() * pz.getReal();

  747.         return new FieldGeodeticPoint<>(DerivativeStructure.atan2(gpz, gpr.multiply(g2)),
  748.                                                                   DerivativeStructure.atan2(p.getY(), p.getX()),
  749.                                                                   DerivativeStructure.hypot(dr, dz).copySign(insideIfNegative));
  750.     }

  752.     /** Compute the azimuth angle from local north between the two points.
  753.      *
  754.      * The angle is calculated clockwise from local north at the origin point
  755.      * and follows the rhumb line to the destination point.
  756.      *
  757.      * @param origin the origin point, at which the azimuth angle will be computed (non-{@code null})
  758.      * @param destination the destination point, to which the angle is defined (non-{@code null})
  759.      * @return the resulting azimuth angle (radians, {@code [0-2pi)})
  760.      * @since 11.3
  761.      */
  762.     public double azimuthBetweenPoints(final GeodeticPoint origin, final GeodeticPoint destination) {
  763.         final double dLon = MathUtils.normalizeAngle(destination.getLongitude(), origin.getLongitude()) - origin.getLongitude();
  764.         final double originIsoLat = geodeticToIsometricLatitude(origin.getLatitude());
  765.         final double destIsoLat = geodeticToIsometricLatitude(destination.getLatitude());

  767.         final double az = FastMath.atan2(dLon, destIsoLat - originIsoLat);
  768.         if (az < 0.) {
  769.             return az + MathUtils.TWO_PI;
  770.         }
  771.         return az;
  772.     }

  774.     /** Compute the azimuth angle from local north between the two points.
  775.      *
  776.      * The angle is calculated clockwise from local north at the origin point
  777.      * and follows the rhumb line to the destination point.
  778.      *
  779.      * @param origin the origin point, at which the azimuth angle will be computed (non-{@code null})
  780.      * @param destination the destination point, to which the angle is defined (non-{@code null})
  781.      * @param <T> the type of field elements
  782.      * @return the resulting azimuth angle (radians, {@code [0-2pi)})
  783.      * @since 11.3
  784.      */
  785.     public <T extends CalculusFieldElement<T>> T azimuthBetweenPoints(final FieldGeodeticPoint<T> origin, final FieldGeodeticPoint<T> destination) {
  786.         final T dLon = MathUtils.normalizeAngle(destination.getLongitude().subtract(origin.getLongitude()), origin.getLongitude().getField().getZero());
  787.         final T originIsoLat = geodeticToIsometricLatitude(origin.getLatitude());
  788.         final T destIsoLat = geodeticToIsometricLatitude(destination.getLatitude());

  790.         final T az = FastMath.atan2(dLon, destIsoLat.subtract(originIsoLat));
  791.         if (az.getReal() < 0.) {
  792.             return az.add(az.getPi().multiply(2));
  793.         }
  794.         return az;
  795.     }

  797.     /** Compute the <a href="https://mathworld.wolfram.com/IsometricLatitude.html">isometric latitude</a>
  798.      *  corresponding to the provided latitude.
  799.      *
  800.      * @param geodeticLatitude the latitude (radians, within interval {@code [-pi/2, +pi/2]})
  801.      * @return the isometric latitude (radians)
  802.      * @since 11.3
  803.      */
  804.     public double geodeticToIsometricLatitude(final double geodeticLatitude) {
  805.         if (FastMath.abs(geodeticLatitude) <= angularThreshold) {
  806.             return 0.;
  807.         }

  809.         final double eSinLat = e * FastMath.sin(geodeticLatitude);

  811.         // first term: ln(tan(pi/4 + lat/2))
  812.         final double a = FastMath.log(FastMath.tan(FastMath.PI / 4. + geodeticLatitude / 2.));
  813.         // second term: (ecc / 2) * ln((1 - ecc*sin(lat)) / (1 + ecc * sin(lat)))
  814.         final double b = (e / 2.) * FastMath.log((1. - eSinLat) / (1. + eSinLat));

  816.         return a + b;
  817.     }

  819.     /** Compute the <a href="https://mathworld.wolfram.com/IsometricLatitude.html">isometric latitude</a>
  820.      *  corresponding to the provided latitude.
  821.      *
  822.      * @param geodeticLatitude the latitude (radians, within interval {@code [-pi/2, +pi/2]})
  823.      * @param <T> the type of field elements
  824.      * @return the isometric latitude (radians)
  825.      * @since 11.3
  826.      */
  827.     public <T extends CalculusFieldElement<T>> T geodeticToIsometricLatitude(final T geodeticLatitude) {
  828.         if (geodeticLatitude.abs().getReal() <= angularThreshold) {
  829.             return geodeticLatitude.getField().getZero();
  830.         }
  831.         final Field<T> field = geodeticLatitude.getField();
  832.         final T ecc = geodeticLatitude.newInstance(e);
  833.         final T eSinLat = ecc.multiply(geodeticLatitude.sin());

  835.         // first term: ln(tan(pi/4 + lat/2))
  836.         final T a = FastMath.log(FastMath.tan(geodeticLatitude.getPi().divide(4.).add(geodeticLatitude.divide(2.))));
  837.         // second term: (ecc / 2) * ln((1 - ecc*sin(lat)) / (1 + ecc * sin(lat)))
  838.         final T b = ecc.divide(2.).multiply(FastMath.log(field.getOne().subtract(eSinLat).divide(field.getOne().add(eSinLat))));

  840.         return a.add(b);
  841.     }

  843.     /** Find intermediate point of lowest altitude along a line between two endpoints.
  844.      * @param endpoint1 first endpoint, in body frame
  845.      * @param endpoint2 second endpoint, in body frame
  846.      * @return point with lowest altitude between {@code endpoint1} and {@code endpoint2}.
  847.      * @since 12.0
  848.      */
  849.     public GeodeticPoint lowestAltitudeIntermediate(final Vector3D endpoint1, final Vector3D endpoint2) {

  851.         final Vector3D delta = endpoint2.subtract(endpoint1);

  853.         // function computing intermediate point above ellipsoid (lambda varying between 0 and 1)
  854.         final DoubleFunction<GeodeticPoint> intermediate =
  855.                         lambda -> transform(new Vector3D(1 - lambda, endpoint1, lambda, endpoint2),
  856.                                             bodyFrame, null);

  858.         // first endpoint
  859.         final GeodeticPoint gp1 = intermediate.apply(0.0);

  861.         if (Vector3D.dotProduct(delta, gp1.getZenith()) >= 0) {
  862.             // the line from first endpoint to second endpoint is going away from central body
  863.             // the minimum altitude is reached at first endpoint
  864.             return gp1;
  865.         } else {
  866.             // the line from first endpoint to second endpoint is closing the central body

  868.             // second endpoint
  869.             final GeodeticPoint gp2 = intermediate.apply(1.0);

  871.             if (Vector3D.dotProduct(delta, gp2.getZenith()) <= 0) {
  872.                 // the line from first endpoint to second endpoint is still decreasing when reaching second endpoint,
  873.                 // the minimum altitude is reached at second endpoint
  874.                 return gp2;
  875.             } else {
  876.                 // the line from first endpoint to second endpoint reaches a minimum between first and second endpoints
  877.                 final double lambdaMin = new BracketingNthOrderBrentSolver(1.0e-14, 5).
  878.                                          solve(1000,
  879.                                                lambda -> Vector3D.dotProduct(delta, intermediate.apply(lambda).getZenith()),
  880.                                                0.0, 1.0);
  881.                 return intermediate.apply(lambdaMin);
  882.             }
  883.         }

  885.     }

  887.     /** Find intermediate point of lowest altitude along a line between two endpoints.
  888.      * @param endpoint1 first endpoint, in body frame
  889.      * @param endpoint2 second endpoint, in body frame
  890.      * @param <T> type of the field elements
  891.      * @return point with lowest altitude between {@code endpoint1} and {@code endpoint2}.
  892.      * @since 12.0
  893.      */
  894.     public <T extends CalculusFieldElement<T>> FieldGeodeticPoint<T> lowestAltitudeIntermediate(final FieldVector3D<T> endpoint1,
  895.                                                                                                 final FieldVector3D<T> endpoint2) {

  897.         final FieldVector3D<T> delta = endpoint2.subtract(endpoint1);

  899.         // function computing intermediate point above ellipsoid (lambda varying between 0 and 1)
  900.         final DoubleFunction<FieldGeodeticPoint<T>> intermediate =
  901.                         lambda -> transform(new FieldVector3D<>(1 - lambda, endpoint1, lambda, endpoint2),
  902.                                             bodyFrame, null);

  904.         // first endpoint
  905.         final FieldGeodeticPoint<T> gp1 = intermediate.apply(0.0);

  907.         if (FieldVector3D.dotProduct(delta, gp1.getZenith()).getReal() >= 0) {
  908.             // the line from first endpoint to second endpoint is going away from central body
  909.             // the minimum altitude is reached at first endpoint
  910.             return gp1;
  911.         } else {
  912.             // the line from first endpoint to second endpoint is closing the central body

  914.             // second endpoint
  915.             final FieldGeodeticPoint<T> gp2 = intermediate.apply(1.0);

  917.             if (FieldVector3D.dotProduct(delta, gp2.getZenith()).getReal() <= 0) {
  918.                 // the line from first endpoint to second endpoint is still decreasing when reaching second endpoint,
  919.                 // the minimum altitude is reached at second endpoint
  920.                 return gp2;
  921.             } else {
  922.                 // the line from first endpoint to second endpoint reaches a minimum between first and second endpoints
  923.                 final double lambdaMin = new BracketingNthOrderBrentSolver(1.0e-14, 5).
  924.                                          solve(1000,
  925.                                                lambda -> FieldVector3D.dotProduct(delta, intermediate.apply(lambda).getZenith()).getReal(),
  926.                                                0.0, 1.0);
  927.                 return intermediate.apply(lambdaMin);
  928.             }
  929.         }

  931.     }

  933.     /** Replace the instance with a data transfer object for serialization.
  934.      * <p>
  935.      * This intermediate class serializes the files supported names, the
  936.      * ephemeris type and the body name.
  937.      * </p>
  938.      * @return data transfer object that will be serialized
  939.      */
  940.     private Object writeReplace() {
  941.         return new DataTransferObject(getA(), f, bodyFrame, angularThreshold);
  942.     }

  944.     /** Internal class used only for serialization. */
  945.     private static class DataTransferObject implements Serializable {

  947.         /** Serializable UID. */
  948.         private static final long serialVersionUID = 20130518L;

  950.         /** Equatorial radius. */
  951.         private final double ae;

  953.         /** Flattening. */
  954.         private final double f;

  956.         /** Body frame related to body shape. */
  957.         private final Frame bodyFrame;

  959.         /** Convergence limit. */
  960.         private final double angularThreshold;

  962.         /** Simple constructor.
  963.          * @param ae equatorial radius
  964.          * @param f the flattening (f = (a-b)/a)
  965.          * @param bodyFrame body frame related to body shape
  966.          * @param angularThreshold convergence limit
  967.          */
  968.         DataTransferObject(final double ae, final double f,
  969.                                   final Frame bodyFrame, final double angularThreshold) {
  970.             this.ae               = ae;
  971.             this.f                = f;
  972.             this.bodyFrame        = bodyFrame;
  973.             this.angularThreshold = angularThreshold;
  974.         }

  976.         /** Replace the deserialized data transfer object with a
  977.          * {@link JPLCelestialBody}.
  978.          * @return replacement {@link JPLCelestialBody}
  979.          */
  980.         private Object readResolve() {
  981.             final OneAxisEllipsoid ellipsoid = new OneAxisEllipsoid(ae, f, bodyFrame);
  982.             ellipsoid.setAngularThreshold(angularThreshold);
  983.             return ellipsoid;
  984.         }

  986.     }

  988. }
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