OneAxisEllipsoid.java

  1. /* Copyright 2002-2018 CS Systèmes d'Information
  2.  * Licensed to CS Systèmes d'Information (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;

  21. import org.hipparchus.RealFieldElement;
  22. import org.hipparchus.analysis.differentiation.DerivativeStructure;
  23. import org.hipparchus.geometry.euclidean.threed.FieldLine;
  24. import org.hipparchus.geometry.euclidean.threed.FieldVector3D;
  25. import org.hipparchus.geometry.euclidean.threed.Line;
  26. import org.hipparchus.geometry.euclidean.threed.Vector3D;
  27. import org.hipparchus.geometry.euclidean.twod.Vector2D;
  28. import org.hipparchus.util.FastMath;
  29. import org.hipparchus.util.MathArrays;
  30. import org.orekit.errors.OrekitException;
  31. import org.orekit.frames.FieldTransform;
  32. import org.orekit.frames.Frame;
  33. import org.orekit.frames.Transform;
  34. import org.orekit.time.AbsoluteDate;
  35. import org.orekit.time.FieldAbsoluteDate;
  36. import org.orekit.utils.PVCoordinates;
  37. import org.orekit.utils.TimeStampedPVCoordinates;


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

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

  47.  * @author Luc Maisonobe
  48.  */
  49. public class OneAxisEllipsoid extends Ellipsoid implements BodyShape {

  51.     /** Serializable UID. */
  52.     private static final long serialVersionUID = 20130518L;

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

  57.     /** Body frame related to body shape. */
  58.     private final Frame bodyFrame;

  60.     /** Equatorial radius power 2. */
  61.     private final double ae2;

  63.     /** Polar radius power 2. */
  64.     private final double ap2;

  66.     /** Flattening. */
  67.     private final double f;

  69.     /** Eccentricity power 2. */
  70.     private final double e2;

  72.     /** 1 minus flatness. */
  73.     private final double g;

  75.     /** g * g. */
  76.     private final double g2;

  78.     /** Convergence limit. */
  79.     private double angularThreshold;

  81.     /** Simple constructor.
  82.      * <p>Standard values for Earth models can be found in the {@link org.orekit.utils.Constants Constants} class:</p>
  83.      * <table border="1" cellpadding="5" style="background-color:#f5f5dc;">
  84.      * <caption>Ellipsoid Models</caption>
  85.      * <tr style="background-color:#c9d5c9;"><th>model</th><th>a<sub>e</sub> (m)</th> <th>f</th></tr>
  86.      * <tr><td style="background-color:#c9d5c9;">GRS 80</td>
  87.      *     <td>{@link org.orekit.utils.Constants#GRS80_EARTH_EQUATORIAL_RADIUS Constants.GRS80_EARTH_EQUATORIAL_RADIUS}</td>
  88.      *     <td>{@link org.orekit.utils.Constants#GRS80_EARTH_FLATTENING Constants.GRS80_EARTH_FLATTENING}</td></tr>
  89.      * <tr><td style="background-color:#c9d5c9;">WGS84</td>
  90.      *     <td>{@link org.orekit.utils.Constants#WGS84_EARTH_EQUATORIAL_RADIUS Constants.WGS84_EARTH_EQUATORIAL_RADIUS}</td>
  91.      *     <td>{@link org.orekit.utils.Constants#WGS84_EARTH_FLATTENING Constants.WGS84_EARTH_FLATTENING}</td></tr>
  92.      * </table summary="">
  93.      * @param ae equatorial radius
  94.      * @param f the flattening (f = (a-b)/a)
  95.      * @param bodyFrame body frame related to body shape
  96.      * @see org.orekit.frames.FramesFactory#getITRF(org.orekit.utils.IERSConventions, boolean)
  97.      */
  98.     public OneAxisEllipsoid(final double ae, final double f,
  99.                             final Frame bodyFrame) {
  100.         super(bodyFrame, ae, ae, ae * (1.0 - f));
  101.         this.f    = f;
  102.         this.ae2  = ae * ae;
  103.         this.e2   = f * (2.0 - f);
  104.         this.g    = 1.0 - f;
  105.         this.g2   = g * g;
  106.         this.ap2  = ae2 * g2;
  107.         setAngularThreshold(1.0e-12);
  108.         this.bodyFrame = bodyFrame;
  109.     }

  111.     /** Set the angular convergence threshold.
  112.      * <p>The angular threshold is used both to identify points close to
  113.      * the ellipse axes and as the convergence threshold used to
  114.      * stop the iterations in the {@link #transform(Vector3D, Frame,
  115.      * AbsoluteDate)} method.</p>
  116.      * <p>If this method is not called, the default value is set to
  117.      * 10<sup>-12</sup>.</p>
  118.      * @param angularThreshold angular convergence threshold (rad)
  119.      */
  120.     public void setAngularThreshold(final double angularThreshold) {
  121.         this.angularThreshold = angularThreshold;
  122.     }

  124.     /** Get the equatorial radius of the body.
  125.      * @return equatorial radius of the body (m)
  126.      */
  127.     public double getEquatorialRadius() {
  128.         return getA();
  129.     }

  131.     /** Get the flattening of the body: f = (a-b)/a.
  132.      * @return the flattening
  133.      */
  134.     public double getFlattening() {
  135.         return f;
  136.     }

  138.     /** {@inheritDoc} */
  139.     public Frame getBodyFrame() {
  140.         return bodyFrame;
  141.     }

  143.     /** {@inheritDoc} */
  144.     public GeodeticPoint getIntersectionPoint(final Line line, final Vector3D close,
  145.                                               final Frame frame, final AbsoluteDate date)
  146.         throws OrekitException {

  148.         // transform line and close to body frame
  149.         final Transform frameToBodyFrame = frame.getTransformTo(bodyFrame, date);
  150.         final Line lineInBodyFrame = frameToBodyFrame.transformLine(line);
  151.         final Vector3D closeInBodyFrame = frameToBodyFrame.transformPosition(close);
  152.         final double closeAbscissa = lineInBodyFrame.getAbscissa(closeInBodyFrame);

  154.         // compute some miscellaneous variables outside of the loop
  155.         final Vector3D point    = lineInBodyFrame.getOrigin();
  156.         final double x          = point.getX();
  157.         final double y          = point.getY();
  158.         final double z          = point.getZ();
  159.         final double z2         = z * z;
  160.         final double r2         = x * x + y * y;

  162.         final Vector3D direction = lineInBodyFrame.getDirection();
  163.         final double dx         = direction.getX();
  164.         final double dy         = direction.getY();
  165.         final double dz         = direction.getZ();
  166.         final double cz2        = dx * dx + dy * dy;

  168.         // abscissa of the intersection as a root of a 2nd degree polynomial :
  169.         // a k^2 - 2 b k + c = 0
  170.         final double a  = 1.0 - e2 * cz2;
  171.         final double b  = -(g2 * (x * dx + y * dy) + z * dz);
  172.         final double c  = g2 * (r2 - ae2) + z2;
  173.         final double b2 = b * b;
  174.         final double ac = a * c;
  175.         if (b2 < ac) {
  176.             return null;
  177.         }
  178.         final double s  = FastMath.sqrt(b2 - ac);
  179.         final double k1 = (b < 0) ? (b - s) / a : c / (b + s);
  180.         final double k2 = c / (a * k1);

  182.         // select the right point
  183.         final double k =
  184.             (FastMath.abs(k1 - closeAbscissa) < FastMath.abs(k2 - closeAbscissa)) ? k1 : k2;
  185.         final Vector3D intersection = lineInBodyFrame.pointAt(k);
  186.         final double ix = intersection.getX();
  187.         final double iy = intersection.getY();
  188.         final double iz = intersection.getZ();

  190.         final double lambda = FastMath.atan2(iy, ix);
  191.         final double phi    = FastMath.atan2(iz, g2 * FastMath.sqrt(ix * ix + iy * iy));
  192.         return new GeodeticPoint(phi, lambda, 0.0);

  194.     }

  196.     /** {@inheritDoc} */
  197.     public <T extends RealFieldElement<T>> FieldGeodeticPoint<T> getIntersectionPoint(final FieldLine<T> line,
  198.                                                                                       final FieldVector3D<T> close,
  199.                                                                                       final Frame frame,
  200.                                                                                       final FieldAbsoluteDate<T> date)
  201.         throws OrekitException {

  203.         // transform line and close to body frame
  204.         final FieldTransform<T> frameToBodyFrame = frame.getTransformTo(bodyFrame, date);
  205.         final FieldLine<T>      lineInBodyFrame  = frameToBodyFrame.transformLine(line);
  206.         final FieldVector3D<T>  closeInBodyFrame = frameToBodyFrame.transformPosition(close);
  207.         final T                 closeAbscissa    = lineInBodyFrame.getAbscissa(closeInBodyFrame);

  209.         // compute some miscellaneous variables outside of the loop
  210.         final FieldVector3D<T> point = lineInBodyFrame.getOrigin();
  211.         final T x  = point.getX();
  212.         final T y  = point.getY();
  213.         final T z  = point.getZ();
  214.         final T z2 = z.multiply(z);
  215.         final T r2 = x.multiply(x).add(y.multiply(y));

  217.         final FieldVector3D<T> direction = lineInBodyFrame.getDirection();
  218.         final T dx  = direction.getX();
  219.         final T dy  = direction.getY();
  220.         final T dz  = direction.getZ();
  221.         final T cz2 = dx.multiply(dx).add(dy.multiply(dy));

  223.         // abscissa of the intersection as a root of a 2nd degree polynomial :
  224.         // a k^2 - 2 b k + c = 0
  225.         final T a  = cz2.multiply(e2).subtract(1.0).negate();
  226.         final T b  = x.multiply(dx).add(y.multiply(dy)).multiply(g2).add(z.multiply(dz)).negate();
  227.         final T c  = r2.subtract(ae2).multiply(g2).add(z2);
  228.         final T b2 = b.multiply(b);
  229.         final T ac = a.multiply(c);
  230.         if (b2.getReal() < ac.getReal()) {
  231.             return null;
  232.         }
  233.         final T s  = b2.subtract(ac).sqrt();
  234.         final T k1 = (b.getReal() < 0) ? b.subtract(s).divide(a) : c.divide(b.add(s));
  235.         final T k2 = c.divide(a.multiply(k1));

  237.         // select the right point
  238.         final T k = (FastMath.abs(k1.getReal() - closeAbscissa.getReal()) < FastMath.abs(k2.getReal() - closeAbscissa.getReal())) ?
  239.                     k1 : k2;
  240.         final FieldVector3D<T> intersection = lineInBodyFrame.pointAt(k);
  241.         final T ix = intersection.getX();
  242.         final T iy = intersection.getY();
  243.         final T iz = intersection.getZ();

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

  249.     }

  251.     /** {@inheritDoc} */
  252.     public Vector3D transform(final GeodeticPoint point) {
  253.         final double longitude = point.getLongitude();
  254.         final double cLambda   = FastMath.cos(longitude);
  255.         final double sLambda   = FastMath.sin(longitude);
  256.         final double latitude  = point.getLatitude();
  257.         final double cPhi      = FastMath.cos(latitude);
  258.         final double sPhi      = FastMath.sin(latitude);
  259.         final double h         = point.getAltitude();
  260.         final double n         = getA() / FastMath.sqrt(1.0 - e2 * sPhi * sPhi);
  261.         final double r         = (n + h) * cPhi;
  262.         return new Vector3D(r * cLambda, r * sLambda, (g2 * n + h) * sPhi);
  263.     }

  265.     /** {@inheritDoc} */
  266.     public <T extends RealFieldElement<T>> FieldVector3D<T> transform(final FieldGeodeticPoint<T> point) {

  268.         final T latitude  = point.getLatitude();
  269.         final T longitude = point.getLongitude();
  270.         final T altitude  = point.getAltitude();

  272.         final T cLambda = longitude.cos();
  273.         final T sLambda = longitude.sin();
  274.         final T cPhi    = latitude.cos();
  275.         final T sPhi    = latitude.sin();
  276.         final T n       = sPhi.multiply(sPhi).multiply(e2).subtract(1.0).negate().sqrt().reciprocal().multiply(getA());
  277.         final T r       = n.add(altitude).multiply(cPhi);

  279.         return new FieldVector3D<>(r.multiply(cLambda),
  280.                                    r.multiply(sLambda),
  281.                                    sPhi.multiply(altitude.add(n.multiply(g2))));
  282.     }

  284.     /** {@inheritDoc} */
  285.     public Vector3D projectToGround(final Vector3D point, final AbsoluteDate date, final Frame frame)
  286.         throws OrekitException {

  288.         // transform point to body frame
  289.         final Transform  toBody    = frame.getTransformTo(bodyFrame, date);
  290.         final Vector3D   p         = toBody.transformPosition(point);
  291.         final double     z         = p.getZ();
  292.         final double     r         = FastMath.hypot(p.getX(), p.getY());

  294.         // set up the 2D meridian ellipse
  295.         final Ellipse meridian = new Ellipse(Vector3D.ZERO,
  296.                                              new Vector3D(p.getX() / r, p.getY() / r, 0),
  297.                                              Vector3D.PLUS_K,
  298.                                              getA(), getC(), bodyFrame);

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

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

  306.     }

  308.     /** {@inheritDoc} */
  309.     public TimeStampedPVCoordinates projectToGround(final TimeStampedPVCoordinates pv, final Frame frame)
  310.         throws OrekitException {

  312.         // transform point to body frame
  313.         final Transform                toBody        = frame.getTransformTo(bodyFrame, pv.getDate());
  314.         final TimeStampedPVCoordinates pvInBodyFrame = toBody.transformPVCoordinates(pv);
  315.         final Vector3D                 p             = pvInBodyFrame.getPosition();
  316.         final double                   r             = FastMath.hypot(p.getX(), p.getY());

  318.         // set up the 2D ellipse corresponding to first principal curvature along meridian
  319.         final Vector3D meridian = new Vector3D(p.getX() / r, p.getY() / r, 0);
  320.         final Ellipse firstPrincipalCurvature =
  321.                 new Ellipse(Vector3D.ZERO, meridian, Vector3D.PLUS_K, getA(), getC(), bodyFrame);

  323.         // project coordinates in the meridian plane
  324.         final TimeStampedPVCoordinates gpFirst = firstPrincipalCurvature.projectToEllipse(pvInBodyFrame);
  325.         final Vector3D                 gpP     = gpFirst.getPosition();
  326.         final double                   gr      = MathArrays.linearCombination(gpP.getX(), meridian.getX(),
  327.                                                                               gpP.getY(), meridian.getY());
  328.         final double                   gz      = gpP.getZ();

  330.         // topocentric frame
  331.         final Vector3D east   = new Vector3D(-meridian.getY(), meridian.getX(), 0);
  332.         final Vector3D zenith = new Vector3D(gr * getC() / getA(), meridian, gz * getA() / getC(), Vector3D.PLUS_K).normalize();
  333.         final Vector3D north  = Vector3D.crossProduct(zenith, east);

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

  339.         final Vector3D gpV = gpFirst.getVelocity().add(gpSecond.getVelocity());
  340.         final Vector3D gpA = gpFirst.getAcceleration().add(gpSecond.getAcceleration());

  342.         // moving projected point
  343.         final TimeStampedPVCoordinates groundPV =
  344.                 new TimeStampedPVCoordinates(pv.getDate(), gpP, gpV, gpA);

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

  349.     }

  351.     /** {@inheritDoc}
  352.      * <p>
  353.      * This method is based on Toshio Fukushima's algorithm which uses Halley's method.
  354.      * <a href="https://www.researchgate.net/publication/227215135_Transformation_from_Cartesian_to_Geodetic_Coordinates_Accelerated_by_Halley's_Method">
  355.      * transformation from Cartesian to Geodetic Coordinates Accelerated by Halley's Method</a>,
  356.      * Toshio Fukushima, Journal of Geodesy 9(12):689-693, February 2006
  357.      * </p>
  358.      * <p>
  359.      * Some changes have been added to the original method:
  360.      * <ul>
  361.      *   <li>in order to handle more accurately corner cases near the pole</li>
  362.      *   <li>in order to handle properly corner cases near the equatorial plane, even far inside the ellipsoid</li>
  363.      *   <li>in order to handle very flat ellipsoids</li>
  364.      * </ul>
  365.      * </p>
  366.      */
  367.     public GeodeticPoint transform(final Vector3D point, final Frame frame, final AbsoluteDate date)
  368.         throws OrekitException {

  370.         // transform point to body frame
  371.         final Vector3D pointInBodyFrame = frame.getTransformTo(bodyFrame, date).transformPosition(point);
  372.         final double   r2               = pointInBodyFrame.getX() * pointInBodyFrame.getX() +
  373.                                           pointInBodyFrame.getY() * pointInBodyFrame.getY();
  374.         final double   r                = FastMath.sqrt(r2);
  375.         final double   z                = pointInBodyFrame.getZ();

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

  379.         double h;
  380.         double phi;
  381.         if (r <= ANGULAR_THRESHOLD * FastMath.abs(z)) {
  382.             // the point is almost on the polar axis, approximate the ellipsoid with
  383.             // the osculating sphere whose center is at evolute cusp along polar axis
  384.             final double osculatingRadius = ae2 / getC();
  385.             final double evoluteCuspZ     = FastMath.copySign(getA() * e2 / g, -z);
  386.             final double deltaZ           = z - evoluteCuspZ;
  387.             // we use π/2 - atan(r/Δz) instead of atan(Δz/r) for accuracy purposes, as r is much smaller than Δz
  388.             phi = FastMath.copySign(0.5 * FastMath.PI - FastMath.atan(r / FastMath.abs(deltaZ)), deltaZ);
  389.             h   = FastMath.hypot(deltaZ, r) - osculatingRadius;
  390.         } else if (FastMath.abs(z) <= ANGULAR_THRESHOLD * r) {
  391.             // the point is almost on the major axis

  393.             final double osculatingRadius = ap2 / getA();
  394.             final double evoluteCuspR     = getA() * e2;
  395.             final double deltaR           = r - evoluteCuspR;
  396.             if (deltaR >= 0) {
  397.                 // the point is outside of the ellipse evolute, approximate the ellipse
  398.                 // with the osculating circle whose center is at evolute cusp along major axis
  399.                 phi = (deltaR == 0) ? 0.0 : FastMath.atan(z / deltaR);
  400.                 h   = FastMath.hypot(deltaR, z) - osculatingRadius;
  401.             } else {
  402.                 // the point is on the part of the major axis within ellipse evolute
  403.                 // we can compute the closest ellipse point analytically, and it is NOT near the equator
  404.                 final double rClose = r / e2;
  405.                 final double zClose = FastMath.copySign(g * FastMath.sqrt(ae2 - rClose * rClose), z);
  406.                 phi = FastMath.atan((zClose - z) / (rClose - r));
  407.                 h   = -FastMath.hypot(r - rClose, z - zClose);
  408.             }

  410.         } else {
  411.             // use Toshio Fukushima method, with several iterations
  412.             final double epsPhi = 1.0e-15;
  413.             final double epsH   = 1.0e-14 * FastMath.max(getA(), FastMath.sqrt(r2 + z * z));
  414.             final double c     = getA() * e2;
  415.             final double absZ  = FastMath.abs(z);
  416.             final double zc    = g * absZ;
  417.             double sn  = absZ;
  418.             double sn2 = sn * sn;
  419.             double cn  = g * r;
  420.             double cn2 = cn * cn;
  421.             double an2 = cn2 + sn2;
  422.             double an  = FastMath.sqrt(an2);
  423.             double bn  = 0;
  424.             phi = Double.POSITIVE_INFINITY;
  425.             h   = Double.POSITIVE_INFINITY;
  426.             for (int i = 0; i < 10; ++i) { // this usually converges in 2 iterations
  427.                 final double oldSn  = sn;
  428.                 final double oldCn  = cn;
  429.                 final double oldPhi = phi;
  430.                 final double oldH   = h;
  431.                 final double an3    = an2 * an;
  432.                 final double csncn  = c * sn * cn;
  433.                 bn    = 1.5 * csncn * ((r * sn - zc * cn) * an - csncn);
  434.                 sn    = (zc * an3 + c * sn2 * sn) * an3 - bn * sn;
  435.                 cn    = (r  * an3 - c * cn2 * cn) * an3 - bn * cn;
  436.                 if (sn * oldSn < 0 || cn < 0) {
  437.                     // the Halley iteration went too far, we restrict it and iterate again
  438.                     while (sn * oldSn < 0 || cn < 0) {
  439.                         sn = (sn + oldSn) / 2;
  440.                         cn = (cn + oldCn) / 2;
  441.                     }
  442.                 } else {

  444.                     // rescale components to avoid overflow when several iterations are used
  445.                     final int exp = (FastMath.getExponent(sn) + FastMath.getExponent(cn)) / 2;
  446.                     sn = FastMath.scalb(sn, -exp);
  447.                     cn = FastMath.scalb(cn, -exp);

  449.                     sn2 = sn * sn;
  450.                     cn2 = cn * cn;
  451.                     an2 = cn2 + sn2;
  452.                     an  = FastMath.sqrt(an2);

  454.                     final double cc = g * cn;
  455.                     h = (r * cc + absZ * sn - getA() * g * an) / FastMath.sqrt(an2 - e2 * cn2);
  456.                     if (FastMath.abs(oldH   - h)   < epsH) {
  457.                         phi = FastMath.copySign(FastMath.atan(sn / cc), z);
  458.                         if (FastMath.abs(oldPhi - phi) < epsPhi) {
  459.                             break;
  460.                         }
  461.                     }

  463.                 }

  465.             }
  466.         }

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

  470.     }

  472.     /** {@inheritDoc}
  473.      * <p>
  474.      * This method is based on Toshio Fukushima's algorithm which uses Halley's method.
  475.      * <a href="https://www.researchgate.net/publication/227215135_Transformation_from_Cartesian_to_Geodetic_Coordinates_Accelerated_by_Halley's_Method">
  476.      * transformation from Cartesian to Geodetic Coordinates Accelerated by Halley's Method</a>,
  477.      * Toshio Fukushima, Journal of Geodesy 9(12):689-693, February 2006
  478.      * </p>
  479.      * <p>
  480.      * Some changes have been added to the original method:
  481.      * <ul>
  482.      *   <li>in order to handle more accurately corner cases near the pole</li>
  483.      *   <li>in order to handle properly corner cases near the equatorial plane, even far inside the ellipsoid</li>
  484.      *   <li>in order to handle very flat ellipsoids</li>
  485.      * </ul>
  486.      * </p>
  487.      */
  488.     public <T extends RealFieldElement<T>> FieldGeodeticPoint<T> transform(final FieldVector3D<T> point,
  489.                                                                            final Frame frame,
  490.                                                                            final FieldAbsoluteDate<T> date)
  491.         throws OrekitException {

  493.         // transform point to body frame
  494.         final FieldVector3D<T> pointInBodyFrame = frame.getTransformTo(bodyFrame, date).transformPosition(point);
  495.         final T   r2                            = pointInBodyFrame.getX().multiply(pointInBodyFrame.getX()).
  496.                                               add(pointInBodyFrame.getY().multiply(pointInBodyFrame.getY()));
  497.         final T   r                             = r2.sqrt();
  498.         final T   z                             = pointInBodyFrame.getZ();

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

  502.         T h;
  503.         T phi;
  504.         if (r.getReal() <= ANGULAR_THRESHOLD * FastMath.abs(z.getReal())) {
  505.             // the point is almost on the polar axis, approximate the ellipsoid with
  506.             // the osculating sphere whose center is at evolute cusp along polar axis
  507.             final double osculatingRadius = ae2 / getC();
  508.             final double evoluteCuspZ     = FastMath.copySign(getA() * e2 / g, -z.getReal());
  509.             final T      deltaZ           = z.subtract(evoluteCuspZ);
  510.             // we use π/2 - atan(r/Δz) instead of atan(Δz/r) for accuracy purposes, as r is much smaller than Δz
  511.             phi = r.divide(deltaZ.abs()).atan().negate().add(0.5 * FastMath.PI).copySign(deltaZ);
  512.             h   = deltaZ.hypot(r).subtract(osculatingRadius);
  513.         } else if (FastMath.abs(z.getReal()) <= ANGULAR_THRESHOLD * r.getReal()) {
  514.             // the point is almost on the major axis

  516.             final double osculatingRadius = ap2 / getA();
  517.             final double evoluteCuspR     = getA() * e2;
  518.             final T      deltaR           = r.subtract(evoluteCuspR);
  519.             if (deltaR.getReal() >= 0) {
  520.                 // the point is outside of the ellipse evolute, approximate the ellipse
  521.                 // with the osculating circle whose center is at evolute cusp along major axis
  522.                 phi = (deltaR.getReal() == 0) ? z.getField().getZero() : z.divide(deltaR).atan();
  523.                 h   = deltaR.hypot(z).subtract(osculatingRadius);
  524.             } else {
  525.                 // the point is on the part of the major axis within ellipse evolute
  526.                 // we can compute the closest ellipse point analytically, and it is NOT near the equator
  527.                 final T rClose = r.divide(e2);
  528.                 final T zClose = rClose.multiply(rClose).negate().add(ae2).sqrt().multiply(g).copySign(z);
  529.                 phi = zClose.subtract(z).divide(rClose.subtract(r)).atan();
  530.                 h   = r.subtract(rClose).hypot(z.subtract(zClose)).negate();
  531.             }

  533.         } else {
  534.             // use Toshio Fukushima method, with several iterations
  535.             final double epsPhi = 1.0e-15;
  536.             final double epsH   = 1.0e-14 * getA();
  537.             final double c      = getA() * e2;
  538.             final T      absZ   = z.abs();
  539.             final T      zc     = absZ.multiply(g);
  540.             T            sn     = absZ;
  541.             T            sn2    = sn.multiply(sn);
  542.             T            cn     = r.multiply(g);
  543.             T            cn2    = cn.multiply(cn);
  544.             T            an2    = cn2.add(sn2);
  545.             T            an     = an2.sqrt();
  546.             T            bn     = an.getField().getZero();
  547.             phi = an.getField().getZero().add(Double.POSITIVE_INFINITY);
  548.             h   = an.getField().getZero().add(Double.POSITIVE_INFINITY);
  549.             for (int i = 0; i < 10; ++i) { // this usually converges in 2 iterations
  550.                 final T oldSn  = sn;
  551.                 final T oldCn  = cn;
  552.                 final T oldPhi = phi;
  553.                 final T oldH   = h;
  554.                 final T an3    = an2.multiply(an);
  555.                 final T csncn  = sn.multiply(cn).multiply(c);
  556.                 bn    = csncn.multiply(1.5).multiply((r.multiply(sn).subtract(zc.multiply(cn))).multiply(an).subtract(csncn));
  557.                 sn    = zc.multiply(an3).add(sn2.multiply(sn).multiply(c)).multiply(an3).subtract(bn.multiply(sn));
  558.                 cn    = r.multiply(an3).subtract(cn2.multiply(cn).multiply(c)).multiply(an3).subtract(bn.multiply(cn));
  559.                 if (sn.getReal() * oldSn.getReal() < 0 || cn.getReal() < 0) {
  560.                     // the Halley iteration went too far, we restrict it and iterate again
  561.                     while (sn.getReal() * oldSn.getReal() < 0 || cn.getReal() < 0) {
  562.                         sn = sn.add(oldSn).multiply(0.5);
  563.                         cn = cn.add(oldCn).multiply(0.5);
  564.                     }
  565.                 } else {

  567.                     // rescale components to avoid overflow when several iterations are used
  568.                     final int exp = (FastMath.getExponent(sn.getReal()) + FastMath.getExponent(cn.getReal())) / 2;
  569.                     sn = sn.scalb(-exp);
  570.                     cn = cn.scalb(-exp);

  572.                     sn2 = sn.multiply(sn);
  573.                     cn2 = cn.multiply(cn);
  574.                     an2 = cn2.add(sn2);
  575.                     an  = an2.sqrt();

  577.                     final T cc = cn.multiply(g);
  578.                     h = r.multiply(cc).add(absZ.multiply(sn)).subtract(an.multiply(getA() * g)).divide(an2.subtract(cn2.multiply(e2)).sqrt());
  579.                     if (FastMath.abs(oldH.getReal()  - h.getReal())   < epsH) {
  580.                         phi = sn.divide(cc).atan().copySign(z);
  581.                         if (FastMath.abs(oldPhi.getReal() - phi.getReal()) < epsPhi) {
  582.                             break;
  583.                         }
  584.                     }

  586.                 }

  588.             }
  589.         }

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

  593.     }

  595.     /** Transform a Cartesian point to a surface-relative point.
  596.      * @param point Cartesian point
  597.      * @param frame frame in which Cartesian point is expressed
  598.      * @param date date of the computation (used for frames conversions)
  599.      * @return point at the same location but as a surface-relative point,
  600.      * using time as the single derivation parameter
  601.      * @exception OrekitException if point cannot be converted to body frame
  602.      */
  603.     public FieldGeodeticPoint<DerivativeStructure> transform(final PVCoordinates point,
  604.                                                              final Frame frame, final AbsoluteDate date)
  605.         throws OrekitException {

  607.         // transform point to body frame
  608.         final Transform toBody = frame.getTransformTo(bodyFrame, date);
  609.         final PVCoordinates pointInBodyFrame = toBody.transformPVCoordinates(point);
  610.         final FieldVector3D<DerivativeStructure> p = pointInBodyFrame.toDerivativeStructureVector(2);
  611.         final DerivativeStructure   pr2 = p.getX().multiply(p.getX()).add(p.getY().multiply(p.getY()));
  612.         final DerivativeStructure   pr  = pr2.sqrt();
  613.         final DerivativeStructure   pz  = p.getZ();

  615.         // project point on the ellipsoid surface
  616.         final TimeStampedPVCoordinates groundPoint = projectToGround(new TimeStampedPVCoordinates(date, pointInBodyFrame),
  617.                                                                      bodyFrame);
  618.         final FieldVector3D<DerivativeStructure> gp = groundPoint.toDerivativeStructureVector(2);
  619.         final DerivativeStructure   gpr2 = gp.getX().multiply(gp.getX()).add(gp.getY().multiply(gp.getY()));
  620.         final DerivativeStructure   gpr  = gpr2.sqrt();
  621.         final DerivativeStructure   gpz  = gp.getZ();

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

  628.         return new FieldGeodeticPoint<>(DerivativeStructure.atan2(gpz, gpr.multiply(g2)),
  629.                                                                   DerivativeStructure.atan2(p.getY(), p.getX()),
  630.                                                                   DerivativeStructure.hypot(dr, dz).copySign(insideIfNegative));
  631.     }

  633.     /** Replace the instance with a data transfer object for serialization.
  634.      * <p>
  635.      * This intermediate class serializes the files supported names, the
  636.      * ephemeris type and the body name.
  637.      * </p>
  638.      * @return data transfer object that will be serialized
  639.      */
  640.     private Object writeReplace() {
  641.         return new DataTransferObject(getA(), f, bodyFrame, angularThreshold);
  642.     }

  644.     /** Internal class used only for serialization. */
  645.     private static class DataTransferObject implements Serializable {

  647.         /** Serializable UID. */
  648.         private static final long serialVersionUID = 20130518L;

  650.         /** Equatorial radius. */
  651.         private final double ae;

  653.         /** Flattening. */
  654.         private final double f;

  656.         /** Body frame related to body shape. */
  657.         private final Frame bodyFrame;

  659.         /** Convergence limit. */
  660.         private final double angularThreshold;

  662.         /** Simple constructor.
  663.          * @param ae equatorial radius
  664.          * @param f the flattening (f = (a-b)/a)
  665.          * @param bodyFrame body frame related to body shape
  666.          * @param angularThreshold convergence limit
  667.          */
  668.         DataTransferObject(final double ae, final double f,
  669.                                   final Frame bodyFrame, final double angularThreshold) {
  670.             this.ae               = ae;
  671.             this.f                = f;
  672.             this.bodyFrame        = bodyFrame;
  673.             this.angularThreshold = angularThreshold;
  674.         }

  676.         /** Replace the deserialized data transfer object with a
  677.          * {@link JPLCelestialBody}.
  678.          * @return replacement {@link JPLCelestialBody}
  679.          */
  680.         private Object readResolve() {
  681.             final OneAxisEllipsoid ellipsoid = new OneAxisEllipsoid(ae, f, bodyFrame);
  682.             ellipsoid.setAngularThreshold(angularThreshold);
  683.             return ellipsoid;
  684.         }

  686.     }

  688. }
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