OneAxisEllipsoid.java

  1. /* Copyright 2002-2022 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;

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


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

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

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

  56.     /** Serializable UID. */
  57.     private static final long serialVersionUID = 20130518L;

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

  62.     /** Body frame related to body shape. */
  63.     private final Frame bodyFrame;

  65.     /** Equatorial radius power 2. */
  66.     private final double ae2;

  68.     /** Polar radius power 2. */
  69.     private final double ap2;

  71.     /** Flattening. */
  72.     private final double f;

  74.     /** Eccentricity. */
  75.     private final double e;

  77.     /** Eccentricity squared. */
  78.     private final double e2;

  80.     /** 1 minus flatness. */
  81.     private final double g;

  83.     /** g squared. */
  84.     private final double g2;

  86.     /** Convergence limit. */
  87.     private double angularThreshold;

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

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

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

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

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

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

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

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

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

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

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

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

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

  232.     }

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

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

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

  250.     }

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

  272.         // transform line and close to body frame
  273.         final FieldTransform<T> frameToBodyFrame = frame.getTransformTo(bodyFrame, date);
  274.         final FieldLine<T>      lineInBodyFrame  = frameToBodyFrame.transformLine(line);

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

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

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

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

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

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

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

  330.     }

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

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

  347.         final T latitude  = point.getLatitude();
  348.         final T longitude = point.getLongitude();
  349.         final T altitude  = point.getAltitude();

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

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

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

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

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

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

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

  386.     }

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

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

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

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

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

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

  418.         final Vector3D gpV = gpFirst.getVelocity().add(gpSecond.getVelocity());
  419.         final Vector3D gpA = gpFirst.getAcceleration().add(gpSecond.getAcceleration());

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

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

  428.     }

  430.     /** {@inheritDoc}
  431.      * <p>
  432.      * This method is based on Toshio Fukushima's algorithm which uses Halley's method.
  433.      * <a href="https://www.researchgate.net/publication/227215135_Transformation_from_Cartesian_to_Geodetic_Coordinates_Accelerated_by_Halley's_Method">
  434.      * transformation from Cartesian to Geodetic Coordinates Accelerated by Halley's Method</a>,
  435.      * Toshio Fukushima, Journal of Geodesy 9(12):689-693, February 2006
  436.      * </p>
  437.      * <p>
  438.      * Some changes have been added to the original method:
  439.      * <ul>
  440.      *   <li>in order to handle more accurately corner cases near the pole</li>
  441.      *   <li>in order to handle properly corner cases near the equatorial plane, even far inside the ellipsoid</li>
  442.      *   <li>in order to handle very flat ellipsoids</li>
  443.      * </ul>
  444.      */
  445.     public GeodeticPoint transform(final Vector3D point, final Frame frame, final AbsoluteDate date) {

  447.         // transform point to body frame
  448.         final Vector3D pointInBodyFrame = frame.getStaticTransformTo(bodyFrame, date)
  449.                 .transformPosition(point);
  450.         final double   r2               = pointInBodyFrame.getX() * pointInBodyFrame.getX() +
  451.                                           pointInBodyFrame.getY() * pointInBodyFrame.getY();
  452.         final double   r                = FastMath.sqrt(r2);
  453.         final double   z                = pointInBodyFrame.getZ();

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

  457.         double h;
  458.         double phi;
  459.         if (r <= ANGULAR_THRESHOLD * FastMath.abs(z)) {
  460.             // the point is almost on the polar axis, approximate the ellipsoid with
  461.             // the osculating sphere whose center is at evolute cusp along polar axis
  462.             final double osculatingRadius = ae2 / getC();
  463.             final double evoluteCuspZ     = FastMath.copySign(getA() * e2 / g, -z);
  464.             final double deltaZ           = z - evoluteCuspZ;
  465.             // we use Ï€/2 - atan(r/Δz) instead of atan(Δz/r) for accuracy purposes, as r is much smaller than Δz
  466.             phi = FastMath.copySign(0.5 * FastMath.PI - FastMath.atan(r / FastMath.abs(deltaZ)), deltaZ);
  467.             h   = FastMath.hypot(deltaZ, r) - osculatingRadius;
  468.         } else if (FastMath.abs(z) <= ANGULAR_THRESHOLD * r) {
  469.             // the point is almost on the major axis

  471.             final double osculatingRadius = ap2 / getA();
  472.             final double evoluteCuspR     = getA() * e2;
  473.             final double deltaR           = r - evoluteCuspR;
  474.             if (deltaR >= 0) {
  475.                 // the point is outside of the ellipse evolute, approximate the ellipse
  476.                 // with the osculating circle whose center is at evolute cusp along major axis
  477.                 phi = (deltaR == 0) ? 0.0 : FastMath.atan(z / deltaR);
  478.                 h   = FastMath.hypot(deltaR, z) - osculatingRadius;
  479.             } else {
  480.                 // the point is on the part of the major axis within ellipse evolute
  481.                 // we can compute the closest ellipse point analytically, and it is NOT near the equator
  482.                 final double rClose = r / e2;
  483.                 final double zClose = FastMath.copySign(g * FastMath.sqrt(ae2 - rClose * rClose), z);
  484.                 phi = FastMath.atan((zClose - z) / (rClose - r));
  485.                 h   = -FastMath.hypot(r - rClose, z - zClose);
  486.             }

  488.         } else {
  489.             // use Toshio Fukushima method, with several iterations
  490.             final double epsPhi = 1.0e-15;
  491.             final double epsH   = 1.0e-14 * FastMath.max(getA(), FastMath.sqrt(r2 + z * z));
  492.             final double c     = getA() * e2;
  493.             final double absZ  = FastMath.abs(z);
  494.             final double zc    = g * absZ;
  495.             double sn  = absZ;
  496.             double sn2 = sn * sn;
  497.             double cn  = g * r;
  498.             double cn2 = cn * cn;
  499.             double an2 = cn2 + sn2;
  500.             double an  = FastMath.sqrt(an2);
  501.             double bn  = 0;
  502.             phi = Double.POSITIVE_INFINITY;
  503.             h   = Double.POSITIVE_INFINITY;
  504.             for (int i = 0; i < 10; ++i) { // this usually converges in 2 iterations
  505.                 final double oldSn  = sn;
  506.                 final double oldCn  = cn;
  507.                 final double oldPhi = phi;
  508.                 final double oldH   = h;
  509.                 final double an3    = an2 * an;
  510.                 final double csncn  = c * sn * cn;
  511.                 bn    = 1.5 * csncn * ((r * sn - zc * cn) * an - csncn);
  512.                 sn    = (zc * an3 + c * sn2 * sn) * an3 - bn * sn;
  513.                 cn    = (r  * an3 - c * cn2 * cn) * an3 - bn * cn;
  514.                 if (sn * oldSn < 0 || cn < 0) {
  515.                     // the Halley iteration went too far, we restrict it and iterate again
  516.                     while (sn * oldSn < 0 || cn < 0) {
  517.                         sn = (sn + oldSn) / 2;
  518.                         cn = (cn + oldCn) / 2;
  519.                     }
  520.                 } else {

  522.                     // rescale components to avoid overflow when several iterations are used
  523.                     final int exp = (FastMath.getExponent(sn) + FastMath.getExponent(cn)) / 2;
  524.                     sn = FastMath.scalb(sn, -exp);
  525.                     cn = FastMath.scalb(cn, -exp);

  527.                     sn2 = sn * sn;
  528.                     cn2 = cn * cn;
  529.                     an2 = cn2 + sn2;
  530.                     an  = FastMath.sqrt(an2);

  532.                     final double cc = g * cn;
  533.                     h = (r * cc + absZ * sn - getA() * g * an) / FastMath.sqrt(an2 - e2 * cn2);
  534.                     if (FastMath.abs(oldH   - h)   < epsH) {
  535.                         phi = FastMath.copySign(FastMath.atan(sn / cc), z);
  536.                         if (FastMath.abs(oldPhi - phi) < epsPhi) {
  537.                             break;
  538.                         }
  539.                     }

  541.                 }

  543.             }
  544.         }

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

  548.     }

  550.     /** {@inheritDoc}
  551.      * <p>
  552.      * This method is based on Toshio Fukushima's algorithm which uses Halley's method.
  553.      * <a href="https://www.researchgate.net/publication/227215135_Transformation_from_Cartesian_to_Geodetic_Coordinates_Accelerated_by_Halley's_Method">
  554.      * transformation from Cartesian to Geodetic Coordinates Accelerated by Halley's Method</a>,
  555.      * Toshio Fukushima, Journal of Geodesy 9(12):689-693, February 2006
  556.      * </p>
  557.      * <p>
  558.      * Some changes have been added to the original method:
  559.      * <ul>
  560.      *   <li>in order to handle more accurately corner cases near the pole</li>
  561.      *   <li>in order to handle properly corner cases near the equatorial plane, even far inside the ellipsoid</li>
  562.      *   <li>in order to handle very flat ellipsoids</li>
  563.      * </ul>
  564.      */
  565.     public <T extends CalculusFieldElement<T>> FieldGeodeticPoint<T> transform(final FieldVector3D<T> point,
  566.                                                                            final Frame frame,
  567.                                                                            final FieldAbsoluteDate<T> date) {

  569.         // transform point to body frame
  570.         final FieldVector3D<T> pointInBodyFrame = frame.getTransformTo(bodyFrame, date).transformPosition(point);
  571.         final T   r2                            = pointInBodyFrame.getX().multiply(pointInBodyFrame.getX()).
  572.                                               add(pointInBodyFrame.getY().multiply(pointInBodyFrame.getY()));
  573.         final T   r                             = r2.sqrt();
  574.         final T   z                             = pointInBodyFrame.getZ();

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

  578.         T h;
  579.         T phi;
  580.         if (r.getReal() <= ANGULAR_THRESHOLD * FastMath.abs(z.getReal())) {
  581.             // the point is almost on the polar axis, approximate the ellipsoid with
  582.             // the osculating sphere whose center is at evolute cusp along polar axis
  583.             final double osculatingRadius = ae2 / getC();
  584.             final double evoluteCuspZ     = FastMath.copySign(getA() * e2 / g, -z.getReal());
  585.             final T      deltaZ           = z.subtract(evoluteCuspZ);
  586.             // we use Ï€/2 - atan(r/Δz) instead of atan(Δz/r) for accuracy purposes, as r is much smaller than Δz
  587.             phi = r.divide(deltaZ.abs()).atan().negate().add(r.getPi().multiply(0.5)).copySign(deltaZ);
  588.             h   = deltaZ.hypot(r).subtract(osculatingRadius);
  589.         } else if (FastMath.abs(z.getReal()) <= ANGULAR_THRESHOLD * r.getReal()) {
  590.             // the point is almost on the major axis

  592.             final double osculatingRadius = ap2 / getA();
  593.             final double evoluteCuspR     = getA() * e2;
  594.             final T      deltaR           = r.subtract(evoluteCuspR);
  595.             if (deltaR.getReal() >= 0) {
  596.                 // the point is outside of the ellipse evolute, approximate the ellipse
  597.                 // with the osculating circle whose center is at evolute cusp along major axis
  598.                 phi = (deltaR.getReal() == 0) ? z.getField().getZero() : z.divide(deltaR).atan();
  599.                 h   = deltaR.hypot(z).subtract(osculatingRadius);
  600.             } else {
  601.                 // the point is on the part of the major axis within ellipse evolute
  602.                 // we can compute the closest ellipse point analytically, and it is NOT near the equator
  603.                 final T rClose = r.divide(e2);
  604.                 final T zClose = rClose.multiply(rClose).negate().add(ae2).sqrt().multiply(g).copySign(z);
  605.                 phi = zClose.subtract(z).divide(rClose.subtract(r)).atan();
  606.                 h   = r.subtract(rClose).hypot(z.subtract(zClose)).negate();
  607.             }

  609.         } else {
  610.             // use Toshio Fukushima method, with several iterations
  611.             final double epsPhi = 1.0e-15;
  612.             final double epsH   = 1.0e-14 * getA();
  613.             final double c      = getA() * e2;
  614.             final T      absZ   = z.abs();
  615.             final T      zc     = absZ.multiply(g);
  616.             T            sn     = absZ;
  617.             T            sn2    = sn.multiply(sn);
  618.             T            cn     = r.multiply(g);
  619.             T            cn2    = cn.multiply(cn);
  620.             T            an2    = cn2.add(sn2);
  621.             T            an     = an2.sqrt();
  622.             T            bn     = an.getField().getZero();
  623.             phi = an.getField().getZero().add(Double.POSITIVE_INFINITY);
  624.             h   = an.getField().getZero().add(Double.POSITIVE_INFINITY);
  625.             for (int i = 0; i < 10; ++i) { // this usually converges in 2 iterations
  626.                 final T oldSn  = sn;
  627.                 final T oldCn  = cn;
  628.                 final T oldPhi = phi;
  629.                 final T oldH   = h;
  630.                 final T an3    = an2.multiply(an);
  631.                 final T csncn  = sn.multiply(cn).multiply(c);
  632.                 bn    = csncn.multiply(1.5).multiply((r.multiply(sn).subtract(zc.multiply(cn))).multiply(an).subtract(csncn));
  633.                 sn    = zc.multiply(an3).add(sn2.multiply(sn).multiply(c)).multiply(an3).subtract(bn.multiply(sn));
  634.                 cn    = r.multiply(an3).subtract(cn2.multiply(cn).multiply(c)).multiply(an3).subtract(bn.multiply(cn));
  635.                 if (sn.getReal() * oldSn.getReal() < 0 || cn.getReal() < 0) {
  636.                     // the Halley iteration went too far, we restrict it and iterate again
  637.                     while (sn.getReal() * oldSn.getReal() < 0 || cn.getReal() < 0) {
  638.                         sn = sn.add(oldSn).multiply(0.5);
  639.                         cn = cn.add(oldCn).multiply(0.5);
  640.                     }
  641.                 } else {

  643.                     // rescale components to avoid overflow when several iterations are used
  644.                     final int exp = (FastMath.getExponent(sn.getReal()) + FastMath.getExponent(cn.getReal())) / 2;
  645.                     sn = sn.scalb(-exp);
  646.                     cn = cn.scalb(-exp);

  648.                     sn2 = sn.multiply(sn);
  649.                     cn2 = cn.multiply(cn);
  650.                     an2 = cn2.add(sn2);
  651.                     an  = an2.sqrt();

  653.                     final T cc = cn.multiply(g);
  654.                     h = r.multiply(cc).add(absZ.multiply(sn)).subtract(an.multiply(getA() * g)).divide(an2.subtract(cn2.multiply(e2)).sqrt());
  655.                     if (FastMath.abs(oldH.getReal()  - h.getReal())   < epsH) {
  656.                         phi = sn.divide(cc).atan().copySign(z);
  657.                         if (FastMath.abs(oldPhi.getReal() - phi.getReal()) < epsPhi) {
  658.                             break;
  659.                         }
  660.                     }

  662.                 }

  664.             }
  665.         }

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

  669.     }

  671.     /** Transform a Cartesian point to a surface-relative point.
  672.      * @param point Cartesian point
  673.      * @param frame frame in which Cartesian point is expressed
  674.      * @param date date of the computation (used for frames conversions)
  675.      * @return point at the same location but as a surface-relative point,
  676.      * using time as the single derivation parameter
  677.      */
  678.     public FieldGeodeticPoint<DerivativeStructure> transform(final PVCoordinates point,
  679.                                                              final Frame frame, final AbsoluteDate date) {

  681.         // transform point to body frame
  682.         final Transform toBody = frame.getTransformTo(bodyFrame, date);
  683.         final PVCoordinates pointInBodyFrame = toBody.transformPVCoordinates(point);
  684.         final FieldVector3D<DerivativeStructure> p = pointInBodyFrame.toDerivativeStructureVector(2);
  685.         final DerivativeStructure   pr2 = p.getX().multiply(p.getX()).add(p.getY().multiply(p.getY()));
  686.         final DerivativeStructure   pr  = pr2.sqrt();
  687.         final DerivativeStructure   pz  = p.getZ();

  689.         // project point on the ellipsoid surface
  690.         final TimeStampedPVCoordinates groundPoint = projectToGround(new TimeStampedPVCoordinates(date, pointInBodyFrame),
  691.                                                                      bodyFrame);
  692.         final FieldVector3D<DerivativeStructure> gp = groundPoint.toDerivativeStructureVector(2);
  693.         final DerivativeStructure   gpr2 = gp.getX().multiply(gp.getX()).add(gp.getY().multiply(gp.getY()));
  694.         final DerivativeStructure   gpr  = gpr2.sqrt();
  695.         final DerivativeStructure   gpz  = gp.getZ();

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

  702.         return new FieldGeodeticPoint<>(DerivativeStructure.atan2(gpz, gpr.multiply(g2)),
  703.                                                                   DerivativeStructure.atan2(p.getY(), p.getX()),
  704.                                                                   DerivativeStructure.hypot(dr, dz).copySign(insideIfNegative));
  705.     }

  707.     /** Compute the azimuth angle from local north between the two points.
  708.      *
  709.      * The angle is calculated clockwise from local north at the origin point
  710.      * and follows the rhumb line to the destination point.
  711.      *
  712.      * @param origin the origin point, at which the azimuth angle will be computed (non-{@code null})
  713.      * @param destination the destination point, to which the angle is defined (non-{@code null})
  714.      * @return the resulting azimuth angle (radians, {@code [0-2pi)})
  715.      * @since 11.3
  716.      */
  717.     public double azimuthBetweenPoints(final GeodeticPoint origin, final GeodeticPoint destination) {
  718.         final double dLon = MathUtils.normalizeAngle(destination.getLongitude(), origin.getLongitude()) - origin.getLongitude();
  719.         final double originIsoLat = geodeticToIsometricLatitude(origin.getLatitude());
  720.         final double destIsoLat = geodeticToIsometricLatitude(destination.getLatitude());

  722.         final double az = FastMath.atan2(dLon, destIsoLat - originIsoLat);
  723.         if (az < 0.) {
  724.             return az + MathUtils.TWO_PI;
  725.         }
  726.         return az;
  727.     }

  729.     /** Compute the azimuth angle from local north between the two points.
  730.      *
  731.      * The angle is calculated clockwise from local north at the origin point
  732.      * and follows the rhumb line to the destination point.
  733.      *
  734.      * @param origin the origin point, at which the azimuth angle will be computed (non-{@code null})
  735.      * @param destination the destination point, to which the angle is defined (non-{@code null})
  736.      * @param <T> the type of field elements
  737.      * @return the resulting azimuth angle (radians, {@code [0-2pi)})
  738.      * @since 11.3
  739.      */
  740.     public <T extends CalculusFieldElement<T>> T azimuthBetweenPoints(final FieldGeodeticPoint<T> origin, final FieldGeodeticPoint<T> destination) {
  741.         final T dLon = MathUtils.normalizeAngle(destination.getLongitude().subtract(origin.getLongitude()), origin.getLongitude().getField().getZero());
  742.         final T originIsoLat = geodeticToIsometricLatitude(origin.getLatitude());
  743.         final T destIsoLat = geodeticToIsometricLatitude(destination.getLatitude());

  745.         final T az = FastMath.atan2(dLon, destIsoLat.subtract(originIsoLat));
  746.         if (az.getReal() < 0.) {
  747.             return az.add(az.getPi().multiply(2));
  748.         }
  749.         return az;
  750.     }

  752.     /** Compute the <a href="https://mathworld.wolfram.com/IsometricLatitude.html">isometric latitude</a>
  753.      *  corresponding to the provided latitude.
  754.      *
  755.      * @param geodeticLatitude the latitude (radians, within interval {@code [-pi/2, +pi/2]})
  756.      * @return the isometric latitude (radians)
  757.      * @since 11.3
  758.      */
  759.     public double geodeticToIsometricLatitude(final double geodeticLatitude) {
  760.         if (FastMath.abs(geodeticLatitude) <= angularThreshold) {
  761.             return 0.;
  762.         }

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

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

  771.         return a + b;
  772.     }

  774.     /** Compute the <a href="https://mathworld.wolfram.com/IsometricLatitude.html">isometric latitude</a>
  775.      *  corresponding to the provided latitude.
  776.      *
  777.      * @param geodeticLatitude the latitude (radians, within interval {@code [-pi/2, +pi/2]})
  778.      * @param <T> the type of field elements
  779.      * @return the isometric latitude (radians)
  780.      * @since 11.3
  781.      */
  782.     public <T extends CalculusFieldElement<T>> T geodeticToIsometricLatitude(final T geodeticLatitude) {
  783.         if (geodeticLatitude.abs().getReal() <= angularThreshold) {
  784.             return geodeticLatitude.getField().getZero();
  785.         }
  786.         final Field<T> field = geodeticLatitude.getField();
  787.         final T ecc = geodeticLatitude.newInstance(e);
  788.         final T eSinLat = ecc.multiply(geodeticLatitude.sin());

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

  795.         return a.add(b);
  796.     }

  798.     /** Replace the instance with a data transfer object for serialization.
  799.      * <p>
  800.      * This intermediate class serializes the files supported names, the
  801.      * ephemeris type and the body name.
  802.      * </p>
  803.      * @return data transfer object that will be serialized
  804.      */
  805.     private Object writeReplace() {
  806.         return new DataTransferObject(getA(), f, bodyFrame, angularThreshold);
  807.     }

  809.     /** Internal class used only for serialization. */
  810.     private static class DataTransferObject implements Serializable {

  812.         /** Serializable UID. */
  813.         private static final long serialVersionUID = 20130518L;

  815.         /** Equatorial radius. */
  816.         private final double ae;

  818.         /** Flattening. */
  819.         private final double f;

  821.         /** Body frame related to body shape. */
  822.         private final Frame bodyFrame;

  824.         /** Convergence limit. */
  825.         private final double angularThreshold;

  827.         /** Simple constructor.
  828.          * @param ae equatorial radius
  829.          * @param f the flattening (f = (a-b)/a)
  830.          * @param bodyFrame body frame related to body shape
  831.          * @param angularThreshold convergence limit
  832.          */
  833.         DataTransferObject(final double ae, final double f,
  834.                                   final Frame bodyFrame, final double angularThreshold) {
  835.             this.ae               = ae;
  836.             this.f                = f;
  837.             this.bodyFrame        = bodyFrame;
  838.             this.angularThreshold = angularThreshold;
  839.         }

  841.         /** Replace the deserialized data transfer object with a
  842.          * {@link JPLCelestialBody}.
  843.          * @return replacement {@link JPLCelestialBody}
  844.          */
  845.         private Object readResolve() {
  846.             final OneAxisEllipsoid ellipsoid = new OneAxisEllipsoid(ae, f, bodyFrame);
  847.             ellipsoid.setAngularThreshold(angularThreshold);
  848.             return ellipsoid;
  849.         }

  851.     }

  853. }
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