/* Copyright 2002-2021 CS GROUP
    * Licensed to CS GROUP (CS) under one or more
    * contributor license agreements.  See the NOTICE file distributed with
    * this work for additional information regarding copyright ownership.
    * CS licenses this file to You under the Apache License, Version 2.0
    * (the "License"); you may not use this file except in compliance with
    * the License.  You may obtain a copy of the License at
    *
    *   http://www.apache.org/licenses/LICENSE-2.0
   *
   * Unless required by applicable law or agreed to in writing, software
   * distributed under the License is distributed on an "AS IS" BASIS,
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  package org.orekit.bodies;
  
  import java.io.Serializable;
  
  import org.hipparchus.CalculusFieldElement;
  import org.hipparchus.analysis.differentiation.DerivativeStructure;
  import org.hipparchus.geometry.euclidean.threed.FieldLine;
  import org.hipparchus.geometry.euclidean.threed.FieldVector3D;
  import org.hipparchus.geometry.euclidean.threed.Line;
  import org.hipparchus.geometry.euclidean.threed.Vector3D;
  import org.hipparchus.geometry.euclidean.twod.Vector2D;
  import org.hipparchus.util.FastMath;
  import org.hipparchus.util.FieldSinCos;
  import org.hipparchus.util.MathArrays;
  import org.hipparchus.util.SinCos;
  import org.orekit.frames.FieldTransform;
  import org.orekit.frames.Frame;
  import org.orekit.frames.Transform;
  import org.orekit.time.AbsoluteDate;
  import org.orekit.time.FieldAbsoluteDate;
  import org.orekit.utils.PVCoordinates;
  import org.orekit.utils.TimeStampedPVCoordinates;
  
  
  /** Modeling of a one-axis ellipsoid.
  
   * <p>One-axis ellipsoids is a good approximate model for most planet-size
   * and larger natural bodies. It is the equilibrium shape reached by
   * a fluid body under its own gravity field when it rotates. The symmetry
   * axis is the rotation or polar axis.</p>
  
   * @author Luc Maisonobe
   * @author Guylaine Prat
   */
  public class OneAxisEllipsoid extends Ellipsoid implements BodyShape {
  
      /** Serializable UID. */
      private static final long serialVersionUID = 20130518L;
  
      /** Threshold for polar and equatorial points detection. */
      private static final double ANGULAR_THRESHOLD = 1.0e-4;
  
      /** Body frame related to body shape. */
      private final Frame bodyFrame;
  
      /** Equatorial radius power 2. */
      private final double ae2;
  
      /** Polar radius power 2. */
      private final double ap2;
  
      /** Flattening. */
      private final double f;
  
      /** Eccentricity power 2. */
      private final double e2;
  
      /** 1 minus flatness. */
      private final double g;
  
      /** g * g. */
      private final double g2;
  
      /** Convergence limit. */
      private double angularThreshold;
  
      /** Simple constructor.
       * <p>Standard values for Earth models can be found in the {@link org.orekit.utils.Constants Constants} class:</p>
       * <table border="1" style="background-color:#f5f5dc;">
       * <caption>Ellipsoid Models</caption>
       * <tr style="background-color:#c9d5c9;"><th>model</th><th>a<sub>e</sub> (m)</th> <th>f</th></tr>
       * <tr><td style="background-color:#c9d5c9; padding:5px">GRS 80</td>
       *     <td>{@link org.orekit.utils.Constants#GRS80_EARTH_EQUATORIAL_RADIUS Constants.GRS80_EARTH_EQUATORIAL_RADIUS}</td>
       *     <td>{@link org.orekit.utils.Constants#GRS80_EARTH_FLATTENING Constants.GRS80_EARTH_FLATTENING}</td></tr>
       * <tr><td style="background-color:#c9d5c9; padding:5px">WGS84</td>
       *     <td>{@link org.orekit.utils.Constants#WGS84_EARTH_EQUATORIAL_RADIUS Constants.WGS84_EARTH_EQUATORIAL_RADIUS}</td>
       *     <td>{@link org.orekit.utils.Constants#WGS84_EARTH_FLATTENING Constants.WGS84_EARTH_FLATTENING}</td></tr>
       * <tr><td style="background-color:#c9d5c9; padding:5px">IERS96</td>
       *     <td>{@link org.orekit.utils.Constants#IERS96_EARTH_EQUATORIAL_RADIUS Constants.IERS96_EARTH_EQUATORIAL_RADIUS}</td>
       *     <td>{@link org.orekit.utils.Constants#IERS96_EARTH_FLATTENING Constants.IERS96_EARTH_FLATTENING}</td></tr>
       * <tr><td style="background-color:#c9d5c9; padding:5px">IERS2003</td>
       *     <td>{@link org.orekit.utils.Constants#IERS2003_EARTH_EQUATORIAL_RADIUS Constants.IERS2003_EARTH_EQUATORIAL_RADIUS}</td>
       *     <td>{@link org.orekit.utils.Constants#IERS2003_EARTH_FLATTENING Constants.IERS2003_EARTH_FLATTENING}</td></tr>
      * <tr><td style="background-color:#c9d5c9; padding:5px">IERS2010</td>
      *     <td>{@link org.orekit.utils.Constants#IERS2010_EARTH_EQUATORIAL_RADIUS Constants.IERS2010_EARTH_EQUATORIAL_RADIUS}</td>
      *     <td>{@link org.orekit.utils.Constants#IERS2010_EARTH_FLATTENING Constants.IERS2010_EARTH_FLATTENING}</td></tr>
      * </table>
      * @param ae equatorial radius
      * @param f the flattening (f = (a-b)/a)
      * @param bodyFrame body frame related to body shape
      * @see org.orekit.frames.FramesFactory#getITRF(org.orekit.utils.IERSConventions, boolean)
      */
     public OneAxisEllipsoid(final double ae, final double f,
                             final Frame bodyFrame) {
         super(bodyFrame, ae, ae, ae * (1.0 - f));
         this.f    = f;
         this.ae2  = ae * ae;
         this.e2   = f * (2.0 - f);
         this.g    = 1.0 - f;
         this.g2   = g * g;
         this.ap2  = ae2 * g2;
         setAngularThreshold(1.0e-12);
         this.bodyFrame = bodyFrame;
     }
 
     /** Set the angular convergence threshold.
      * <p>The angular threshold is used both to identify points close to
      * the ellipse axes and as the convergence threshold used to
      * stop the iterations in the {@link #transform(Vector3D, Frame,
      * AbsoluteDate)} method.</p>
      * <p>If this method is not called, the default value is set to
      * 10<sup>-12</sup>.</p>
      * @param angularThreshold angular convergence threshold (rad)
      */
     public void setAngularThreshold(final double angularThreshold) {
         this.angularThreshold = angularThreshold;
     }
 
     /** Get the equatorial radius of the body.
      * @return equatorial radius of the body (m)
      */
     public double getEquatorialRadius() {
         return getA();
     }
 
     /** Get the flattening of the body: f = (a-b)/a.
      * @return the flattening
      */
     public double getFlattening() {
         return f;
     }
 
     /** {@inheritDoc} */
     public Frame getBodyFrame() {
         return bodyFrame;
     }
 
     /** Get the intersection point of a line with the surface of the body.
      * <p>A line may have several intersection points with a closed
      * surface (we consider the one point case as a degenerated two
      * points case). The close parameter is used to select which of
      * these points should be returned. The selected point is the one
      * that is closest to the close point.</p>
      * @param line test line (may intersect the body or not)
      * @param close point used for intersections selection
      * @param frame frame in which line is expressed
      * @param date date of the line in given frame
      * @return intersection point at altitude zero or null if the line does
      * not intersect the surface
      * @since 9.3
      */
     public Vector3D getCartesianIntersectionPoint(final Line line, final Vector3D close,
                                                   final Frame frame, final AbsoluteDate date) {
 
         // transform line and close to body frame
         final Transform frameToBodyFrame = frame.getTransformTo(bodyFrame, date);
         final Line lineInBodyFrame = frameToBodyFrame.transformLine(line);
 
         // compute some miscellaneous variables
         final Vector3D point    = lineInBodyFrame.getOrigin();
         final double x          = point.getX();
         final double y          = point.getY();
         final double z          = point.getZ();
         final double z2         = z * z;
         final double r2         = x * x + y * y;
 
         final Vector3D direction = lineInBodyFrame.getDirection();
         final double dx         = direction.getX();
         final double dy         = direction.getY();
         final double dz         = direction.getZ();
         final double cz2        = dx * dx + dy * dy;
 
         // abscissa of the intersection as a root of a 2nd degree polynomial :
         // a k^2 - 2 b k + c = 0
         final double a  = 1.0 - e2 * cz2;
         final double b  = -(g2 * (x * dx + y * dy) + z * dz);
         final double c  = g2 * (r2 - ae2) + z2;
         final double b2 = b * b;
         final double ac = a * c;
         if (b2 < ac) {
             return null;
         }
         final double s  = FastMath.sqrt(b2 - ac);
         final double k1 = (b < 0) ? (b - s) / a : c / (b + s);
         final double k2 = c / (a * k1);
 
         // select the right point
         final Vector3D closeInBodyFrame = frameToBodyFrame.transformPosition(close);
         final double   closeAbscissa    = lineInBodyFrame.getAbscissa(closeInBodyFrame);
         final double k =
             (FastMath.abs(k1 - closeAbscissa) < FastMath.abs(k2 - closeAbscissa)) ? k1 : k2;
         return lineInBodyFrame.pointAt(k);
 
     }
 
     /** {@inheritDoc} */
     public GeodeticPoint getIntersectionPoint(final Line line, final Vector3D close,
                                               final Frame frame, final AbsoluteDate date) {
 
         final Vector3D intersection = getCartesianIntersectionPoint(line, close, frame, date);
         if (intersection == null) {
             return null;
         }
         final double ix = intersection.getX();
         final double iy = intersection.getY();
         final double iz = intersection.getZ();
 
         final double lambda = FastMath.atan2(iy, ix);
         final double phi    = FastMath.atan2(iz, g2 * FastMath.sqrt(ix * ix + iy * iy));
         return new GeodeticPoint(phi, lambda, 0.0);
 
     }
 
     /** Get the intersection point of a line with the surface of the body.
      * <p>A line may have several intersection points with a closed
      * surface (we consider the one point case as a degenerated two
      * points case). The close parameter is used to select which of
      * these points should be returned. The selected point is the one
      * that is closest to the close point.</p>
      * @param line test line (may intersect the body or not)
      * @param close point used for intersections selection
      * @param frame frame in which line is expressed
      * @param date date of the line in given frame
      * @param <T> type of the field elements
      * @return intersection point at altitude zero or null if the line does
      * not intersect the surface
      * @since 9.3
      */
     public <T extends CalculusFieldElement<T>> FieldVector3D<T> getCartesianIntersectionPoint(final FieldLine<T> line,
                                                                                           final FieldVector3D<T> close,
                                                                                           final Frame frame,
                                                                                           final FieldAbsoluteDate<T> date) {
 
         // transform line and close to body frame
         final FieldTransform<T> frameToBodyFrame = frame.getTransformTo(bodyFrame, date);
         final FieldLine<T>      lineInBodyFrame  = frameToBodyFrame.transformLine(line);
 
         // compute some miscellaneous variables
         final FieldVector3D<T> point = lineInBodyFrame.getOrigin();
         final T x  = point.getX();
         final T y  = point.getY();
         final T z  = point.getZ();
         final T z2 = z.multiply(z);
         final T r2 = x.multiply(x).add(y.multiply(y));
 
         final FieldVector3D<T> direction = lineInBodyFrame.getDirection();
         final T dx  = direction.getX();
         final T dy  = direction.getY();
         final T dz  = direction.getZ();
         final T cz2 = dx.multiply(dx).add(dy.multiply(dy));
 
         // abscissa of the intersection as a root of a 2nd degree polynomial :
         // a k^2 - 2 b k + c = 0
         final T a  = cz2.multiply(e2).subtract(1.0).negate();
         final T b  = x.multiply(dx).add(y.multiply(dy)).multiply(g2).add(z.multiply(dz)).negate();
         final T c  = r2.subtract(ae2).multiply(g2).add(z2);
         final T b2 = b.multiply(b);
         final T ac = a.multiply(c);
         if (b2.getReal() < ac.getReal()) {
             return null;
         }
         final T s  = b2.subtract(ac).sqrt();
         final T k1 = (b.getReal() < 0) ? b.subtract(s).divide(a) : c.divide(b.add(s));
         final T k2 = c.divide(a.multiply(k1));
 
         // select the right point
         final FieldVector3D<T>  closeInBodyFrame = frameToBodyFrame.transformPosition(close);
         final T                 closeAbscissa    = lineInBodyFrame.getAbscissa(closeInBodyFrame);
         final T k = (FastMath.abs(k1.getReal() - closeAbscissa.getReal()) < FastMath.abs(k2.getReal() - closeAbscissa.getReal())) ?
                     k1 : k2;
         return lineInBodyFrame.pointAt(k);
     }
 
     /** {@inheritDoc} */
     public <T extends CalculusFieldElement<T>> FieldGeodeticPoint<T> getIntersectionPoint(final FieldLine<T> line,
                                                                                       final FieldVector3D<T> close,
                                                                                       final Frame frame,
                                                                                       final FieldAbsoluteDate<T> date) {
 
         final FieldVector3D<T> intersection = getCartesianIntersectionPoint(line, close, frame, date);
         if (intersection == null) {
             return null;
         }
         final T ix = intersection.getX();
         final T iy = intersection.getY();
         final T iz = intersection.getZ();
 
         final T lambda = iy.atan2(ix);
         final T phi    = iz.atan2(ix.multiply(ix).add(iy.multiply(iy)).sqrt().multiply(g2));
         return new FieldGeodeticPoint<>(phi, lambda, phi.getField().getZero());
 
     }
 
     /** {@inheritDoc} */
     public Vector3D transform(final GeodeticPoint point) {
         final double longitude = point.getLongitude();
         final SinCos scLambda  = FastMath.sinCos(longitude);
         final double latitude  = point.getLatitude();
         final SinCos scPhi     = FastMath.sinCos(latitude);
         final double h         = point.getAltitude();
         final double n         = getA() / FastMath.sqrt(1.0 - e2 * scPhi.sin() * scPhi.sin());
         final double r         = (n + h) * scPhi.cos();
         return new Vector3D(r * scLambda.cos(), r * scLambda.sin(), (g2 * n + h) * scPhi.sin());
     }
 
     /** {@inheritDoc} */
     public <T extends CalculusFieldElement<T>> FieldVector3D<T> transform(final FieldGeodeticPoint<T> point) {
 
         final T latitude  = point.getLatitude();
         final T longitude = point.getLongitude();
         final T altitude  = point.getAltitude();
 
         final FieldSinCos<T> scLambda = FastMath.sinCos(longitude);
         final FieldSinCos<T> scPhi    = FastMath.sinCos(latitude);
         final T cLambda = scLambda.cos();
         final T sLambda = scLambda.sin();
         final T cPhi    = scPhi.cos();
         final T sPhi    = scPhi.sin();
         final T n       = sPhi.multiply(sPhi).multiply(e2).subtract(1.0).negate().sqrt().reciprocal().multiply(getA());
         final T r       = n.add(altitude).multiply(cPhi);
 
         return new FieldVector3D<>(r.multiply(cLambda),
                                    r.multiply(sLambda),
                                    sPhi.multiply(altitude.add(n.multiply(g2))));
     }
 
     /** {@inheritDoc} */
     public Vector3D projectToGround(final Vector3D point, final AbsoluteDate date, final Frame frame) {
 
         // transform point to body frame
         final Transform  toBody    = frame.getTransformTo(bodyFrame, date);
         final Vector3D   p         = toBody.transformPosition(point);
         final double     z         = p.getZ();
         final double     r         = FastMath.hypot(p.getX(), p.getY());
 
         // set up the 2D meridian ellipse
         final Ellipse meridian = new Ellipse(Vector3D.ZERO,
                                              r == 0 ? Vector3D.PLUS_I : new Vector3D(p.getX() / r, p.getY() / r, 0),
                                              Vector3D.PLUS_K,
                                              getA(), getC(), bodyFrame);
 
         // find the closest point in the meridian plane
         final Vector3D groundPoint = meridian.toSpace(meridian.projectToEllipse(new Vector2D(r, z)));
 
         // transform point back to initial frame
         return toBody.getInverse().transformPosition(groundPoint);
 
     }
 
     /** {@inheritDoc} */
     public TimeStampedPVCoordinates projectToGround(final TimeStampedPVCoordinates pv, final Frame frame) {
 
         // transform point to body frame
         final Transform                toBody        = frame.getTransformTo(bodyFrame, pv.getDate());
         final TimeStampedPVCoordinates pvInBodyFrame = toBody.transformPVCoordinates(pv);
         final Vector3D                 p             = pvInBodyFrame.getPosition();
         final double                   r             = FastMath.hypot(p.getX(), p.getY());
 
         // set up the 2D ellipse corresponding to first principal curvature along meridian
         final Vector3D meridian = r == 0 ? Vector3D.PLUS_I : new Vector3D(p.getX() / r, p.getY() / r, 0);
         final Ellipse firstPrincipalCurvature =
                 new Ellipse(Vector3D.ZERO, meridian, Vector3D.PLUS_K, getA(), getC(), bodyFrame);
 
         // project coordinates in the meridian plane
         final TimeStampedPVCoordinates gpFirst = firstPrincipalCurvature.projectToEllipse(pvInBodyFrame);
         final Vector3D                 gpP     = gpFirst.getPosition();
         final double                   gr      = MathArrays.linearCombination(gpP.getX(), meridian.getX(),
                                                                               gpP.getY(), meridian.getY());
         final double                   gz      = gpP.getZ();
 
         // topocentric frame
         final Vector3D east   = new Vector3D(-meridian.getY(), meridian.getX(), 0);
         final Vector3D zenith = new Vector3D(gr * getC() / getA(), meridian, gz * getA() / getC(), Vector3D.PLUS_K).normalize();
         final Vector3D north  = Vector3D.crossProduct(zenith, east);
 
         // set up the ellipse corresponding to second principal curvature in the zenith/east plane
         final Ellipse secondPrincipalCurvature  = getPlaneSection(gpP, north);
         final TimeStampedPVCoordinates gpSecond = secondPrincipalCurvature.projectToEllipse(pvInBodyFrame);
 
         final Vector3D gpV = gpFirst.getVelocity().add(gpSecond.getVelocity());
         final Vector3D gpA = gpFirst.getAcceleration().add(gpSecond.getAcceleration());
 
         // moving projected point
         final TimeStampedPVCoordinates groundPV =
                 new TimeStampedPVCoordinates(pv.getDate(), gpP, gpV, gpA);
 
         // transform moving projected point back to initial frame
         return toBody.getInverse().transformPVCoordinates(groundPV);
 
     }
 
     /** {@inheritDoc}
      * <p>
      * This method is based on Toshio Fukushima's algorithm which uses Halley's method.
      * <a href="https://www.researchgate.net/publication/227215135_Transformation_from_Cartesian_to_Geodetic_Coordinates_Accelerated_by_Halley's_Method">
      * transformation from Cartesian to Geodetic Coordinates Accelerated by Halley's Method</a>,
      * Toshio Fukushima, Journal of Geodesy 9(12):689-693, February 2006
      * </p>
      * <p>
      * Some changes have been added to the original method:
      * <ul>
      *   <li>in order to handle more accurately corner cases near the pole</li>
      *   <li>in order to handle properly corner cases near the equatorial plane, even far inside the ellipsoid</li>
      *   <li>in order to handle very flat ellipsoids</li>
      * </ul>
      */
     public GeodeticPoint transform(final Vector3D point, final Frame frame, final AbsoluteDate date) {
 
         // transform point to body frame
         final Vector3D pointInBodyFrame = frame.getTransformTo(bodyFrame, date).transformPosition(point);
         final double   r2               = pointInBodyFrame.getX() * pointInBodyFrame.getX() +
                                           pointInBodyFrame.getY() * pointInBodyFrame.getY();
         final double   r                = FastMath.sqrt(r2);
         final double   z                = pointInBodyFrame.getZ();
 
         final double   lambda           = FastMath.atan2(pointInBodyFrame.getY(), pointInBodyFrame.getX());
 
         double h;
         double phi;
         if (r <= ANGULAR_THRESHOLD * FastMath.abs(z)) {
             // the point is almost on the polar axis, approximate the ellipsoid with
             // the osculating sphere whose center is at evolute cusp along polar axis
             final double osculatingRadius = ae2 / getC();
             final double evoluteCuspZ     = FastMath.copySign(getA() * e2 / g, -z);
             final double deltaZ           = z - evoluteCuspZ;
             // we use π/2 - atan(r/Δz) instead of atan(Δz/r) for accuracy purposes, as r is much smaller than Δz
             phi = FastMath.copySign(0.5 * FastMath.PI - FastMath.atan(r / FastMath.abs(deltaZ)), deltaZ);
             h   = FastMath.hypot(deltaZ, r) - osculatingRadius;
         } else if (FastMath.abs(z) <= ANGULAR_THRESHOLD * r) {
             // the point is almost on the major axis
 
             final double osculatingRadius = ap2 / getA();
             final double evoluteCuspR     = getA() * e2;
             final double deltaR           = r - evoluteCuspR;
             if (deltaR >= 0) {
                 // the point is outside of the ellipse evolute, approximate the ellipse
                 // with the osculating circle whose center is at evolute cusp along major axis
                 phi = (deltaR == 0) ? 0.0 : FastMath.atan(z / deltaR);
                 h   = FastMath.hypot(deltaR, z) - osculatingRadius;
             } else {
                 // the point is on the part of the major axis within ellipse evolute
                 // we can compute the closest ellipse point analytically, and it is NOT near the equator
                 final double rClose = r / e2;
                 final double zClose = FastMath.copySign(g * FastMath.sqrt(ae2 - rClose * rClose), z);
                 phi = FastMath.atan((zClose - z) / (rClose - r));
                 h   = -FastMath.hypot(r - rClose, z - zClose);
             }
 
         } else {
             // use Toshio Fukushima method, with several iterations
             final double epsPhi = 1.0e-15;
             final double epsH   = 1.0e-14 * FastMath.max(getA(), FastMath.sqrt(r2 + z * z));
             final double c     = getA() * e2;
             final double absZ  = FastMath.abs(z);
             final double zc    = g * absZ;
             double sn  = absZ;
             double sn2 = sn * sn;
             double cn  = g * r;
             double cn2 = cn * cn;
             double an2 = cn2 + sn2;
             double an  = FastMath.sqrt(an2);
             double bn  = 0;
             phi = Double.POSITIVE_INFINITY;
             h   = Double.POSITIVE_INFINITY;
             for (int i = 0; i < 10; ++i) { // this usually converges in 2 iterations
                 final double oldSn  = sn;
                 final double oldCn  = cn;
                 final double oldPhi = phi;
                 final double oldH   = h;
                 final double an3    = an2 * an;
                 final double csncn  = c * sn * cn;
                 bn    = 1.5 * csncn * ((r * sn - zc * cn) * an - csncn);
                 sn    = (zc * an3 + c * sn2 * sn) * an3 - bn * sn;
                 cn    = (r  * an3 - c * cn2 * cn) * an3 - bn * cn;
                 if (sn * oldSn < 0 || cn < 0) {
                     // the Halley iteration went too far, we restrict it and iterate again
                     while (sn * oldSn < 0 || cn < 0) {
                         sn = (sn + oldSn) / 2;
                         cn = (cn + oldCn) / 2;
                     }
                 } else {
 
                     // rescale components to avoid overflow when several iterations are used
                     final int exp = (FastMath.getExponent(sn) + FastMath.getExponent(cn)) / 2;
                     sn = FastMath.scalb(sn, -exp);
                     cn = FastMath.scalb(cn, -exp);
 
                     sn2 = sn * sn;
                     cn2 = cn * cn;
                     an2 = cn2 + sn2;
                     an  = FastMath.sqrt(an2);
 
                     final double cc = g * cn;
                     h = (r * cc + absZ * sn - getA() * g * an) / FastMath.sqrt(an2 - e2 * cn2);
                     if (FastMath.abs(oldH   - h)   < epsH) {
                         phi = FastMath.copySign(FastMath.atan(sn / cc), z);
                         if (FastMath.abs(oldPhi - phi) < epsPhi) {
                             break;
                         }
                     }
 
                 }
 
             }
         }
 
         return new GeodeticPoint(phi, lambda, h);
 
     }
 
     /** {@inheritDoc}
      * <p>
      * This method is based on Toshio Fukushima's algorithm which uses Halley's method.
      * <a href="https://www.researchgate.net/publication/227215135_Transformation_from_Cartesian_to_Geodetic_Coordinates_Accelerated_by_Halley's_Method">
      * transformation from Cartesian to Geodetic Coordinates Accelerated by Halley's Method</a>,
      * Toshio Fukushima, Journal of Geodesy 9(12):689-693, February 2006
      * </p>
      * <p>
      * Some changes have been added to the original method:
      * <ul>
      *   <li>in order to handle more accurately corner cases near the pole</li>
      *   <li>in order to handle properly corner cases near the equatorial plane, even far inside the ellipsoid</li>
      *   <li>in order to handle very flat ellipsoids</li>
      * </ul>
      */
     public <T extends CalculusFieldElement<T>> FieldGeodeticPoint<T> transform(final FieldVector3D<T> point,
                                                                            final Frame frame,
                                                                            final FieldAbsoluteDate<T> date) {
 
         // transform point to body frame
         final FieldVector3D<T> pointInBodyFrame = frame.getTransformTo(bodyFrame, date).transformPosition(point);
         final T   r2                            = pointInBodyFrame.getX().multiply(pointInBodyFrame.getX()).
                                               add(pointInBodyFrame.getY().multiply(pointInBodyFrame.getY()));
         final T   r                             = r2.sqrt();
         final T   z                             = pointInBodyFrame.getZ();
 
         final T   lambda                        = pointInBodyFrame.getY().atan2(pointInBodyFrame.getX());
 
         T h;
         T phi;
         if (r.getReal() <= ANGULAR_THRESHOLD * FastMath.abs(z.getReal())) {
             // the point is almost on the polar axis, approximate the ellipsoid with
             // the osculating sphere whose center is at evolute cusp along polar axis
             final double osculatingRadius = ae2 / getC();
             final double evoluteCuspZ     = FastMath.copySign(getA() * e2 / g, -z.getReal());
             final T      deltaZ           = z.subtract(evoluteCuspZ);
             // we use π/2 - atan(r/Δz) instead of atan(Δz/r) for accuracy purposes, as r is much smaller than Δz
             phi = r.divide(deltaZ.abs()).atan().negate().add(r.getPi().multiply(0.5)).copySign(deltaZ);
             h   = deltaZ.hypot(r).subtract(osculatingRadius);
         } else if (FastMath.abs(z.getReal()) <= ANGULAR_THRESHOLD * r.getReal()) {
             // the point is almost on the major axis
 
             final double osculatingRadius = ap2 / getA();
             final double evoluteCuspR     = getA() * e2;
             final T      deltaR           = r.subtract(evoluteCuspR);
             if (deltaR.getReal() >= 0) {
                 // the point is outside of the ellipse evolute, approximate the ellipse
                 // with the osculating circle whose center is at evolute cusp along major axis
                 phi = (deltaR.getReal() == 0) ? z.getField().getZero() : z.divide(deltaR).atan();
                 h   = deltaR.hypot(z).subtract(osculatingRadius);
             } else {
                 // the point is on the part of the major axis within ellipse evolute
                 // we can compute the closest ellipse point analytically, and it is NOT near the equator
                 final T rClose = r.divide(e2);
                 final T zClose = rClose.multiply(rClose).negate().add(ae2).sqrt().multiply(g).copySign(z);
                 phi = zClose.subtract(z).divide(rClose.subtract(r)).atan();
                 h   = r.subtract(rClose).hypot(z.subtract(zClose)).negate();
             }
 
         } else {
             // use Toshio Fukushima method, with several iterations
             final double epsPhi = 1.0e-15;
             final double epsH   = 1.0e-14 * getA();
             final double c      = getA() * e2;
             final T      absZ   = z.abs();
             final T      zc     = absZ.multiply(g);
             T            sn     = absZ;
             T            sn2    = sn.multiply(sn);
             T            cn     = r.multiply(g);
             T            cn2    = cn.multiply(cn);
             T            an2    = cn2.add(sn2);
             T            an     = an2.sqrt();
             T            bn     = an.getField().getZero();
             phi = an.getField().getZero().add(Double.POSITIVE_INFINITY);
             h   = an.getField().getZero().add(Double.POSITIVE_INFINITY);
             for (int i = 0; i < 10; ++i) { // this usually converges in 2 iterations
                 final T oldSn  = sn;
                 final T oldCn  = cn;
                 final T oldPhi = phi;
                 final T oldH   = h;
                 final T an3    = an2.multiply(an);
                 final T csncn  = sn.multiply(cn).multiply(c);
                 bn    = csncn.multiply(1.5).multiply((r.multiply(sn).subtract(zc.multiply(cn))).multiply(an).subtract(csncn));
                 sn    = zc.multiply(an3).add(sn2.multiply(sn).multiply(c)).multiply(an3).subtract(bn.multiply(sn));
                 cn    = r.multiply(an3).subtract(cn2.multiply(cn).multiply(c)).multiply(an3).subtract(bn.multiply(cn));
                 if (sn.getReal() * oldSn.getReal() < 0 || cn.getReal() < 0) {
                     // the Halley iteration went too far, we restrict it and iterate again
                     while (sn.getReal() * oldSn.getReal() < 0 || cn.getReal() < 0) {
                         sn = sn.add(oldSn).multiply(0.5);
                         cn = cn.add(oldCn).multiply(0.5);
                     }
                 } else {
 
                     // rescale components to avoid overflow when several iterations are used
                     final int exp = (FastMath.getExponent(sn.getReal()) + FastMath.getExponent(cn.getReal())) / 2;
                     sn = sn.scalb(-exp);
                     cn = cn.scalb(-exp);
 
                     sn2 = sn.multiply(sn);
                     cn2 = cn.multiply(cn);
                     an2 = cn2.add(sn2);
                     an  = an2.sqrt();
 
                     final T cc = cn.multiply(g);
                     h = r.multiply(cc).add(absZ.multiply(sn)).subtract(an.multiply(getA() * g)).divide(an2.subtract(cn2.multiply(e2)).sqrt());
                     if (FastMath.abs(oldH.getReal()  - h.getReal())   < epsH) {
                         phi = sn.divide(cc).atan().copySign(z);
                         if (FastMath.abs(oldPhi.getReal() - phi.getReal()) < epsPhi) {
                             break;
                         }
                     }
 
                 }
 
             }
         }
 
         return new FieldGeodeticPoint<>(phi, lambda, h);
 
     }
 
     /** Transform a Cartesian point to a surface-relative point.
      * @param point Cartesian point
      * @param frame frame in which Cartesian point is expressed
      * @param date date of the computation (used for frames conversions)
      * @return point at the same location but as a surface-relative point,
      * using time as the single derivation parameter
      */
     public FieldGeodeticPoint<DerivativeStructure> transform(final PVCoordinates point,
                                                              final Frame frame, final AbsoluteDate date) {
 
         // transform point to body frame
         final Transform toBody = frame.getTransformTo(bodyFrame, date);
         final PVCoordinates pointInBodyFrame = toBody.transformPVCoordinates(point);
         final FieldVector3D<DerivativeStructure> p = pointInBodyFrame.toDerivativeStructureVector(2);
         final DerivativeStructure   pr2 = p.getX().multiply(p.getX()).add(p.getY().multiply(p.getY()));
         final DerivativeStructure   pr  = pr2.sqrt();
         final DerivativeStructure   pz  = p.getZ();
 
         // project point on the ellipsoid surface
         final TimeStampedPVCoordinates groundPoint = projectToGround(new TimeStampedPVCoordinates(date, pointInBodyFrame),
                                                                      bodyFrame);
         final FieldVector3D<DerivativeStructure> gp = groundPoint.toDerivativeStructureVector(2);
         final DerivativeStructure   gpr2 = gp.getX().multiply(gp.getX()).add(gp.getY().multiply(gp.getY()));
         final DerivativeStructure   gpr  = gpr2.sqrt();
         final DerivativeStructure   gpz  = gp.getZ();
 
         // relative position of test point with respect to its ellipse sub-point
         final DerivativeStructure dr  = pr.subtract(gpr);
         final DerivativeStructure dz  = pz.subtract(gpz);
         final double insideIfNegative = g2 * (pr2.getReal() - ae2) + pz.getReal() * pz.getReal();
 
         return new FieldGeodeticPoint<>(DerivativeStructure.atan2(gpz, gpr.multiply(g2)),
                                                                   DerivativeStructure.atan2(p.getY(), p.getX()),
                                                                   DerivativeStructure.hypot(dr, dz).copySign(insideIfNegative));
     }
 
     /** Replace the instance with a data transfer object for serialization.
      * <p>
      * This intermediate class serializes the files supported names, the
      * ephemeris type and the body name.
      * </p>
      * @return data transfer object that will be serialized
      */
     private Object writeReplace() {
         return new DataTransferObject(getA(), f, bodyFrame, angularThreshold);
     }
 
     /** Internal class used only for serialization. */
     private static class DataTransferObject implements Serializable {
 
         /** Serializable UID. */
         private static final long serialVersionUID = 20130518L;
 
         /** Equatorial radius. */
         private final double ae;
 
         /** Flattening. */
         private final double f;
 
         /** Body frame related to body shape. */
         private final Frame bodyFrame;
 
         /** Convergence limit. */
         private final double angularThreshold;
 
         /** Simple constructor.
          * @param ae equatorial radius
          * @param f the flattening (f = (a-b)/a)
          * @param bodyFrame body frame related to body shape
          * @param angularThreshold convergence limit
          */
         DataTransferObject(final double ae, final double f,
                                   final Frame bodyFrame, final double angularThreshold) {
             this.ae               = ae;
             this.f                = f;
             this.bodyFrame        = bodyFrame;
             this.angularThreshold = angularThreshold;
         }
 
         /** Replace the deserialized data transfer object with a
          * {@link JPLCelestialBody}.
          * @return replacement {@link JPLCelestialBody}
          */
         private Object readResolve() {
             final OneAxisEllipsoid ellipsoid = new OneAxisEllipsoid(ae, f, bodyFrame);
             ellipsoid.setAngularThreshold(angularThreshold);
             return ellipsoid;
         }
 
     }
 
 }