/* Copyright 2002-2015 CS Systèmes d'Information
    * Licensed to CS Systèmes d'Information (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,
   * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
   * See the License for the specific language governing permissions and
   * limitations under the License.
   */
  package org.orekit.frames;
  
  import java.io.Serializable;
  import java.util.ArrayList;
  import java.util.Arrays;
  import java.util.Collection;
  import java.util.List;
  
  import org.apache.commons.math3.RealFieldElement;
  import org.apache.commons.math3.geometry.euclidean.threed.FieldRotation;
  import org.apache.commons.math3.geometry.euclidean.threed.FieldVector3D;
  import org.apache.commons.math3.geometry.euclidean.threed.Line;
  import org.apache.commons.math3.geometry.euclidean.threed.Rotation;
  import org.apache.commons.math3.geometry.euclidean.threed.Vector3D;
  import org.orekit.errors.OrekitException;
  import org.orekit.time.AbsoluteDate;
  import org.orekit.time.TimeInterpolable;
  import org.orekit.time.TimeShiftable;
  import org.orekit.time.TimeStamped;
  import org.orekit.utils.AngularCoordinates;
  import org.orekit.utils.AngularDerivativesFilter;
  import org.orekit.utils.CartesianDerivativesFilter;
  import org.orekit.utils.FieldPVCoordinates;
  import org.orekit.utils.PVCoordinates;
  import org.orekit.utils.TimeStampedAngularCoordinates;
  import org.orekit.utils.TimeStampedFieldPVCoordinates;
  import org.orekit.utils.TimeStampedPVCoordinates;
  
  
  /** Transformation class in three dimensional space.
   *
   * <p>This class represents the transformation engine between {@link Frame frames}.
   * It is used both to define the relationship between each frame and its
   * parent frame and to gather all individual transforms into one
   * operation when converting between frames far away from each other.</p>
   * <p> The convention used in OREKIT is vectorial transformation. It means
   * that a transformation is defined as a transform to apply to the
   * coordinates of a vector expressed in the old frame to obtain the
   * same vector expressed in the new frame.<p>
   *
   * <p>Instances of this class are guaranteed to be immutable.</p>
   *
   *  <h5> Example </h5>
   *
   * <pre>
   *
   * 1 ) Example of translation from R<sub>A</sub> to R<sub>B</sub>:
   * We want to transform the {@link PVCoordinates} PV<sub>A</sub> to PV<sub>B</sub>.
   *
   * With :  PV<sub>A</sub> = ({1, 0, 0}, {2, 0, 0}, {3, 0, 0});
   * and  :  PV<sub>B</sub> = ({0, 0, 0}, {0, 0, 0}, {0, 0, 0});
   *
   * The transform to apply then is defined as follows :
   *
   * Vector3D translation  = new Vector3D(-1, 0, 0);
   * Vector3D velocity     = new Vector3D(-2, 0, 0);
   * Vector3D acceleration = new Vector3D(-3, 0, 0);
   *
   * Transform R1toR2 = new Transform(date, translation, velocity, acceleration);
   *
   * PV<sub>B</sub> = R1toR2.transformPVCoordinates(PV<sub>A</sub>);
   *
   *
   * 2 ) Example of rotation from R<sub>A</sub> to R<sub>B</sub>:
   * We want to transform the {@link PVCoordinates} PV<sub>A</sub> to PV<sub>B</sub>.
   *
   * With :  PV<sub>A</sub> = ({1, 0, 0}, {1, 0, 0});
   * and  :  PV<sub>B</sub> = ({0, 1, 0}, {-2, 1, 0});
   *
   * The transform to apply then is defined as follows :
   *
   * Rotation rotation = new Rotation(Vector3D.PLUS_K, FastMath.PI / 2);
   * Vector3D rotationRate = new Vector3D(0, 0, -2);
   *
   * Transform R1toR2 = new Transform(rotation, rotationRate);
   *
   * PV<sub>B</sub> = R1toR2.transformPVCoordinates(PV<sub>A</sub>);
   *
   * </pre>
   *
   * @author Luc Maisonobe
   * @author Fabien Maussion
   */
 public class Transform
     implements TimeStamped, TimeShiftable<Transform>, TimeInterpolable<Transform>, Serializable {
 
     /** Identity transform. */
     public static final Transform IDENTITY = new IdentityTransform();
 
     /** Serializable UID. */
     private static final long serialVersionUID = 210140410L;
 
     /** Date of the transform. */
     private final AbsoluteDate date;
 
     /** Cartesian coordinates of the target frame with respect to the original frame. */
     private final PVCoordinates cartesian;
 
     /** Angular coordinates of the target frame with respect to the original frame. */
     private final AngularCoordinates angular;
 
     /** Build a transform from its primitive operations.
      * @param date date of the transform
      * @param cartesian Cartesian coordinates of the target frame with respect to the original frame
      * @param angular angular coordinates of the target frame with respect to the original frame
      */
     private Transform(final AbsoluteDate date,
                       final PVCoordinates cartesian, final AngularCoordinates angular) {
         this.date      = date;
         this.cartesian = cartesian;
         this.angular   = angular;
     }
 
     /** Build a translation transform.
      * @param date date of the transform
      * @param translation translation to apply (i.e. coordinates of
      * the transformed origin, or coordinates of the origin of the
      * old frame in the new frame)
      */
     public Transform(final AbsoluteDate date, final Vector3D translation) {
         this(date,
              new PVCoordinates(translation, Vector3D.ZERO, Vector3D.ZERO),
              AngularCoordinates.IDENTITY);
     }
 
     /** Build a rotation transform.
      * @param date date of the transform
      * @param rotation rotation to apply ( i.e. rotation to apply to the
      * coordinates of a vector expressed in the old frame to obtain the
      * same vector expressed in the new frame )
      */
     public Transform(final AbsoluteDate date, final Rotation rotation) {
         this(date,
              PVCoordinates.ZERO,
              new AngularCoordinates(rotation, Vector3D.ZERO));
     }
 
     /** Build a translation transform, with its first time derivative.
      * @param date date of the transform
      * @param translation translation to apply (i.e. coordinates of
      * the transformed origin, or coordinates of the origin of the
      * old frame in the new frame)
      * @param velocity the velocity of the translation (i.e. origin
      * of the old frame velocity in the new frame)
      */
     public Transform(final AbsoluteDate date, final Vector3D translation,
                      final Vector3D velocity) {
         this(date,
              new PVCoordinates(translation, velocity, Vector3D.ZERO),
              AngularCoordinates.IDENTITY);
     }
 
     /** Build a translation transform, with its first and second time derivatives.
      * @param date date of the transform
      * @param translation translation to apply (i.e. coordinates of
      * the transformed origin, or coordinates of the origin of the
      * old frame in the new frame)
      * @param velocity the velocity of the translation (i.e. origin
      * of the old frame velocity in the new frame)
      * @param acceleration the acceleration of the translation (i.e. origin
      * of the old frame acceleration in the new frame)
      */
     public Transform(final AbsoluteDate date, final Vector3D translation,
                      final Vector3D velocity, final Vector3D acceleration) {
         this(date,
              new PVCoordinates(translation, velocity, acceleration),
              AngularCoordinates.IDENTITY);
     }
 
     /** Build a translation transform, with its first time derivative.
      * @param date date of the transform
      * @param cartesian cartesian part of the transformation to apply (i.e. coordinates of
      * the transformed origin, or coordinates of the origin of the
      * old frame in the new frame, with their derivatives)
      */
     public Transform(final AbsoluteDate date, final PVCoordinates cartesian) {
         this(date,
              cartesian,
              AngularCoordinates.IDENTITY);
     }
 
     /** Build a rotation transform.
      * @param date date of the transform
      * @param rotation rotation to apply ( i.e. rotation to apply to the
      * coordinates of a vector expressed in the old frame to obtain the
      * same vector expressed in the new frame )
      * @param rotationRate the axis of the instant rotation
      * expressed in the new frame. (norm representing angular rate)
      */
     public Transform(final AbsoluteDate date, final Rotation rotation, final Vector3D rotationRate) {
         this(date,
              PVCoordinates.ZERO,
              new AngularCoordinates(rotation, rotationRate, Vector3D.ZERO));
     }
 
     /** Build a rotation transform.
      * @param date date of the transform
      * @param rotation rotation to apply ( i.e. rotation to apply to the
      * coordinates of a vector expressed in the old frame to obtain the
      * same vector expressed in the new frame )
      * @param rotationRate the axis of the instant rotation
      * @param rotationAcceleration the axis of the instant rotation
      * expressed in the new frame. (norm representing angular rate)
      */
     public Transform(final AbsoluteDate date, final Rotation rotation, final Vector3D rotationRate,
                      final Vector3D rotationAcceleration) {
         this(date,
              PVCoordinates.ZERO,
              new AngularCoordinates(rotation, rotationRate, rotationAcceleration));
     }
 
     /** Build a rotation transform.
      * @param date date of the transform
      * @param angular angular part of the transformation to apply (i.e. rotation to
      * apply to the coordinates of a vector expressed in the old frame to obtain the
      * same vector expressed in the new frame, with its rotation rate)
      */
     public Transform(final AbsoluteDate date, final AngularCoordinates angular) {
         this(date, PVCoordinates.ZERO, angular);
     }
 
     /** Build a transform by combining two existing ones.
      * <p>
      * Note that the dates of the two existing transformed are <em>ignored</em>,
      * and the combined transform date is set to the date supplied in this constructor
      * without any attempt to shift the raw transforms. This is a design choice allowing
      * user full control of the combination.
      * </p>
      * @param date date of the transform
      * @param first first transform applied
      * @param second second transform applied
      */
     public Transform(final AbsoluteDate date, final Transform first, final Transform second) {
         this(date,
              new PVCoordinates(compositeTranslation(first, second),
                                compositeVelocity(first, second),
                                compositeAcceleration(first, second)),
              new AngularCoordinates(compositeRotation(first, second),
                                     compositeRotationRate(first, second),
                                     compositeRotationAcceleration(first, second)));
     }
 
     /** Compute a composite translation.
      * @param first first applied transform
      * @param second second applied transform
      * @return translation part of the composite transform
      */
     private static Vector3D compositeTranslation(final Transform first, final Transform second) {
 
         final Vector3D p1 = first.cartesian.getPosition();
         final Rotation r1 = first.angular.getRotation();
         final Vector3D p2 = second.cartesian.getPosition();
 
         return p1.add(r1.applyInverseTo(p2));
 
     }
 
     /** Compute a composite velocity.
      * @param first first applied transform
      * @param second second applied transform
      * @return velocity part of the composite transform
      */
     private static Vector3D compositeVelocity(final Transform first, final Transform second) {
 
         final Vector3D v1 = first.cartesian.getVelocity();
         final Rotation r1 = first.angular.getRotation();
         final Vector3D o1 = first.angular.getRotationRate();
         final Vector3D p2 = second.cartesian.getPosition();
         final Vector3D v2 = second.cartesian.getVelocity();
 
         final Vector3D crossP = Vector3D.crossProduct(o1, p2);
 
         return v1.add(r1.applyInverseTo(v2.add(crossP)));
 
     }
 
     /** Compute a composite acceleration.
      * @param first first applied transform
      * @param second second applied transform
      * @return acceleration part of the composite transform
      */
     private static Vector3D compositeAcceleration(final Transform first, final Transform second) {
 
         final Vector3D a1    = first.cartesian.getAcceleration();
         final Rotation r1    = first.angular.getRotation();
         final Vector3D o1    = first.angular.getRotationRate();
         final Vector3D oDot1 = first.angular.getRotationAcceleration();
         final Vector3D p2    = second.cartesian.getPosition();
         final Vector3D v2    = second.cartesian.getVelocity();
         final Vector3D a2    = second.cartesian.getAcceleration();
 
         final Vector3D crossCrossP = Vector3D.crossProduct(o1,    Vector3D.crossProduct(o1, p2));
         final Vector3D crossV      = Vector3D.crossProduct(o1,    v2);
         final Vector3D crossDotP   = Vector3D.crossProduct(oDot1, p2);
 
         return a1.add(r1.applyInverseTo(new Vector3D(1, a2, 2, crossV, 1, crossCrossP, 1, crossDotP)));
 
     }
 
     /** Compute a composite rotation.
      * @param first first applied transform
      * @param second second applied transform
      * @return rotation part of the composite transform
      */
     private static Rotation compositeRotation(final Transform first, final Transform second) {
 
         final Rotation r1 = first.angular.getRotation();
         final Rotation r2 = second.angular.getRotation();
 
         return r2.applyTo(r1);
 
     }
 
     /** Compute a composite rotation rate.
      * @param first first applied transform
      * @param second second applied transform
      * @return rotation rate part of the composite transform
      */
     private static Vector3D compositeRotationRate(final Transform first, final Transform second) {
 
         final Vector3D o1 = first.angular.getRotationRate();
         final Rotation r2 = second.angular.getRotation();
         final Vector3D o2 = second.angular.getRotationRate();
 
         return o2.add(r2.applyTo(o1));
 
     }
 
     /** Compute a composite rotation acceleration.
      * @param first first applied transform
      * @param second second applied transform
      * @return rotation acceleration part of the composite transform
      */
     private static Vector3D compositeRotationAcceleration(final Transform first, final Transform second) {
 
         final Vector3D o1    = first.angular.getRotationRate();
         final Vector3D oDot1 = first.angular.getRotationAcceleration();
         final Rotation r2    = second.angular.getRotation();
         final Vector3D o2    = second.angular.getRotationRate();
         final Vector3D oDot2 = second.angular.getRotationAcceleration();
 
         return new Vector3D( 1, oDot2,
                              1, r2.applyTo(oDot1),
                             -1, Vector3D.crossProduct(o2, r2.applyTo(o1)));
 
     }
 
     /** {@inheritDoc} */
     public AbsoluteDate getDate() {
         return date;
     }
 
     /** {@inheritDoc} */
     public Transform shiftedBy(final double dt) {
         return new Transform(date.shiftedBy(dt), cartesian.shiftedBy(dt), angular.shiftedBy(dt));
     };
 
     /** {@inheritDoc}
      * <p>
      * Calling this method is equivalent to call {@link #interpolate(AbsoluteDate,
      * CartesianDerivativesFilter, AngularDerivativesFilter, Collection)} with {@code cFilter}
      * set to {@link CartesianDerivativesFilter#USE_PVA} and {@code aFilter} set to
      * {@link AngularDerivativesFilter#USE_RRA}
      * set to true.
      * </p>
      * @exception OrekitException if the number of point is too small for interpolating
      */
     public Transform interpolate(final AbsoluteDate interpolationDate, final Collection<Transform> sample)
         throws OrekitException {
         return interpolate(interpolationDate,
                            CartesianDerivativesFilter.USE_PVA, AngularDerivativesFilter.USE_RRA,
                            sample);
     }
 
     /** Interpolate a transform from a sample set of existing transforms.
      * <p>
      * Note that even if first time derivatives (velocities and rotation rates)
      * from sample can be ignored, the interpolated instance always includes
      * interpolated derivatives. This feature can be used explicitly to
      * compute these derivatives when it would be too complex to compute them
      * from an analytical formula: just compute a few sample points from the
      * explicit formula and set the derivatives to zero in these sample points,
      * then use interpolation to add derivatives consistent with the positions
      * and rotations.
      * </p>
      * <p>
      * As this implementation of interpolation is polynomial, it should be used only
      * with small samples (about 10-20 points) in order to avoid <a
      * href="http://en.wikipedia.org/wiki/Runge%27s_phenomenon">Runge's phenomenon</a>
      * and numerical problems (including NaN appearing).
      * </p>
      * @param date interpolation date
      * @param useVelocities if true, use sample transforms velocities,
      * otherwise ignore them and use only positions
      * @param useRotationRates if true, use sample points rotation rates,
      * otherwise ignore them and use only rotations
      * @param sample sample points on which interpolation should be done
      * @return a new instance, interpolated at specified date
      * @exception OrekitException if the number of point is too small for interpolating
      * @deprecated as of 7.0, replaced with {@link #interpolate(AbsoluteDate, CartesianDerivativesFilter, AngularDerivativesFilter, Collection)}
      */
     @Deprecated
     public static Transform interpolate(final AbsoluteDate date,
                                         final boolean useVelocities, final boolean useRotationRates,
                                         final Collection<Transform> sample)
         throws OrekitException {
         return interpolate(date,
                            useVelocities    ? CartesianDerivativesFilter.USE_PV : CartesianDerivativesFilter.USE_P,
                            useRotationRates ? AngularDerivativesFilter.USE_RR   : AngularDerivativesFilter.USE_R,
                            sample);
     }
 
     /** Interpolate a transform from a sample set of existing transforms.
      * <p>
      * Note that even if first time derivatives (velocities and rotation rates)
      * from sample can be ignored, the interpolated instance always includes
      * interpolated derivatives. This feature can be used explicitly to
      * compute these derivatives when it would be too complex to compute them
      * from an analytical formula: just compute a few sample points from the
      * explicit formula and set the derivatives to zero in these sample points,
      * then use interpolation to add derivatives consistent with the positions
      * and rotations.
      * </p>
      * <p>
      * As this implementation of interpolation is polynomial, it should be used only
      * with small samples (about 10-20 points) in order to avoid <a
      * href="http://en.wikipedia.org/wiki/Runge%27s_phenomenon">Runge's phenomenon</a>
      * and numerical problems (including NaN appearing).
      * </p>
      * @param date interpolation date
      * @param cFilter filter for derivatives from the sample to use in interpolation
      * @param aFilter filter for derivatives from the sample to use in interpolation
      * @param sample sample points on which interpolation should be done
      * @return a new instance, interpolated at specified date
      * @exception OrekitException if the number of point is too small for interpolating
      * @since 7.0
      */
     public static Transform interpolate(final AbsoluteDate date,
                                         final CartesianDerivativesFilter cFilter,
                                         final AngularDerivativesFilter aFilter,
                                         final Collection<Transform> sample)
         throws OrekitException {
         final List<TimeStampedPVCoordinates>      datedPV = new ArrayList<TimeStampedPVCoordinates>(sample.size());
         final List<TimeStampedAngularCoordinates> datedAC = new ArrayList<TimeStampedAngularCoordinates>(sample.size());
         for (final Transform t : sample) {
             datedPV.add(new TimeStampedPVCoordinates(t.getDate(), t.getTranslation(), t.getVelocity(), t.getAcceleration()));
             datedAC.add(new TimeStampedAngularCoordinates(t.getDate(), t.getRotation(), t.getRotationRate(), t.getRotationAcceleration()));
         }
         final TimeStampedPVCoordinates      interpolatedPV = TimeStampedPVCoordinates.interpolate(date, cFilter, datedPV);
         final TimeStampedAngularCoordinates interpolatedAC = TimeStampedAngularCoordinates.interpolate(date, aFilter, datedAC);
         return new Transform(date, interpolatedPV, interpolatedAC);
     }
 
     /** Get the inverse transform of the instance.
      * @return inverse transform of the instance
      */
     public Transform getInverse() {
 
         final Rotation r    = angular.getRotation();
         final Vector3D o    = angular.getRotationRate();
         final Vector3D oDot = angular.getRotationAcceleration();
         final Vector3D rp   = r.applyTo(cartesian.getPosition());
         final Vector3D rv   = r.applyTo(cartesian.getVelocity());
         final Vector3D ra   = r.applyTo(cartesian.getAcceleration());
 
         final Vector3D pInv        = rp.negate();
         final Vector3D crossP      = Vector3D.crossProduct(o, rp);
         final Vector3D vInv        = crossP.subtract(rv);
         final Vector3D crossV      = Vector3D.crossProduct(o, rv);
         final Vector3D crossDotP   = Vector3D.crossProduct(oDot, rp);
         final Vector3D crossCrossP = Vector3D.crossProduct(o, crossP);
         final Vector3D aInv        = new Vector3D(-1, ra,
                                                    2, crossV,
                                                    1, crossDotP,
                                                   -1, crossCrossP);
 
         return new Transform(date, new PVCoordinates(pInv, vInv, aInv), angular.revert());
 
     }
 
     /** Get a frozen transform.
      * <p>
      * This method creates a copy of the instance but frozen in time,
      * i.e. with velocity, acceleration and rotation rate forced to zero.
      * </p>
      * @return a new transform, without any time-dependent parts
      */
     public Transform freeze() {
         return new Transform(date,
                              new PVCoordinates(cartesian.getPosition(), Vector3D.ZERO, Vector3D.ZERO),
                              new AngularCoordinates(angular.getRotation(), Vector3D.ZERO, Vector3D.ZERO));
     }
 
     /** Transform a position vector (including translation effects).
      * @param position vector to transform
      * @return transformed position
      */
     public Vector3D transformPosition(final Vector3D position) {
         return angular.getRotation().applyTo(cartesian.getPosition().add(position));
     }
 
     /** Transform a position vector (including translation effects).
      * @param position vector to transform
      * @param <T> the type of the field elements
      * @return transformed position
      */
     public <T extends RealFieldElement<T>> FieldVector3D<T> transformPosition(final FieldVector3D<T> position) {
         return FieldRotation.applyTo(angular.getRotation(), position.add(cartesian.getPosition()));
     }
 
     /** Transform a vector (ignoring translation effects).
      * @param vector vector to transform
      * @return transformed vector
      */
     public Vector3D transformVector(final Vector3D vector) {
         return angular.getRotation().applyTo(vector);
     }
 
     /** Transform a vector (ignoring translation effects).
      * @param vector vector to transform
      * @param <T> the type of the field elements
      * @return transformed vector
      */
     public <T extends RealFieldElement<T>> FieldVector3D<T> transformVector(final FieldVector3D<T> vector) {
         return FieldRotation.applyTo(angular.getRotation(), vector);
     }
 
     /** Transform a line.
      * @param line to transform
      * @return transformed line
      */
     public Line transformLine(final Line line) {
         final Vector3D transformedP0 = transformPosition(line.getOrigin());
         final Vector3D transformedP1 = transformPosition(line.pointAt(1.0e6));
         return new Line(transformedP0, transformedP1, 1.0e-10);
     }
 
     /** Transform {@link PVCoordinates} including kinematic effects.
      * @param pva the position-velocity-acceleration triplet to transform.
      * @return transformed position-velocity-acceleration
      */
     public PVCoordinates transformPVCoordinates(final PVCoordinates pva) {
         return angular.applyTo(new PVCoordinates(1, pva, 1, cartesian));
     }
 
     /** Transform {@link TimeStampedPVCoordinates} including kinematic effects.
      * <p>
      * In order to allow the user more flexibility, this method does <em>not</em> check for
      * consistency between the transform {@link #getDate() date} and the time-stamped
      * position-velocity {@link TimeStampedPVCoordinates#getDate() date}. The returned
      * value will always have the same {@link TimeStampedPVCoordinates#getDate() date} as
      * the input argument, regardless of the instance {@link #getDate() date}.
      * </p>
      * @param pv time-stamped  position-velocity to transform.
      * @return transformed time-stamped position-velocity
      * @since 7.0
      */
     public TimeStampedPVCoordinates transformPVCoordinates(final TimeStampedPVCoordinates pv) {
         return angular.applyTo(new TimeStampedPVCoordinates(pv.getDate(), 1, pv, 1, cartesian));
     }
 
     /** Transform {@link FieldPVCoordinates} including kinematic effects.
      * @param pv position-velocity to transform.
      * @param <T> type of the field elements
      * @return transformed position-velocity
      */
     public <T extends RealFieldElement<T>> FieldPVCoordinates<T> transformPVCoordinates(final FieldPVCoordinates<T> pv) {
 
         // apply translation
         final FieldVector3D<T> intermediateP = pv.getPosition().add(cartesian.getPosition());
         final FieldVector3D<T> intermediateV = pv.getVelocity().add(cartesian.getVelocity());
         final FieldVector3D<T> intermediateA = pv.getAcceleration().add(cartesian.getAcceleration());
 
         // apply rotation
         final FieldVector3D<T> transformedP = FieldRotation.applyTo(angular.getRotation(), intermediateP);
         final FieldVector3D<T> crossP       = FieldVector3D.crossProduct(angular.getRotationRate(), transformedP);
         final FieldVector3D<T> transformedV = FieldRotation.applyTo(angular.getRotation(), intermediateV).subtract(crossP);
         final FieldVector3D<T> crossV       = FieldVector3D.crossProduct(angular.getRotationRate(), transformedV);
         final FieldVector3D<T> crossCrossP  = FieldVector3D.crossProduct(angular.getRotationRate(), crossP);
         final FieldVector3D<T> crossDotP    = FieldVector3D.crossProduct(angular.getRotationAcceleration(), transformedP);
         final FieldVector3D<T> transformedA =
                 new FieldVector3D<T>( 1, FieldRotation.applyTo(angular.getRotation(), intermediateA),
                                      -2, crossV,
                                      -1, crossCrossP,
                                      -1, crossDotP);
 
         // build transformed object
         return new FieldPVCoordinates<T>(transformedP, transformedV, transformedA);
 
     }
 
     /** Transform {@link TimeStampedFieldPVCoordinates} including kinematic effects.
      * <p>
      * In order to allow the user more flexibility, this method does <em>not</em> check for
      * consistency between the transform {@link #getDate() date} and the time-stamped
      * position-velocity {@link TimeStampedFieldPVCoordinates#getDate() date}. The returned
      * value will always have the same {@link TimeStampedFieldPVCoordinates#getDate() date} as
      * the input argument, regardless of the instance {@link #getDate() date}.
      * </p>
      * @param pv time-stamped position-velocity to transform.
      * @param <T> type of the field elements
      * @return transformed time-stamped position-velocity
      * @since 7.0
      */
     public <T extends RealFieldElement<T>> TimeStampedFieldPVCoordinates<T> transformPVCoordinates(final TimeStampedFieldPVCoordinates<T> pv) {
 
         // apply translation
         final FieldVector3D<T> intermediateP = pv.getPosition().add(cartesian.getPosition());
         final FieldVector3D<T> intermediateV = pv.getVelocity().add(cartesian.getVelocity());
         final FieldVector3D<T> intermediateA = pv.getAcceleration().add(cartesian.getAcceleration());
 
         // apply rotation
         final FieldVector3D<T> transformedP = FieldRotation.applyTo(angular.getRotation(), intermediateP);
         final FieldVector3D<T> crossP       = FieldVector3D.crossProduct(angular.getRotationRate(), transformedP);
         final FieldVector3D<T> transformedV = FieldRotation.applyTo(angular.getRotation(), intermediateV).subtract(crossP);
         final FieldVector3D<T> crossV       = FieldVector3D.crossProduct(angular.getRotationRate(), transformedV);
         final FieldVector3D<T> crossCrossP  = FieldVector3D.crossProduct(angular.getRotationRate(), crossP);
         final FieldVector3D<T> crossDotP    = FieldVector3D.crossProduct(angular.getRotationAcceleration(), transformedP);
         final FieldVector3D<T> transformedA =
                 new FieldVector3D<T>( 1, FieldRotation.applyTo(angular.getRotation(), intermediateA),
                                      -2, crossV,
                                      -1, crossCrossP,
                                      -1, crossDotP);
 
         // build transformed object
         return new TimeStampedFieldPVCoordinates<T>(pv.getDate(), transformedP, transformedV, transformedA);
 
     }
 
     /** Compute the Jacobian of the {@link #transformPVCoordinates(PVCoordinates)}
      * method of the transform.
      * <p>
      * Element {@code jacobian[i][j]} is the derivative of Cartesian coordinate i
      * of the transformed {@link PVCoordinates} with respect to Cartesian coordinate j
      * of the input {@link PVCoordinates} in method {@link #transformPVCoordinates(PVCoordinates)}.
      * </p>
      * <p>
      * This definition implies that if we define position-velocity coordinates
      * <pre>
      * PV₁ = transform.transformPVCoordinates(PV₀), then
      * </pre>
      * their differentials dPV₁ and dPV₀ will obey the following relation
      * where J is the matrix computed by this method:<br/>
      * <pre>
      * dPV₁ = J &times; dPV₀
      * </pre>
      * </p>
      * @param jacobian placeholder 6x6 (or larger) matrix to be filled with
      * the Jacobian, only the upper left 6x6 corner will be modified
      * @deprecated as of 7.0, replaced with {@link #getJacobian(CartesianDerivativesFilter, double[][])}
      */
     @Deprecated
     public void getJacobian(final double[][] jacobian) {
         getJacobian(CartesianDerivativesFilter.USE_PV, jacobian);
     }
 
     /** Compute the Jacobian of the {@link #transformPVCoordinates(PVCoordinates)}
      * method of the transform.
      * <p>
      * Element {@code jacobian[i][j]} is the derivative of Cartesian coordinate i
      * of the transformed {@link PVCoordinates} with respect to Cartesian coordinate j
      * of the input {@link PVCoordinates} in method {@link #transformPVCoordinates(PVCoordinates)}.
      * </p>
      * <p>
      * This definition implies that if we define position-velocity coordinates
      * <pre>
      * PV₁ = transform.transformPVCoordinates(PV₀), then
      * </pre>
      * their differentials dPV₁ and dPV₀ will obey the following relation
      * where J is the matrix computed by this method:<br/>
      * <pre>
      * dPV₁ = J &times; dPV₀
      * </pre>
      * </p>
      * @param selector selector specifying the size of the upper left corner that must be filled
      * (either 3x3 for positions only, 6x6 for positions and velocities, 9x9 for positions,
      * velocities and accelerations)
      * @param jacobian placeholder matrix whose upper-left corner is to be filled with
      * the Jacobian, the rest of the matrix remaining untouched
      */
     public void getJacobian(final CartesianDerivativesFilter selector, final double[][] jacobian) {
 
         // elementary matrix for rotation
         final double[][] mData = angular.getRotation().getMatrix();
 
         // dP1/dP0
         System.arraycopy(mData[0], 0, jacobian[0], 0, 3);
         System.arraycopy(mData[1], 0, jacobian[1], 0, 3);
         System.arraycopy(mData[2], 0, jacobian[2], 0, 3);
 
         if (selector.getMaxOrder() >= 1) {
 
             // dP1/dV0
             Arrays.fill(jacobian[0], 3, 6, 0.0);
             Arrays.fill(jacobian[1], 3, 6, 0.0);
             Arrays.fill(jacobian[2], 3, 6, 0.0);
 
             // dV1/dP0
             final Vector3D o = angular.getRotationRate();
             final double ox = o.getX();
             final double oy = o.getY();
             final double oz = o.getZ();
             for (int i = 0; i < 3; ++i) {
                 jacobian[3][i] = -(oy * mData[2][i] - oz * mData[1][i]);
                 jacobian[4][i] = -(oz * mData[0][i] - ox * mData[2][i]);
                 jacobian[5][i] = -(ox * mData[1][i] - oy * mData[0][i]);
             }
 
             // dV1/dV0
             System.arraycopy(mData[0], 0, jacobian[3], 3, 3);
             System.arraycopy(mData[1], 0, jacobian[4], 3, 3);
             System.arraycopy(mData[2], 0, jacobian[5], 3, 3);
 
             if (selector.getMaxOrder() >= 2) {
 
                 // dP1/dA0
                 Arrays.fill(jacobian[0], 6, 9, 0.0);
                 Arrays.fill(jacobian[1], 6, 9, 0.0);
                 Arrays.fill(jacobian[2], 6, 9, 0.0);
 
                 // dV1/dA0
                 Arrays.fill(jacobian[3], 6, 9, 0.0);
                 Arrays.fill(jacobian[4], 6, 9, 0.0);
                 Arrays.fill(jacobian[5], 6, 9, 0.0);
 
                 // dA1/dP0
                 final Vector3D oDot = angular.getRotationAcceleration();
                 final double oDotx  = oDot.getX();
                 final double oDoty  = oDot.getY();
                 final double oDotz  = oDot.getZ();
                 for (int i = 0; i < 3; ++i) {
                     jacobian[6][i] = -(oDoty * mData[2][i] - oDotz * mData[1][i]) - (oy * jacobian[5][i] - oz * jacobian[4][i]);
                     jacobian[7][i] = -(oDotz * mData[0][i] - oDotx * mData[2][i]) - (oz * jacobian[3][i] - ox * jacobian[5][i]);
                     jacobian[8][i] = -(oDotx * mData[1][i] - oDoty * mData[0][i]) - (ox * jacobian[4][i] - oy * jacobian[3][i]);
                 }
 
                 // dA1/dV0
                 for (int i = 0; i < 3; ++i) {
                     jacobian[6][i + 3] = -2 * (oy * mData[2][i] - oz * mData[1][i]);
                     jacobian[7][i + 3] = -2 * (oz * mData[0][i] - ox * mData[2][i]);
                     jacobian[8][i + 3] = -2 * (ox * mData[1][i] - oy * mData[0][i]);
                 }
 
                 // dA1/dA0
                 System.arraycopy(mData[0], 0, jacobian[6], 6, 3);
                 System.arraycopy(mData[1], 0, jacobian[7], 6, 3);
                 System.arraycopy(mData[2], 0, jacobian[8], 6, 3);
 
             }
 
         }
 
     }
 
     /** Get the underlying elementary cartesian part.
      * <p>A transform can be uniquely represented as an elementary
      * translation followed by an elementary rotation. This method
      * returns this unique elementary translation with its derivative.</p>
      * @return underlying elementary cartesian part
      * @see #getTranslation()
      * @see #getVelocity()
      */
     public PVCoordinates getCartesian() {
         return cartesian;
     }
 
     /** Get the underlying elementary translation.
      * <p>A transform can be uniquely represented as an elementary
      * translation followed by an elementary rotation. This method
      * returns this unique elementary translation.</p>
      * @return underlying elementary translation
      * @see #getCartesian()
      * @see #getVelocity()
      * @see #getAcceleration()
      */
     public Vector3D getTranslation() {
         return cartesian.getPosition();
     }
 
     /** Get the first time derivative of the translation.
      * @return first time derivative of the translation
      * @see #getCartesian()
      * @see #getTranslation()
      * @see #getAcceleration()
      */
     public Vector3D getVelocity() {
         return cartesian.getVelocity();
     }
 
     /** Get the second time derivative of the translation.
      * @return second time derivative of the translation
      * @see #getCartesian()
      * @see #getTranslation()
      * @see #getVelocity()
      */
     public Vector3D getAcceleration() {
         return cartesian.getAcceleration();
     }
 
     /** Get the underlying elementary angular part.
      * <p>A transform can be uniquely represented as an elementary
      * translation followed by an elementary rotation. This method
      * returns this unique elementary rotation with its derivative.</p>
      * @return underlying elementary angular part
      * @see #getRotation()
      * @see #getRotationRate()
      * @see #getRotationAcceleration()
      */
     public AngularCoordinates getAngular() {
         return angular;
     }
 
     /** Get the underlying elementary rotation.
      * <p>A transform can be uniquely represented as an elementary
      * translation followed by an elementary rotation. This method
      * returns this unique elementary rotation.</p>
      * @return underlying elementary rotation
      * @see #getAngular()
      * @see #getRotationRate()
      * @see #getRotationAcceleration()
      */
     public Rotation getRotation() {
         return angular.getRotation();
     }
 
     /** Get the first time derivative of the rotation.
      * <p>The norm represents the angular rate.</p>
      * @return First time derivative of the rotation
      * @see #getAngular()
      * @see #getRotation()
      * @see #getRotationAcceleration()
      */
     public Vector3D getRotationRate() {
         return angular.getRotationRate();
     }
 
     /** Get the second time derivative of the rotation.
      * @return Second time derivative of the rotation
      * @see #getAngular()
      * @see #getRotation()
      * @see #getRotationRate()
      */
     public Vector3D getRotationAcceleration() {
         return angular.getRotationAcceleration();
     }
 
     /** Specialized class for identity transform. */
     private static class IdentityTransform extends Transform {
 
         /** Serializable UID. */
         private static final long serialVersionUID = -9042082036141830517L;
 
         /** Simple constructor. */
         public IdentityTransform() {
             super(AbsoluteDate.J2000_EPOCH, PVCoordinates.ZERO, AngularCoordinates.IDENTITY);
         }
 
         /** {@inheritDoc} */
         @Override
         public Transform shiftedBy(final double dt) {
             return this;
         }
 
         /** {@inheritDoc} */
         @Override
         public Transform getInverse() {
             return this;
         };
 
         /** {@inheritDoc} */
         @Override
         public Vector3D transformPosition(final Vector3D position) {
             return position;
         }
 
         /** {@inheritDoc} */
         @Override
         public Vector3D transformVector(final Vector3D vector) {
             return vector;
         }
 
         /** {@inheritDoc} */
         @Override
         public Line transformLine(final Line line) {
             return line;
         }
 
         /** {@inheritDoc} */
         @Override
         public PVCoordinates transformPVCoordinates(final PVCoordinates pv) {
             return pv;
         }
 
         /** {@inheritDoc} */
         @Override
         public void getJacobian(final CartesianDerivativesFilter selector, final double[][] jacobian) {
             final int n = 3 * (selector.getMaxOrder() + 1);
             for (int i = 0; i < n; ++i) {
                 Arrays.fill(jacobian[i], 0, n, 0.0);
                 jacobian[i][i] = 1.0;
             }
         }
 
     }
 
 }