/*
    * Licensed to the Apache Software Foundation (ASF) under one or more
    * contributor license agreements.  See the NOTICE file distributed with
    * this work for additional information regarding copyright ownership.
    * The ASF 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
    *
    *      https://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.
   */
  
  /*
   * This is not the original file distributed by the Apache Software Foundation
   * It has been modified by the Hipparchus project
   */
  
  package org.hipparchus.geometry.euclidean.threed;
  
  import java.io.Serializable;
  
  import org.hipparchus.CalculusFieldElement;
  import org.hipparchus.Field;
  import org.hipparchus.exception.MathIllegalArgumentException;
  import org.hipparchus.exception.MathRuntimeException;
  import org.hipparchus.geometry.LocalizedGeometryFormats;
  import org.hipparchus.util.FastMath;
  import org.hipparchus.util.FieldSinCos;
  import org.hipparchus.util.MathArrays;
  
  /**
   * This class is a re-implementation of {@link Rotation} using {@link CalculusFieldElement}.
   * <p>Instance of this class are guaranteed to be immutable.</p>
   *
   * @param <T> the type of the field elements
   * @see FieldVector3D
   * @see RotationOrder
   */
  
  public class FieldRotation<T extends CalculusFieldElement<T>> implements Serializable {
  
      /** Serializable version identifier */
      private static final long serialVersionUID = 20130224L;
  
      /** Scalar coordinate of the quaternion. */
      private final T q0;
  
      /** First coordinate of the vectorial part of the quaternion. */
      private final T q1;
  
      /** Second coordinate of the vectorial part of the quaternion. */
      private final T q2;
  
      /** Third coordinate of the vectorial part of the quaternion. */
      private final T q3;
  
      /** Build a rotation from the quaternion coordinates.
       * <p>A rotation can be built from a <em>normalized</em> quaternion,
       * i.e. a quaternion for which q<sub>0</sub><sup>2</sup> +
       * q<sub>1</sub><sup>2</sup> + q<sub>2</sub><sup>2</sup> +
       * q<sub>3</sub><sup>2</sup> = 1. If the quaternion is not normalized,
       * the constructor can normalize it in a preprocessing step.</p>
       * <p>Note that some conventions put the scalar part of the quaternion
       * as the 4<sup>th</sup> component and the vector part as the first three
       * components. This is <em>not</em> our convention. We put the scalar part
       * as the first component.</p>
       * @param q0 scalar part of the quaternion
       * @param q1 first coordinate of the vectorial part of the quaternion
       * @param q2 second coordinate of the vectorial part of the quaternion
       * @param q3 third coordinate of the vectorial part of the quaternion
       * @param needsNormalization if true, the coordinates are considered
       * not to be normalized, a normalization preprocessing step is performed
       * before using them
       */
      public FieldRotation(final T q0, final T q1, final T q2, final T q3, final boolean needsNormalization) {
  
          if (needsNormalization) {
              // normalization preprocessing
              final T inv =
                      q0.square().add(q1.square()).add(q2.square()).add(q3.square()).sqrt().reciprocal();
              this.q0 = inv.multiply(q0);
              this.q1 = inv.multiply(q1);
              this.q2 = inv.multiply(q2);
              this.q3 = inv.multiply(q3);
          } else {
              this.q0 = q0;
              this.q1 = q1;
              this.q2 = q2;
              this.q3 = q3;
          }
  
      }
  
      /** Build a rotation from an axis and an angle.
      * <p>We use the convention that angles are oriented according to
      * the effect of the rotation on vectors around the axis. That means
      * that if (i, j, k) is a direct frame and if we first provide +k as
      * the axis and &pi;/2 as the angle to this constructor, and then
      * {@link #applyTo(FieldVector3D) apply} the instance to +i, we will get
      * +j.</p>
      * <p>Another way to represent our convention is to say that a rotation
      * of angle &theta; about the unit vector (x, y, z) is the same as the
      * rotation build from quaternion components { cos(-&theta;/2),
      * x * sin(-&theta;/2), y * sin(-&theta;/2), z * sin(-&theta;/2) }.
      * Note the minus sign on the angle!</p>
      * <p>On the one hand this convention is consistent with a vectorial
      * perspective (moving vectors in fixed frames), on the other hand it
      * is different from conventions with a frame perspective (fixed vectors
      * viewed from different frames) like the ones used for example in spacecraft
      * attitude community or in the graphics community.</p>
      * @param axis axis around which to rotate
      * @param angle rotation angle.
      * @param convention convention to use for the semantics of the angle
      * @exception MathIllegalArgumentException if the axis norm is zero
      */
     public FieldRotation(final FieldVector3D<T> axis, final T angle, final RotationConvention convention)
         throws MathIllegalArgumentException {
 
         final T norm = axis.getNorm();
         if (norm.getReal() == 0) {
             throw new MathIllegalArgumentException(LocalizedGeometryFormats.ZERO_NORM_FOR_ROTATION_AXIS);
         }
 
         final T halfAngle = angle.multiply(convention == RotationConvention.VECTOR_OPERATOR ? -0.5 : 0.5);
         final FieldSinCos<T> sinCos = FastMath.sinCos(halfAngle);
         final T coeff = sinCos.sin().divide(norm);
 
         q0 = sinCos.cos();
         q1 = coeff.multiply(axis.getX());
         q2 = coeff.multiply(axis.getY());
         q3 = coeff.multiply(axis.getZ());
 
     }
 
     /** Build a {@link FieldRotation} from a {@link Rotation}.
      * @param field field for the components
      * @param r rotation to convert
      */
     public FieldRotation(final Field<T> field, final Rotation r) {
         this.q0 = field.getZero().add(r.getQ0());
         this.q1 = field.getZero().add(r.getQ1());
         this.q2 = field.getZero().add(r.getQ2());
         this.q3 = field.getZero().add(r.getQ3());
     }
 
     /** Build a rotation from a 3X3 matrix.
 
      * <p>Rotation matrices are orthogonal matrices, i.e. unit matrices
      * (which are matrices for which m.m<sup>T</sup> = I) with real
      * coefficients. The module of the determinant of unit matrices is
      * 1, among the orthogonal 3X3 matrices, only the ones having a
      * positive determinant (+1) are rotation matrices.</p>
 
      * <p>When a rotation is defined by a matrix with truncated values
      * (typically when it is extracted from a technical sheet where only
      * four to five significant digits are available), the matrix is not
      * orthogonal anymore. This constructor handles this case
      * transparently by using a copy of the given matrix and applying a
      * correction to the copy in order to perfect its orthogonality. If
      * the Frobenius norm of the correction needed is above the given
      * threshold, then the matrix is considered to be too far from a
      * true rotation matrix and an exception is thrown.</p>
 
      * @param m rotation matrix
      * @param threshold convergence threshold for the iterative
      * orthogonality correction (convergence is reached when the
      * difference between two steps of the Frobenius norm of the
      * correction is below this threshold)
 
      * @exception MathIllegalArgumentException if the matrix is not a 3X3
      * matrix, or if it cannot be transformed into an orthogonal matrix
      * with the given threshold, or if the determinant of the resulting
      * orthogonal matrix is negative
 
      */
     public FieldRotation(final T[][] m, final double threshold)
         throws MathIllegalArgumentException {
 
         // dimension check
         if ((m.length != 3) || (m[0].length != 3) ||
                 (m[1].length != 3) || (m[2].length != 3)) {
             throw new MathIllegalArgumentException(LocalizedGeometryFormats.ROTATION_MATRIX_DIMENSIONS,
                                                    m.length, m[0].length);
         }
 
         // compute a "close" orthogonal matrix
         final T[][] ort = orthogonalizeMatrix(m, threshold);
 
         // check the sign of the determinant
         final T d0 = ort[1][1].multiply(ort[2][2]).subtract(ort[2][1].multiply(ort[1][2]));
         final T d1 = ort[0][1].multiply(ort[2][2]).subtract(ort[2][1].multiply(ort[0][2]));
         final T d2 = ort[0][1].multiply(ort[1][2]).subtract(ort[1][1].multiply(ort[0][2]));
         final T det =
                 ort[0][0].multiply(d0).subtract(ort[1][0].multiply(d1)).add(ort[2][0].multiply(d2));
         if (det.getReal() < 0.0) {
             throw new MathIllegalArgumentException(LocalizedGeometryFormats.CLOSEST_ORTHOGONAL_MATRIX_HAS_NEGATIVE_DETERMINANT,
                                                    det);
         }
 
         final T[] quat = mat2quat(ort);
         q0 = quat[0];
         q1 = quat[1];
         q2 = quat[2];
         q3 = quat[3];
 
     }
 
     /** Build the rotation that transforms a pair of vectors into another pair.
 
      * <p>Except for possible scale factors, if the instance were applied to
      * the pair (u<sub>1</sub>, u<sub>2</sub>) it will produce the pair
      * (v<sub>1</sub>, v<sub>2</sub>).</p>
 
      * <p>If the angular separation between u<sub>1</sub> and u<sub>2</sub> is
      * not the same as the angular separation between v<sub>1</sub> and
      * v<sub>2</sub>, then a corrected v'<sub>2</sub> will be used rather than
      * v<sub>2</sub>, the corrected vector will be in the (±v<sub>1</sub>,
      * +v<sub>2</sub>) half-plane.</p>
      * @param u1 first vector of the origin pair
      * @param u2 second vector of the origin pair
      * @param v1 desired image of u1 by the rotation
      * @param v2 desired image of u2 by the rotation
      * @exception MathRuntimeException if the norm of one of the vectors is zero,
      * or if one of the pair is degenerated (i.e. the vectors of the pair are collinear)
      */
     public FieldRotation(FieldVector3D<T> u1, FieldVector3D<T> u2, FieldVector3D<T> v1, FieldVector3D<T> v2)
         throws MathRuntimeException {
 
         // build orthonormalized base from u1, u2
         // this fails when vectors are null or collinear, which is forbidden to define a rotation
         final FieldVector3D<T> u3 = FieldVector3D.crossProduct(u1, u2).normalize();
         u2 = FieldVector3D.crossProduct(u3, u1).normalize();
         u1 = u1.normalize();
 
         // build an orthonormalized base from v1, v2
         // this fails when vectors are null or collinear, which is forbidden to define a rotation
         final FieldVector3D<T> v3 = FieldVector3D.crossProduct(v1, v2).normalize();
         v2 = FieldVector3D.crossProduct(v3, v1).normalize();
         v1 = v1.normalize();
 
         // buid a matrix transforming the first base into the second one
         final T[][] array = MathArrays.buildArray(u1.getX().getField(), 3, 3);
         array[0][0] = u1.getX().multiply(v1.getX()).add(u2.getX().multiply(v2.getX())).add(u3.getX().multiply(v3.getX()));
         array[0][1] = u1.getY().multiply(v1.getX()).add(u2.getY().multiply(v2.getX())).add(u3.getY().multiply(v3.getX()));
         array[0][2] = u1.getZ().multiply(v1.getX()).add(u2.getZ().multiply(v2.getX())).add(u3.getZ().multiply(v3.getX()));
         array[1][0] = u1.getX().multiply(v1.getY()).add(u2.getX().multiply(v2.getY())).add(u3.getX().multiply(v3.getY()));
         array[1][1] = u1.getY().multiply(v1.getY()).add(u2.getY().multiply(v2.getY())).add(u3.getY().multiply(v3.getY()));
         array[1][2] = u1.getZ().multiply(v1.getY()).add(u2.getZ().multiply(v2.getY())).add(u3.getZ().multiply(v3.getY()));
         array[2][0] = u1.getX().multiply(v1.getZ()).add(u2.getX().multiply(v2.getZ())).add(u3.getX().multiply(v3.getZ()));
         array[2][1] = u1.getY().multiply(v1.getZ()).add(u2.getY().multiply(v2.getZ())).add(u3.getY().multiply(v3.getZ()));
         array[2][2] = u1.getZ().multiply(v1.getZ()).add(u2.getZ().multiply(v2.getZ())).add(u3.getZ().multiply(v3.getZ()));
 
         T[] quat = mat2quat(array);
         q0 = quat[0];
         q1 = quat[1];
         q2 = quat[2];
         q3 = quat[3];
 
     }
 
     /** Build one of the rotations that transform one vector into another one.
 
      * <p>Except for a possible scale factor, if the instance were
      * applied to the vector u it will produce the vector v. There is an
      * infinite number of such rotations, this constructor choose the
      * one with the smallest associated angle (i.e. the one whose axis
      * is orthogonal to the (u, v) plane). If u and v are collinear, an
      * arbitrary rotation axis is chosen.</p>
 
      * @param u origin vector
      * @param v desired image of u by the rotation
      * @exception MathRuntimeException if the norm of one of the vectors is zero
      */
     public FieldRotation(final FieldVector3D<T> u, final FieldVector3D<T> v) throws MathRuntimeException {
 
         final T normProduct = u.getNorm().multiply(v.getNorm());
         if (normProduct.getReal() == 0) {
             throw new MathRuntimeException(LocalizedGeometryFormats.ZERO_NORM_FOR_ROTATION_DEFINING_VECTOR);
         }
 
         final T dot = FieldVector3D.dotProduct(u, v);
 
         if (dot.getReal() < ((2.0e-15 - 1.0) * normProduct.getReal())) {
             // special case u = -v: we select a PI angle rotation around
             // an arbitrary vector orthogonal to u
             final FieldVector3D<T> w = u.orthogonal();
             q0 = normProduct.getField().getZero();
             q1 = w.getX().negate();
             q2 = w.getY().negate();
             q3 = w.getZ().negate();
         } else {
             // general case: (u, v) defines a plane, we select
             // the shortest possible rotation: axis orthogonal to this plane
             q0 = dot.divide(normProduct).add(1.0).multiply(0.5).sqrt();
             final T coeff = q0.multiply(normProduct).multiply(2.0).reciprocal();
             final FieldVector3D<T> q = FieldVector3D.crossProduct(v, u);
             q1 = coeff.multiply(q.getX());
             q2 = coeff.multiply(q.getY());
             q3 = coeff.multiply(q.getZ());
         }
 
     }
 
     /** Build a rotation from three Cardan or Euler elementary rotations.
 
      * <p>Cardan rotations are three successive rotations around the
      * canonical axes X, Y and Z, each axis being used once. There are
      * 6 such sets of rotations (XYZ, XZY, YXZ, YZX, ZXY and ZYX). Euler
      * rotations are three successive rotations around the canonical
      * axes X, Y and Z, the first and last rotations being around the
      * same axis. There are 6 such sets of rotations (XYX, XZX, YXY,
      * YZY, ZXZ and ZYZ), the most popular one being ZXZ.</p>
      * <p>Beware that many people routinely use the term Euler angles even
      * for what really are Cardan angles (this confusion is especially
      * widespread in the aerospace business where Roll, Pitch and Yaw angles
      * are often wrongly tagged as Euler angles).</p>
 
      * @param order order of rotations to compose, from left to right
      * (i.e. we will use {@code r1.compose(r2.compose(r3, convention), convention)})
      * @param convention convention to use for the semantics of the angle
      * @param alpha1 angle of the first elementary rotation
      * @param alpha2 angle of the second elementary rotation
      * @param alpha3 angle of the third elementary rotation
      */
     public FieldRotation(final RotationOrder order, final RotationConvention convention,
                          final T alpha1, final T alpha2, final T alpha3) {
         final Field<T> field = alpha1.getField();
         final FieldRotation<T> r1 = new FieldRotation<>(new FieldVector3D<>(field, order.getA1()), alpha1, convention);
         final FieldRotation<T> r2 = new FieldRotation<>(new FieldVector3D<>(field, order.getA2()), alpha2, convention);
         final FieldRotation<T> r3 = new FieldRotation<>(new FieldVector3D<>(field, order.getA3()), alpha3, convention);
         final FieldRotation<T> composed = r1.compose(r2.compose(r3, convention), convention);
         q0 = composed.q0;
         q1 = composed.q1;
         q2 = composed.q2;
         q3 = composed.q3;
     }
 
     /** Get identity rotation.
      * @param field field for the components
      * @return a new rotation
      * @param <T> the type of the field elements
      */
     public static <T extends CalculusFieldElement<T>> FieldRotation<T> getIdentity(final Field<T> field) {
         return new FieldRotation<>(field, Rotation.IDENTITY);
     }
 
     /** Convert an orthogonal rotation matrix to a quaternion.
      * @param ort orthogonal rotation matrix
      * @return quaternion corresponding to the matrix
      */
     private T[] mat2quat(final T[][] ort) {
 
         final T[] quat = MathArrays.buildArray(ort[0][0].getField(), 4);
 
         // There are different ways to compute the quaternions elements
         // from the matrix. They all involve computing one element from
         // the diagonal of the matrix, and computing the three other ones
         // using a formula involving a division by the first element,
         // which unfortunately can be zero. Since the norm of the
         // quaternion is 1, we know at least one element has an absolute
         // value greater or equal to 0.5, so it is always possible to
         // select the right formula and avoid division by zero and even
         // numerical inaccuracy. Checking the elements in turn and using
         // the first one greater than 0.45 is safe (this leads to a simple
         // test since qi = 0.45 implies 4 qi^2 - 1 = -0.19)
         T s = ort[0][0].add(ort[1][1]).add(ort[2][2]);
         if (s.getReal() > -0.19) {
             // compute q0 and deduce q1, q2 and q3
             quat[0] = s.add(1.0).sqrt().multiply(0.5);
             T inv = quat[0].reciprocal().multiply(0.25);
             quat[1] = inv.multiply(ort[1][2].subtract(ort[2][1]));
             quat[2] = inv.multiply(ort[2][0].subtract(ort[0][2]));
             quat[3] = inv.multiply(ort[0][1].subtract(ort[1][0]));
         } else {
             s = ort[0][0].subtract(ort[1][1]).subtract(ort[2][2]);
             if (s.getReal() > -0.19) {
                 // compute q1 and deduce q0, q2 and q3
                 quat[1] = s.add(1.0).sqrt().multiply(0.5);
                 T inv = quat[1].reciprocal().multiply(0.25);
                 quat[0] = inv.multiply(ort[1][2].subtract(ort[2][1]));
                 quat[2] = inv.multiply(ort[0][1].add(ort[1][0]));
                 quat[3] = inv.multiply(ort[0][2].add(ort[2][0]));
             } else {
                 s = ort[1][1].subtract(ort[0][0]).subtract(ort[2][2]);
                 if (s.getReal() > -0.19) {
                     // compute q2 and deduce q0, q1 and q3
                     quat[2] = s.add(1.0).sqrt().multiply(0.5);
                     T inv = quat[2].reciprocal().multiply(0.25);
                     quat[0] = inv.multiply(ort[2][0].subtract(ort[0][2]));
                     quat[1] = inv.multiply(ort[0][1].add(ort[1][0]));
                     quat[3] = inv.multiply(ort[2][1].add(ort[1][2]));
                 } else {
                     // compute q3 and deduce q0, q1 and q2
                     s = ort[2][2].subtract(ort[0][0]).subtract(ort[1][1]);
                     quat[3] = s.add(1.0).sqrt().multiply(0.5);
                     T inv = quat[3].reciprocal().multiply(0.25);
                     quat[0] = inv.multiply(ort[0][1].subtract(ort[1][0]));
                     quat[1] = inv.multiply(ort[0][2].add(ort[2][0]));
                     quat[2] = inv.multiply(ort[2][1].add(ort[1][2]));
                 }
             }
         }
 
         return quat;
 
     }
 
     /** Revert a rotation.
      * Build a rotation which reverse the effect of another
      * rotation. This means that if r(u) = v, then r.revert(v) = u. The
      * instance is not changed.
      * @return a new rotation whose effect is the reverse of the effect
      * of the instance
      */
     public FieldRotation<T> revert() {
         return new FieldRotation<>(q0.negate(), q1, q2, q3, false);
     }
 
     /** Get the scalar coordinate of the quaternion.
      * @return scalar coordinate of the quaternion
      */
     public T getQ0() {
         return q0;
     }
 
     /** Get the first coordinate of the vectorial part of the quaternion.
      * @return first coordinate of the vectorial part of the quaternion
      */
     public T getQ1() {
         return q1;
     }
 
     /** Get the second coordinate of the vectorial part of the quaternion.
      * @return second coordinate of the vectorial part of the quaternion
      */
     public T getQ2() {
         return q2;
     }
 
     /** Get the third coordinate of the vectorial part of the quaternion.
      * @return third coordinate of the vectorial part of the quaternion
      */
     public T getQ3() {
         return q3;
     }
 
     /** Get the normalized axis of the rotation.
      * <p>
      * Note that as {@link #getAngle()} always returns an angle
      * between 0 and &pi;, changing the convention changes the
      * direction of the axis, not the sign of the angle.
      * </p>
      * @param convention convention to use for the semantics of the angle
      * @return normalized axis of the rotation
      * @see #FieldRotation(FieldVector3D, CalculusFieldElement, RotationConvention)
      */
     public FieldVector3D<T> getAxis(final RotationConvention convention) {
         final T squaredSine = q1.square().add(q2.square()).add(q3.square());
         if (squaredSine.getReal() == 0) {
             final Field<T> field = squaredSine.getField();
             return new FieldVector3D<>(convention == RotationConvention.VECTOR_OPERATOR ? field.getOne(): field.getOne().negate(),
                                        field.getZero(),
                                        field.getZero());
         } else {
             final double sgn = convention == RotationConvention.VECTOR_OPERATOR ? +1 : -1;
             if (q0.getReal() < 0) {
                 T inverse = squaredSine.sqrt().reciprocal().multiply(sgn);
                 return new FieldVector3D<>(q1.multiply(inverse), q2.multiply(inverse), q3.multiply(inverse));
             }
             final T inverse = squaredSine.sqrt().reciprocal().negate().multiply(sgn);
             return new FieldVector3D<>(q1.multiply(inverse), q2.multiply(inverse), q3.multiply(inverse));
         }
     }
 
     /** Get the angle of the rotation.
      * @return angle of the rotation (between 0 and &pi;)
      * @see #FieldRotation(FieldVector3D, CalculusFieldElement, RotationConvention)
      */
     public T getAngle() {
         if ((q0.getReal() < -0.1) || (q0.getReal() > 0.1)) {
             return q1.square().add(q2.square()).add(q3.square()).sqrt().asin().multiply(2);
         } else if (q0.getReal() < 0) {
             return q0.negate().acos().multiply(2);
         }
         return q0.acos().multiply(2);
     }
 
     /** Get the Cardan or Euler angles corresponding to the instance.
 
      * <p>The equations show that each rotation can be defined by two
      * different values of the Cardan or Euler angles set. For example
      * if Cardan angles are used, the rotation defined by the angles
      * a<sub>1</sub>, a<sub>2</sub> and a<sub>3</sub> is the same as
      * the rotation defined by the angles &pi; + a<sub>1</sub>, &pi;
      * - a<sub>2</sub> and &pi; + a<sub>3</sub>. This method implements
      * the following arbitrary choices:</p>
      * <ul>
      *   <li>for Cardan angles, the chosen set is the one for which the
      *   second angle is between -&pi;/2 and &pi;/2 (i.e its cosine is
      *   positive),</li>
      *   <li>for Euler angles, the chosen set is the one for which the
      *   second angle is between 0 and &pi; (i.e its sine is positive).</li>
      * </ul>
 
      * <p>Cardan and Euler angle have a very disappointing drawback: all
      * of them have singularities. This means that if the instance is
      * too close to the singularities corresponding to the given
      * rotation order, it will be impossible to retrieve the angles. For
      * Cardan angles, this is often called gimbal lock. There is
      * <em>nothing</em> to do to prevent this, it is an intrinsic problem
      * with Cardan and Euler representation (but not a problem with the
      * rotation itself, which is perfectly well defined). For Cardan
      * angles, singularities occur when the second angle is close to
      * -&pi;/2 or +&pi;/2, for Euler angle singularities occur when the
      * second angle is close to 0 or &pi;, this implies that the identity
      * rotation is always singular for Euler angles!</p>
 
      * @param order rotation order to use
      * @param convention convention to use for the semantics of the angle
      * @return an array of three angles, in the order specified by the set
      */
     public T[] getAngles(final RotationOrder order, RotationConvention convention) {
         return order.getAngles(this, convention);
     }
 
     /** Get the 3X3 matrix corresponding to the instance
      * @return the matrix corresponding to the instance
      */
     public T[][] getMatrix() {
 
         // products
         final T q0q0  = q0.square();
         final T q0q1  = q0.multiply(q1);
         final T q0q2  = q0.multiply(q2);
         final T q0q3  = q0.multiply(q3);
         final T q1q1  = q1.square();
         final T q1q2  = q1.multiply(q2);
         final T q1q3  = q1.multiply(q3);
         final T q2q2  = q2.square();
         final T q2q3  = q2.multiply(q3);
         final T q3q3  = q3.square();
 
         // create the matrix
         final T[][] m = MathArrays.buildArray(q0.getField(), 3, 3);
 
         m [0][0] = q0q0.add(q1q1).multiply(2).subtract(1);
         m [1][0] = q1q2.subtract(q0q3).multiply(2);
         m [2][0] = q1q3.add(q0q2).multiply(2);
 
         m [0][1] = q1q2.add(q0q3).multiply(2);
         m [1][1] = q0q0.add(q2q2).multiply(2).subtract(1);
         m [2][1] = q2q3.subtract(q0q1).multiply(2);
 
         m [0][2] = q1q3.subtract(q0q2).multiply(2);
         m [1][2] = q2q3.add(q0q1).multiply(2);
         m [2][2] = q0q0.add(q3q3).multiply(2).subtract(1);
 
         return m;
 
     }
 
     /** Convert to a constant vector without derivatives.
      * @return a constant vector
      */
     public Rotation toRotation() {
         return new Rotation(q0.getReal(), q1.getReal(), q2.getReal(), q3.getReal(), false);
     }
 
     /** Apply the rotation to a vector.
      * @param u vector to apply the rotation to
      * @return a new vector which is the image of u by the rotation
      */
     public FieldVector3D<T> applyTo(final FieldVector3D<T> u) {
 
         final T x = u.getX();
         final T y = u.getY();
         final T z = u.getZ();
 
         final T s = q1.multiply(x).add(q2.multiply(y)).add(q3.multiply(z));
 
         return new FieldVector3D<>(q0.multiply(x.multiply(q0).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x),
                                    q0.multiply(y.multiply(q0).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y),
                                    q0.multiply(z.multiply(q0).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z));
 
     }
 
     /** Apply the rotation to a vector.
      * @param u vector to apply the rotation to
      * @return a new vector which is the image of u by the rotation
      */
     public FieldVector3D<T> applyTo(final Vector3D u) {
 
         final double x = u.getX();
         final double y = u.getY();
         final double z = u.getZ();
 
         final T s = q1.multiply(x).add(q2.multiply(y)).add(q3.multiply(z));
 
         return new FieldVector3D<>(q0.multiply(q0.multiply(x).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x),
                                    q0.multiply(q0.multiply(y).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y),
                                    q0.multiply(q0.multiply(z).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z));
 
     }
 
     /** Apply the rotation to a vector stored in an array.
      * @param in an array with three items which stores vector to rotate
      * @param out an array with three items to put result to (it can be the same
      * array as in)
      */
     public void applyTo(final T[] in, final T[] out) {
 
         final T x = in[0];
         final T y = in[1];
         final T z = in[2];
 
         final T s = q1.multiply(x).add(q2.multiply(y)).add(q3.multiply(z));
 
         out[0] = q0.multiply(x.multiply(q0).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x);
         out[1] = q0.multiply(y.multiply(q0).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y);
         out[2] = q0.multiply(z.multiply(q0).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z);
 
     }
 
     /** Apply the rotation to a vector stored in an array.
      * @param in an array with three items which stores vector to rotate
      * @param out an array with three items to put result to
      */
     public void applyTo(final double[] in, final T[] out) {
 
         final double x = in[0];
         final double y = in[1];
         final double z = in[2];
 
         final T s = q1.multiply(x).add(q2.multiply(y)).add(q3.multiply(z));
 
         out[0] = q0.multiply(q0.multiply(x).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x);
         out[1] = q0.multiply(q0.multiply(y).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y);
         out[2] = q0.multiply(q0.multiply(z).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z);
 
     }
 
     /** Apply a rotation to a vector.
      * @param r rotation to apply
      * @param u vector to apply the rotation to
      * @param <T> the type of the field elements
      * @return a new vector which is the image of u by the rotation
      */
     public static <T extends CalculusFieldElement<T>> FieldVector3D<T> applyTo(final Rotation r, final FieldVector3D<T> u) {
 
         final T x = u.getX();
         final T y = u.getY();
         final T z = u.getZ();
 
         final T s = x.multiply(r.getQ1()).add(y.multiply(r.getQ2())).add(z.multiply(r.getQ3()));
 
         return new FieldVector3D<>(x.multiply(r.getQ0()).subtract(z.multiply(r.getQ2()).subtract(y.multiply(r.getQ3()))).multiply(r.getQ0()).add(s.multiply(r.getQ1())).multiply(2).subtract(x),
                                    y.multiply(r.getQ0()).subtract(x.multiply(r.getQ3()).subtract(z.multiply(r.getQ1()))).multiply(r.getQ0()).add(s.multiply(r.getQ2())).multiply(2).subtract(y),
                                    z.multiply(r.getQ0()).subtract(y.multiply(r.getQ1()).subtract(x.multiply(r.getQ2()))).multiply(r.getQ0()).add(s.multiply(r.getQ3())).multiply(2).subtract(z));
 
     }
 
     /** Apply the inverse of the rotation to a vector.
      * @param u vector to apply the inverse of the rotation to
      * @return a new vector which such that u is its image by the rotation
      */
     public FieldVector3D<T> applyInverseTo(final FieldVector3D<T> u) {
 
         final T x = u.getX();
         final T y = u.getY();
         final T z = u.getZ();
 
         final T s  = q1.multiply(x).add(q2.multiply(y)).add(q3.multiply(z));
         final T m0 = q0.negate();
 
         return new FieldVector3D<>(m0.multiply(x.multiply(m0).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x),
                                    m0.multiply(y.multiply(m0).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y),
                                    m0.multiply(z.multiply(m0).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z));
 
     }
 
     /** Apply the inverse of the rotation to a vector.
      * @param u vector to apply the inverse of the rotation to
      * @return a new vector which such that u is its image by the rotation
      */
     public FieldVector3D<T> applyInverseTo(final Vector3D u) {
 
         final double x = u.getX();
         final double y = u.getY();
         final double z = u.getZ();
 
         final T s  = q1.multiply(x).add(q2.multiply(y)).add(q3.multiply(z));
         final T m0 = q0.negate();
 
         return new FieldVector3D<>(m0.multiply(m0.multiply(x).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x),
                                    m0.multiply(m0.multiply(y).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y),
                                    m0.multiply(m0.multiply(z).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z));
 
     }
 
     /** Apply the inverse of the rotation to a vector stored in an array.
      * @param in an array with three items which stores vector to rotate
      * @param out an array with three items to put result to (it can be the same
      * array as in)
      */
     public void applyInverseTo(final T[] in, final T[] out) {
 
         final T x = in[0];
         final T y = in[1];
         final T z = in[2];
 
         final T s = q1.multiply(x).add(q2.multiply(y)).add(q3.multiply(z));
         final T m0 = q0.negate();
 
         out[0] = m0.multiply(x.multiply(m0).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x);
         out[1] = m0.multiply(y.multiply(m0).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y);
         out[2] = m0.multiply(z.multiply(m0).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z);
 
     }
 
     /** Apply the inverse of the rotation to a vector stored in an array.
      * @param in an array with three items which stores vector to rotate
      * @param out an array with three items to put result to
      */
     public void applyInverseTo(final double[] in, final T[] out) {
 
         final double x = in[0];
         final double y = in[1];
         final double z = in[2];
 
         final T s = q1.multiply(x).add(q2.multiply(y)).add(q3.multiply(z));
         final T m0 = q0.negate();
 
         out[0] = m0.multiply(m0.multiply(x).subtract(q2.multiply(z).subtract(q3.multiply(y)))).add(s.multiply(q1)).multiply(2).subtract(x);
         out[1] = m0.multiply(m0.multiply(y).subtract(q3.multiply(x).subtract(q1.multiply(z)))).add(s.multiply(q2)).multiply(2).subtract(y);
         out[2] = m0.multiply(m0.multiply(z).subtract(q1.multiply(y).subtract(q2.multiply(x)))).add(s.multiply(q3)).multiply(2).subtract(z);
 
     }
 
     /** Apply the inverse of a rotation to a vector.
      * @param r rotation to apply
      * @param u vector to apply the inverse of the rotation to
      * @param <T> the type of the field elements
      * @return a new vector which such that u is its image by the rotation
      */
     public static <T extends CalculusFieldElement<T>> FieldVector3D<T> applyInverseTo(final Rotation r, final FieldVector3D<T> u) {
 
         final T x = u.getX();
         final T y = u.getY();
         final T z = u.getZ();
 
         final T s  = x.multiply(r.getQ1()).add(y.multiply(r.getQ2())).add(z.multiply(r.getQ3()));
         final double m0 = -r.getQ0();
 
         return new FieldVector3D<>(x.multiply(m0).subtract(z.multiply(r.getQ2()).subtract(y.multiply(r.getQ3()))).multiply(m0).add(s.multiply(r.getQ1())).multiply(2).subtract(x),
                                    y.multiply(m0).subtract(x.multiply(r.getQ3()).subtract(z.multiply(r.getQ1()))).multiply(m0).add(s.multiply(r.getQ2())).multiply(2).subtract(y),
                                    z.multiply(m0).subtract(y.multiply(r.getQ1()).subtract(x.multiply(r.getQ2()))).multiply(m0).add(s.multiply(r.getQ3())).multiply(2).subtract(z));
 
     }
 
     /** Apply the instance to another rotation.
      * <p>
      * Calling this method is equivalent to call
      * {@link #compose(FieldRotation, RotationConvention)
      * compose(r, RotationConvention.VECTOR_OPERATOR)}.
      * </p>
      * @param r rotation to apply the rotation to
      * @return a new rotation which is the composition of r by the instance
      */
     public FieldRotation<T> applyTo(final FieldRotation<T> r) {
         return compose(r, RotationConvention.VECTOR_OPERATOR);
     }
 
     /** Compose the instance with another rotation.
      * <p>
      * If the semantics of the rotations composition corresponds to a
      * {@link RotationConvention#VECTOR_OPERATOR vector operator} convention,
      * applying the instance to a rotation is computing the composition
      * in an order compliant with the following rule : let {@code u} be any
      * vector and {@code v} its image by {@code r1} (i.e.
      * {@code r1.applyTo(u) = v}). Let {@code w} be the image of {@code v} by
      * rotation {@code r2} (i.e. {@code r2.applyTo(v) = w}). Then
      * {@code w = comp.applyTo(u)}, where
      * {@code comp = r2.compose(r1, RotationConvention.VECTOR_OPERATOR)}.
      * </p>
      * <p>
      * If the semantics of the rotations composition corresponds to a
      * {@link RotationConvention#FRAME_TRANSFORM frame transform} convention,
      * the application order will be reversed. So keeping the exact same
      * meaning of all {@code r1}, {@code r2}, {@code u}, {@code v}, {@code w}
      * and  {@code comp} as above, {@code comp} could also be computed as
      * {@code comp = r1.compose(r2, RotationConvention.FRAME_TRANSFORM)}.
      * </p>
      * @param r rotation to apply the rotation to
      * @param convention convention to use for the semantics of the angle
      * @return a new rotation which is the composition of r by the instance
      */
     public FieldRotation<T> compose(final FieldRotation<T> r, final RotationConvention convention) {
         return convention == RotationConvention.VECTOR_OPERATOR ?
                              composeInternal(r) : r.composeInternal(this);
     }
 
     /** Compose the instance with another rotation using vector operator convention.
      * @param r rotation to apply the rotation to
      * @return a new rotation which is the composition of r by the instance
      * using vector operator convention
      */
     private FieldRotation<T> composeInternal(final FieldRotation<T> r) {
         return new FieldRotation<>(r.q0.multiply(q0).subtract(r.q1.multiply(q1).add(r.q2.multiply(q2)).add(r.q3.multiply(q3))),
                                    r.q1.multiply(q0).add(r.q0.multiply(q1)).add(r.q2.multiply(q3).subtract(r.q3.multiply(q2))),
                                    r.q2.multiply(q0).add(r.q0.multiply(q2)).add(r.q3.multiply(q1).subtract(r.q1.multiply(q3))),
                                    r.q3.multiply(q0).add(r.q0.multiply(q3)).add(r.q1.multiply(q2).subtract(r.q2.multiply(q1))),
                                    false);
     }
 
     /** Apply the instance to another rotation.
      * <p>
      * Calling this method is equivalent to call
      * {@link #compose(Rotation, RotationConvention)
      * compose(r, RotationConvention.VECTOR_OPERATOR)}.
      * </p>
      * @param r rotation to apply the rotation to
      * @return a new rotation which is the composition of r by the instance
      */
     public FieldRotation<T> applyTo(final Rotation r) {
         return compose(r, RotationConvention.VECTOR_OPERATOR);
     }
 
     /** Compose the instance with another rotation.
      * <p>
      * If the semantics of the rotations composition corresponds to a
      * {@link RotationConvention#VECTOR_OPERATOR vector operator} convention,
      * applying the instance to a rotation is computing the composition
      * in an order compliant with the following rule : let {@code u} be any
      * vector and {@code v} its image by {@code r1} (i.e.
      * {@code r1.applyTo(u) = v}). Let {@code w} be the image of {@code v} by
      * rotation {@code r2} (i.e. {@code r2.applyTo(v) = w}). Then
      * {@code w = comp.applyTo(u)}, where
      * {@code comp = r2.compose(r1, RotationConvention.VECTOR_OPERATOR)}.
      * </p>
      * <p>
      * If the semantics of the rotations composition corresponds to a
      * {@link RotationConvention#FRAME_TRANSFORM frame transform} convention,
      * the application order will be reversed. So keeping the exact same
      * meaning of all {@code r1}, {@code r2}, {@code u}, {@code v}, {@code w}
      * and  {@code comp} as above, {@code comp} could also be computed as
      * {@code comp = r1.compose(r2, RotationConvention.FRAME_TRANSFORM)}.
      * </p>
      * @param r rotation to apply the rotation to
      * @param convention convention to use for the semantics of the angle
      * @return a new rotation which is the composition of r by the instance
      */
     public FieldRotation<T> compose(final Rotation r, final RotationConvention convention) {
         return convention == RotationConvention.VECTOR_OPERATOR ?
                              composeInternal(r) : applyTo(r, this);
     }
 
     /** Compose the instance with another rotation using vector operator convention.
      * @param r rotation to apply the rotation to
      * @return a new rotation which is the composition of r by the instance
      * using vector operator convention
      */
     private FieldRotation<T> composeInternal(final Rotation r) {
         return new FieldRotation<>(q0.multiply(r.getQ0()).subtract(q1.multiply(r.getQ1()).add(q2.multiply(r.getQ2())).add(q3.multiply(r.getQ3()))),
                                    q0.multiply(r.getQ1()).add(q1.multiply(r.getQ0())).add(q3.multiply(r.getQ2()).subtract(q2.multiply(r.getQ3()))),
                                    q0.multiply(r.getQ2()).add(q2.multiply(r.getQ0())).add(q1.multiply(r.getQ3()).subtract(q3.multiply(r.getQ1()))),
                                    q0.multiply(r.getQ3()).add(q3.multiply(r.getQ0())).add(q2.multiply(r.getQ1()).subtract(q1.multiply(r.getQ2()))),
                                    false);
     }
 
     /** Apply a rotation to another rotation.
      * Applying a rotation to another rotation is computing the composition
      * in an order compliant with the following rule : let u be any
      * vector and v its image by rInner (i.e. rInner.applyTo(u) = v), let w be the image
      * of v by rOuter (i.e. rOuter.applyTo(v) = w), then w = comp.applyTo(u),
      * where comp = applyTo(rOuter, rInner).
      * @param r1 rotation to apply
      * @param rInner rotation to apply the rotation to
      * @param <T> the type of the field elements
      * @return a new rotation which is the composition of r by the instance
      */
     public static <T extends CalculusFieldElement<T>> FieldRotation<T> applyTo(final Rotation r1, final FieldRotation<T> rInner) {
         return new FieldRotation<>(rInner.q0.multiply(r1.getQ0()).subtract(rInner.q1.multiply(r1.getQ1()).add(rInner.q2.multiply(r1.getQ2())).add(rInner.q3.multiply(r1.getQ3()))),
                                    rInner.q1.multiply(r1.getQ0()).add(rInner.q0.multiply(r1.getQ1())).add(rInner.q2.multiply(r1.getQ3()).subtract(rInner.q3.multiply(r1.getQ2()))),
                                    rInner.q2.multiply(r1.getQ0()).add(rInner.q0.multiply(r1.getQ2())).add(rInner.q3.multiply(r1.getQ1()).subtract(rInner.q1.multiply(r1.getQ3()))),
                                    rInner.q3.multiply(r1.getQ0()).add(rInner.q0.multiply(r1.getQ3())).add(rInner.q1.multiply(r1.getQ2()).subtract(rInner.q2.multiply(r1.getQ1()))),
                                    false);
     }
 
     /** Apply the inverse of the instance to another rotation.
      * <p>
      * Calling this method is equivalent to call
      * {@link #composeInverse(FieldRotation, RotationConvention)
      * composeInverse(r, RotationConvention.VECTOR_OPERATOR)}.
      * </p>
      * @param r rotation to apply the rotation to
      * @return a new rotation which is the composition of r by the inverse
      * of the instance
      */
     public FieldRotation<T> applyInverseTo(final FieldRotation<T> r) {
         return composeInverse(r, RotationConvention.VECTOR_OPERATOR);
     }
 
     /** Compose the inverse of the instance with another rotation.
      * <p>
      * If the semantics of the rotations composition corresponds to a
      * {@link RotationConvention#VECTOR_OPERATOR vector operator} convention,
      * applying the inverse of the instance to a rotation is computing
      * the composition in an order compliant with the following rule :
      * let {@code u} be any vector and {@code v} its image by {@code r1}
      * (i.e. {@code r1.applyTo(u) = v}). Let {@code w} be the inverse image
      * of {@code v} by {@code r2} (i.e. {@code r2.applyInverseTo(v) = w}).
      * Then {@code w = comp.applyTo(u)}, where
      * {@code comp = r2.composeInverse(r1)}.
      * </p>
      * <p>
      * If the semantics of the rotations composition corresponds to a
      * {@link RotationConvention#FRAME_TRANSFORM frame transform} convention,
      * the application order will be reversed, which means it is the
      * <em>innermost</em> rotation that will be reversed. So keeping the exact same
      * meaning of all {@code r1}, {@code r2}, {@code u}, {@code v}, {@code w}
      * and  {@code comp} as above, {@code comp} could also be computed as
      * {@code comp = r1.revert().composeInverse(r2.revert(), RotationConvention.FRAME_TRANSFORM)}.
      * </p>
      * @param r rotation to apply the rotation to
      * @param convention convention to use for the semantics of the angle
      * @return a new rotation which is the composition of r by the inverse
      * of the instance
      */
     public FieldRotation<T> composeInverse(final FieldRotation<T> r, final RotationConvention convention) {
         return convention == RotationConvention.VECTOR_OPERATOR ?
                              composeInverseInternal(r) : r.composeInternal(revert());
     }
 
     /** Compose the inverse of the instance with another rotation
      * using vector operator convention.
      * @param r rotation to apply the rotation to
      * @return a new rotation which is the composition of r by the inverse
      * of the instance using vector operator convention
      */
     private FieldRotation<T> composeInverseInternal(FieldRotation<T> r) {
         return new FieldRotation<>(r.q0.multiply(q0).add(r.q1.multiply(q1)).add(r.q2.multiply(q2)).add(r.q3.multiply(q3)).negate(),
                                    r.q0.multiply(q1).add(r.q2.multiply(q3).subtract(r.q3.multiply(q2))).subtract(r.q1.multiply(q0)),
                                    r.q0.multiply(q2).add(r.q3.multiply(q1).subtract(r.q1.multiply(q3))).subtract(r.q2.multiply(q0)),
                                    r.q0.multiply(q3).add(r.q1.multiply(q2).subtract(r.q2.multiply(q1))).subtract(r.q3.multiply(q0)),
                                    false);
     }
 
     /** Apply the inverse of the instance to another rotation.
      * <p>
      * Calling this method is equivalent to call
      * {@link #composeInverse(Rotation, RotationConvention)
      * composeInverse(r, RotationConvention.VECTOR_OPERATOR)}.
      * </p>
      * @param r rotation to apply the rotation to
      * @return a new rotation which is the composition of r by the inverse
      * of the instance
      */
     public FieldRotation<T> applyInverseTo(final Rotation r) {
         return composeInverse(r, RotationConvention.VECTOR_OPERATOR);
     }
 
     /** Compose the inverse of the instance with another rotation.
      * <p>
      * If the semantics of the rotations composition corresponds to a
      * {@link RotationConvention#VECTOR_OPERATOR vector operator} convention,
      * applying the inverse of the instance to a rotation is computing
      * the composition in an order compliant with the following rule :
      * let {@code u} be any vector and {@code v} its image by {@code r1}
      * (i.e. {@code r1.applyTo(u) = v}). Let {@code w} be the inverse image
      * of {@code v} by {@code r2} (i.e. {@code r2.applyInverseTo(v) = w}).
      * Then {@code w = comp.applyTo(u)}, where
      * {@code comp = r2.composeInverse(r1)}.
      * </p>
      * <p>
      * If the semantics of the rotations composition corresponds to a
      * {@link RotationConvention#FRAME_TRANSFORM frame transform} convention,
      * the application order will be reversed, which means it is the
      * <em>innermost</em> rotation that will be reversed. So keeping the exact same
      * meaning of all {@code r1}, {@code r2}, {@code u}, {@code v}, {@code w}
      * and  {@code comp} as above, {@code comp} could also be computed as
      * {@code comp = r1.revert().composeInverse(r2.revert(), RotationConvention.FRAME_TRANSFORM)}.
      * </p>
      * @param r rotation to apply the rotation to
      * @param convention convention to use for the semantics of the angle
      * @return a new rotation which is the composition of r by the inverse
      * of the instance
      */
     public FieldRotation<T> composeInverse(final Rotation r, final RotationConvention convention) {
         return convention == RotationConvention.VECTOR_OPERATOR ?
                              composeInverseInternal(r) : applyTo(r, revert());
     }
 
     /** Compose the inverse of the instance with another rotation
      * using vector operator convention.
      * @param r rotation to apply the rotation to
      * @return a new rotation which is the composition of r by the inverse
      * of the instance using vector operator convention
      */
     private FieldRotation<T> composeInverseInternal(Rotation r) {
         return new FieldRotation<>(q0.multiply(r.getQ0()).add(q1.multiply(r.getQ1()).add(q2.multiply(r.getQ2())).add(q3.multiply(r.getQ3()))).negate(),
                                    q1.multiply(r.getQ0()).add(q3.multiply(r.getQ2()).subtract(q2.multiply(r.getQ3()))).subtract(q0.multiply(r.getQ1())),
                                    q2.multiply(r.getQ0()).add(q1.multiply(r.getQ3()).subtract(q3.multiply(r.getQ1()))).subtract(q0.multiply(r.getQ2())),
                                    q3.multiply(r.getQ0()).add(q2.multiply(r.getQ1()).subtract(q1.multiply(r.getQ2()))).subtract(q0.multiply(r.getQ3())),
                                    false);
     }
 
     /** Apply the inverse of a rotation to another rotation.
      * Applying the inverse of a rotation to another rotation is computing
      * the composition in an order compliant with the following rule :
      * let u be any vector and v its image by rInner (i.e. rInner.applyTo(u) = v),
      * let w be the inverse image of v by rOuter
      * (i.e. rOuter.applyInverseTo(v) = w), then w = comp.applyTo(u), where
      * comp = applyInverseTo(rOuter, rInner).
      * @param rOuter rotation to apply the rotation to
      * @param rInner rotation to apply the rotation to
      * @param <T> the type of the field elements
      * @return a new rotation which is the composition of r by the inverse
      * of the instance
      */
     public static <T extends CalculusFieldElement<T>> FieldRotation<T> applyInverseTo(final Rotation rOuter, final FieldRotation<T> rInner) {
         return new FieldRotation<>(rInner.q0.multiply(rOuter.getQ0()).add(rInner.q1.multiply(rOuter.getQ1()).add(rInner.q2.multiply(rOuter.getQ2())).add(rInner.q3.multiply(rOuter.getQ3()))).negate(),
                                    rInner.q0.multiply(rOuter.getQ1()).add(rInner.q2.multiply(rOuter.getQ3()).subtract(rInner.q3.multiply(rOuter.getQ2()))).subtract(rInner.q1.multiply(rOuter.getQ0())),
                                    rInner.q0.multiply(rOuter.getQ2()).add(rInner.q3.multiply(rOuter.getQ1()).subtract(rInner.q1.multiply(rOuter.getQ3()))).subtract(rInner.q2.multiply(rOuter.getQ0())),
                                    rInner.q0.multiply(rOuter.getQ3()).add(rInner.q1.multiply(rOuter.getQ2()).subtract(rInner.q2.multiply(rOuter.getQ1()))).subtract(rInner.q3.multiply(rOuter.getQ0())),
                                    false);
     }
 
     /** Perfect orthogonality on a 3X3 matrix.
      * @param m initial matrix (not exactly orthogonal)
      * @param threshold convergence threshold for the iterative
      * orthogonality correction (convergence is reached when the
      * difference between two steps of the Frobenius norm of the
      * correction is below this threshold)
      * @return an orthogonal matrix close to m
      * @exception MathIllegalArgumentException if the matrix cannot be
      * orthogonalized with the given threshold after 10 iterations
      */
     private T[][] orthogonalizeMatrix(final T[][] m, final double threshold)
         throws MathIllegalArgumentException {
 
         T x00 = m[0][0];
         T x01 = m[0][1];
         T x02 = m[0][2];
         T x10 = m[1][0];
         T x11 = m[1][1];
         T x12 = m[1][2];
         T x20 = m[2][0];
         T x21 = m[2][1];
         T x22 = m[2][2];
         double fn = 0;
         double fn1;
 
         final T[][] o = MathArrays.buildArray(m[0][0].getField(), 3, 3);
 
         // iterative correction: Xn+1 = Xn - 0.5 * (Xn.Mt.Xn - M)
         int i;
         for (i = 0; i < 11; ++i) {
 
             // Mt.Xn
             final T mx00 = m[0][0].multiply(x00).add(m[1][0].multiply(x10)).add(m[2][0].multiply(x20));
             final T mx10 = m[0][1].multiply(x00).add(m[1][1].multiply(x10)).add(m[2][1].multiply(x20));
             final T mx20 = m[0][2].multiply(x00).add(m[1][2].multiply(x10)).add(m[2][2].multiply(x20));
             final T mx01 = m[0][0].multiply(x01).add(m[1][0].multiply(x11)).add(m[2][0].multiply(x21));
             final T mx11 = m[0][1].multiply(x01).add(m[1][1].multiply(x11)).add(m[2][1].multiply(x21));
             final T mx21 = m[0][2].multiply(x01).add(m[1][2].multiply(x11)).add(m[2][2].multiply(x21));
             final T mx02 = m[0][0].multiply(x02).add(m[1][0].multiply(x12)).add(m[2][0].multiply(x22));
             final T mx12 = m[0][1].multiply(x02).add(m[1][1].multiply(x12)).add(m[2][1].multiply(x22));
             final T mx22 = m[0][2].multiply(x02).add(m[1][2].multiply(x12)).add(m[2][2].multiply(x22));
 
             // Xn+1
             o[0][0] = x00.subtract(x00.multiply(mx00).add(x01.multiply(mx10)).add(x02.multiply(mx20)).subtract(m[0][0]).multiply(0.5));
             o[0][1] = x01.subtract(x00.multiply(mx01).add(x01.multiply(mx11)).add(x02.multiply(mx21)).subtract(m[0][1]).multiply(0.5));
             o[0][2] = x02.subtract(x00.multiply(mx02).add(x01.multiply(mx12)).add(x02.multiply(mx22)).subtract(m[0][2]).multiply(0.5));
             o[1][0] = x10.subtract(x10.multiply(mx00).add(x11.multiply(mx10)).add(x12.multiply(mx20)).subtract(m[1][0]).multiply(0.5));
             o[1][1] = x11.subtract(x10.multiply(mx01).add(x11.multiply(mx11)).add(x12.multiply(mx21)).subtract(m[1][1]).multiply(0.5));
             o[1][2] = x12.subtract(x10.multiply(mx02).add(x11.multiply(mx12)).add(x12.multiply(mx22)).subtract(m[1][2]).multiply(0.5));
             o[2][0] = x20.subtract(x20.multiply(mx00).add(x21.multiply(mx10)).add(x22.multiply(mx20)).subtract(m[2][0]).multiply(0.5));
             o[2][1] = x21.subtract(x20.multiply(mx01).add(x21.multiply(mx11)).add(x22.multiply(mx21)).subtract(m[2][1]).multiply(0.5));
             o[2][2] = x22.subtract(x20.multiply(mx02).add(x21.multiply(mx12)).add(x22.multiply(mx22)).subtract(m[2][2]).multiply(0.5));
 
             // correction on each elements
             final double corr00 = o[0][0].getReal() - m[0][0].getReal();
             final double corr01 = o[0][1].getReal() - m[0][1].getReal();
             final double corr02 = o[0][2].getReal() - m[0][2].getReal();
             final double corr10 = o[1][0].getReal() - m[1][0].getReal();
             final double corr11 = o[1][1].getReal() - m[1][1].getReal();
             final double corr12 = o[1][2].getReal() - m[1][2].getReal();
             final double corr20 = o[2][0].getReal() - m[2][0].getReal();
             final double corr21 = o[2][1].getReal() - m[2][1].getReal();
             final double corr22 = o[2][2].getReal() - m[2][2].getReal();
 
             // Frobenius norm of the correction
             fn1 = corr00 * corr00 + corr01 * corr01 + corr02 * corr02 +
                   corr10 * corr10 + corr11 * corr11 + corr12 * corr12 +
                   corr20 * corr20 + corr21 * corr21 + corr22 * corr22;
 
             // convergence test
             if (FastMath.abs(fn1 - fn) <= threshold) {
                 return o;
             }
 
             // prepare next iteration
             x00 = o[0][0];
             x01 = o[0][1];
             x02 = o[0][2];
             x10 = o[1][0];
             x11 = o[1][1];
             x12 = o[1][2];
             x20 = o[2][0];
             x21 = o[2][1];
             x22 = o[2][2];
             fn  = fn1;
 
         }
 
         // the algorithm did not converge after 10 iterations
         throw new MathIllegalArgumentException(LocalizedGeometryFormats.UNABLE_TO_ORTHOGONOLIZE_MATRIX,
                                                i - 1);
 
     }
 
     /** Compute the <i>distance</i> between two rotations.
      * <p>The <i>distance</i> is intended here as a way to check if two
      * rotations are almost similar (i.e. they transform vectors the same way)
      * or very different. It is mathematically defined as the angle of
      * the rotation r that prepended to one of the rotations gives the other
      * one: \(r_1(r) = r_2\)
      * </p>
      * <p>This distance is an angle between 0 and &pi;. Its value is the smallest
      * possible upper bound of the angle in radians between r<sub>1</sub>(v)
      * and r<sub>2</sub>(v) for all possible vectors v. This upper bound is
      * reached for some v. The distance is equal to 0 if and only if the two
      * rotations are identical.</p>
      * <p>Comparing two rotations should always be done using this value rather
      * than for example comparing the components of the quaternions. It is much
      * more stable, and has a geometric meaning. Also comparing quaternions
      * components is error prone since for example quaternions (0.36, 0.48, -0.48, -0.64)
      * and (-0.36, -0.48, 0.48, 0.64) represent exactly the same rotation despite
      * their components are different (they are exact opposites).</p>
      * @param r1 first rotation
      * @param r2 second rotation
      * @param <T> the type of the field elements
      * @return <i>distance</i> between r1 and r2
      */
     public static <T extends CalculusFieldElement<T>> T distance(final FieldRotation<T> r1, final FieldRotation<T> r2) {
         return r1.composeInverseInternal(r2).getAngle();
     }
 
 }