/*
 * 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
 *
 *      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.apache.commons.math.geometry;

import java.io.Serializable;

import org.apache.commons.math.MathRuntimeException;

/**
 * This class implements rotations in a three-dimensional space.
 *
 * <p>
 * Rotations can be represented by several different mathematical entities
 * (matrices, axe and angle, Cardan or Euler angles, quaternions). This class
 * presents an higher level abstraction, more user-oriented and hiding this
 * implementation details. Well, for the curious, we use quaternions for the
 * internal representation. The user can build a rotation from any of these
 * representations, and any of these representations can be retrieved from a
 * <code>Rotation</code> instance (see the various constructors and getters). In
 * addition, a rotation can also be built implicitely from a set of vectors and
 * their image.
 * </p>
 * <p>
 * This implies that this class can be used to convert from one representation
 * to another one. For example, converting a rotation matrix into a set of
 * Cardan angles from can be done using the followong single line of code:
 * </p>
 * 
 * <pre>
 * double[] angles = new Rotation(matrix, 1.0e-10).getAngles(RotationOrder.XYZ);
 * </pre>
 * <p>
 * Focus is oriented on what a rotation <em>do</em> rather than on its
 * underlying representation. Once it has been built, and regardless of its
 * internal representation, a rotation is an <em>operator</em> which basically
 * transforms three dimensional {@link Vector3D vectors} into other three
 * dimensional {@link Vector3D vectors}. Depending on the application, the
 * meaning of these vectors may vary and the semantics of the rotation also.
 * </p>
 * <p>
 * For example in an spacecraft attitude simulation tool, users will often
 * consider the vectors are fixed (say the Earth direction for example) and the
 * rotation transforms the coordinates coordinates of this vector in inertial
 * frame into the coordinates of the same vector in satellite frame. In this
 * case, the rotation implicitely defines the relation between the two frames.
 * Another example could be a telescope control application, where the rotation
 * would transform the sighting direction at rest into the desired observing
 * direction when the telescope is pointed towards an object of interest. In
 * this case the rotation transforms the directionf at rest in a topocentric
 * frame into the sighting direction in the same topocentric frame. In many
 * case, both approaches will be combined, in our telescope example, we will
 * probably also need to transform the observing direction in the topocentric
 * frame into the observing direction in inertial frame taking into account the
 * observatory location and the Earth rotation.
 * </p>
 *
 * <p>
 * These examples show that a rotation is what the user wants it to be, so this
 * class does not push the user towards one specific definition and hence does
 * not provide methods like <code>projectVectorIntoDestinationFrame</code> or
 * <code>computeTransformedDirection</code>. It provides simpler and more
 * generic methods: {@link #applyTo(Vector3D) applyTo(Vector3D)} and
 * {@link #applyInverseTo(Vector3D) applyInverseTo(Vector3D)}.
 * </p>
 *
 * <p>
 * Since a rotation is basically a vectorial operator, several rotations can be
 * composed together and the composite operation <code>r = r<sub>1</sub> o
 * r<sub>2</sub></code> (which means that for each vector <code>u</code>,
 * <code>r(u) = r<sub>1</sub>(r<sub>2</sub>(u))</code>) is also a rotation.
 * Hence we can consider that in addition to vectors, a rotation can be applied
 * to other rotations as well (or to itself). With our previous notations, we
 * would say we can apply <code>r<sub>1</sub></code> to
 * <code>r<sub>2</sub></code> and the result we get is
 * <code>r = r<sub>1</sub> o r<sub>2</sub></code>. For this purpose, the class
 * provides the methods: {@link #applyTo(Rotation) applyTo(Rotation)} and
 * {@link #applyInverseTo(Rotation) applyInverseTo(Rotation)}.
 * </p>
 *
 * <p>
 * Rotations are guaranteed to be immutable objects.
 * </p>
 *
 * @version $Revision$ $Date$
 * @see Vector3D
 * @see RotationOrder
 * @since 1.2
 */

public class Rotation implements Serializable {

	/** Identity rotation. */
	public static final Rotation IDENTITY = new Rotation(1.0, 0.0, 0.0, 0.0, false);

	/** Serializable version identifier */
	private static final long serialVersionUID = -2153622329907944313L;

	/** Scalar coordinate of the quaternion. */
	private final double q0;

	/** First coordinate of the vectorial part of the quaternion. */
	private final double q1;

	/** Second coordinate of the vectorial part of the quaternion. */
	private final double q2;

	/** Third coordinate of the vectorial part of the quaternion. */
	private final double 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>
	 * 
	 * @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 Rotation(double q0, double q1, double q2, double q3, boolean needsNormalization) {

		if (needsNormalization) {
			// normalization preprocessing
			double inv = 1.0 / Math.sqrt(q0 * q0 + q1 * q1 + q2 * q2 + q3 * q3);
			q0 *= inv;
			q1 *= inv;
			q2 *= inv;
			q3 *= inv;
		}

		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(Vector3D) apply} the
	 * instance to +i, we will get +j.
	 * </p>
	 * 
	 * @param axis
	 *            axis around which to rotate
	 * @param angle
	 *            rotation angle.
	 * @exception ArithmeticException
	 *                if the axis norm is zero
	 */
	public Rotation(Vector3D axis, double angle) {

		double norm = axis.getNorm();
		if (norm == 0) {
			throw MathRuntimeException.createArithmeticException("zero norm for rotation axis");
		}

		double halfAngle = -0.5 * angle;
		double coeff = Math.sin(halfAngle) / norm;

		q0 = Math.cos(halfAngle);
		q1 = coeff * axis.getX();
		q2 = coeff * axis.getY();
		q3 = coeff * axis.getZ();

	}

	/**
	 * 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 NotARotationMatrixException
	 *                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 Rotation(double[][] m, double threshold) throws NotARotationMatrixException {

		// dimension check
		if ((m.length != 3) || (m[0].length != 3) || (m[1].length != 3) || (m[2].length != 3)) {
			throw new NotARotationMatrixException("a {0}x{1} matrix cannot be a rotation matrix", m.length,
					m[0].length);
		}

		// compute a "close" orthogonal matrix
		double[][] ort = orthogonalizeMatrix(m, threshold);

		// check the sign of the determinant
		double det = ort[0][0] * (ort[1][1] * ort[2][2] - ort[2][1] * ort[1][2])
				- ort[1][0] * (ort[0][1] * ort[2][2] - ort[2][1] * ort[0][2])
				+ ort[2][0] * (ort[0][1] * ort[1][2] - ort[1][1] * ort[0][2]);
		if (det < 0.0) {
			throw new NotARotationMatrixException("the closest orthogonal matrix has a negative determinant {0}", det);
		}

		// 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)
		double s = ort[0][0] + ort[1][1] + ort[2][2];
		if (s > -0.19) {
			// compute q0 and deduce q1, q2 and q3
			q0 = 0.5 * Math.sqrt(s + 1.0);
			double inv = 0.25 / q0;
			q1 = inv * (ort[1][2] - ort[2][1]);
			q2 = inv * (ort[2][0] - ort[0][2]);
			q3 = inv * (ort[0][1] - ort[1][0]);
		} else {
			s = ort[0][0] - ort[1][1] - ort[2][2];
			if (s > -0.19) {
				// compute q1 and deduce q0, q2 and q3
				q1 = 0.5 * Math.sqrt(s + 1.0);
				double inv = 0.25 / q1;
				q0 = inv * (ort[1][2] - ort[2][1]);
				q2 = inv * (ort[0][1] + ort[1][0]);
				q3 = inv * (ort[0][2] + ort[2][0]);
			} else {
				s = ort[1][1] - ort[0][0] - ort[2][2];
				if (s > -0.19) {
					// compute q2 and deduce q0, q1 and q3
					q2 = 0.5 * Math.sqrt(s + 1.0);
					double inv = 0.25 / q2;
					q0 = inv * (ort[2][0] - ort[0][2]);
					q1 = inv * (ort[0][1] + ort[1][0]);
					q3 = inv * (ort[2][1] + ort[1][2]);
				} else {
					// compute q3 and deduce q0, q1 and q2
					s = ort[2][2] - ort[0][0] - ort[1][1];
					q3 = 0.5 * Math.sqrt(s + 1.0);
					double inv = 0.25 / q3;
					q0 = inv * (ort[0][1] - ort[1][0]);
					q1 = inv * (ort[0][2] + ort[2][0]);
					q2 = inv * (ort[2][1] + ort[1][2]);
				}
			}
		}

	}

	/**
	 * Build the rotation that transforms a pair of vector 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>) 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 IllegalArgumentException
	 *                if the norm of one of the vectors is zero
	 */
	public Rotation(Vector3D u1, Vector3D u2, Vector3D v1, Vector3D v2) {

		// norms computation
		double u1u1 = Vector3D.dotProduct(u1, u1);
		double u2u2 = Vector3D.dotProduct(u2, u2);
		double v1v1 = Vector3D.dotProduct(v1, v1);
		double v2v2 = Vector3D.dotProduct(v2, v2);
		if ((u1u1 == 0) || (u2u2 == 0) || (v1v1 == 0) || (v2v2 == 0)) {
			throw MathRuntimeException.createIllegalArgumentException("zero norm for rotation defining vector");
		}

		double u1x = u1.getX();
		double u1y = u1.getY();
		double u1z = u1.getZ();

		double u2x = u2.getX();
		double u2y = u2.getY();
		double u2z = u2.getZ();

		// normalize v1 in order to have (v1'|v1') = (u1|u1)
		double coeff = Math.sqrt(u1u1 / v1v1);
		double v1x = coeff * v1.getX();
		double v1y = coeff * v1.getY();
		double v1z = coeff * v1.getZ();
		v1 = new Vector3D(v1x, v1y, v1z);

		// adjust v2 in order to have (u1|u2) = (v1|v2) and (v2'|v2') = (u2|u2)
		double u1u2 = Vector3D.dotProduct(u1, u2);
		double v1v2 = Vector3D.dotProduct(v1, v2);
		double coeffU = u1u2 / u1u1;
		double coeffV = v1v2 / u1u1;
		double beta = Math.sqrt((u2u2 - u1u2 * coeffU) / (v2v2 - v1v2 * coeffV));
		double alpha = coeffU - beta * coeffV;
		double v2x = alpha * v1x + beta * v2.getX();
		double v2y = alpha * v1y + beta * v2.getY();
		double v2z = alpha * v1z + beta * v2.getZ();
		v2 = new Vector3D(v2x, v2y, v2z);

		// preliminary computation (we use explicit formulation instead
		// of relying on the Vector3D class in order to avoid building lots
		// of temporary objects)
		Vector3D uRef = u1;
		Vector3D vRef = v1;
		double dx1 = v1x - u1.getX();
		double dy1 = v1y - u1.getY();
		double dz1 = v1z - u1.getZ();
		double dx2 = v2x - u2.getX();
		double dy2 = v2y - u2.getY();
		double dz2 = v2z - u2.getZ();
		Vector3D k = new Vector3D(dy1 * dz2 - dz1 * dy2, dz1 * dx2 - dx1 * dz2, dx1 * dy2 - dy1 * dx2);
		double c = k.getX() * (u1y * u2z - u1z * u2y) + k.getY() * (u1z * u2x - u1x * u2z)
				+ k.getZ() * (u1x * u2y - u1y * u2x);

		if (c == 0) {
			// the (q1, q2, q3) vector is in the (u1, u2) plane
			// we try other vectors
			Vector3D u3 = Vector3D.crossProduct(u1, u2);
			Vector3D v3 = Vector3D.crossProduct(v1, v2);
			double u3x = u3.getX();
			double u3y = u3.getY();
			double u3z = u3.getZ();
			double v3x = v3.getX();
			double v3y = v3.getY();
			double v3z = v3.getZ();

			double dx3 = v3x - u3x;
			double dy3 = v3y - u3y;
			double dz3 = v3z - u3z;
			k = new Vector3D(dy1 * dz3 - dz1 * dy3, dz1 * dx3 - dx1 * dz3, dx1 * dy3 - dy1 * dx3);
			c = k.getX() * (u1y * u3z - u1z * u3y) + k.getY() * (u1z * u3x - u1x * u3z)
					+ k.getZ() * (u1x * u3y - u1y * u3x);

			if (c == 0) {
				// the (q1, q2, q3) vector is aligned with u1:
				// we try (u2, u3) and (v2, v3)
				k = new Vector3D(dy2 * dz3 - dz2 * dy3, dz2 * dx3 - dx2 * dz3, dx2 * dy3 - dy2 * dx3);
				c = k.getX() * (u2y * u3z - u2z * u3y) + k.getY() * (u2z * u3x - u2x * u3z)
						+ k.getZ() * (u2x * u3y - u2y * u3x);

				if (c == 0) {
					// the (q1, q2, q3) vector is aligned with everything
					// this is really the identity rotation
					q0 = 1.0;
					q1 = 0.0;
					q2 = 0.0;
					q3 = 0.0;
					return;
				}

				// we will have to use u2 and v2 to compute the scalar part
				uRef = u2;
				vRef = v2;

			}

		}

		// compute the vectorial part
		c = Math.sqrt(c);
		double inv = 1.0 / (c + c);
		q1 = inv * k.getX();
		q2 = inv * k.getY();
		q3 = inv * k.getZ();

		// compute the scalar part
		k = new Vector3D(uRef.getY() * q3 - uRef.getZ() * q2, uRef.getZ() * q1 - uRef.getX() * q3,
				uRef.getX() * q2 - uRef.getY() * q1);
		c = Vector3D.dotProduct(k, k);
		q0 = Vector3D.dotProduct(vRef, k) / (c + c);

	}

	/**
	 * 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 colinear, an arbitrary rotation axis is chosen.
	 * </p>
	 * 
	 * @param u
	 *            origin vector
	 * @param v
	 *            desired image of u by the rotation
	 * @exception IllegalArgumentException
	 *                if the norm of one of the vectors is zero
	 */
	public Rotation(Vector3D u, Vector3D v) {

		double normProduct = u.getNorm() * v.getNorm();
		if (normProduct == 0) {
			throw MathRuntimeException.createIllegalArgumentException("zero norm for rotation defining vector");
		}

		double dot = Vector3D.dotProduct(u, v);

		if (dot < ((2.0e-15 - 1.0) * normProduct)) {
			// special case u = -v: we select a PI angle rotation around
			// an arbitrary vector orthogonal to u
			Vector3D w = u.orthogonal();
			q0 = 0.0;
			q1 = -w.getX();
			q2 = -w.getY();
			q3 = -w.getZ();
		} else {
			// general case: (u, v) defines a plane, we select
			// the shortest possible rotation: axis orthogonal to this plane
			q0 = Math.sqrt(0.5 * (1.0 + dot / normProduct));
			double coeff = 1.0 / (2.0 * q0 * normProduct);
			q1 = coeff * (v.getY() * u.getZ() - v.getZ() * u.getY());
			q2 = coeff * (v.getZ() * u.getX() - v.getX() * u.getZ());
			q3 = coeff * (v.getX() * u.getY() - v.getY() * u.getX());
		}

	}

	/**
	 * 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 use
	 * @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 Rotation(RotationOrder order, double alpha1, double alpha2, double alpha3) {
		Rotation r1 = new Rotation(order.getA1(), alpha1);
		Rotation r2 = new Rotation(order.getA2(), alpha2);
		Rotation r3 = new Rotation(order.getA3(), alpha3);
		Rotation composed = r1.applyTo(r2.applyTo(r3));
		q0 = composed.q0;
		q1 = composed.q1;
		q2 = composed.q2;
		q3 = composed.q3;
	}

	/**
	 * 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 Rotation revert() {
		return new Rotation(-q0, q1, q2, q3, false);
	}

	/**
	 * Get the scalar coordinate of the quaternion.
	 * 
	 * @return scalar coordinate of the quaternion
	 */
	public double 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 double 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 double 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 double getQ3() {
		return q3;
	}

	/**
	 * Get the normalized axis of the rotation.
	 * 
	 * @return normalized axis of the rotation
	 */
	public Vector3D getAxis() {
		double squaredSine = q1 * q1 + q2 * q2 + q3 * q3;
		if (squaredSine == 0) {
			return new Vector3D(1, 0, 0);
		} else if (q0 < 0) {
			double inverse = 1 / Math.sqrt(squaredSine);
			return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
		}
		double inverse = -1 / Math.sqrt(squaredSine);
		return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
	}

	/**
	 * Get the angle of the rotation.
	 * 
	 * @return angle of the rotation (between 0 and &pi;)
	 */
	public double getAngle() {
		if ((q0 < -0.1) || (q0 > 0.1)) {
			return 2 * Math.asin(Math.sqrt(q1 * q1 + q2 * q2 + q3 * q3));
		} else if (q0 < 0) {
			return 2 * Math.acos(-q0);
		}
		return 2 * Math.acos(q0);
	}

	/**
	 * 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
	 * @return an array of three angles, in the order specified by the set
	 * @exception CardanEulerSingularityException
	 *                if the rotation is singular with respect to the angles set
	 *                specified
	 */
	public double[] getAngles(RotationOrder order) throws CardanEulerSingularityException {

		if (order == RotationOrder.XYZ) {

			// r (Vector3D.plusK) coordinates are :
			// sin (theta), -cos (theta) sin (phi), cos (theta) cos (phi)
			// (-r) (Vector3D.plusI) coordinates are :
			// cos (psi) cos (theta), -sin (psi) cos (theta), sin (theta)
			// and we can choose to have theta in the interval [-PI/2 ; +PI/2]
			Vector3D v1 = applyTo(Vector3D.PLUS_K);
			Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
			if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
				throw new CardanEulerSingularityException(true);
			}
			return new double[] { Math.atan2(-(v1.getY()), v1.getZ()), Math.asin(v2.getZ()),
					Math.atan2(-(v2.getY()), v2.getX()) };

		} else if (order == RotationOrder.XZY) {

			// r (Vector3D.plusJ) coordinates are :
			// -sin (psi), cos (psi) cos (phi), cos (psi) sin (phi)
			// (-r) (Vector3D.plusI) coordinates are :
			// cos (theta) cos (psi), -sin (psi), sin (theta) cos (psi)
			// and we can choose to have psi in the interval [-PI/2 ; +PI/2]
			Vector3D v1 = applyTo(Vector3D.PLUS_J);
			Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
			if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
				throw new CardanEulerSingularityException(true);
			}
			return new double[] { Math.atan2(v1.getZ(), v1.getY()), -Math.asin(v2.getY()),
					Math.atan2(v2.getZ(), v2.getX()) };

		} else if (order == RotationOrder.YXZ) {

			// r (Vector3D.plusK) coordinates are :
			// cos (phi) sin (theta), -sin (phi), cos (phi) cos (theta)
			// (-r) (Vector3D.plusJ) coordinates are :
			// sin (psi) cos (phi), cos (psi) cos (phi), -sin (phi)
			// and we can choose to have phi in the interval [-PI/2 ; +PI/2]
			Vector3D v1 = applyTo(Vector3D.PLUS_K);
			Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
			if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
				throw new CardanEulerSingularityException(true);
			}
			return new double[] { Math.atan2(v1.getX(), v1.getZ()), -Math.asin(v2.getZ()),
					Math.atan2(v2.getX(), v2.getY()) };

		} else if (order == RotationOrder.YZX) {

			// r (Vector3D.plusI) coordinates are :
			// cos (psi) cos (theta), sin (psi), -cos (psi) sin (theta)
			// (-r) (Vector3D.plusJ) coordinates are :
			// sin (psi), cos (phi) cos (psi), -sin (phi) cos (psi)
			// and we can choose to have psi in the interval [-PI/2 ; +PI/2]
			Vector3D v1 = applyTo(Vector3D.PLUS_I);
			Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
			if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
				throw new CardanEulerSingularityException(true);
			}
			return new double[] { Math.atan2(-(v1.getZ()), v1.getX()), Math.asin(v2.getX()),
					Math.atan2(-(v2.getZ()), v2.getY()) };

		} else if (order == RotationOrder.ZXY) {

			// r (Vector3D.plusJ) coordinates are :
			// -cos (phi) sin (psi), cos (phi) cos (psi), sin (phi)
			// (-r) (Vector3D.plusK) coordinates are :
			// -sin (theta) cos (phi), sin (phi), cos (theta) cos (phi)
			// and we can choose to have phi in the interval [-PI/2 ; +PI/2]
			Vector3D v1 = applyTo(Vector3D.PLUS_J);
			Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
			if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
				throw new CardanEulerSingularityException(true);
			}
			return new double[] { Math.atan2(-(v1.getX()), v1.getY()), Math.asin(v2.getY()),
					Math.atan2(-(v2.getX()), v2.getZ()) };

		} else if (order == RotationOrder.ZYX) {

			// r (Vector3D.plusI) coordinates are :
			// cos (theta) cos (psi), cos (theta) sin (psi), -sin (theta)
			// (-r) (Vector3D.plusK) coordinates are :
			// -sin (theta), sin (phi) cos (theta), cos (phi) cos (theta)
			// and we can choose to have theta in the interval [-PI/2 ; +PI/2]
			Vector3D v1 = applyTo(Vector3D.PLUS_I);
			Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
			if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
				throw new CardanEulerSingularityException(true);
			}
			return new double[] { Math.atan2(v1.getY(), v1.getX()), -Math.asin(v2.getX()),
					Math.atan2(v2.getY(), v2.getZ()) };

		} else if (order == RotationOrder.XYX) {

			// r (Vector3D.plusI) coordinates are :
			// cos (theta), sin (phi1) sin (theta), -cos (phi1) sin (theta)
			// (-r) (Vector3D.plusI) coordinates are :
			// cos (theta), sin (theta) sin (phi2), sin (theta) cos (phi2)
			// and we can choose to have theta in the interval [0 ; PI]
			Vector3D v1 = applyTo(Vector3D.PLUS_I);
			Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
			if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
				throw new CardanEulerSingularityException(false);
			}
			return new double[] { Math.atan2(v1.getY(), -v1.getZ()), Math.acos(v2.getX()),
					Math.atan2(v2.getY(), v2.getZ()) };

		} else if (order == RotationOrder.XZX) {

			// r (Vector3D.plusI) coordinates are :
			// cos (psi), cos (phi1) sin (psi), sin (phi1) sin (psi)
			// (-r) (Vector3D.plusI) coordinates are :
			// cos (psi), -sin (psi) cos (phi2), sin (psi) sin (phi2)
			// and we can choose to have psi in the interval [0 ; PI]
			Vector3D v1 = applyTo(Vector3D.PLUS_I);
			Vector3D v2 = applyInverseTo(Vector3D.PLUS_I);
			if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) {
				throw new CardanEulerSingularityException(false);
			}
			return new double[] { Math.atan2(v1.getZ(), v1.getY()), Math.acos(v2.getX()),
					Math.atan2(v2.getZ(), -v2.getY()) };

		} else if (order == RotationOrder.YXY) {

			// r (Vector3D.plusJ) coordinates are :
			// sin (theta1) sin (phi), cos (phi), cos (theta1) sin (phi)
			// (-r) (Vector3D.plusJ) coordinates are :
			// sin (phi) sin (theta2), cos (phi), -sin (phi) cos (theta2)
			// and we can choose to have phi in the interval [0 ; PI]
			Vector3D v1 = applyTo(Vector3D.PLUS_J);
			Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
			if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
				throw new CardanEulerSingularityException(false);
			}
			return new double[] { Math.atan2(v1.getX(), v1.getZ()), Math.acos(v2.getY()),
					Math.atan2(v2.getX(), -v2.getZ()) };

		} else if (order == RotationOrder.YZY) {

			// r (Vector3D.plusJ) coordinates are :
			// -cos (theta1) sin (psi), cos (psi), sin (theta1) sin (psi)
			// (-r) (Vector3D.plusJ) coordinates are :
			// sin (psi) cos (theta2), cos (psi), sin (psi) sin (theta2)
			// and we can choose to have psi in the interval [0 ; PI]
			Vector3D v1 = applyTo(Vector3D.PLUS_J);
			Vector3D v2 = applyInverseTo(Vector3D.PLUS_J);
			if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) {
				throw new CardanEulerSingularityException(false);
			}
			return new double[] { Math.atan2(v1.getZ(), -v1.getX()), Math.acos(v2.getY()),
					Math.atan2(v2.getZ(), v2.getX()) };

		} else if (order == RotationOrder.ZXZ) {

			// r (Vector3D.plusK) coordinates are :
			// sin (psi1) sin (phi), -cos (psi1) sin (phi), cos (phi)
			// (-r) (Vector3D.plusK) coordinates are :
			// sin (phi) sin (psi2), sin (phi) cos (psi2), cos (phi)
			// and we can choose to have phi in the interval [0 ; PI]
			Vector3D v1 = applyTo(Vector3D.PLUS_K);
			Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
			if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
				throw new CardanEulerSingularityException(false);
			}
			return new double[] { Math.atan2(v1.getX(), -v1.getY()), Math.acos(v2.getZ()),
					Math.atan2(v2.getX(), v2.getY()) };

		} else { // last possibility is ZYZ

			// r (Vector3D.plusK) coordinates are :
			// cos (psi1) sin (theta), sin (psi1) sin (theta), cos (theta)
			// (-r) (Vector3D.plusK) coordinates are :
			// -sin (theta) cos (psi2), sin (theta) sin (psi2), cos (theta)
			// and we can choose to have theta in the interval [0 ; PI]
			Vector3D v1 = applyTo(Vector3D.PLUS_K);
			Vector3D v2 = applyInverseTo(Vector3D.PLUS_K);
			if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) {
				throw new CardanEulerSingularityException(false);
			}
			return new double[] { Math.atan2(v1.getY(), v1.getX()), Math.acos(v2.getZ()),
					Math.atan2(v2.getY(), -v2.getX()) };

		}

	}

	/**
	 * Get the 3X3 matrix corresponding to the instance
	 * 
	 * @return the matrix corresponding to the instance
	 */
	public double[][] getMatrix() {

		// products
		double q0q0 = q0 * q0;
		double q0q1 = q0 * q1;
		double q0q2 = q0 * q2;
		double q0q3 = q0 * q3;
		double q1q1 = q1 * q1;
		double q1q2 = q1 * q2;
		double q1q3 = q1 * q3;
		double q2q2 = q2 * q2;
		double q2q3 = q2 * q3;
		double q3q3 = q3 * q3;

		// create the matrix
		double[][] m = new double[3][];
		m[0] = new double[3];
		m[1] = new double[3];
		m[2] = new double[3];

		m[0][0] = 2.0 * (q0q0 + q1q1) - 1.0;
		m[1][0] = 2.0 * (q1q2 - q0q3);
		m[2][0] = 2.0 * (q1q3 + q0q2);

		m[0][1] = 2.0 * (q1q2 + q0q3);
		m[1][1] = 2.0 * (q0q0 + q2q2) - 1.0;
		m[2][1] = 2.0 * (q2q3 - q0q1);

		m[0][2] = 2.0 * (q1q3 - q0q2);
		m[1][2] = 2.0 * (q2q3 + q0q1);
		m[2][2] = 2.0 * (q0q0 + q3q3) - 1.0;

		return m;

	}

	/**
	 * 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 Vector3D applyTo(Vector3D u) {

		double x = u.getX();
		double y = u.getY();
		double z = u.getZ();

		double s = q1 * x + q2 * y + q3 * z;

		return new Vector3D(2 * (q0 * (x * q0 - (q2 * z - q3 * y)) + s * q1) - x,
				2 * (q0 * (y * q0 - (q3 * x - q1 * z)) + s * q2) - y,
				2 * (q0 * (z * q0 - (q1 * y - q2 * x)) + s * q3) - 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 Vector3D applyInverseTo(Vector3D u) {

		double x = u.getX();
		double y = u.getY();
		double z = u.getZ();

		double s = q1 * x + q2 * y + q3 * z;
		double m0 = -q0;

		return new Vector3D(2 * (m0 * (x * m0 - (q2 * z - q3 * y)) + s * q1) - x,
				2 * (m0 * (y * m0 - (q3 * x - q1 * z)) + s * q2) - y,
				2 * (m0 * (z * m0 - (q1 * y - q2 * x)) + s * q3) - z);

	}

	/**
	 * Apply the instance to another rotation. Applying the instance to a
	 * rotation is computing the composition in an order compliant with the
	 * following rule : let u be any vector and v its image by r (i.e.
	 * r.applyTo(u) = v), let w be the image of v by the instance (i.e.
	 * applyTo(v) = w), then w = comp.applyTo(u), where comp = applyTo(r).
	 * 
	 * @param r
	 *            rotation to apply the rotation to
	 * @return a new rotation which is the composition of r by the instance
	 */
	public Rotation applyTo(Rotation r) {
		return new Rotation(r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3),
				r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2), r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),
				r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1), false);
	}

	/**
	 * Apply the inverse of the instance to another rotation. Applying the
	 * inverse of the instance to a rotation is computing the composition in an
	 * order compliant with the following rule : let u be any vector and v its
	 * image by r (i.e. r.applyTo(u) = v), let w be the inverse image of v by
	 * the instance (i.e. applyInverseTo(v) = w), then w = comp.applyTo(u),
	 * where comp = applyInverseTo(r).
	 * 
	 * @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 Rotation applyInverseTo(Rotation r) {
		return new Rotation(-r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3),
				-r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2), -r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3),
				-r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1), 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 NotARotationMatrixException
	 *                if the matrix cannot be orthogonalized with the given
	 *                threshold after 10 iterations
	 */
	private double[][] orthogonalizeMatrix(double[][] m, double threshold) throws NotARotationMatrixException {
		double[] m0 = m[0];
		double[] m1 = m[1];
		double[] m2 = m[2];
		double x00 = m0[0];
		double x01 = m0[1];
		double x02 = m0[2];
		double x10 = m1[0];
		double x11 = m1[1];
		double x12 = m1[2];
		double x20 = m2[0];
		double x21 = m2[1];
		double x22 = m2[2];
		double fn = 0;
		double fn1;

		double[][] o = new double[3][3];
		double[] o0 = o[0];
		double[] o1 = o[1];
		double[] o2 = o[2];

		// iterative correction: Xn+1 = Xn - 0.5 * (Xn.Mt.Xn - M)
		int i = 0;
		while (++i < 11) {

			// Mt.Xn
			double mx00 = m0[0] * x00 + m1[0] * x10 + m2[0] * x20;
			double mx10 = m0[1] * x00 + m1[1] * x10 + m2[1] * x20;
			double mx20 = m0[2] * x00 + m1[2] * x10 + m2[2] * x20;
			double mx01 = m0[0] * x01 + m1[0] * x11 + m2[0] * x21;
			double mx11 = m0[1] * x01 + m1[1] * x11 + m2[1] * x21;
			double mx21 = m0[2] * x01 + m1[2] * x11 + m2[2] * x21;
			double mx02 = m0[0] * x02 + m1[0] * x12 + m2[0] * x22;
			double mx12 = m0[1] * x02 + m1[1] * x12 + m2[1] * x22;
			double mx22 = m0[2] * x02 + m1[2] * x12 + m2[2] * x22;

			// Xn+1
			o0[0] = x00 - 0.5 * (x00 * mx00 + x01 * mx10 + x02 * mx20 - m0[0]);
			o0[1] = x01 - 0.5 * (x00 * mx01 + x01 * mx11 + x02 * mx21 - m0[1]);
			o0[2] = x02 - 0.5 * (x00 * mx02 + x01 * mx12 + x02 * mx22 - m0[2]);
			o1[0] = x10 - 0.5 * (x10 * mx00 + x11 * mx10 + x12 * mx20 - m1[0]);
			o1[1] = x11 - 0.5 * (x10 * mx01 + x11 * mx11 + x12 * mx21 - m1[1]);
			o1[2] = x12 - 0.5 * (x10 * mx02 + x11 * mx12 + x12 * mx22 - m1[2]);
			o2[0] = x20 - 0.5 * (x20 * mx00 + x21 * mx10 + x22 * mx20 - m2[0]);
			o2[1] = x21 - 0.5 * (x20 * mx01 + x21 * mx11 + x22 * mx21 - m2[1]);
			o2[2] = x22 - 0.5 * (x20 * mx02 + x21 * mx12 + x22 * mx22 - m2[2]);

			// correction on each elements
			double corr00 = o0[0] - m0[0];
			double corr01 = o0[1] - m0[1];
			double corr02 = o0[2] - m0[2];
			double corr10 = o1[0] - m1[0];
			double corr11 = o1[1] - m1[1];
			double corr12 = o1[2] - m1[2];
			double corr20 = o2[0] - m2[0];
			double corr21 = o2[1] - m2[1];
			double corr22 = o2[2] - m2[2];

			// 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 (Math.abs(fn1 - fn) <= threshold)
				return o;

			// prepare next iteration
			x00 = o0[0];
			x01 = o0[1];
			x02 = o0[2];
			x10 = o1[0];
			x11 = o1[1];
			x12 = o1[2];
			x20 = o2[0];
			x21 = o2[1];
			x22 = o2[2];
			fn = fn1;

		}

		// the algorithm did not converge after 10 iterations
		throw new NotARotationMatrixException("unable to orthogonalize matrix in {0} iterations", 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:
	 * </p>
	 * 
	 * <pre>
	 *        r<sub>1</sub>(r) = r<sub>2</sub>
	 * </pre>
	 * <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
	 * @return <i>distance</i> between r1 and r2
	 */
	public static double distance(Rotation r1, Rotation r2) {
		return r1.applyInverseTo(r2).getAngle();
	}

}
