/*
* 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.math3.geometry.euclidean.threed;
import java.io.Serializable;
import org.apache.commons.math3.RealFieldElement;
import org.apache.commons.math3.Field;
import org.apache.commons.math3.exception.MathArithmeticException;
import org.apache.commons.math3.exception.MathIllegalArgumentException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.util.FastMath;
import org.apache.commons.math3.util.MathArrays;
/**
* This class is a re-implementation of {@link Rotation} using {@link RealFieldElement}.
* <p>Instance of this class are guaranteed to be immutable.</p>
*
* @param <T> the type of the field elements
* @see FieldVector3D
* @see RotationOrder
* @since 3.2
*/
public class FieldRotation<T extends RealFieldElement<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.multiply(q0).add(q1.multiply(q1)).add(q2.multiply(q2)).add(q3.multiply(q3)).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 π/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 θ about the unit vector (x, y, z) is the same as the
* rotation build from quaternion components { cos(-θ/2),
* x * sin(-θ/2), y * sin(-θ/2), z * sin(-θ/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.
* @exception MathIllegalArgumentException if the axis norm is zero
* @deprecated as of 3.6, replaced with {@link
* #FieldRotation(FieldVector3D, RealFieldElement, RotationConvention)}
*/
@Deprecated
public FieldRotation(final FieldVector3D<T> axis, final T angle)
throws MathIllegalArgumentException {
this(axis, angle, RotationConvention.VECTOR_OPERATOR);
}
/** 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 π/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 θ about the unit vector (x, y, z) is the same as the
* rotation build from quaternion components { cos(-θ/2),
* x * sin(-θ/2), y * sin(-θ/2), z * sin(-θ/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
* @since 3.6
*/
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(LocalizedFormats.ZERO_NORM_FOR_ROTATION_AXIS);
}
final T halfAngle = angle.multiply(convention == RotationConvention.VECTOR_OPERATOR ? -0.5 : 0.5);
final T coeff = halfAngle.sin().divide(norm);
q0 = halfAngle.cos();
q1 = coeff.multiply(axis.getX());
q2 = coeff.multiply(axis.getY());
q3 = coeff.multiply(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 FieldRotation(final T[][] m, final 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(
LocalizedFormats.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 NotARotationMatrixException(
LocalizedFormats.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 MathArithmeticException 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 MathArithmeticException {
// 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 MathArithmeticException if the norm of one of the vectors is zero
*/
public FieldRotation(final FieldVector3D<T> u, final FieldVector3D<T> v) throws MathArithmeticException {
final T normProduct = u.getNorm().multiply(v.getNorm());
if (normProduct.getReal() == 0) {
throw new MathArithmeticException(LocalizedFormats.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 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
* @deprecated as of 3.6, replaced with {@link
* #FieldRotation(RotationOrder, RotationConvention,
* RealFieldElement, RealFieldElement, RealFieldElement)}
*/
@Deprecated
public FieldRotation(final RotationOrder order, final T alpha1, final T alpha2, final T alpha3) {
this(order, RotationConvention.VECTOR_OPERATOR, alpha1, alpha2, alpha3);
}
/** 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
* @since 3.6
*/
public FieldRotation(final RotationOrder order, final RotationConvention convention,
final T alpha1, final T alpha2, final T alpha3) {
final T one = alpha1.getField().getOne();
final FieldRotation<T> r1 = new FieldRotation<T>(new FieldVector3D<T>(one, order.getA1()), alpha1, convention);
final FieldRotation<T> r2 = new FieldRotation<T>(new FieldVector3D<T>(one, order.getA2()), alpha2, convention);
final FieldRotation<T> r3 = new FieldRotation<T>(new FieldVector3D<T>(one, 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;
}
/** 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<T>(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.
* @return normalized axis of the rotation
* @see #FieldRotation(FieldVector3D, RealFieldElement)
* @deprecated as of 3.6, replaced with {@link #getAxis(RotationConvention)}
*/
@Deprecated
public FieldVector3D<T> getAxis() {
return getAxis(RotationConvention.VECTOR_OPERATOR);
}
/** Get the normalized axis of the rotation.
* <p>
* Note that as {@link #getAngle()} always returns an angle
* between 0 and π, 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, RealFieldElement)
* @since 3.6
*/
public FieldVector3D<T> getAxis(final RotationConvention convention) {
final T squaredSine = q1.multiply(q1).add(q2.multiply(q2)).add(q3.multiply(q3));
if (squaredSine.getReal() == 0) {
final Field<T> field = squaredSine.getField();
return new FieldVector3D<T>(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<T>(q1.multiply(inverse), q2.multiply(inverse), q3.multiply(inverse));
}
final T inverse = squaredSine.sqrt().reciprocal().negate().multiply(sgn);
return new FieldVector3D<T>(q1.multiply(inverse), q2.multiply(inverse), q3.multiply(inverse));
}
}
/** Get the angle of the rotation.
* @return angle of the rotation (between 0 and π)
* @see #FieldRotation(FieldVector3D, RealFieldElement)
*/
public T getAngle() {
if ((q0.getReal() < -0.1) || (q0.getReal() > 0.1)) {
return q1.multiply(q1).add(q2.multiply(q2)).add(q3.multiply(q3)).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 π + a<sub>1</sub>, π
* - a<sub>2</sub> and π + 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 -π/2 and π/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 π (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
* -π/2 or +π/2, for Euler angle singularities occur when the
* second angle is close to 0 or π, 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
* @deprecated as of 3.6, replaced with {@link #getAngles(RotationOrder, RotationConvention)}
*/
@Deprecated
public T[] getAngles(final RotationOrder order)
throws CardanEulerSingularityException {
return getAngles(order, RotationConvention.VECTOR_OPERATOR);
}
/** 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 π + a<sub>1</sub>, π
* - a<sub>2</sub> and π + 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 -π/2 and π/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 π (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
* -π/2 or +π/2, for Euler angle singularities occur when the
* second angle is close to 0 or π, 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
* @exception CardanEulerSingularityException if the rotation is
* singular with respect to the angles set specified
* @since 3.6
*/
public T[] getAngles(final RotationOrder order, RotationConvention convention)
throws CardanEulerSingularityException {
if (convention == RotationConvention.VECTOR_OPERATOR) {
if (order == RotationOrder.XYZ) {
// r (+K) coordinates are :
// sin (theta), -cos (theta) sin (phi), cos (theta) cos (phi)
// (-r) (+I) coordinates are :
// cos (psi) cos (theta), -sin (psi) cos (theta), sin (theta)
final // and we can choose to have theta in the interval [-PI/2 ; +PI/2]
FieldVector3D<T> v1 = applyTo(vector(0, 0, 1));
final FieldVector3D<T> v2 = applyInverseTo(vector(1, 0, 0));
if ((v2.getZ().getReal() < -0.9999999999) || (v2.getZ().getReal() > 0.9999999999)) {
throw new CardanEulerSingularityException(true);
}
return buildArray(v1.getY().negate().atan2(v1.getZ()),
v2.getZ().asin(),
v2.getY().negate().atan2(v2.getX()));
} else if (order == RotationOrder.XZY) {
// r (+J) coordinates are :
// -sin (psi), cos (psi) cos (phi), cos (psi) sin (phi)
// (-r) (+I) 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]
final FieldVector3D<T> v1 = applyTo(vector(0, 1, 0));
final FieldVector3D<T> v2 = applyInverseTo(vector(1, 0, 0));
if ((v2.getY().getReal() < -0.9999999999) || (v2.getY().getReal() > 0.9999999999)) {
throw new CardanEulerSingularityException(true);
}
return buildArray(v1.getZ().atan2(v1.getY()),
v2.getY().asin().negate(),
v2.getZ().atan2(v2.getX()));
} else if (order == RotationOrder.YXZ) {
// r (+K) coordinates are :
// cos (phi) sin (theta), -sin (phi), cos (phi) cos (theta)
// (-r) (+J) 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]
final FieldVector3D<T> v1 = applyTo(vector(0, 0, 1));
final FieldVector3D<T> v2 = applyInverseTo(vector(0, 1, 0));
if ((v2.getZ().getReal() < -0.9999999999) || (v2.getZ().getReal() > 0.9999999999)) {
throw new CardanEulerSingularityException(true);
}
return buildArray(v1.getX().atan2(v1.getZ()),
v2.getZ().asin().negate(),
v2.getX().atan2(v2.getY()));
} else if (order == RotationOrder.YZX) {
// r (+I) coordinates are :
// cos (psi) cos (theta), sin (psi), -cos (psi) sin (theta)
// (-r) (+J) 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]
final FieldVector3D<T> v1 = applyTo(vector(1, 0, 0));
final FieldVector3D<T> v2 = applyInverseTo(vector(0, 1, 0));
if ((v2.getX().getReal() < -0.9999999999) || (v2.getX().getReal() > 0.9999999999)) {
throw new CardanEulerSingularityException(true);
}
return buildArray(v1.getZ().negate().atan2(v1.getX()),
v2.getX().asin(),
v2.getZ().negate().atan2(v2.getY()));
} else if (order == RotationOrder.ZXY) {
// r (+J) coordinates are :
// -cos (phi) sin (psi), cos (phi) cos (psi), sin (phi)
// (-r) (+K) 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]
final FieldVector3D<T> v1 = applyTo(vector(0, 1, 0));
final FieldVector3D<T> v2 = applyInverseTo(vector(0, 0, 1));
if ((v2.getY().getReal() < -0.9999999999) || (v2.getY().getReal() > 0.9999999999)) {
throw new CardanEulerSingularityException(true);
}
return buildArray(v1.getX().negate().atan2(v1.getY()),
v2.getY().asin(),
v2.getX().negate().atan2(v2.getZ()));
} else if (order == RotationOrder.ZYX) {
// r (+I) coordinates are :
// cos (theta) cos (psi), cos (theta) sin (psi), -sin (theta)
// (-r) (+K) 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]
final FieldVector3D<T> v1 = applyTo(vector(1, 0, 0));
final FieldVector3D<T> v2 = applyInverseTo(vector(0, 0, 1));
if ((v2.getX().getReal() < -0.9999999999) || (v2.getX().getReal() > 0.9999999999)) {
throw new CardanEulerSingularityException(true);
}
return buildArray(v1.getY().atan2(v1.getX()),
v2.getX().asin().negate(),
v2.getY().atan2(v2.getZ()));
} else if (order == RotationOrder.XYX) {
// r (+I) coordinates are :
// cos (theta), sin (phi1) sin (theta), -cos (phi1) sin (theta)
// (-r) (+I) coordinates are :
// cos (theta), sin (theta) sin (phi2), sin (theta) cos (phi2)
// and we can choose to have theta in the interval [0 ; PI]
final FieldVector3D<T> v1 = applyTo(vector(1, 0, 0));
final FieldVector3D<T> v2 = applyInverseTo(vector(1, 0, 0));
if ((v2.getX().getReal() < -0.9999999999) || (v2.getX().getReal() > 0.9999999999)) {
throw new CardanEulerSingularityException(false);
}
return buildArray(v1.getY().atan2(v1.getZ().negate()),
v2.getX().acos(),
v2.getY().atan2(v2.getZ()));
} else if (order == RotationOrder.XZX) {
// r (+I) coordinates are :
// cos (psi), cos (phi1) sin (psi), sin (phi1) sin (psi)
// (-r) (+I) coordinates are :
// cos (psi), -sin (psi) cos (phi2), sin (psi) sin (phi2)
// and we can choose to have psi in the interval [0 ; PI]
final FieldVector3D<T> v1 = applyTo(vector(1, 0, 0));
final FieldVector3D<T> v2 = applyInverseTo(vector(1, 0, 0));
if ((v2.getX().getReal() < -0.9999999999) || (v2.getX().getReal() > 0.9999999999)) {
throw new CardanEulerSingularityException(false);
}
return buildArray(v1.getZ().atan2(v1.getY()),
v2.getX().acos(),
v2.getZ().atan2(v2.getY().negate()));
} else if (order == RotationOrder.YXY) {
// r (+J) coordinates are :
// sin (theta1) sin (phi), cos (phi), cos (theta1) sin (phi)
// (-r) (+J) coordinates are :
// sin (phi) sin (theta2), cos (phi), -sin (phi) cos (theta2)
// and we can choose to have phi in the interval [0 ; PI]
final FieldVector3D<T> v1 = applyTo(vector(0, 1, 0));
final FieldVector3D<T> v2 = applyInverseTo(vector(0, 1, 0));
if ((v2.getY().getReal() < -0.9999999999) || (v2.getY().getReal() > 0.9999999999)) {
throw new CardanEulerSingularityException(false);
}
return buildArray(v1.getX().atan2(v1.getZ()),
v2.getY().acos(),
v2.getX().atan2(v2.getZ().negate()));
} else if (order == RotationOrder.YZY) {
// r (+J) coordinates are :
// -cos (theta1) sin (psi), cos (psi), sin (theta1) sin (psi)
// (-r) (+J) coordinates are :
// sin (psi) cos (theta2), cos (psi), sin (psi) sin (theta2)
// and we can choose to have psi in the interval [0 ; PI]
final FieldVector3D<T> v1 = applyTo(vector(0, 1, 0));
final FieldVector3D<T> v2 = applyInverseTo(vector(0, 1, 0));
if ((v2.getY().getReal() < -0.9999999999) || (v2.getY().getReal() > 0.9999999999)) {
throw new CardanEulerSingularityException(false);
}
return buildArray(v1.getZ().atan2(v1.getX().negate()),
v2.getY().acos(),
v2.getZ().atan2(v2.getX()));
} else if (order == RotationOrder.ZXZ) {
// r (+K) coordinates are :
// sin (psi1) sin (phi), -cos (psi1) sin (phi), cos (phi)
// (-r) (+K) coordinates are :
// sin (phi) sin (psi2), sin (phi) cos (psi2), cos (phi)
// and we can choose to have phi in the interval [0 ; PI]
final FieldVector3D<T> v1 = applyTo(vector(0, 0, 1));
final FieldVector3D<T> v2 = applyInverseTo(vector(0, 0, 1));
if ((v2.getZ().getReal() < -0.9999999999) || (v2.getZ().getReal() > 0.9999999999)) {
throw new CardanEulerSingularityException(false);
}
return buildArray(v1.getX().atan2(v1.getY().negate()),
v2.getZ().acos(),
v2.getX().atan2(v2.getY()));
} else { // last possibility is ZYZ
// r (+K) coordinates are :
// cos (psi1) sin (theta), sin (psi1) sin (theta), cos (theta)
// (-r) (+K) coordinates are :
// -sin (theta) cos (psi2), sin (theta) sin (psi2), cos (theta)
// and we can choose to have theta in the interval [0 ; PI]
final FieldVector3D<T> v1 = applyTo(vector(0, 0, 1));
final FieldVector3D<T> v2 = applyInverseTo(vector(0, 0, 1));
if ((v2.getZ().getReal() < -0.9999999999) || (v2.getZ().getReal() > 0.9999999999)) {
throw new CardanEulerSingularityException(false);
}
return buildArray(v1.getY().atan2(v1.getX()),
v2.getZ().acos(),
v2.getY().atan2(v2.getX().negate()));
}
} else {
if (order == RotationOrder.XYZ) {
// 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]
FieldVector3D<T> v1 = applyTo(Vector3D.PLUS_I);
FieldVector3D<T> v2 = applyInverseTo(Vector3D.PLUS_K);
if ((v2.getX().getReal() < -0.9999999999) || (v2.getX().getReal() > 0.9999999999)) {
throw new CardanEulerSingularityException(true);
}
return buildArray(v2.getY().negate().atan2(v2.getZ()),
v2.getX().asin(),
v1.getY().negate().atan2(v1.getX()));
} else if (order == RotationOrder.XZY) {
// 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]
FieldVector3D<T> v1 = applyTo(Vector3D.PLUS_I);
FieldVector3D<T> v2 = applyInverseTo(Vector3D.PLUS_J);
if ((v2.getX().getReal() < -0.9999999999) || (v2.getX().getReal() > 0.9999999999)) {
throw new CardanEulerSingularityException(true);
}
return buildArray(v2.getZ().atan2(v2.getY()),
v2.getX().asin().negate(),
v1.getZ().atan2(v1.getX()));
} else if (order == RotationOrder.YXZ) {
// 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]
FieldVector3D<T> v1 = applyTo(Vector3D.PLUS_J);
FieldVector3D<T> v2 = applyInverseTo(Vector3D.PLUS_K);
if ((v2.getY().getReal() < -0.9999999999) || (v2.getY().getReal() > 0.9999999999)) {
throw new CardanEulerSingularityException(true);
}
return buildArray(v2.getX().atan2(v2.getZ()),
v2.getY().asin().negate(),
v1.getX().atan2(v1.getY()));
} else if (order == RotationOrder.YZX) {
// 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]
FieldVector3D<T> v1 = applyTo(Vector3D.PLUS_J);
FieldVector3D<T> v2 = applyInverseTo(Vector3D.PLUS_I);
if ((v2.getY().getReal() < -0.9999999999) || (v2.getY().getReal() > 0.9999999999)) {
throw new CardanEulerSingularityException(true);
}
return buildArray(v2.getZ().negate().atan2(v2.getX()),
v2.getY().asin(),
v1.getZ().negate().atan2(v1.getY()));
} else if (order == RotationOrder.ZXY) {
// 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]
FieldVector3D<T> v1 = applyTo(Vector3D.PLUS_K);
FieldVector3D<T> v2 = applyInverseTo(Vector3D.PLUS_J);
if ((v2.getZ().getReal() < -0.9999999999) || (v2.getZ().getReal() > 0.9999999999)) {
throw new CardanEulerSingularityException(true);
}
return buildArray(v2.getX().negate().atan2(v2.getY()),
v2.getZ().asin(),
v1.getX().negate().atan2(v1.getZ()));
} else if (order == RotationOrder.ZYX) {
// 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]
FieldVector3D<T> v1 = applyTo(Vector3D.PLUS_K);
FieldVector3D<T> v2 = applyInverseTo(Vector3D.PLUS_I);
if ((v2.getZ().getReal() < -0.9999999999) || (v2.getZ().getReal() > 0.9999999999)) {
throw new CardanEulerSingularityException(true);
}
return buildArray(v2.getY().atan2(v2.getX()),
v2.getZ().asin().negate(),
v1.getY().atan2(v1.getZ()));
} else if (order == RotationOrder.XYX) {
// r (Vector3D.plusI) coordinates are :
// cos (theta), sin (phi2) sin (theta), cos (phi2) sin (theta)
// (-r) (Vector3D.plusI) coordinates are :
// cos (theta), sin (theta) sin (phi1), -sin (theta) cos (phi1)
// and we can choose to have theta in the interval [0 ; PI]
FieldVector3D<T> v1 = applyTo(Vector3D.PLUS_I);
FieldVector3D<T> v2 = applyInverseTo(Vector3D.PLUS_I);
if ((v2.getX().getReal() < -0.9999999999) || (v2.getX().getReal() > 0.9999999999)) {
throw new CardanEulerSingularityException(false);
}
return buildArray(v2.getY().atan2(v2.getZ().negate()),
v2.getX().acos(),
v1.getY().atan2(v1.getZ()));
} else if (order == RotationOrder.XZX) {
// r (Vector3D.plusI) coordinates are :
// cos (psi), -cos (phi2) sin (psi), sin (phi2) sin (psi)
// (-r) (Vector3D.plusI) coordinates are :
// cos (psi), sin (psi) cos (phi1), sin (psi) sin (phi1)
// and we can choose to have psi in the interval [0 ; PI]
FieldVector3D<T> v1 = applyTo(Vector3D.PLUS_I);
FieldVector3D<T> v2 = applyInverseTo(Vector3D.PLUS_I);
if ((v2.getX().getReal() < -0.9999999999) || (v2.getX().getReal() > 0.9999999999)) {
throw new CardanEulerSingularityException(false);
}
return buildArray(v2.getZ().atan2(v2.getY()),
v2.getX().acos(),
v1.getZ().atan2(v1.getY().negate()));
} else if (order == RotationOrder.YXY) {
// r (Vector3D.plusJ) coordinates are :
// sin (phi) sin (theta2), cos (phi), -sin (phi) cos (theta2)
// (-r) (Vector3D.plusJ) coordinates are :
// sin (theta1) sin (phi), cos (phi), cos (theta1) sin (phi)
// and we can choose to have phi in the interval [0 ; PI]
FieldVector3D<T> v1 = applyTo(Vector3D.PLUS_J);
FieldVector3D<T> v2 = applyInverseTo(Vector3D.PLUS_J);
if ((v2.getY().getReal() < -0.9999999999) || (v2.getY().getReal() > 0.9999999999)) {
throw new CardanEulerSingularityException(false);
}
return buildArray(v2.getX().atan2(v2.getZ()),
v2.getY().acos(),
v1.getX().atan2(v1.getZ().negate()));
} else if (order == RotationOrder.YZY) {
// r (Vector3D.plusJ) coordinates are :
// sin (psi) cos (theta2), cos (psi), sin (psi) sin (theta2)
// (-r) (Vector3D.plusJ) coordinates are :
// -cos (theta1) sin (psi), cos (psi), sin (theta1) sin (psi)
// and we can choose to have psi in the interval [0 ; PI]
FieldVector3D<T> v1 = applyTo(Vector3D.PLUS_J);
FieldVector3D<T> v2 = applyInverseTo(Vector3D.PLUS_J);
if ((v2.getY().getReal() < -0.9999999999) || (v2.getY().getReal() > 0.9999999999)) {
throw new CardanEulerSingularityException(false);
}
return buildArray(v2.getZ().atan2(v2.getX().negate()),
v2.getY().acos(),
v1.getZ().atan2(v1.getX()));
} else if (order == RotationOrder.ZXZ) {
// r (Vector3D.plusK) coordinates are :
// sin (phi) sin (psi2), sin (phi) cos (psi2), cos (phi)
// (-r) (Vector3D.plusK) coordinates are :
// sin (psi1) sin (phi), -cos (psi1) sin (phi), cos (phi)
// and we can choose to have phi in the interval [0 ; PI]
FieldVector3D<T> v1 = applyTo(Vector3D.PLUS_K);
FieldVector3D<T> v2 = applyInverseTo(Vector3D.PLUS_K);
if ((v2.getZ().getReal() < -0.9999999999) || (v2.getZ().getReal() > 0.9999999999)) {
throw new CardanEulerSingularityException(false);
}
return buildArray(v2.getX().atan2(v2.getY().negate()),
v2.getZ().acos(),
v1.getX().atan2(v1.getY()));
} else { // last possibility is ZYZ
// r (Vector3D.plusK) coordinates are :
// -sin (theta) cos (psi2), sin (theta) sin (psi2), cos (theta)
// (-r) (Vector3D.plusK) coordinates are :
// cos (psi1) sin (theta), sin (psi1) sin (theta), cos (theta)
// and we can choose to have theta in the interval [0 ; PI]
FieldVector3D<T> v1 = applyTo(Vector3D.PLUS_K);
FieldVector3D<T> v2 = applyInverseTo(Vector3D.PLUS_K);
if ((v2.getZ().getReal() < -0.9999999999) || (v2.getZ().getReal() > 0.9999999999)) {
throw new CardanEulerSingularityException(false);
}
return buildArray(v2.getY().atan2(v2.getX()),
v2.getZ().acos(),
v1.getY().atan2(v1.getX().negate()));
}
}
}
/** Create a dimension 3 array.
* @param a0 first array element
* @param a1 second array element
* @param a2 third array element
* @return new array
*/
private T[] buildArray(final T a0, final T a1, final T a2) {
final T[] array = MathArrays.buildArray(a0.getField(), 3);
array[0] = a0;
array[1] = a1;
array[2] = a2;
return array;
}
/** Create a constant vector.
* @param x abscissa
* @param y ordinate
* @param z height
* @return a constant vector
*/
private FieldVector3D<T> vector(final double x, final double y, final double z) {
final T zero = q0.getField().getZero();
return new FieldVector3D<T>(zero.add(x), zero.add(y), zero.add(z));
}
/** Get the 3X3 matrix corresponding to the instance
* @return the matrix corresponding to the instance
*/
public T[][] getMatrix() {
// products
final T q0q0 = q0.multiply(q0);
final T q0q1 = q0.multiply(q1);
final T q0q2 = q0.multiply(q2);
final T q0q3 = q0.multiply(q3);
final T q1q1 = q1.multiply(q1);
final T q1q2 = q1.multiply(q2);
final T q1q3 = q1.multiply(q3);
final T q2q2 = q2.multiply(q2);
final T q2q3 = q2.multiply(q3);
final T q3q3 = q3.multiply(q3);
// 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<T>(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<T>(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 RealFieldElement<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<T>(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<T>(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<T>(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 RealFieldElement<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<T>(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<T>(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<T>(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 RealFieldElement<T>> FieldRotation<T> applyTo(final Rotation r1, final FieldRotation<T> rInner) {
return new FieldRotation<T>(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<T>(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<T>(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 RealFieldElement<T>> FieldRotation<T> applyInverseTo(final Rotation rOuter, final FieldRotation<T> rInner) {
return new FieldRotation<T>(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 NotARotationMatrixException if the matrix cannot be
* orthogonalized with the given threshold after 10 iterations
*/
private T[][] orthogonalizeMatrix(final T[][] m, final double threshold)
throws NotARotationMatrixException {
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 = 0;
while (++i < 11) {
// 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 NotARotationMatrixException(LocalizedFormats.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:</p>
* <pre>
* r<sub>1</sub>(r) = r<sub>2</sub>
* </pre>
* <p>This distance is an angle between 0 and π. 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 RealFieldElement<T>> T distance(final FieldRotation<T> r1, final FieldRotation<T> r2) {
return r1.composeInverseInternal(r2).getAngle();
}
}