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Rotation
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IDENTITY: Rotation
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serialVersionUID: long
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q0: double
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q1: double
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q2: double
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q3: double
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Rotation(double, double, double, double, boolean): void
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Rotation(Vector3D, double): void
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Rotation(Vector3D, double, RotationConvention): void
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Rotation(double[][], double): void
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Rotation(Vector3D, Vector3D, Vector3D, Vector3D): void
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Rotation(Vector3D, Vector3D): void
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Rotation(RotationOrder, double, double, double): void
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Rotation(RotationOrder, RotationConvention, double, double, double): void
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mat2quat(double[][]): double[]
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revert(): Rotation
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getQ0(): double
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getQ1(): double
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getQ2(): double
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getQ3(): double
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getAxis(): Vector3D
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getAxis(RotationConvention): Vector3D
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getAngle(): double
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getAngles(RotationOrder): double[]
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getAngles(RotationOrder, RotationConvention): double[]
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getMatrix(): double[][]
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applyTo(Vector3D): Vector3D
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applyTo(double[], double[]): void
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applyInverseTo(Vector3D): Vector3D
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applyInverseTo(double[], double[]): void
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applyTo(Rotation): Rotation
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compose(Rotation, RotationConvention): Rotation
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composeInternal(Rotation): Rotation
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applyInverseTo(Rotation): Rotation
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composeInverse(Rotation, RotationConvention): Rotation
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composeInverseInternal(Rotation): Rotation
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orthogonalizeMatrix(double[][], double): double[][]
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distance(Rotation, Rotation): double
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/*
* 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.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 implements rotations in a three-dimensional space.
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
Rotation instance (see the various constructors and
getters). In addition, a rotation can also be built implicitly
from a set of vectors and their image.
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
following single line of code:
double[] angles = new Rotation(matrix, 1.0e-10).getAngles(RotationOrder.XYZ);
Focus is oriented on what a rotation do rather than on its
underlying representation. Once it has been built, and regardless of its
internal representation, a rotation is an operator which basically transforms three dimensional vectors into other three dimensional vectors. Depending on the application, the meaning of these vectors may vary and the semantics of the rotation also.
For example in an spacecraft attitude simulation tool, users will often
consider the vectors are fixed (say the Earth direction for example) and the
frames change. The rotation transforms the coordinates of the vector in inertial
frame into the coordinates of the same vector in satellite frame. In this
case, the rotation implicitly 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 direction at rest in a topocentric frame
into the sighting direction in the same topocentric frame. This implies in this
case the frame is fixed and the vector moves.
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, which would essentially be an application of the
first approach.
These examples show that a rotation is what the user wants it to be. This
class does not push the user towards one specific definition and hence does not
provide methods like projectVectorIntoDestinationFrame or
computeTransformedDirection. It provides simpler and more generic methods: applyTo(Vector3D) and applyInverseTo(Vector3D).
Since a rotation is basically a vectorial operator, several rotations can be
composed together and the composite operation r = r1 o
r2 (which means that for each vector u,
r(u) = r1(r2(u))) 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 r1 to r2 and the result
we get is r = r1 o r2. For this purpose, the class provides the methods: applyTo(Rotation) and applyInverseTo(Rotation).
Rotations are guaranteed to be immutable objects.
See Also: Since: 1.2
/**
* 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 implicitly
* 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
* following 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
* frames change. The rotation transforms the coordinates of the vector in inertial
* frame into the coordinates of the same vector in satellite frame. In this
* case, the rotation implicitly defines the relation between the two frames.</p>
* <p>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 direction at rest in a topocentric frame
* into the sighting direction in the same topocentric frame. This implies in this
* case the frame is fixed and the vector moves.</p>
* <p>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, which would essentially be an application of the
* first approach.</p>
*
* <p>These examples show that a rotation is what the user wants it to be. 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>
*
* @see Vector3D
* @see RotationOrder
* @since 1.2
*/
public class Rotation implements Serializable {
Identity rotation. /** Identity rotation. */
public static final Rotation IDENTITY = new Rotation(1.0, 0.0, 0.0, 0.0, false);
Serializable version identifier /** Serializable version identifier */
private static final long serialVersionUID = -2153622329907944313L;
Scalar coordinate of the quaternion. /** Scalar coordinate of the quaternion. */
private final double q0;
First coordinate of the vectorial part of the quaternion. /** First coordinate of the vectorial part of the quaternion. */
private final double q1;
Second coordinate of the vectorial part of the quaternion. /** Second coordinate of the vectorial part of the quaternion. */
private final double q2;
Third coordinate of the vectorial part of the quaternion. /** Third coordinate of the vectorial part of the quaternion. */
private final double q3;
Build a rotation from the quaternion coordinates.
A rotation can be built from a normalized quaternion,
i.e. a quaternion for which q02 +
q12 + q22 +
q32 = 1. If the quaternion is not normalized,
the constructor can normalize it in a preprocessing step.
Note that some conventions put the scalar part of the quaternion
as the 4th component and the vector part as the first three
components. This is not our convention. We put the scalar part
as the first component.
Params: - q0 – scalar part of the quaternion
- q1 – first coordinate of the vectorial part of the quaternion
- q2 – second coordinate of the vectorial part of the quaternion
- q3 – third coordinate of the vectorial part of the quaternion
- needsNormalization – if true, the coordinates are considered
not to be normalized, a normalization preprocessing step is performed
before using them
/** 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 Rotation(double q0, double q1, double q2, double q3,
boolean needsNormalization) {
if (needsNormalization) {
// normalization preprocessing
double inv = 1.0 / FastMath.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.
Calling this constructor is equivalent to call
new Rotation(axis, angle, RotationConvention.VECTOR_OPERATOR)
Params: - axis – axis around which to rotate
- angle – rotation angle.
Throws: - MathIllegalArgumentException – if the axis norm is zero
Deprecated: as of 3.6, replaced with Rotation(Vector3D, double, RotationConvention)
/** Build a rotation from an axis and an angle.
* <p>
* Calling this constructor is equivalent to call
* {@link #Rotation(Vector3D, double, RotationConvention)
* new Rotation(axis, angle, RotationConvention.VECTOR_OPERATOR)}
* </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 #Rotation(Vector3D, double, RotationConvention)}
*/
@Deprecated
public Rotation(Vector3D axis, double angle) throws MathIllegalArgumentException {
this(axis, angle, RotationConvention.VECTOR_OPERATOR);
}
Build a rotation from an axis and an angle.
Params: - axis – axis around which to rotate
- angle – rotation angle
- convention – convention to use for the semantics of the angle
Throws: - MathIllegalArgumentException – if the axis norm is zero
Since: 3.6
/** Build a rotation from an axis and an angle.
* @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 Rotation(final Vector3D axis, final double angle, final RotationConvention convention)
throws MathIllegalArgumentException {
double norm = axis.getNorm();
if (norm == 0) {
throw new MathIllegalArgumentException(LocalizedFormats.ZERO_NORM_FOR_ROTATION_AXIS);
}
double halfAngle = convention == RotationConvention.VECTOR_OPERATOR ? -0.5 * angle : +0.5 * angle;
double coeff = FastMath.sin(halfAngle) / norm;
q0 = FastMath.cos (halfAngle);
q1 = coeff * axis.getX();
q2 = coeff * axis.getY();
q3 = coeff * axis.getZ();
}
Build a rotation from a 3X3 matrix.
Rotation matrices are orthogonal matrices, i.e. unit matrices
(which are matrices for which m.mT = 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.
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.
Params: - m – rotation matrix
- 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)
Throws: - 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
/** 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(
LocalizedFormats.ROTATION_MATRIX_DIMENSIONS,
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(
LocalizedFormats.CLOSEST_ORTHOGONAL_MATRIX_HAS_NEGATIVE_DETERMINANT,
det);
}
double[] 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.
Except for possible scale factors, if the instance were applied to
the pair (u1, u2) it will produce the pair
(v1, v2).
If the angular separation between u1 and u2 is
not the same as the angular separation between v1 and
v2, then a corrected v'2 will be used rather than
v2, the corrected vector will be in the (±v1,
+v2) half-plane.
Params: - u1 – first vector of the origin pair
- u2 – second vector of the origin pair
- v1 – desired image of u1 by the rotation
- v2 – desired image of u2 by the rotation
Throws: - 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)
/** 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 Rotation(Vector3D u1, Vector3D u2, Vector3D v1, Vector3D 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 Vector3D u3 = u1.crossProduct(u2).normalize();
u2 = u3.crossProduct(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 Vector3D v3 = v1.crossProduct(v2).normalize();
v2 = v3.crossProduct(v1).normalize();
v1 = v1.normalize();
// buid a matrix transforming the first base into the second one
final double[][] m = new double[][] {
{
MathArrays.linearCombination(u1.getX(), v1.getX(), u2.getX(), v2.getX(), u3.getX(), v3.getX()),
MathArrays.linearCombination(u1.getY(), v1.getX(), u2.getY(), v2.getX(), u3.getY(), v3.getX()),
MathArrays.linearCombination(u1.getZ(), v1.getX(), u2.getZ(), v2.getX(), u3.getZ(), v3.getX())
},
{
MathArrays.linearCombination(u1.getX(), v1.getY(), u2.getX(), v2.getY(), u3.getX(), v3.getY()),
MathArrays.linearCombination(u1.getY(), v1.getY(), u2.getY(), v2.getY(), u3.getY(), v3.getY()),
MathArrays.linearCombination(u1.getZ(), v1.getY(), u2.getZ(), v2.getY(), u3.getZ(), v3.getY())
},
{
MathArrays.linearCombination(u1.getX(), v1.getZ(), u2.getX(), v2.getZ(), u3.getX(), v3.getZ()),
MathArrays.linearCombination(u1.getY(), v1.getZ(), u2.getY(), v2.getZ(), u3.getY(), v3.getZ()),
MathArrays.linearCombination(u1.getZ(), v1.getZ(), u2.getZ(), v2.getZ(), u3.getZ(), v3.getZ())
}
};
double[] quat = mat2quat(m);
q0 = quat[0];
q1 = quat[1];
q2 = quat[2];
q3 = quat[3];
}
Build one of the rotations that transform one vector into another one.
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.
Params: - u – origin vector
- v – desired image of u by the rotation
Throws: - MathArithmeticException – if the norm of one of the vectors is zero
/** 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 Rotation(Vector3D u, Vector3D v) throws MathArithmeticException {
double normProduct = u.getNorm() * v.getNorm();
if (normProduct == 0) {
throw new MathArithmeticException(LocalizedFormats.ZERO_NORM_FOR_ROTATION_DEFINING_VECTOR);
}
double dot = u.dotProduct(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 = FastMath.sqrt(0.5 * (1.0 + dot / normProduct));
double coeff = 1.0 / (2.0 * q0 * normProduct);
Vector3D q = v.crossProduct(u);
q1 = coeff * q.getX();
q2 = coeff * q.getY();
q3 = coeff * q.getZ();
}
}
Build a rotation from three Cardan or Euler elementary rotations.
Calling this constructor is equivalent to call
new Rotation(order, RotationConvention.VECTOR_OPERATOR, alpha1, alpha2, alpha3)
Params: - order – order of rotations to use
- alpha1 – angle of the first elementary rotation
- alpha2 – angle of the second elementary rotation
- alpha3 – angle of the third elementary rotation
Deprecated: as of 3.6, replaced with Rotation(RotationOrder, RotationConvention, double, double, double)
/** Build a rotation from three Cardan or Euler elementary rotations.
* <p>
* Calling this constructor is equivalent to call
* {@link #Rotation(RotationOrder, RotationConvention, double, double, double)
* new Rotation(order, RotationConvention.VECTOR_OPERATOR, alpha1, alpha2, alpha3)}
* </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
* #Rotation(RotationOrder, RotationConvention, double, double, double)}
*/
@Deprecated
public Rotation(RotationOrder order,
double alpha1, double alpha2, double alpha3) {
this(order, RotationConvention.VECTOR_OPERATOR, alpha1, alpha2, alpha3);
}
Build a rotation from three Cardan or Euler elementary rotations.
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.
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).
Params: - order – order of rotations to compose, from left to right (i.e. we will use
r1.compose(r2.compose(r3, convention), convention)) - convention – convention to use for the semantics of the angle
- alpha1 – angle of the first elementary rotation
- alpha2 – angle of the second elementary rotation
- alpha3 – angle of the third elementary rotation
Since: 3.6
/** 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 Rotation(RotationOrder order, RotationConvention convention,
double alpha1, double alpha2, double alpha3) {
Rotation r1 = new Rotation(order.getA1(), alpha1, convention);
Rotation r2 = new Rotation(order.getA2(), alpha2, convention);
Rotation r3 = new Rotation(order.getA3(), alpha3, convention);
Rotation 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.
Params: - ort – orthogonal rotation matrix
Returns: quaternion corresponding to the matrix
/** Convert an orthogonal rotation matrix to a quaternion.
* @param ort orthogonal rotation matrix
* @return quaternion corresponding to the matrix
*/
private static double[] mat2quat(final double[][] ort) {
final double[] quat = new double[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)
double s = ort[0][0] + ort[1][1] + ort[2][2];
if (s > -0.19) {
// compute q0 and deduce q1, q2 and q3
quat[0] = 0.5 * FastMath.sqrt(s + 1.0);
double inv = 0.25 / quat[0];
quat[1] = inv * (ort[1][2] - ort[2][1]);
quat[2] = inv * (ort[2][0] - ort[0][2]);
quat[3] = 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
quat[1] = 0.5 * FastMath.sqrt(s + 1.0);
double inv = 0.25 / quat[1];
quat[0] = inv * (ort[1][2] - ort[2][1]);
quat[2] = inv * (ort[0][1] + ort[1][0]);
quat[3] = 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
quat[2] = 0.5 * FastMath.sqrt(s + 1.0);
double inv = 0.25 / quat[2];
quat[0] = inv * (ort[2][0] - ort[0][2]);
quat[1] = inv * (ort[0][1] + ort[1][0]);
quat[3] = 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];
quat[3] = 0.5 * FastMath.sqrt(s + 1.0);
double inv = 0.25 / quat[3];
quat[0] = inv * (ort[0][1] - ort[1][0]);
quat[1] = inv * (ort[0][2] + ort[2][0]);
quat[2] = inv * (ort[2][1] + 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.
Returns: a new rotation whose effect is the reverse of the effect
of the instance
/** 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.
Returns: scalar coordinate of the quaternion
/** 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.
Returns: first coordinate of the vectorial part of the quaternion
/** 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.
Returns: second coordinate of the vectorial part of the quaternion
/** 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.
Returns: third coordinate of the vectorial part of the quaternion
/** 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.
Calling this method is equivalent to call getAxis(RotationConvention.VECTOR_OPERATOR)
See Also: Returns: normalized axis of the rotation Deprecated: as of 3.6, replaced with getAxis(RotationConvention)
/** Get the normalized axis of the rotation.
* <p>
* Calling this method is equivalent to call
* {@link #getAxis(RotationConvention) getAxis(RotationConvention.VECTOR_OPERATOR)}
* </p>
* @return normalized axis of the rotation
* @see #Rotation(Vector3D, double, RotationConvention)
* @deprecated as of 3.6, replaced with {@link #getAxis(RotationConvention)}
*/
@Deprecated
public Vector3D getAxis() {
return getAxis(RotationConvention.VECTOR_OPERATOR);
}
Get the normalized axis of the rotation.
Note that as getAngle() always returns an angle between 0 and π, changing the convention changes the direction of the axis, not the sign of the angle.
Params: - convention – convention to use for the semantics of the angle
See Also: Returns: normalized axis of the rotation Since: 3.6
/** 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 #Rotation(Vector3D, double, RotationConvention)
* @since 3.6
*/
public Vector3D getAxis(final RotationConvention convention) {
final double squaredSine = q1 * q1 + q2 * q2 + q3 * q3;
if (squaredSine == 0) {
return convention == RotationConvention.VECTOR_OPERATOR ? Vector3D.PLUS_I : Vector3D.MINUS_I;
} else {
final double sgn = convention == RotationConvention.VECTOR_OPERATOR ? +1 : -1;
if (q0 < 0) {
final double inverse = sgn / FastMath.sqrt(squaredSine);
return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
}
final double inverse = -sgn / FastMath.sqrt(squaredSine);
return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse);
}
}
Get the angle of the rotation.
See Also: Returns: angle of the rotation (between 0 and π)
/** Get the angle of the rotation.
* @return angle of the rotation (between 0 and π)
* @see #Rotation(Vector3D, double)
*/
public double getAngle() {
if ((q0 < -0.1) || (q0 > 0.1)) {
return 2 * FastMath.asin(FastMath.sqrt(q1 * q1 + q2 * q2 + q3 * q3));
} else if (q0 < 0) {
return 2 * FastMath.acos(-q0);
}
return 2 * FastMath.acos(q0);
}
Get the Cardan or Euler angles corresponding to the instance.
Calling this method is equivalent to call
getAngles(order, RotationConvention.VECTOR_OPERATOR)
Params: - order – rotation order to use
Throws: - CardanEulerSingularityException – if the rotation is
singular with respect to the angles set specified
Returns: an array of three angles, in the order specified by the set Deprecated: as of 3.6, replaced with getAngles(RotationOrder, RotationConvention)
/** Get the Cardan or Euler angles corresponding to the instance.
* <p>
* Calling this method is equivalent to call
* {@link #getAngles(RotationOrder, RotationConvention)
* getAngles(order, RotationConvention.VECTOR_OPERATOR)}
* </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 double[] getAngles(RotationOrder order)
throws CardanEulerSingularityException {
return getAngles(order, RotationConvention.VECTOR_OPERATOR);
}
Get the Cardan or Euler angles corresponding to the instance.
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
a1, a2 and a3 is the same as
the rotation defined by the angles π + a1, π
- a2 and π + a3. This method implements
the following arbitrary choices:
- 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),
- 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).
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
nothing 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!
Params: - order – rotation order to use
- convention – convention to use for the semantics of the angle
Throws: - CardanEulerSingularityException – if the rotation is
singular with respect to the angles set specified
Returns: an array of three angles, in the order specified by the set Since: 3.6
/** 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 double[] getAngles(RotationOrder order, RotationConvention convention)
throws CardanEulerSingularityException {
if (convention == RotationConvention.VECTOR_OPERATOR) {
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[] {
FastMath.atan2(-(v1.getY()), v1.getZ()),
FastMath.asin(v2.getZ()),
FastMath.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[] {
FastMath.atan2(v1.getZ(), v1.getY()),
-FastMath.asin(v2.getY()),
FastMath.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[] {
FastMath.atan2(v1.getX(), v1.getZ()),
-FastMath.asin(v2.getZ()),
FastMath.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[] {
FastMath.atan2(-(v1.getZ()), v1.getX()),
FastMath.asin(v2.getX()),
FastMath.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[] {
FastMath.atan2(-(v1.getX()), v1.getY()),
FastMath.asin(v2.getY()),
FastMath.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[] {
FastMath.atan2(v1.getY(), v1.getX()),
-FastMath.asin(v2.getX()),
FastMath.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[] {
FastMath.atan2(v1.getY(), -v1.getZ()),
FastMath.acos(v2.getX()),
FastMath.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[] {
FastMath.atan2(v1.getZ(), v1.getY()),
FastMath.acos(v2.getX()),
FastMath.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[] {
FastMath.atan2(v1.getX(), v1.getZ()),
FastMath.acos(v2.getY()),
FastMath.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[] {
FastMath.atan2(v1.getZ(), -v1.getX()),
FastMath.acos(v2.getY()),
FastMath.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[] {
FastMath.atan2(v1.getX(), -v1.getY()),
FastMath.acos(v2.getZ()),
FastMath.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[] {
FastMath.atan2(v1.getY(), v1.getX()),
FastMath.acos(v2.getZ()),
FastMath.atan2(v2.getY(), -v2.getX())
};
}
} 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]
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[] {
FastMath.atan2(-v2.getY(), v2.getZ()),
FastMath.asin(v2.getX()),
FastMath.atan2(-v1.getY(), 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]
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[] {
FastMath.atan2(v2.getZ(), v2.getY()),
-FastMath.asin(v2.getX()),
FastMath.atan2(v1.getZ(), 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]
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[] {
FastMath.atan2(v2.getX(), v2.getZ()),
-FastMath.asin(v2.getY()),
FastMath.atan2(v1.getX(), 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]
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[] {
FastMath.atan2(-v2.getZ(), v2.getX()),
FastMath.asin(v2.getY()),
FastMath.atan2(-v1.getZ(), 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]
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[] {
FastMath.atan2(-v2.getX(), v2.getY()),
FastMath.asin(v2.getZ()),
FastMath.atan2(-v1.getX(), 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]
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[] {
FastMath.atan2(v2.getY(), v2.getX()),
-FastMath.asin(v2.getZ()),
FastMath.atan2(v1.getY(), 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]
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[] {
FastMath.atan2(v2.getY(), -v2.getZ()),
FastMath.acos(v2.getX()),
FastMath.atan2(v1.getY(), 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]
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[] {
FastMath.atan2(v2.getZ(), v2.getY()),
FastMath.acos(v2.getX()),
FastMath.atan2(v1.getZ(), -v1.getY())
};
} 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]
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[] {
FastMath.atan2(v2.getX(), v2.getZ()),
FastMath.acos(v2.getY()),
FastMath.atan2(v1.getX(), -v1.getZ())
};
} 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]
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[] {
FastMath.atan2(v2.getZ(), -v2.getX()),
FastMath.acos(v2.getY()),
FastMath.atan2(v1.getZ(), 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]
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[] {
FastMath.atan2(v2.getX(), -v2.getY()),
FastMath.acos(v2.getZ()),
FastMath.atan2(v1.getX(), 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]
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[] {
FastMath.atan2(v2.getY(), v2.getX()),
FastMath.acos(v2.getZ()),
FastMath.atan2(v1.getY(), -v1.getX())
};
}
}
}
Get the 3X3 matrix corresponding to the instance
Returns: the matrix corresponding to the instance
/** 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.
Params: - u – vector to apply the rotation to
Returns: a new vector which is the image of u by the rotation
/** 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 rotation to a vector stored in an array.
Params: - in – an array with three items which stores vector to rotate
- out – an array with three items to put result to (it can be the same
array as in)
/** 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 double[] in, final double[] out) {
final double x = in[0];
final double y = in[1];
final double z = in[2];
final double s = q1 * x + q2 * y + q3 * z;
out[0] = 2 * (q0 * (x * q0 - (q2 * z - q3 * y)) + s * q1) - x;
out[1] = 2 * (q0 * (y * q0 - (q3 * x - q1 * z)) + s * q2) - y;
out[2] = 2 * (q0 * (z * q0 - (q1 * y - q2 * x)) + s * q3) - z;
}
Apply the inverse of the rotation to a vector.
Params: - u – vector to apply the inverse of the rotation to
Returns: a new vector which such that u is its image by the rotation
/** 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 inverse of the rotation to a vector stored in an array.
Params: - in – an array with three items which stores vector to rotate
- out – an array with three items to put result to (it can be the same
array as in)
/** 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 double[] in, final double[] out) {
final double x = in[0];
final double y = in[1];
final double z = in[2];
final double s = q1 * x + q2 * y + q3 * z;
final double m0 = -q0;
out[0] = 2 * (m0 * (x * m0 - (q2 * z - q3 * y)) + s * q1) - x;
out[1] = 2 * (m0 * (y * m0 - (q3 * x - q1 * z)) + s * q2) - y;
out[2] = 2 * (m0 * (z * m0 - (q1 * y - q2 * x)) + s * q3) - z;
}
Apply the instance to another rotation.
Calling this method is equivalent to call
compose(r, RotationConvention.VECTOR_OPERATOR).
Params: - r – rotation to apply the rotation to
Returns: a new rotation which is the composition of r by the instance
/** 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 Rotation applyTo(Rotation r) {
return compose(r, RotationConvention.VECTOR_OPERATOR);
}
Compose the instance with another rotation.
If the semantics of the rotations composition corresponds to a vector operator convention, 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 r1 (i.e. r1.applyTo(u) = v). Let w be the image of v by rotation r2 (i.e. r2.applyTo(v) = w). Then w = comp.applyTo(u), where comp = r2.compose(r1, RotationConvention.VECTOR_OPERATOR).
If the semantics of the rotations composition corresponds to a frame transform convention, the application order will be reversed. So keeping the exact same meaning of all r1, r2, u, v, w and comp as above, comp could also be computed as comp = r1.compose(r2, RotationConvention.FRAME_TRANSFORM).
Params: - r – rotation to apply the rotation to
- convention – convention to use for the semantics of the angle
Returns: a new rotation which is the composition of r by the instance
/** 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 Rotation compose(final Rotation r, final RotationConvention convention) {
return convention == RotationConvention.VECTOR_OPERATOR ?
composeInternal(r) : r.composeInternal(this);
}
Compose the instance with another rotation using vector operator convention.
Params: - r – rotation to apply the rotation to
Returns: a new rotation which is the composition of r by the instance
using vector operator convention
/** 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 Rotation composeInternal(final 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.
Calling this method is equivalent to call
composeInverse(r, RotationConvention.VECTOR_OPERATOR).
Params: - r – rotation to apply the rotation to
Returns: a new rotation which is the composition of r by the inverse
of the instance
/** 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 Rotation applyInverseTo(Rotation r) {
return composeInverse(r, RotationConvention.VECTOR_OPERATOR);
}
Compose the inverse of the instance with another rotation.
If the semantics of the rotations composition corresponds to a 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 u be any vector and v its image by r1 (i.e. r1.applyTo(u) = v). Let w be the inverse image of v by r2 (i.e. r2.applyInverseTo(v) = w). Then w = comp.applyTo(u), where comp = r2.composeInverse(r1).
If the semantics of the rotations composition corresponds to a frame transform convention, the application order will be reversed, which means it is the innermost rotation that will be reversed. So keeping the exact same meaning of all r1, r2, u, v, w and comp as above, comp could also be computed as comp = r1.revert().composeInverse(r2.revert(), RotationConvention.FRAME_TRANSFORM).
Params: - r – rotation to apply the rotation to
- convention – convention to use for the semantics of the angle
Returns: a new rotation which is the composition of r by the inverse
of the instance
/** 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 Rotation composeInverse(final Rotation 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.
Params: - r – rotation to apply the rotation to
Returns: a new rotation which is the composition of r by the inverse
of the instance using vector operator convention
/** 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 Rotation composeInverseInternal(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.
Params: - m – initial matrix (not exactly orthogonal)
- 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)
Throws: - NotARotationMatrixException – if the matrix cannot be
orthogonalized with the given threshold after 10 iterations
Returns: an orthogonal matrix close to m
/** 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 (FastMath.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(
LocalizedFormats.UNABLE_TO_ORTHOGONOLIZE_MATRIX,
i - 1);
}
Compute the distance between two rotations.
The distance 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:
r1(r) = r2
This distance is an angle between 0 and π. Its value is the smallest
possible upper bound of the angle in radians between r1(v)
and r2(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.
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).
Params: - r1 – first rotation
- r2 – second rotation
Returns: distance between r1 and r2
/** 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
* @return <i>distance</i> between r1 and r2
*/
public static double distance(Rotation r1, Rotation r2) {
return r1.composeInverseInternal(r2).getAngle();
}
}